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Monetary policy analysis in a small open economy : development and evaluation of quantitative tools Vitek, Francis 2007

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Monetary Policy Analysis in a Small Open Economy: Development and Evaluation of Quantitative Tools  by  Francis Vitek B . S c , The University o f Victoria, 2001 M . A . , Queen's University, 2002  A thesis submitted in partial fulfillment o f the requirements for the degree o f  Doctor of Philosophy in  The Faculty of Graduate Studies (Economics)  The University of British Columbia March 2007  © Francis Vitek, 2007  11  ABSTRACT  This doctoral thesis consists o f four papers, the unifying theme o f which is the development and evaluation o f quantitative tools for purposes o f monetary policy analysis and inflation targeting i n a small open economy. These tools consist o f alternative macroeconometric models of small open economies which either provide a quantitative description o f the monetary transmission mechanism, or yield a mutually consistent set o f indicators o f inflationary pressure together with confidence intervals, or both.  The models vary considerably with regards to  theoretical structure, and are estimated with novel Bayesian procedures. In all cases, parameters and trend components are jointly estimated, conditional on prior information concerning the values of parameters or trend components. The first paper develops and estimates a dynamic stochastic general equilibrium or D S G E model o f a small open economy which approximately accounts for the empirical evidence concerning the monetary transmission mechanism, as summarized by impulse response functions derived from an estimated structural vector autoregressive or S V A R model, while dominating that S V A R model in terms o f predictive accuracy. The primary contribution o f this first paper is the joint modeling o f cyclical and trend components as unobserved components while imposing theoretical restrictions derived from the approximate multivariate linear rational expectations representation o f a D S G E model. The second paper develops and estimates an unobserved components model for purposes o f monetary policy analysis and inflation targeting in a small open economy.  The primary  contribution o f this second paper is the development o f a procedure to estimate a linear state space model conditional on prior information concerning the values o f unobserved state variables. The third paper develops and estimates a D S G E model o f a small open economy for purposes of monetary policy analysis and inflation targeting which provides a quantitative description o f the monetary transmission mechanism, yields a mutually consistent set o f indicators o f inflationary pressure together with confidence intervals, and facilitates the generation of relatively accurate forecasts. The primary contribution o f this third paper is the development o f a Bayesian procedure to estimate the levels o f the flexible price and wage equilibrium components of endogenous variables while imposing relatively weak identifying restrictions on their trend components. The fourth paper evaluates the finite sample properties o f the procedure proposed in the third paper for the measurement o f the stance o f monetary policy in a small open economy with a  iii  Monte Carlo experiment.  This Bayesian estimation procedure is found to yield reasonably  accurate and precise results in samples o f currently available size.  iv  T A B L E OF CONTENTS  ABSTRACT  •  T A B L E OF C O N T E N T S  -  LIST OF T A B L E S  iv  :  LIST OF F I G U R E S PREFACE  u  .' •.  ACKNOWLEDGEMENTS  CHAPTER 1 Monetary Policy Analysis in a Small Open Economy: A Dynamic Stochastic  ix •  x xii xviii  1 General  Equilibrium Approach  1.1. Introduction  1  1.2. Model Development  4  1.2.1. The Utility Maximization Problem of the Representative Household  4  1.2.1.1. Consumption and Saving Behaviour  4  1.2.1.2. Labour Supply and Wage Setting Behaviour  7  1.2.2. The Value Maximization Problem of the Representative F i r m 1.2.2.1. Employment and Investment Behaviour 1.2.2.2. Output Supply and Price Setting Behaviour 1.2.3. The Value Maximization Problem of the Representative Importer  9 9 12 14  1.2.3.1. The Real Exchange Rate and the Terms of Trade  14  1.2.3.2. Import Supply and Price Setting Behaviour  16  1.2.4. Monetary and Fiscal Policy 1.2.4.1. The Monetary Authority  18 19  V  1.2.4.2. T h e F i s c a l A u t h o r i t y  19  1.2.5. M a r k e t C l e a r i n g C o n d i t i o n s  20  1.2.6. T h e A p p r o x i m a t e L i n e a r M o d e l  ,  22  1.2.6.1. C y c l i c a l C o m p o n e n t s  22  1.2.6.2. T r e n d C o m p o n e n t s  =  1.3. Estimation, Inference and Forecasting 1.3.1. E s t i m a t i o n  29 31  '  1.3.1.1. E s t i m a t i o n P r o c e d u r e  32 :  33  1.3.1.2. E s t i m a t i o n R e s u l t s  36  1.3.2. Inference  39  1.3.2.1. E m p i r i c a l I m p u l s e R e s p o n s e A n a l y s i s  39  1.3.2.2. T h e o r e t i c a l I m p u l s e R e s p o n s e A n a l y s i s  44  1.3.3. F o r e c a s t i n g  45  1.4. Conclusion  46  Appendix l . A . Description of the Data Set  47  Appendix l . B . Tables and Figures  48  References  CHAPTER 2  •  •  69  72  A n U n o b s e r v e d C o m p o n e n t s M o d e l o f the M o n e t a r y T r a n s m i s s i o n M e c h a n i s m i n a S m a l l Open E c o n o m y  2.1. Introduction  72  2.2. The Unobserved Components Model .,  74  2.2.1. C y c l i c a l C o m p o n e n t s  75  2.2.2. T r e n d C o m p o n e n t s  78  2.3. Estimation of Unobserved State Variables  79  2.3.1. U n r e s t r i c t e d E s t i m a t i o n o f U n o b s e r v e d State V a r i a b l e s  80  2.3.2. R e s t r i c t e d E s t i m a t i o n o f U n o b s e r v e d State V a r i a b l e s  82  vi  2.4. Estimation, Inference and Forecasting 2.4.1. Estimation  85  ;  85  2.4.1.1. Estimation Procedure  85  2.4.1.2. Estimation Results  '.  -89  2.4.2. Inference  90  2.4.2.1. Quantifying Inflationary Pressure  .'90.  2.4.2.2. Quantifying the Monetary Transmission Mechanism  92  2.4.3. Forecasting 2.5. Conclusion  95  '.  97  Appendix 2.A. Description df the Data Set  98  Appendix 2.B. Tables and Figures  99  References  113  CHAPTER 3  •'  H5  Measuring the Stance o f Monetary Policy in a Small Open Economy: A Dynamic Stochastic General Equilibrium Approach 3.1. Introduction  115  3.2. Model Development  119  3.2.1. The Utility Maximization Problem of the Representative Household  119  3.2.1.1. Consumption, Saving and Investment Behaviour  119  3.2.1.2. Labour Supply and Wage Setting Behaviour  123  3.2.2. The Value Maximization Problem of the Representative F i r m  125  3.2.2.1. Employment and Investment Behaviour  125  3.2.2.2. Output Supply and Price Setting Behaviour  128  3.2.3. The Value Maximization Problem of the Representative Importer  131  3.2.3.1. The Real Exchange Rate and the Terms of Trade  ..131  3.2.3.2. Import Supply and Price Setting Behaviour  133  3.2.4. Monetary and Fiscal Policy 3.2.4.1. The Monetary Authority  • '..  136 136  3.2.4.2. The Fiscal Authority  136  3.2.5. Market Clearing Conditions  137  3.2.6. The Approximate Linear M o d e l  139  3.2.6.1. Cyclical Components  139  3.2.6.2. Trend Components  148  3.3. Estimation, Inference and Forecasting  150  3.3.1. Estimation  151  3.3.1.1. Estimation Procedure 3.3.1.2. Estimation Results  151 '.  3.3.2. Inference  155 158  3.3.2.1. Quantifying the Stance o f Monetary Policy  159  3.3.2.2. Quantifying the Monetary Transmission Mechanism  162  3.3.3. Forecasting  168  3.4. Conclusion  169  Appendix 3.A. Description of the Data Set  170  Appendix 3.B. Tables and Figures  171  References  CHAPTER 4  ....193  196  Measuring the Stance o f Monetary Policy in a Small Open Economy: A Monte Carlo Evaluation  4.1. Introduction  196  4.2. Model Development  199  4.2.1. The Utility Maximization Problem o f the Representative Household  199  4.2.2. The Value Maximization Problem o f the Representative F i r m  201  4.2.2.1. Employment Behaviour 4.2.2.2. Output Supply and Price Setting Behaviour 4.2.3. International Trade and Financial Linkages 4.2.3.1. International Trade Linkages  ....202 203 205 205  Vlll  4.2.3.2. International Financial Linkages  207  4.2.4. Monetary Policy  ....207  4.2.5. Market Clearing Conditions  208  4.3. The Approximate Linear Model  208  4.3.1. First Best Approximation  209  4.3.1.1. Cyclical Components  209  4.3.1.2. Trend Components  ,  4.3.2. Second Best Approximation 4.3.2.1. Cyclical Components 4.3.2.2. Trend Components 4.4. Estimation and Inference  211 :  :  212 212 214 215  4.4.1. Estimation  216  4.4.2. Inference  220  4.5. Conclusion  227  References  228  ix  LIST OF T A B L E S  CHAPTER 1 Table 1.1. Deterministic steady state equilibrium values o f great ratios  37  Table 1.2. Model selection criterion function values  40  Table 1.3. Results o f tests o f overidentifying restrictions  41  Table 1.4. Bayesian estimation results  48  CHAPTER 2 Table 2.1. Full information maximum likelihood estimation results, domestic economy Table 2.2. Full information maximum likelihood estimation results, foreign economy  99 100  CHAPTER 3 Table 3.1. Deterministic steady state equilibrium values o f great ratios  156  Table 3.2. Model selection criterion function values  163  Table 3.3. Results o f tests o f overidentifying restrictions  164  Table 3.4. Bayesian estimation results  171  CHAPTER 4 Table 4.1. True values o f parameters  221  Table 4.2. Experimental results under deterministic trend specification, parameters  223  Table 4.3. Experimental results under stochastic trend specification, parameters  224  Table 4.4. Experimental results under deterministic trend specification, natural rate o f interest  225  Table 4.5. Experimental results under stochastic trend specification, natural rate of interest ...1  226  Table 4.6. Experimental results under deterministic trend specification, natural exchange rate  227  Table 4.7. Experimental results under stochastic trend specification, natural exchange rate  227  X  LIST OF FIGURES  CHAPTER 1 Figure 1.1. Theoretical versus empirical impulse responses to a domestic monetary policy shock  42  Figure 1.2. Theoretical versus empirical impulse responses to a foreign monetary policy shock  43  Figure 1.3. Predicted cyclical components o f observed endogenous variables  50  Figure 1.4. Filtered cyclical components of observed endogenous variables  51  Figure 1.5. Smoothed cyclical components of observed endogenous variables  52  Figure 1.6. Predicted trend components of observed endogenous variables  53  Figure 1.7. Filtered trend components o f observed endogenous variables  54  Figure 1.8. Smoothed trend components o f observed endogenous variables  55  Figure 1.9. Theoretical impulse responses to a domestic output technology shock  56  Figure 1.10. Theoretical impulse responses to a domestic monetary policy shock  57  Figure 1.11. Theoretical impulse responses to a domestic fiscal expenditure shock  58  Figure 1.12. Theoretical impulse responses to a foreign output technology shock  59  Figure 1.13. Theoretical impulse responses to a foreign monetary policy shock  60  Figure 1.14. Theoretical impulse responses to a foreign fiscal expenditure shock  61  Figure 1.15. Theoretical forecast error variance decompositions  62  Figure 1.16. Mean squared prediction error differentials for levels  63  Figure 1.17. Mean squared prediction error differentials for ordinary differences  64  Figure 1.18. Mean squared prediction error differentials for seasonal differences  65  Figure 1.19. Dynamic forecasts o f levels o f observed endogenous variables  66  Figure 1.20. Dynamic forecasts of ordinary differences of observed endogenous variables  67  Figure 1.21. Dynamic forecasts of seasonal differences of observed endogenous variables  68  CHAPTER 2 Figure 2.1. Predicted, filtered and smoothed estimates of the natural rate of interest  91  Figure 2.2. Predicted, filtered and smoothed estimates of the natural exchange rate  92  Figure 2.3. Estimated impulse responses to a domestic monetary policy shock  93  Figure 2.4. Estimated impulse responses to a foreign monetary policy shock  95  Figure 2.5. Predicted cyclical components o f observed endogenous variables  101  Figure 2.6. Filtered cyclical components of observed endogenous variables  102  Figure 2.7. Smoothed cyclical components o f observed endogenous variables  103  Figure 2.8. Predicted trend components of observed endogenous variables  104  Figure 2.9. Filtered trend components of observed endogenous variables  105  Figure 2.10. Smoothed trend components of observed endogenous variables  106  Figure 2.11. Mean squared prediction error differentials for levels  107  Figure 2.12. Mean squared prediction error differentials for ordinary differences  108  Figure 2.13. Mean squared prediction error differentials for seasonal differences  109  Figure 2.14. Dynamic forecasts o f levels of observed endogenous variables  110  Figure 2.15. Dynamic forecasts o f ordinary differences o f observed endogenous variables  111  Figure 2.16. Dynamic forecasts o f seasonal differences o f observed endogenous variables  112  CHAPTER 3  Figure 3.1. Predicted, fdtered and smoothed estimates o f the natural rate o f interest  160  Figure 3.2. Predicted, fdtered and smoothed estimates o f the natural exchange rate  161  Figure 3.3. Theoretical versus empirical impulse responses to a domestic monetary policy shock  165  Figure 3.4. Theoretical versus empirical impulse responses to a foreign monetary policy shock  167  Figure 3.5. Predicted cyclical components o f observed endogenous variables  173  Figure 3.6. Filtered cyclical components o f observed endogenous variables  174  Figure 3.7. Smoothed cyclical components o f observed endogenous variables  175  Figure 3.8. Predicted trend components o f observed endogenous variables  176  Figure 3.9. Filtered trend components o f observed endogenous variables  177  Figure 3.10. Smoothed trend components o f observed endogenous variables  178  Figure 3.11. Theoretical impulse responses to a domestic output technology shock  179  Figure 3.12. Theoretical impulse responses to a domestic monetary policy shock  180  Figure 3.13. Theoretical impulse responses to a domestic fiscal expenditure shock  181  Figure 3.14. Theoretical impulse responses to a foreign output technology shock  182  Figure 3.15. Theoretical impulse responses to a foreign monetary policy shock  183  Figure 3.16. Theoretical impulse responses to a foreign fiscal expenditure shock  184  Figure 3.17. Theoretical forecast error variance decompositions under sticky price and wage equilibrium ... 185 Figure 3.18. Theoretical forecast error variance decompositions under flexible price and wage equilibrium 186 Figure 3.19. Mean squared prediction error differentials for levels  :  Figure 3.20. Mean squared prediction error differentials for ordinary differences  187 188  Figure 3.21. Mean squared prediction error differentials for seasonal differences  189  Figure 3.22. Dynamic forecasts o f levels o f observed endogenous variables  190  Figure 3.23. Dynamic forecasts o f ordinary differences o f observed endogenous variables  191  Figure 3.24. Dynamic forecasts o f seasonal differences o f observed endogenous variables  192  xn  PREFACE The last decade has witnessed a revival of academic interest in monetary policy analysis, stimulated by revolutionary developments in theoretical and empirical macroeconomics. From the theoretical perspective, the incorporation of short run nominal price and wage rigidities into dynamic stochastic general equilibrium or D S G E models based on rigorous microeconomic foundations has provided an internally consistent framework for the analysis of the monetary transmission mechanism, which describes the dynamic effects of unsystematic variation in the instrument of monetary policy on indicators and targets, and the optimal conduct of monetary policy. From the empirical perspective, the development of Bayesian procedures to accurately and precisely estimate D S G E models has legitimized their emerging role as quantitative monetary policy analysis tools. Although the quantitative monetary policy analysis literature is advancing rapidly, significant theoretical and empirical problems remain unsolved. On the theoretical front, D S G E models which yield empirically adequate predictions at all frequencies, as opposed to only cyclical frequencies, remain to be developed. On the empirical front, Bayesian procedures which fully exploit the information content of the levels of observed endogenous  variables, while  emphasizing the predictions of D S G E models at cyclical frequencies, and deemphasizing them at trend frequencies, are required. Recent developments in the analysis of monetary policy in open economies have to some extent lagged behind those in the analysis of monetary policy in closed economies, particularly from the empirical perspective. In an open economy, the existence of international trade and financial linkages introduces additional channels through which variation in the instrument of monetary policy affects indicators and targets, complicating the analysis of the monetary transmission mechanism and the optimal conduct of monetary policy. Yet the recent adoption of explicit quantitative inflation targets by the central banks of many economies, particularly those of relatively small and open economies, calls for accurate and precise indicators of inflationary pressure in such economies, together with accurate and precise quantitative descriptions of the monetary transmission mechanism. This doctoral thesis consists of four papers, the unifying theme of which is the development and evaluation of quantitative tools for purposes of monetary policy analysis and inflation targeting in a small open economy. These tools consist of alternative macroeconometric models of small open economies which either provide a quantitative description of the monetary transmission mechanism, or yield a mutually consistent set of indicators of inflationary pressure  Xlll  together with confidence intervals, or both.  The models vary considerably with regards to  theoretical structure, and are estimated with novel Bayesian procedures. In all cases, parameters and trend components are jointly estimated, conditional on prior information concerning the values of parameters or trend components. The first paper develops and estimates a D S G E model of a small open economy which approximately accounts for the empirical evidence concerning the monetary transmission mechanism, as summarized by impulse response functions derived from an estimated structural vector autoregressive or S V A R model, while dominating that S V A R model in terms of predictive accuracy. The model features short run nominal price and wage rigidities generated by monopolistic competition and staggered reoptimization in output and labour markets.  The  resultant inertia in inflation and persistence in output is enhanced with other features such as habit persistence in consumption, adjustment costs in investment, and variable capital utilization. Incomplete exchange rate pass through is generated by short run nominal price rigidities in the import market, with monopolistically competitive importers setting the domestic currency prices of differentiated  intermediate  import goods subject  to randomly arriving reoptimization  opportunities. Cyclical components are modeled by linearizing equilibrium conditions around a stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path.  Parameters and trend components are jointly estimated with a  novel Bayesian procedure, conditional on prior information concerning the values of parameters and trend components. The primary contribution of this first paper is the joint modeling of cyclical and trend components as unobserved components while imposing theoretical restrictions derived from the approximate multivariate linear rational expectations representation of a D S G E model. This merging of modeling paradigms drawn from the theoretical and empirical macroeconomics literatures confers a number of important benefits. First, the joint estimation of parameters and trend components ensures their mutual consistency, as estimates of parameters appropriately reflect estimates of trend components, and vice versa.  It has been shown that decomposing  integrated observed endogenous variables into cyclical and trend components with atheoretic deterministic polynomial functions or low pass filters may induce spurious cyclical dynamics, invalidating subsequent estimation, inference and forecasting. Second, basing estimation on the levels as opposed to differences of observed endogenous variables may be expected to yield efficiency gains. A central result of the voluminous cointegration literature is that, i f there exist cointegrating relationships, then differencing all integrated observed endogenous variables prior to the conduct of estimation, inference and forecasting results in the loss of information. Third, the proposed unobserved components framework ensures stochastic nonsingularity of the  xiv resulting approximate linear state space representation of the D S G E model, as associated with each observed endogenous variable is at least one exogenous stochastic process.  Stochastic  nonsingularity requires that the number of observed endogenous variables used to construct the loglikelihood function associated with the approximate linear state space representation of a D S G E model not exceed the number of exogenous stochastic processes, with efficiency losses incurred i f this constraint binds.  Fourth, the proposed unobserved components framework  facilitates the direct generation of forecasts of the levels of endogenous variables as opposed to their cyclical components together with confidence intervals, while ensuring that these forecasts satisfy the stability restrictions associated with balanced growth. These stability restrictions are necessary but not sufficient for full cointegration, as along a balanced growth path, great ratios and trend growth rates are time independent but state dependent, robustifying forecasts to intermittent structural breaks that occur within sample. The second paper develops and estimates an unobserved components model for purposes of monetary policy analysis and inflation targeting in a small open economy. Cyclical components are modeled as a multivariate linear rational expectations model of the monetary transmission mechanism, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path. Although not derived from microeconomic foundations, this unobserved components model of the monetary transmission mechanism in a small open economy arguably provides a closer approximation to the data generating process than existing D S G E models, as fewer cross-coefficient restrictions are imposed. Full information maximum likelihood estimation of this unobserved components model, conditional on prior information concerning the values of trend components, provides a quantitative description of the monetary transmission mechanism in a small open economy, yields a mutually consistent set of indicators of inflationary pressure together with confidence intervals, and facilitates the generation of relatively accurate forecasts. The primary contribution of this second paper is the development of a procedure to estimate a linear state space model conditional on prior information concerning the values of unobserved state variables. This prior information assumes the form of a set of deterministic or stochastic restrictions on linear combinations of unobserved state variables.  In addition to mitigating  potential model misspecification and identification problems, exploiting such prior information may be expected to yield efficiency gains in estimation. The third paper develops and estimates a D S G E model of a small open economy for purposes of monetary policy analysis and inflation targeting.  This estimated D S G E model provides a  quantitative description of the monetary transmission mechanism in a small open economy, yields a mutually consistent set of indicators of inflationary pressure together with confidence intervals, and facilitates the generation of relatively accurate forecasts.  In an extension and  XV  refinement of the D S G E model developed in the first paper, cyclical components are decomposed into subcomponents identified by the presence or absence of short run nominal price and wage rigidities, while investment in housing and investment in capital are separately modeled. In addition to being a necessary step towards providing a more detailed quantitative description of the monetary transmission mechanism in a small open economy, separately modeling investment in housing and investment in capital has implications for the measurement of the stance of monetary policy. Cyclical components are modeled by linearizing equilibrium conditions around a stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path. Parameters and unobserved components are jointly estimated with a novel Bayesian procedure, conditional on prior information concerning the values of parameters and trend components. The primary contribution of this third paper is the development of a procedure to estimate the levels of the flexible price and wage equilibrium components of endogenous variables while imposing relatively weak, and hence relatively credible, identifying restrictions on their trend components. Based on an extension and refinement of the unobserved components framework proposed in the first paper, this estimation procedure confers a number of benefits of particular importance to the conduct of monetary policy. First, the levels of the flexible price and wage equilibrium components of various observed and unobserved endogenous variables are important inputs into the optimal conduct of monetary policy. In particular, the level of the natural rate of interest, defined as that short term real interest rate consistent with price and wage flexibility, provides a measure of the neutral stance of monetary policy, with deviations of the real interest rate from the natural rate of interest generating inflationary pressure. The proposed unobserved components framework facilitates estimation of the levels as opposed to cyclical components of the flexible price and wage equilibrium components of endogenous variables, while ensuring that they satisfy the stability restrictions associated with balanced growth. Second, given an interest rate smoothing objective derived from a concern with financial market stability, variation in the natural rate of interest caused by shocks having permanent effects may call for larger monetary policy responses than variation caused by shocks having temporary effects.  The proposed  unobserved components framework yields a decomposition of the levels of the flexible price and wage equilibrium components of endogenous variables into cyclical and trend components, together with confidence intervals which account for uncertainty associated with the detrending procedure. Third, accommodating the existence of intermittent structural breaks requires flexible trend component specifications. However, the joint derivation of empirically adequate cyclical and trend component specifications from microeconomic foundations is a formidable task. The proposed unobserved components framework facilitates estimation of the levels of the flexible  xvi price and wage equilibrium components of endogenous variables while allowing for the possibility that the determinants of their trend components are unknown but persistent. The fourth paper evaluates the finite sample properties of the procedure proposed in the third paper for the measurement of the stance of monetary policy in a small open economy.  In  particular, the accuracy and precision of the Bayesian procedure proposed for the estimation of the levels of the flexible price equilibrium components of various observed and unobserved endogenous variables is analyzed with a Monte Carlo experiment, with an emphasis on the levels of the natural rate of interest and natural exchange rate.  The data generating process is a  calibrated D S G E model of a small open economy featuring long run balanced growth driven by trend inflation, productivity growth, and population growth. Alternative versions of this D S G E model incorporating common deterministic or stochastic trends are considered. Given a large number of artificial data sets generated under these alternative trend component specifications, estimation of the levels of the flexible price equilibrium components of various observed and unobserved endogenous variables is based on a linear state space representation of an approximate unobserved components representation of this D S G E model of a small open economy, in which cyclical components are modeled by linearizing equilibrium conditions around a stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path. Repeated joint estimation of the parameters and unobserved components of this linear state space representation of this approximate unobserved components representation of this D S G E model with the Bayesian procedure under consideration facilitates simulation of the finite sample distributions of estimators of the levels of flexible price equilibrium components, with respect to which accuracy and precision are measured in terms of bias and root mean squared error. The primary contribution of this fourth paper is the evaluation of the accuracy and precision of the procedure proposed in the third paper for the estimation of the levels of the flexible price equilibrium components of various observed and unobserved endogenous variables, with an emphasis on the levels of the natural rate of interest and natural exchange rate. This Bayesian estimation procedure is found to yield reasonably accurate and precise results in samples of currently available size. In particular, estimates of the levels of the natural rate of interest and natural exchange rate conditional on alternative information sets are approximately unbiased, while root mean squared errors are relatively small, irrespective of whether the data generating process features common deterministic or stochastic trends.  Moreover, analytical root mean  squared errors appropriately account for uncertainty surrounding estimates of the levels of the natural rate of interest and natural exchange rate.  XVII  The remainder of this doctoral thesis consists of four papers. The first paper develops and estimates a D S G E model of a small open economy. The second paper develops and estimates an unobserved components model of the monetary transmission mechanism in a small open economy. The third paper considers the measurement of the stance of monetary policy in a small open economy within a D S G E framework. The fourth paper evaluates the procedure proposed for the measurement of the stance of monetary policy in a small open economy. economy  versions  of these  grad.econ.ubc.ca/fvitek.  papers  have  also  been  written,  and  are  Closed  available  at  XV111  ACKNOWLEDGMENTS  The author gratefully acknowledges advice provided by Paul Beaudry, Michael Devereux, Shinichi Sakata and Henry Siu, in addition to comments and suggestions received from seminar participants at the Bank of Canada and the University of British Columbia. The author thanks the Social Sciences and Humanities Research Council of Canada for financial support.  1 CHAPTER 1  Monetary  Policy Analysis in a Small  D y n a m i c Stochastic General  Open Economy:  Equilibrium  A  Approach  1.1. Introduction Estimated dynamic stochastic general equilibrium or D S G E models have recently emerged as quantitative monetary policy analysis and inflation targeting tools.  A s extensions of real  business cycle models, D S G E models explicitly specify the objectives and constraints faced by optimizing households and firms, which interact in an uncertain environment to determine equilibrium prices and quantities. The existence of short run nominal price and wage rigidities generated by monopolistic competition and staggered reoptimization in output and labour markets permits a cyclical stabilization role for monetary policy, which is generally implemented through control of the nominal interest rate according to a monetary policy rule. The persistence of the effects of monetary policy shocks on output and inflation is often enhanced with other features such as habit persistence in consumption, adjustment costs in investment, and variable capital utilization.  Early examples of closed economy D S G E models incorporating some of  these features include those of Y u n (1996), Goodfriend and King (1997), Rotemberg and Woodford (1995, 1997), and McCallum and Nelson (1999), while recent examples of closed economy D S G E models incorporating all of these features include those of Christiano, Eichenbaum and Evans (2005), Altig, Christiano, Eichenbaum and Linde (2005), and Smets and Wouters (2003, 2005). Open economy D S G E models extend their closed economy counterparts to allow for international trade and financial linkages, implying that the monetary transmission mechanism features both interest rate and exchange rate channels. Building on the seminal work of Obstfeld and Rogoff (1995, 1996), these open economy D S G E models determine trade and current account balances through both intratemporal and intertemporal optimization, while the nominal exchange rate is determined by an uncovered interest parity condition. Existing open economy D S G E models differ primarily with respect to the degree of exchange rate pass through. Models in which exchange rate pass through is complete include those of Benigno and Benigno (2002), McCallum and Nelson (2000), Clarida, Gali and Gertler (2001, 2002), and Gertler, Gilchrist and  2 Natalucci (2001), while models in which exchange rate pass through is incomplete include those of Adolfson (2001), Betts and Devereux (2000), Kollman (2001), Corsetti and Pesenti (2002), and Monacelli (2005). In an empirical investigation of the degree of exchange rate pass through among developed economies, Campa and Goldberg (2002) find that short run exchange rate pass through is incomplete, while long run exchange rate pass through is complete.  This empirical evidence  rejects both local currency pricing, under which the domestic currency prices of imports are invariant to exchange rate fluctuations in the short run, and producer currency pricing, under which the domestic currency prices of imports fully reflect exchange rate fluctuations in the short run. In response to this empirical evidence, Monacelli (2005) incorporates short run import price rigidities into an open economy D S G E model by allowing for monopolistic competition and staggered reoptimization in the import market. These import price rigidities generate incomplete exchange rate pass through in the short run, while exchange rate pass through is complete in the long run. The economy is complex, and any model of it is necessarily misspecified to some extent. A n operational substitute for the concept of a correctly specified model is the concept of an empirically adequate model. A model is empirically adequate i f it approximately accounts for the existing empirical evidence in all measurable respects, which as discussed in Clements and Hendry (1998) does not require that it be correctly specified. A s argued by Diebold and Mariano (1995), a necessary condition for empirical adequacy is predictive accuracy, which must be measured in relative terms. Quantitative monetary policy analysis and inflation targeting should be based on empirically adequate models of the economy. Thus far, empirical evaluations of D S G E models have generally focused on unconditional second moment and impulse response properties. While empirically valid unconditional second moment and impulse response properties are necessary conditions for empirical adequacy, they are not sufficient. Moreover, empirical evaluations of unconditional second moment properties are generally conditional on atheoretic estimates of trend components, while empirical evaluations of impulse response properties are generally conditional on controversial identifying restrictions.  It follows that the empirical evaluation of predictive accuracy is a necessary  precursor to a well informed judgment regarding the extent to which any D S G E model can and should contribute to quantitative monetary policy analysis and inflation targeting. Existing D S G E models featuring long run balanced growth driven by trend inflation, productivity growth, and population growth generally predict the existence of common deterministic or stochastic trends. Estimated D S G E models incorporating common deterministic trends include those of Ireland (1997) and Smets and Wouters (2005), while estimated D S G E models incorporating common stochastic trends include those of Altig, Christiano, Eichenbaum  3 and Linde (2005) and Del Negro, Schorfheide, Smets and Wouters (2005).  However, as  discussed in Clements and Hendry (1999) and Maddala and K i m (1998), intermittent structural breaks  render  such common deterministic or stochastic  trends empirically inadequate  representations of low frequency variation in observed macroeconomic variables.  For this  reason, it is common to remove trend components from observed macroeconomic variables with deterministic polynomial functions or linear filters, such as the difference filter or the low pass fdter described in Hodrick and Prescott (1997), prior to the conduct of estimation, inference and forecasting. Decomposing observed macroeconomic variables into cyclical and trend components prior to the conduct of estimation, inference and forecasting reflects an emphasis on the predictions of D S G E models at business cycle frequencies.  Since such decompositions are additive, given  observed macroeconomic variables, predictions at business cycle frequencies imply predictions at lower frequencies.  A s argued by Harvey (1997), the removal of trend components from  observed macroeconomic variables with atheoretic deterministic polynomial functions or linear fdters ignores these predictions, potentially invalidating subsequent estimation, inference and forecasting.  A s an alternative, this paper proposes jointly modeling cyclical and trend  components as unobserved components while imposing theoretical restrictions derived from the approximate multivariate linear rational expectations representation of a D S G E model. The development of empirically adequate D S G E models for purposes of quantitative monetary policy analysis and inflation targeting in a small open economy is currently an active area of research.  Nevertheless, an estimated D S G E model of a small open economy which  approximately accounts for the empirical evidence concerning the monetary transmission mechanism, as summarized by impulse response functions derived from an estimated structural vector autoregressive or S V A R model, while dominating that S V A R model in terms of predictive accuracy, has yet to be developed. This paper develops and estimates a D S G E model of a small open economy which satisfies these impulse response and predictive accuracy criteria. The model features short run nominal price and wage rigidities generated by monopolistic competition and staggered reoptimization in output and labour markets. The resultant inertia in inflation and persistence in output is enhanced with other features such as habit persistence in consumption, adjustment costs in investment, and variable capital utilization.  Incomplete  exchange rate pass through is generated by short run nominal price rigidities in the import market, with monopolistically competitive importers setting the domestic currency prices of differentiated  intermediate  import  goods  subject  to  randomly  arriving reoptimization  opportunities. Cyclical components are modeled by linearizing equilibrium conditions around a stationary deterministic steady state equilibrium, while trend components are modeled as random  4  walks while ensuring the existence of a well defined balanced growth path. Parameters and trend components are jointly estimated with a novel Bayesian procedure. The organization of this paper is as follows. The next section develops a D S G E model of a small open economy. Estimation, inference and forecasting within the framework of a linear state space representation of an approximate unobserved components representation of this D S G E model are the subjects of section three.  Finally, section four offers conclusions and  recommendations for further research.  1.2. Model Development Consider two open economies which are asymmetric in size, but are otherwise identical. The domestic economy is of negligible size relative to the foreign economy.  1.2.1. The Utility Maximization Problem of the Representative Household There exists a continuum of households indexed by  ?'e[0,l].  Households supply  differentiated intermediate labour services, but are otherwise identical.  1.2.1.1. Consumption and Saving Behaviour The representative infinitely lived household has preferences defined over consumption C and labour supply f  s  s  represented by intertemporal utility function  CO  (1) where subjective discount factor B satisfies 0<B<1.  The intratemporal utility function is  additively separable and represents external habit formation preferences in consumption,  (C„-gC_ ) 1  <C, ,L, ) s  s  where 0 < a < 1.  >c  = v'  l  s  1-1/tT  | / g  {Lj 1+  1/7  (2)  This intratemporal utility function is strictly increasing with respect to  consumption i f and only i f v  c s  > 0 , and given this parameter restriction is strictly decreasing with  5  respect to labour supply i f and only i f v > 0.  Given these parameter restrictions, this  L  intratemporal utility function is strictly concave i f a > 0 and rj > 0. The representative household enters period 5 in possession of previously purchased domestic currency denominated bonds Bff which yield interest at risk free rate * ._,, and foreign currency 4  denominated bonds Bf/ which yield interest at risk free rate i/. portfolio of shares {xjj }  l  s  J=0  in domestic intermediate good firms which pay dividends {TJf}\ , s  and a diversified portfolio of shares {xf }[ k s  pay dividends {n/ }\ . s  It also holds a diversified  =0  in domestic intermediate good importers which  The representative household supplies differentiated  =0  =0  intermediate  labour service L. , earning labour income at nominal wage W . Households pool their labour js  income, and the government levies a tax on pooled labour income at rate r . These sources of s  private wealth are summed in household dynamic budget constraint:  b:/+£X./- \vjXj^+  K  C  ^  ^  I  H  J  C  ^  O  ^  ,  )  ^  (3) k=0  7=0  1=0  According to this dynamic budget constraint, at the end o f period s, the representative household purchases domestic bonds Bf/, and foreign bonds Bf/ at price £ . It also s  purchases a diversified portfolio o f shares {x]j }  X  S+X  {y] } o> >d a diversified portfolio of shares {x/ l  tS  ar  J=  k J+1  J=0  }[  =0  in intermediate good firms at prices in intermediate good importers at prices  {V } . Finally, the representative household purchases final consumption good C M  k  ]  s  P -  k=0  ls  at price  '  C  S  In  period  t, the representative  household  chooses  state contingent  sequences for  consumption {C,.^}™,, domestic bond holdings {Bff }™ , foreign bond holdings {Bf/}^,, share +]  holdings in intermediate good firms {{xl } }™ , l  JiS+l  J=0  =t  =l  and share holdings in intermediate good  importers {{x^ y }^, to maximize intertemporal utility function (1) subject to dynamic ks+]  k=0  budget constraint (3) and terminal nonnegativity constraints Bfj > 0 , Bf/ > 0 , xj'. h  +i  xf  k r + 1  X  r + 1  > 0 and  > 0 for T - > co . In equilibrium, selected necessary first order conditions associated with  this utility maximization problem may be stated as u (C ,L.,) = P A  (4)  c  c  l  l  n  4=/?Cl + ' , ) E , A ,  + 1  (5)  ,  £,A, = B(l + if)E,S A , l+l  l+l  (6)  6  0=/rc,(/tf, , +  +  C.H..  () 8  +  denotes the Lagrange multiplier associated with the period 5 household dynamic  where k  is  budget constraint.  In equilibrium, necessary complementary slackness conditions associated  with the terminal nonnegativity constraints may be stated as: l i m £ ^ / &  = 0 ,  (9)  hm £^£, X;L=o,  (10)  +  I  +  hm £ ^ y !  li ^  +  T  X  yM  m  T  i  r  =,  (11)  0  U  Q  ( L 2  )  A,  Provided that the intertemporal utility function is bounded and strictly concave, together with all necessary first order conditions, these transversality conditions are sufficient for the unique utility maximizing state contingent intertemporal household allocation. Combination of necessary first order conditions (4) and (5) yields intertemporal optimality condition  u ( C , L. ) = BE, (1 + /,) |1 u (C, , L. ), c  t  c  +1  l+l  (13)  which ensures that at a utility maximum, the representative household cannot benefit from feasible intertemporal consumption reallocations. Finally, combination of necessary first order conditions (4), (5) and (6) yields intratemporal optimality condition  E , W ^ j ) . £ , ^ ^ (14) W  which equates the expected present discounted values of the gross real returns on domestic and foreign bonds.  7  1.2.1.2. Labour Supply and Wage Setting Behaviour There exist a large number of perfectly competitive firms which combine differentiated intermediate labour services L  i:  supplied by households in a monopolistically competitive  labour market to produce final labour service L according to constant elasticity of substitution t  production function  \(L.,) '  di  e  where Of > 1.  (15)  The representative final labour service firm maximizes profits derived from  production of the final labour service n, =w,L,-  jW.,L.,di,  L  (16)  with respect to inputs of intermediate labour services, subject to production function (15). The necessary  first order conditions associated with this profit maximization problem yield  intermediate labour service demand functions:  A, =  (17)  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final labour service firm earns zero profit, implying aggregate wage index:  \{Wj~ 'di  (18)  e  As the wage elasticity of demand for intermediate labour services  increases, they become  closer substitutes, and individual households have less market power. In an extension of the model of nominal wage rigidity proposed by Erceg, Henderson and Levin (2000) along the lines of Smets and Wouters (2003, 2005), each period a randomly selected fraction l-eo  L  of households adjust their wage optimally. The remaining fraction a>  L  8 of households adjust their wage to account for past consumption price inflation according to partial indexation rule f  n C  Y " f pC  \ ' X  r  W,. =  pC \ t-2 r  w,i,t-\'  (19)  J  where 0 < y < 1. Under this specification, although households adjust their wage every period, L  they infrequently adjust their wage optimally, and the interval between optimal wage adjustments is a random variable. If the representative household can adjust its wage optimally in period t, then it does so to maximize intertemporal utility function (1) subject to dynamic budget constraint (3), intermediate labour service demand function (17), and the assumed form of nominal wage rigidity.  Since all households that adjust their wage optimally in period t solve an identical  utility maximization problem, in equilibrium they all choose a common wage W* given by necessary first order condition:  i-l  r  '  c( n ;,,)  u  C  L  c(*>A>)  M  C  w.  Y  (p }  J c r  (p ^ pC  c  i-r"  i-\  r  pc (  pC*) \-yi-\ pc  w, w,  (K) (20)  r  w,  This necessary first order condition equates the expected present discounted value of the consumption benefit generated by an additional unit of labour supply to the expected present discounted value of its leisure cost. Aggregate wage index (18) equals an average of the wage set by the fraction 1 - co of households that adjust their wage optimally in period t, and the L  average of the wages set by the remaining fraction co of households that adjust their wage L  according to partial indexation rule (19):  r  W. =(\-a> w;y- L  e  +co  L  l-\ pc r  t-\ pC r  w.( - 1  (21)  Since those households able to adjust their wage optimally in period t are selected randomly from among all households, the average wage set by the remaining households equals the value of the aggregate wage index that prevailed during period t -1, rescaled to account for past consumption price inflation.  9  1.2.2. The Value Maximization Problem of the Representative Firm  There exists a continuum of intermediate goodfirmsindexed by j e [0,1]. Intermediate good firms supply differentiated intermediate output goods, but are otherwise identical. Entry into and exit from the monopolistically competitive intermediate output good sector is prohibited.  1.2.2.1. Employment and Investment Behaviour  The representative intermediate good firm sells shares {xj }J V+1  price V . Recursive forward substitution for Vj Y  }f  t+S  =0  to domestic households at  with s > 0 in necessaryfirstorder condition  (7) applying the law of iterated expectations reveals that the post-dividend stock market value of the representative intermediate good firm equals the expected present discounted value of future dividend payments:  K  ^  t ^ K s=l+\  -  (22)  Acting in the interests of its shareholders, the representative intermediate goodfirmmaximizes its pre-dividend stock market value, equal to the expected present discounted value of current and future dividend payments:  The derivation of result (22) imposes transversality condition (11), which rules out self-fulfilling speculative asset price bubbles. Shares entitle households to dividend payments equal to net profits  TlJ , s  defined as after tax  earnings less investment expenditures:  n], = a - *, )(pI j, ~ s j,s ) - p!i, • y  w  (24)  l  Earnings are defined as revenues derived from sales of differentiated intermediate output good Y  js  at price Pj less expenditures onfinallabour service L . The government levies a tax on s  j s  earnings at rate r , and negative dividend payments are a theoretical possibility. t  10 The representative intermediate good firm utilizes capital K at rate u s  labour service L  js  and rents final  given labour augmenting technology coefficient A to produce differentiated  js  s  intermediate output good Y according to constant elasticity of substitution production function js  9-\  9-\  ((py(u K )° hs  where 0 <  9-\  +(\-(py(A L )°  s  s  (25)  Js  1, & >0 and A > 0. This constant elasticity of substitution production function s  exhibits constant returns to scale, and nests the production function proposed by Cobb and Douglas (1928) under constant returns to scale for 3 = 1 In utilizing capital to produce output, the representative intermediate good firm incurs a cost G(Uj ,K ) denominated in terms of output: s  s  Y. =F(u. K ,A L. )-G{u. ,K ). s  s  s  s  s  s  (26)  s  Following Christiano, Eichenbaum and Evans (2005), this capital utilization cost is increasing in the rate of capital utilization at an increasing rate, G(u. ,K ) = s  s  M  [e^-l]K  (27)  s  where /j. > 0 and k > 0 . In deterministic steady state equilibrium, the rate of capital utilization is normalized to one, and the cost of utilizing capital equals zero. Capital is endogenous but not firm-specific, and the representative intermediate good firm enters period s with access to previously accumulated capital stock K , which subsequently s  evolves according to accumulation function K^={\-S)K H{I„I,_ \ I+  (28)  X  where depreciation rate parameter 8 satisfies 0 < S < 1. Following Christiano, Eichenbaum and Evans (2005), effective investment function H(I ,I _ ) incorporates convex adjustment costs, S  i-i  f  n(i ,i _ ) s  s  l  =v  !  S  X  v  (29)  s  V  J  *.v-l  where ^ > 0 and v' > 0 . In deterministic steady state equilibrium, these adjustment costs equal s  zero, and effective investment equals actual investment. ' Invoking  [.'Hospital's rule yields  Mm T(u j A , Aj  =  l i m InT(u ,K , A L ) h  s  is  <p-*(\-py'^ (u^KJ(A,L J''. )  j  js  = q>ln(i/^/(,) + (1 -<p)\n(A„L )-<p]n<p-([-ip)\n(\-<p), u  which implies that  11 In period t, the representative intermediate good firm chooses state contingent sequences for employment {L }™ , is  capital utilization {u }™ ,  =l  js  investment  =l  a  ° d the capital stock  maximize pre-dividend stock market value (23) subject to net production function  t 0  (26), capital accumulation function (28), and terminal nonnegativity constraint K  T+l  T -> co.  > 0 for  In equilibrium, demand for the final labour service satisfies necessary first order  condition  ^M , K ,A L ,)0.^(\-T )^-, j t  t  l  where P <t> Y  s  j  (30)  l  denotes the Lagrange multiplier associated with the period s production  JS  technology constraint. This necessary first order condition equates real marginal cost <P., to the ratio of the after tax real wage to the marginal product of labour. In equilibrium, the rate of capital utilization satisfies necessary first order condition  ^  h  l  K , A L ^  = ^f^-,  (31) A,  which equates the marginal product of utilized capital to its marginal cost.  In equilibrium,  demand for the final investment good satisfies necessary first order condition Qft (I,,/,_,) + E, ^  Q ( 7 , , 1 , ) = P/, + 1  (32)  + 1  which equates the expected present discounted value of an additional unit of investment to its price, where Q  js  denotes the Lagrange multiplier associated with the period s capital  accumulation function. In equilibrium, this shadow price of capital satisfies necessary first order condition  Q, = E, ^ { / , > , , , + ,  ,  )-  Qk ( " , V , > ) ] +  + (!-  }  >  which equates it to the expected present discounted value of the sum of the future marginal cost of capital, and the future shadow price of capital net of depreciation.  In equilibrium, the  necessary complementary slackness condition associated with the terminal nonnegativity constraint may be stated as:  lim ^±LQ K i+T  =0.  l+T+l  (34)  (33)  12  Provided that the pre-dividend stock market value of the representative intermediate good firm is bounded  and strictly concave, together with all necessary first order  transversality  condition is sufficient for the unique  conditions, this  value maximizing state  contingent  intertemporal firm allocation.  1.2.2.2. Output Supply and Price Setting Behaviour There exist a large number of perfectly competitive firms which combine differentiated intermediate output goods Y  jt  supplied by intermediate good firms in a monopolistically  competitive output market to produce final output good Y according to constant elasticity of t  substitution production function  e,-\ (35)  Y =  where 0j > 1.  The representative final output good firm maximizes profits derived from  production of the final output good n]=PjY,-  \pj^,dj,  (36)  j=0 with respect to inputs of intermediate output goods, subject to production function (35).  The  necessary first order conditions associated with this profit maximization problem yield intermediate output good demand functions:  Y.  (37)  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final output good firm earns zero profit, implying aggregate output price index:  1-0/  (38)  13 As the price elasticity of demand for intermediate output goods 6  Y  increases, they become closer  substitutes, and individual intermediate good firms have less market power. In an extension of the model of nominal output price rigidity proposed by Calvo (1983) along the lines of Smets and Wouters (2003, 2005), each period a randomly selected fraction l-co  Y  of  intermediate good firms adjust their price optimally. The remaining fraction co of intermediate Y  good firms adjust their price to account for past output price inflation according to partial indexation rule ( nr K  i-r  Y  r  =  1  where 0 < y < 1. Y  (39)  j,t-\'  Under this specification, optimal price adjustment opportunities arrive  randomly, and the interval between optimal price adjustments is a random variable. If the representative intermediate good firm can adjust its price optimally in period t, then it does so to maximize to maximize pre-dividend stock market value (23) subject to net production function (26), capital accumulation function (28), intermediate output good demand function (37), and the assumed form of nominal output price rigidity. Since all intermediate good firms that adjust their price optimally in period t solve an identical value maximization problem, in equilibrium they all choose a common price P '  given by necessary first order condition:  Y  E,I(*/) E,I(*/)  ( y >r ( W ^ t-\ i-\ P P P r  X,  '  '  J s  Y  x,  (0 -W-T,) Y  y  p  Y  Y  ( pY ^ t-\ P r  J  ( p *\  r  Y  t~\ p  f  r  Y  P ''} 1 P  p'  (40)  Y  Y  PjY.  This necessary first order condition equates the expected present discounted value of the after tax revenue benefit generated by an additional unit of output supply to the expected present discounted value of its production cost. Aggregate output price index (38) equals an average of the price set by the fraction 1 - co of intermediate good firms that adjust their price optimally in Y  period t, and the average of the prices set by the remaining fraction co of intermediate good Y  firms that adjust their price according to partial indexation rule (39):  P  Y  =  n - ^ x p r r -  +co  y  f nY \  f nY  (41) V<-2  J  V''-2  J  14  Since those intermediate good firms able to adjust their price optimally in period t are selected randomly from among all intermediate good firms, the average price set by the remaining intermediate good firms equals the value of the aggregate output price index that prevailed during period t -1, rescaled to account for past output price inflation.  1.2.3. The Value Maximization Problem of the Representative Importer There exists a continuum of intermediate good importers indexed by k e [0,1]. Intermediate good importers supply differentiated intermediate import goods, but are otherwise identical. Entry into and exit from the monopolistically competitive intermediate import good sector is prohibited.  1.2.3.1. The Real Exchange Rate and the Terms of Trade The representative intermediate good importer sells shares {xf }) kl+l  at price V . kl  Recursive forward substitution for V  M k  l+s  =0  to domestic households  with s > 0 in necessary first order  condition (8) applying the law of iterated expectations reveals that the post-dividend stock market value of the representative intermediate good importer equals the expected present discounted value of future dividend payments:  ^=E,I^<.  (42)  Acting in the interests of its shareholders, the representative intermediate good importer maximizes its pre-dividend stock market value, equal to the expected present discounted value of current and future dividend payments:  The derivation of result (42) imposes transversality condition (12), which rules out self-fulfilling speculative asset price bubbles. Shares entitle households to dividend payments equal to gross profits TI , ks  earnings less fixed costs:  defined as  15  K  =P ,M  - £ Pj M  M  k  - r,.  J  Ks  s  ktS  (44)  Earnings are defined as revenues derived from sales o f differentiated intermediate import good at price P  M  ks  less expenditures o n foreign final output good M .  M k  s  T h e representative  k<s  intermediate good importer purchases the foreign final output good at domestic currency price £ Pj'  f  s  and differentiates it, generating zero gross profits on average.  The l a w o f one price asserts that arbitrage transactions equalize the domestic currency prices o f domestic imports and foreign exports. Define the real exchange rate,  £  P  YJ  Q  (  =  4  >  5  w h i c h measures the price o f foreign output i n terms o f domestic output. A l s o define the terms o f trade, pM  7>-^F>  (46)  S  w h i c h measures the price o f imports i n terms o f exports. V i o l a t i o n o f the l a w o f one price drives a wedge *F = £ Pj'  IP/  f  S  s  between the real exchange rate and the terms o f trade,  S>=^.  (47)  where the domestic currency price o f exports satisfies Pf = Pj . U n d e r the l a w o f one price and the real exchange rate and terms o f trade coincide.  *F =\, S  There exist a large number o f perfectly competitive firms w h i c h combine a domestic intermediate good Z  e {C ,I ,G }  ht  hl  hJ  hJ  and a foreign intermediate good Z  fJ  e {C ,I ,G } ft  ft  fl  to  produce final good Z , e {C,, /,, G,} according to constant elasticity o f substitution production function  (48)  Z, =  where 0 < ^< 1 , \f/ >\ and v/ Z  >0.  T h e representative final good firm maximizes profits  derived from production o f the final good  n?=P?Z,-P?Z -P Z , M  h  l  fJ  (49)  16 with respect to inputs of domestic and foreign intermediate goods, subject to production function (48).  The necessary first order conditions associated with this profit maximization problem  imply intermediate good demand functions: ( pY  Y*  I  "h,l  (50)  f  pM  \  I  Z =(\-c/> ) z  ft  \  I  MpZ  I  (51)  ,M '  t J  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final good firm earns zero profit, implying aggregate price index: f  r>M\  \  t  P =  (52)  z  J  Combination of this aggregate price index with intermediate good demand functions (50) and (51) yields:  (53) \t  J  v  f  Z =(\-<t> ) z  f  (1-^ ) + ^ Z  Y~  l  T  Z  ..M \i v  (54)  J  These demand functions for domestic and foreign intermediate goods are directly proportional to final good demand, with a proportionality coefficient that varies with the terms of trade.  1.2.3.2. Import Supply and Price Setting Behaviour There exist a large number of perfectly competitive firms which combine differentiated intermediate import goods M  kl  supplied by intermediate good importers in a monopolistically  competitive import market to produce final import good M , according to constant elasticity of substitution production function  17  M. =  where <9 > 1.  (55)  Jit  J(^*,)~  The representative final import good firm maximizes profits derived from  w  (  production of the final import good  n?=p, M,M  \p M dk, M  k  t  (56)  kl  with respect to inputs of intermediate import goods, subject to production function (55). necessary first order conditions associated  The  with this profit maximization problem yield  intermediate import good demand functions: k,l  r  M,.  (57)  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final import good firm earns zero profit, implying aggregate import price index: 1  \(P"V-dk  (58)  k=0 As the price elasticity of demand for intermediate import goods $  M t  increases, they become  closer substitutes, and individual intermediate good importers have less market power. In an extension of the model of nominal import price rigidity proposed by Monacelli (2005) along the lines of Smets and Wouters (2003, 2005), each period a randomly selected fraction 1 - co  M  of intermediate good importers adjust their price optimally. The remaining fraction &>  M  of intermediate good importers adjust their price to account for past import price inflation according to partial indexation rule  pM where 0 < y  M  1  *,/-!»  (59)  < 1. Under this specification, the probability that an intermediate good importer  has adjusted its price optimally is time dependent but state independent. If the representative intermediate good importer can adjust its price optimally in period t, then it does so to maximize to maximize pre-dividend stock market value (43) subject to  18 intermediate import good demand function (57), and the assumed form of nominal import price rigidity. Since all intermediate good importers that adjust their price optimally in period t solve an identical value maximization problem, in equilibrium they all choose a common price  P ' M  t  given by necessary first order condition  E,Z(^) E,Z(^) s-l  pM,* I pM  where W = S P '  Y f  s  s  s  I Pf  'A  (  P  M  ^  QMy  pM  Vr i - i f  pM t-\  r  )  Y  p ")  (  M  pM  \  s-l )  pM pM 1  V  (  pM* 1  \  P M.. M  pM \  >  J  r  p M * \ I  (  r  V  pM  measures real marginal cost.  -, (60)  pM  V i  J  P M, M  J  r  This necessary first order condition  equates the expected present discounted value of the revenue benefit generated by an additional unit of import supply to the expected present discounted value of its production cost. Aggregate import price index (58) equals an average of the price set by the fraction \-co  M  of intermediate  good importers that adjust their price optimally in period t, and the average of the prices set by the remaining fraction co  of intermediate good importers that adjust their price according to  M  partial indexation rule (59): 1_  1-0," f  pM  _  (\-a) )(P, ') - ' M  M  1  0  r>M\  +o)"  (61)  Since those intermediate good importers able to adjust their price optimally in period t are selected randomly from among all intermediate good importers, the average price set by the remaining intermediate good importers equals the value of the aggregate import price index that prevailed during period r - 1 , rescaled to account for past import price inflation.  1.2.4. Monetary and Fiscal Policy The government consists of a monetary authority and a fiscal authority.  The monetary  authority implements monetary policy, while the fiscal authority implements fiscal policy.  19 1.2.4.1. The Monetary A uthority The monetary authority implements monetary policy through control of the nominal interest rate according to monetary policy rule i,-i,= where  - Jcf) + ? (In Y- In  r  > 1 and % > 0 . Y  deterministic  steady  contemporaneous  state  + V,  (62)  As specified, the deviation of the nominal interest rate from its equilibrium value  is  a  linear  increasing  function  of  the  deviation of consumption price inflation from its target value, and the  contemporaneous proportional deviation of output from its deterministic steady state equilibrium value. Persistent departures from this monetary policy rule are captured by serially correlated monetary policy shock v\.  1.2.4.2. The Fiscal Authority The fiscal authority implements fiscal policy through control of nominal government consumption and the tax rate applicable to the pooled labour income of households and the earnings of intermediate good firms. In equilibrium, this distortionary tax collection framework corresponds to proportional output taxation. The ratio of nominal government consumption to nominal output satisfies fiscal expenditure rule  (63)  where C, <0. G  As specified, the proportional deviation of the ratio of nominal government  consumption to nominal output from its deterministic steady state equilibrium value is a linear decreasing function of the contemporaneous proportional deviation of the ratio of net foreign debt to nominal output from its target value. This fiscal expenditure rule is well defined only i f the net foreign debt is positive.  Persistent departures from this fiscal expenditure rule are  captured by serially correlated fiscal expenditure shock v , . c  The tax rate applicable to the pooled labour income of households and the earnings of intermediate good firms satisfies fiscal revenue rule  20  In r, - In  where £" >0. r  t, =  5,a  In  \ (64)  -In  v  A s specified, the proportional deviation of the tax rate from its deterministic  steady state equilibrium value is a linear increasing function of the contemporaneous proportional deviation o f the ratio of net government debt to nominal output from its target value.  This fiscal revenue rule is well defined only i f the net government debt is positive.  Persistent departures from this fiscal revenue rule are captured by serially correlated fiscal revenue shock v]. The fiscal authority enters period t holding previously purchased domestic currency denominated bonds Bf'  which yield interest at risk free rate /,_,, and foreign currency  denominated bonds Bf'  which yield interest at risk free rate if_ . It also levies taxes on the  h  f  {  pooled labour income of households and the earnings o f intermediate good firms at rate r,. These sources of public wealth are summed in government dynamic budget constraint: + Erf'* = (1 + / )Bf* + £, (1 + ) B ? - ' M  \  i  i  (65)  =0/=0  According to this dynamic budget constraint, at the end of period t, the fiscal authority purchases domestic bonds Bf f, and foreign bonds B°\  f  +  government consumption good G, at price  at price £ . It also purchases final t  .  1.2.5. Market Clearing Conditions A rational expectations equilibrium in this D S G E model of a small open economy consists of state contingent intertemporal allocations for domestic and foreign households and firms which solve their constrained optimization problems given prices and policy, together with state contingent intertemporal allocations for domestic and foreign governments which satisfy their policy rules and constraints given prices, with supporting prices such that all markets clear. Since the domestic economy is of negligible size relative to the foreign economy, in equilibrium prj  =  CJ .J= o,f = J and £= Mf_  P  =P  p  pf  *t  =  ^  =  I  Q  1 t  Clearing of the final output good market requires that exports X  t  equal production o f the  domestic final output good less the cumulative demands of domestic households, firms, and the government,  21 (66)  X,=*-Cn-ln-G , hJ  where X = MJ . Clearing of the final import good market requires that imports M satisfy the t  t  cumulative demands of domestic households, firms, and the government for the foreign final output good,  (67)  M,=C +I + G , fJ  where M = Xf.  fJ  fJ  In equilibrium, combination of these final output and import good market  t  clearing conditions yields aggregate resource constraint:  PY y  i  i  +P,'I,+ F?G  = tfc,  +P, X,- Pt  M  (68)  M  X  t  r  The trade balance equals export revenues less import expenditures, or equivalently nominal output less domestic demand. Let B  denote the net foreign asset position of the economy, which in equilibrium equals  l+]  the sum of the domestic currency values of private sector bond holdings B^ = x  + S B^{ t  and  public sector bond holdings Bf = Bf f + £,B°'/ , since domestic bond holdings cancel out when +l  +  the private and public sectors are consolidated:  * =2£ *, .  (69)  C  L+1  1+  +1  The imposition of equilibrium conditions on household dynamic budget constraint (3) reveals that the expected present discounted value of the net increase in private sector asset holdings equals the expected present discounted value of private saving less domestic investment:  HI-t,)P/Y, -Ifq-^I,].  - *,') = E,_,  EM l~\  (70)  t-\  A  A  The imposition of equilibrium conditions on government dynamic budget constraint (65) reveals that the expected present discounted value of the net increase in public sector asset holdings equals the expected present discounted value of public saving:  E  M  ^ V i  (  ^  i  -  ^  C  )  =  E . ^(i . flf+r ^i;-/fG ). l  f  1  I  f  f  (71)  V i  Combination of these household and government dynamic budget constraints with aggregate resource constraint (68) reveals that the expected present discounted value of the net increase in foreign asset holdings equals the expected present discounted value of the sum of net  22 international investment income and the trade balance, or equivalently the expected present discounted value of national saving less domestic investment: E,-,  -B,) = E,_,  (/,_,*, + P, X, - P/M,).  (72)  X  In equilibrium, the current account balance is determined by both intratemporal and intertemporal optimization.  1.2.6. The Approximate Linear Model Estimation, inference and forecasting are based on a linear state space representation of an approximate unobserved components representation of this D S G E model o f a small open economy.  Cyclical components are modeled by linearizing equilibrium conditions around a  stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path. In what follows, E, x  denotes the rational expectation of variable x , conditional on  l+s  l+s  information available at time t. Also, jc, denotes the cyclical component of variable x,, while x denotes the trend component of variable x . Cyclical and trend components are additively t  (  separable, that is x - x + x . t  t  t  1.2.6.1. Cyclical Components The cyclical component of output price inflation depends on a linear combination of past and expected future cyclical components of output price inflation driven by the contemporaneous cyclical components of real marginal cost and the tax rate according to output price Phillips curve  l + y'fi  \+yB Y  ' '  +l  co (\ + y p) Y  Y  XnO, + '  T ,l n.f , - — I  l - r  '  \nO 0 -I '  Y  (73)  Y  0 —l where 0 = (l - t)-^- . The persistence of the cyclical component of output price inflation is increasing in indexation parameter y , while the sensitivity of the cyclical component of output Y  price inflation to changes in the cyclical components of real marginal cost and the tax rate is  23  decreasing in nominal rigidity parameter of and indexation parameter y . Y  This output price  Phillips curve is subject to output price markup shocks. The cyclical component of output depends on the contemporaneous cyclical components of utilized capital and effective labour according to approximate linear net production function 6  WL  Y  InK = 1  6 WL Hu,K,) + 7^7—j—HA,L,), Y  G -\PY Y  where y = j ^ J ^ ~ 'pj)' output technology shocks.  a  PP  r o x  i  (74)  linear net production function is subject to  m a t e  The cyclical component of the rate of capital utilization depends on the contemporaneous cyclical component of the ratio of capital to effective labour according to approximate linear implicit capital utilization function: 0  WL  1  lnw, = —  k9 + -  0 -\PY  9  WL  Y  6  Y  -\PY  Y  t:  Xn  ( 7 5 )  The sensitivity of the cyclical component of the rate of capital utilization to changes in the cyclical component of the ratio of capital to effective labour is decreasing in capital utilization cost parameter k and elasticity of substitution parameter &. This approximate linear implicit capital utilization function is subject to output technology shocks. The cyclical component of consumption, investment, or government consumption price inflation depends on a linear combination of past and expected future cyclical components of consumption,  investment,  or  government  consumption  price  inflation  driven  by  the  contemporaneous cyclical components of real marginal cost and the tax rate according to Phillips curves:  I  /l  , .  Y  \ +yB Y  n  i-\  n  +  .\  7 T 7 ^  A  r  1  Y i-t  B  r +  » t  '  +  co {\  M  Y  <  ,  ^  )  A  1  p)  Y +  r  " # - T W  ln<P,+ — l n f , ' 1-r '  E  '  A  l  ^—\nO 6 -\ '  Y  Y  (76)  n  Reflecting the entry of the price of imports into the aggregate consumption, investment, or government consumption price index, the cyclical component of consumption, investment, or government consumption price inflation also depends on past, contemporaneous, and expected future proportional changes in the cyclical component of the terms o f trade. These Phillips curves are subject to output price markup and import technology shocks.  24 The cyclical component of consumption depends on a linear combination of past and expected future cyclical components of consumption driven by the contemporaneous cyclical component of the real interest rate according to approximate linear consumption Euler equation: InC, = 7 ^ 0 ,  +  ^-E, l +a  l n C , — cr——— r, + E, l n - ^ \ +a v. c  (77)  +l  The persistence of the cyclical component of consumption is increasing in habit persistence parameter a , while the sensitivity of the cyclical component of consumption to changes in the cyclical component of the real interest rate is increasing in intertemporal elasticity of substitution parameter  a  and decreasing in habit persistence parameter  a.  This approximate linear  consumption Euler equation is subject to preference shocks. The cyclical component of investment depends on a linear combination of past and expected future cyclical components of investment driven by the contemporaneous cyclical component of the relative shadow price of capital according to approximate linear investment demand function: In/,  1 \ +P  ln/^,+  J3  6 ^  (  E,ln/,  1 + /?  + 1  +  -In V  (78)  P>' i J  r  The sensitivity of the cyclical component of investment to changes in the cyclical component of the relative shadow price of capital is decreasing in investment adjustment cost parameter x • This approximate linear investment demand function is subject to investment technology shocks. The cyclical component of the relative shadow price of capital depends on the expected future cyclical component of the relative shadow price of capital, the contemporaneous cyclical component of the real interest rate, the expected future cyclical component of real marginal cost, and the expected future cyclical component of the marginal product of capital according to approximate linear investment Euler equation: l n 4 | = /?(l-<f)E,ln OH  + [1-/?(!-!$)] E, In <P, • +1  \-B{\-8) 3  (79)  e  Y  0-1  PY  4+i 4+i  The sensitivity of the cyclical component of the relative shadow price of capital to changes in the cyclical component of the ratio of utilized capital to effective labour is decreasing in elasticity of substitution parameter &. This approximate linear investment Euler equation is subject to output technology shocks.  25 The cyclical component of the capital stock depends on the past cyclical component of the capital stock and the contemporaneous  cyclical component  of investment  according to  approximate linear capital accumulation function mtf,  + 1  =(l-^ln£,+£ta(i?,7,),  where ~ = 5.  (80)  This approximate linear capital accumulation function is subject to investment  technology shocks. The cyclical component of the ratio of nominal government consumption to nominal output depends on the contemporaneous cyclical component of the ratio of net foreign debt to nominal output according to fiscal expenditure rule:  In  P G,  ••c In G  pYy  B,  (81)  pry  This fiscal expenditure rule ensures convergence of the level of the ratio of net foreign debt to nominal output to its target value in deterministic steady state equilibrium, and is subject to fiscal expenditure shocks. The cyclical component o f the tax rate depends on the contemporaneous cyclical component of the ratio of net government debt to nominal output according to fiscal revenue rule: lnr, =^ ln r  5, pry 1 I  + V .  (82)  1 iJ  This fiscal revenue rule ensures convergence of the level of the ratio of net government debt to nominal output to its target value in deterministic steady state equilibrium, and is subject to fiscal revenue shocks. The cyclical component of import price inflation depends on a linear combination of past and expected future cyclical components of import price inflation driven by the contemporaneous cyclical component of the deviation of the domestic currency price of foreign output from the price of imports according to import price Phillips curve: Y  •71, , +•  (\-m )(\-co B) M  P \  B  M +  r  M  co (l + y {3) M  M  £ P l n ^ —  1  YJ  6>  A  -lnfl  A  .(83)  The persistence of the cyclical component of import price inflation is increasing in indexation parameter y , M  while the sensitivity of the cyclical component of import price inflation to  changes in the cyclical component of real marginal cost is decreasing in nominal rigidity  26 parameter a> and indexation parameter y . M  This import price Phillips curve is subject to  M  import price markup shocks. The cyclical component of exports depends on the contemporaneous cyclical components of foreign consumption, investment, government consumption, and the terms of trade according to approximate linear export demand function X Y I n J r Y Y F  -v x Y  f  C C = ( l - ^ ) ^ t a ^  4> {\ CJ  +  (  +  If l - ^ ) ± t a ^  - *' )U J  0 - ^ l n ^  +  G f ' d - <t> ) In I L  (84)  GJ  G G <> / y . The sensitivity of the cyclical component of exports to changes in the cyclical component of the foreign terms of trade is increasing in elasticity of where  Y  Y  u> Y  substitution parameter y/. This approximate linear export demand function is subject to foreign import technology shocks. The cyclical component of imports depends on the contemporaneous cyclical components of consumption, investment, government consumption, and the terms of trade according to approximate linear import demand function  (85) In  -V  where y = ( l - ^ ) y + ( l - ^ ' ) y + ( l - ^ ) y . c  G  The sensitivity of the cyclical component of  imports to changes in the cyclical component of the terms of trade is increasing in elasticity of substitution parameter \f/. This approximate linear import demand function is subject to import technology shocks. The cyclical component of the real wage depends on a linear combination of past and expected future cyclical components of the real wage driven by the contemporaneous cyclical component of the deviation o f the marginal rate of substitution between leisure and consumption from the after tax real wage according to wage Phillips curve:  27  Reflecting the existence of partial wage indexation, the cyclical component of the real wage also depends on past, contemporaneous, and expected future cyclical components of consumption price inflation.  The sensitivity of the cyclical component of the real wage to changes in the  cyclical component of consumption price inflation is increasing in indexation parameter y , to L  changes in the cyclical component of the deviation of the marginal rate o f substitution between leisure and consumption from the after tax real wage is decreasing in nominal rigidity parameter co , and to changes in the cyclical component of employment is decreasing in elasticity o f L  substitution parameter n. This wage Phillips curve is subject to wage markup shocks. The cyclical component of real marginal cost depends on the contemporaneous cyclical component of the deviation of the after tax real wage from the marginal product o f labour according to approximate linear implicit labour demand function: 1 ^ , , ln<2> = l n ^  r  w  P A, Y  . .  lnr,  1-r  1  '3  (  wl^ ,  e d -\PY Y  Y  l  n  ii.K. TfAL, it  (87)  The sensitivity o f the cyclical component of real marginal cost to changes in the cyclical component of the ratio of utilized capital to effective labour is decreasing in elasticity of substitution parameter 9. This approximate linear implicit labour demand function is subject to output technology shocks. The cyclical component of the nominal interest rate depends on the contemporaneous cyclical components of consumption price inflation and output according to monetary policy rule: /,=r^ +^ln^+i?;.  (88)  C  This monetary policy rule ensures convergence of the level of consumption price inflation to its target value in deterministic steady state equilibrium, and is subject to monetary policy shocks. The cyclical component of the output based real interest rate satisfies rj = i - E k  , while the  y  t  (  +]  cyclical component of the consumption based real interest rate satisfies r = i - E ;r, . c  t  c  t  (  +l  The cyclical component of the nominal exchange rate depends on the expected future cyclical component of the nominal exchange rate and the contemporaneous cyclical component of the nominal interest rate differential according to approximate linear uncovered interest parity condition: ln5,=E,ln£,  + 1  -(/,-//).  (89)  28 The cyclical component of the real exchange rate satisfies I n Q = In £ + \nPf  J  t  -\nPf,  while the  cyclical component of the terms of trade satisfies In T = In Pf - In Pf , where In Pf = In Pj . t  The cyclical component of nominal output depends on the contemporaneous cyclical components of nominal consumption, investment, government consumption, exports, and imports according to approximate linear aggregate resource constraint: \n(PfY ) = j\n(PfC ) l  l  + Un(P/l )  +^  l  (90)  In equilibrium, the cyclical component of output is determined by the cumulative demands o f domestic and foreign households, firms, and governments. The cyclical component o f the net government debt depends on the past cyclical component of the net government debt, the past cyclical component of the nominal interest rate, the contemporaneous cyclical component of tax revenues, and the contemporaneous  cyclical  component o f nominal government consumption according to approximate linear government dynamic budget constraint  E,-,ln(-5, )  r o\ B  G  +l  where  +  :  py \-p\ ~~y)' t  ^  (91)  tH^P/YJ-jH^G,)  PY  approximate linear government dynamic budget constraint is  s  well defined only i f the level of the net government debt is positive. The cyclical component of the net foreign debt depends on the past cyclical component of the net foreign debt, the past cyclical component of the nominal interest rate, the contemporaneous cyclical component o f export revenues, and the contemporaneous cyclical component o f import expenditures according to approximate linear national dynamic budget constraint _l_r  where  py  H-B,)  ~~^z^{y~y)'  + l  ^  s  +  a  PP  B  |ln(^i,)-^ln(^M,)  PY r o x U T i a t e  (92)  linear national dynamic budget constraint is well  defined only if tbe level of the net foreign debt is positive. Variation in cyclical components is driven by ten exogenous stochastic processes.  The  cyclical components of the preference, output technology, investment technology, import technology, output price markup, import price markup, wage markup, monetary policy, fiscal expenditure, and fiscal revenue shocks follow stationary first order autoregressive processes: lni?, =p c  Inift + < , ef ~ iid Af(0,(T ,), 2  (93)  29  In A, = p l n 4 _ , + ef  ef ~ iid  A  In v\ =  P y l  \nvf  In vf =  2  Indf+ef,  ef - i i d  Af(0,a ), 2  (97)  gt  M(Q,af),  +  l n # = / v ln4i, +  (98)  , ef ~ iid jV(0,oJ ),  (99)  < ~ i i d A/-(O,<T ,),  v;= ,v;_ ;\ Pv  (96)  M  \n0f=p ln0f ef,ef~M 9U  (95)  ~ iid JV(0, < ),  w  PfjV  (94)  + < ' , < ' ~ iid A/"(0, cr ,),  In v, , + < " , ef  \n9j=  Af(0,a]),  2  l+£  vf = P vf  + ef,  yG  v ,=p JU+ef  ef  ,ef  T  v  ~ iid ~M  (100)  Af{Q,af,),  (101)  yV(0,cx ). 2  (102)  The innovations driving these exogenous stochastic processes are assumed to be independent, which combined with our distributional assumptions implies multivariate normality. deterministic steady state equilibrium, v =v' =v c  <V U  =  = V  = °" 2  v  V  M  =1  and cr =a =cr 2  2  c  2  =a  2  In  =<j .= 2  = cr r = 0 . 2  V  1.2.6.2. Trend Components The trend components of the prices of output, consumption, investment, government consumption, and imports follow random walks with time varying drift n : In Pf = it, + In Pf + ef , ef In Pf = n, + In Pf + ef , ef  ~ iid A/"(0,a ,), 2  p  ~ iid  a , ), 2  In Pf = n, + In Pf + ef , ef ~ iid M(0,a ), pl  ef  ~ iid M(0,af  (104) (105)  2  In Pf = n, + In Pf + ef,  (103)  ),  (106)  30 In P  M t  = n, + In P™ + ef,  ef  ~ iid Af(0,a  2 ptl  ).  (107)  It follows that the trend components of the relative prices of consumption, investment, government consumption, and imports follow random walks without drifts. This implies that along a balanced growth path, the levels of these relative prices are time independent but state dependent. The trend components of output, consumption, investment, government consumption, exports, and imports follow random walks with time varying drift g, +n : l  lnF, = g, + n, + In l  + ef, ef ~ iid JV(0, a\),  7  InC, = g, + n, + In C,_, + ef, ef ~ iid M(0,  inT; = = g, + n, + In InG, lnX, InM,  (108)  a\),  (109)  + ef, ef ~ iid AA(0, a]),  (110) (111)  = g, + n, + In G,_, + ef, ef ~ iid A^(0,4),  (112)  = g, + «, + In + ^, ef ~ iid W(0,4),  (113)  jV(0,4).  = g , + « , + l n M , _ , + ^ , *~iid It follows that the trend components of the ratios of consumption, investment, government ff  consumption, exports, and imports to output follow random walks without drifts. This implies that along a balanced growth path, the levels of these great ratios are time independent but state dependent. The trend component of the nominal wage follows a random walk with time varying drift 7r,+ g , while the trend component of employment follows a random walk with time varying t  drift n : t  In W,=n +g + t  In W _ + ef,  t  InL,=n,+ I n I  t  M  x  + ef, ef ~ iid  ef ~ iid JV(0,a£), A  T  (114)  (  0  ,  (  1  1  5  )  It follows that the trend component of the income share of labour follows a random walk without drift. This implies that along a balanced growth path, the level of the income share of labour is time independent but state dependent.  The trend component of real marginal cost satisfies  ln<2> = ln<2>, while the trend component of the rate of capital utilization satisfies lnw, = 0 . The trend component of the shadow price of capital satisfies In Q = In Pf , while the trend component t  of the capital stock satisfies In ^p- = In % .  31 The trend components o f the nominal interest rate, tax rate, and nominal exchange rate follow random walks without drifts: T =J_ + ej, ej ~ iid A/"(0, a]), l  In  (116)  l  t, = In t,_  {  + ej, ej ~ iid Af(0, o\),  (117)  In £ = In S _ + ef, ef ~ iid JV(0, o £ ) . t  (118)  x  It follows that along a balanced growth path, the levels of the nominal interest rate, tax rate, and nominal exchange rate are time independent but state dependent. output based real interest rate satisfies r = /' - E nf, t  ;  consumption based real interest rate satisfies rf =i -E t  exchange rate satisfies InQ = l n £ , +lnP  rj l  x  x t  government debt satisfies l n l - ^ - l = lnl debt satisfies In I -  xf . The trend component of the real +i  - InFf, while the trend component of the terms of  trade satisfies In7^ = In Pf -\nP, , where lnP 1  t  The trend component of the  while the trend component of the  Y  = In Pf.  The trend component of the net  1, while the trend component of the net foreign  1 = In I - — I.  Long run balanced growth is driven by three common stochastic trends.  Trend inflation,  productivity growth, and population growth follow random walks without drifts: * , = * , _ , + < , < ~ i i d Af(0,crl),  (119)  g , = g , _ , + < , < ~ i i d AT(0,CT;),  (120)  n,=n,_ +e% e~  (121)  x  iid M(0, a]).  It follows that along a balanced growth path, growth rates are time independent but state dependent.  A l l innovations driving variation in trend components are assumed to be  independent,  which combined with  our distributional assumptions  implies multivariate  normality.  1.3. Estimation, Inference and Forecasting Unobserved components models feature prominently in the empirical macroeconomics literature, while D S G E models are pervasive in the theoretical macroeconomics literature. The primary contribution of this paper is the joint modeling of cyclical and trend components as  32 unobserved components while imposing theoretical restrictions derived from the approximate multivariate linear rational expectations representation of a D S G E model. This  merging  of modeling paradigms  drawn  from  the  theoretical  and empirical  macroeconomics literatures confers a number of important benefits. First, the joint estimation of parameters and trend components ensures their mutual consistency, as estimates of parameters appropriately reflect estimates of trend components, and vice versa. A s shown by Nelson and Kang (1981) and Harvey and Jaeger (1993), decomposing integrated observed endogenous variables into cyclical and trend components with atheoretic deterministic polynomial functions or low pass fdters may induce spurious cyclical dynamics, invalidating subsequent estimation, inference and forecasting. Second, basing estimation on the levels as opposed to differences of observed endogenous variables may be expected to yield efficiency gains. A central result of the voluminous cointegration literature surveyed by Maddala and K i m (1998) is that, i f there exist cointegrating relationships, then differencing all integrated observed endogenous variables prior to the conduct of estimation, inference and forecasting results in the loss of information. Third, the proposed unobserved components framework ensures stochastic nonsingularity of the resulting approximate linear state space representation of the D S G E model, as associated with each observed endogenous variable is at least one exogenous stochastic process. As discussed in Ruge-Murcia (2003), stochastic nonsingularity requires that the number of observed endogenous variables used to construct the loglikelihood function associated with the approximate linear state space representation of a D S G E model not exceed the number of exogenous stochastic processes, with efficiency losses incurred i f this constraint binds.  Fourth, the proposed unobserved  components framework facilitates the direct generation of forecasts o f the levels o f endogenous variables as opposed to their cyclical components together with confidence intervals, while ensuring that these forecasts satisfy the stability restrictions associated with balanced growth. These stability restrictions are necessary but not sufficient for full cointegration, as along a balanced growth path, great ratios and trend growth rates are time independent but state dependent, robustifying forecasts to intermittent structural breaks that occur within sample.  1.3.1. Estimation  The traditional econometric interpretation of macroeconometric models regards them as representations of the joint probability distribution of the data.  Adopting this traditional  econometric interpretation, Bayesian estimation of a linear state space representation of an approximate unobserved components representation of this D S G E model of a small open economy, conditional on prior information concerning the values of parameters and trend  33 components, facilitates an empirical evaluation of its impulse response and predictive accuracy properties.  1.3.1.1. Estimation Procedure Let x denote a vector stochastic process consisting of the levels of N nonpredetermined t  endogenous variables, of which M are observed.  The cyclical components of this vector  stochastic process satisfy second order stochastic linear difference equation AqX, =A\x,_ +A E,x x  2  +A v,,  l+l  (122)  3  where vector stochastic process v, consists of the cyclical components of K variables.  exogenous  This vector stochastic process satisfies stationary first order stochastic linear  difference equation (123)  v,  where s  — iid A/"(0,27,). The trend components of vector stochastic process x, satisfy first  u  order stochastic linear difference equation  C x, = C, + C u, + C *,_, + e „ Q  2  3  where e , - iid A/X0,27 ). 2  2  (124)  2  Vector stochastic process u consists of the levels of L common t  stochastic trends, and satisfies nonstationary first order stochastic linear difference equation =  where £ JC,  3 (  w  /-i +  xn  ( )  £  125  ~ iid A/"(0,27 ). 3  Cyclical and trend components are additively separable, that is  = Jc, + X, . If there exists a unique stationary solution to multivariate linear rational expectations model  (122), then it may be expressed as: x,=D,x +D v tX  2  (126)  r  This unique stationary solution is calculated with the matrix decomposition based algorithm due to Klein (2000). Let y,  denote a vector stochastic process consisting o f the levels of M  nonpredetermined endogenous variables.  observed  Also, let z, denote a vector stochastic process  34  consisting of the levels of N - M unobserved nonpredetermined endogenous variables, the cyclical components of N nonpredetermined endogenous variables, the trend components of N nonpredetermined endogenous variables, the cyclical components of K exogenous variables, and the levels of L common stochastic trends. Given unique stationary solution (126), these vector stochastic processes have linear state space representation (127) z, = (7, + G z,_ 2  where s  4j  +G e ,  l  3  ~ iid Af(0,Z ) 4  (128)  4j  and z ~ Af(z ,P ). 0  a]0  Let w, denote a vector stochastic process  QlQ  consisting of preliminary estimates of the trend components of M observed nonpredetermined endogenous variables. Suppose that this vector stochastic process satisfies (129) where s  Sj  ~ iid J\f(0,Z ). 5  Conditional on known parameter values, this signal equation defines  a set of stochastic restrictions on selected unobserved state variables.  The signal and state  innovation vectors are assumed to be independent, while the initial state vector is assumed to be independent from the signal and state innovation vectors, which combined with our distributional assumptions implies multivariate normality. Conditional on the parameters associated with these signal and state equations, estimates of unobserved state vector z, and its mean squared error matrix P may be calculated with the filter t  proposed by Vitek (2006a, 2006b), which adapts the filter due to Kalman (1960) to incorporate prior information. Given initial conditions z  0[Q  and P , estimates conditional on information 0|0  available at time t -1 satisfy prediction equations: t\t~\ -G  + G z,_\\i_ ,  z  x  2  l\i-\ 2 ,-\\ -\ l  P  =G  P  x  ^G^fi],  G  l  (130) (131) (132)  = FP F \ t\i-\ i ' 1  1  r  l  (133) (134)  R,  (135)  35 Given these predictions, under the assumption of multivariate normally distributed signal and state innovation vectors, together with conditionally contemporaneously uncorrelated signal vectors, estimates conditional on information available at time / satisfy updating equations Z<\< = V i  y (y> - JV-i)  +  K  +  », ( / - "V-i  K  w  )'  (136)  P ^ P ^ - K ^ P ^ - K ^ P ^ , where K  yi  = P Ff  (137)  Q^_ and K^ = P ^ H j .  l]/A  Given terminal conditions z  x  T]T  and P  7  obtained from the final evaluation of these prediction and updating equations, estimates conditional on information available at time T satisfy smoothing equations t\T ~ t\t Jt( i+\\T <v+l|r)'  (138)  P,T=P v+J iP^T-P ^)Jf  (139)  Z  Z  +  !  z  -  t  l  where J, = P^GjP/^  . Under our distributional assumptions, these estimators of the unobserved  state vector are mean squared error optimal. Let 0 e 0 cz K  y  denote a J dimensional vector containing the parameters associated with  the signal and state equations of this linear state space model. The Bayesian estimator of this parameter vector has posterior density function f(0\l )Kf(I \0)f(9), T  (140)  T  where I , = {{j }' ,{H> }', }. Under the assumption of multivariate normally distributed signal t  =1  v  =1  and state innovation vectors, together with conditionally contemporaneously uncorrelated signal vectors, conditional density function / ( 1  /(i  T  Under  10)=n r u i  t=\  T  z;-.. * ) • n  10) satisfies:  I  i=\  our distributional assumptions,  z,-> m  ( H O  conditional density  functions  f(yAX ,0) lA  and  / ( w , | 2 ~ , 0 ) satisfy: M  f(y,\l^,0)  = (2n)  f(w,\I,_ ,e)  = {27v)  x  2  ia | exp|--U-^. ) fi V. U-J',H)[. 2  T  | M  2  1  |J?, | H  2  f  (142)  1  exp\--(yv,-w^ f x  R^(w-w^)  .  (143)  36 Prior information concerning parameter vector 0 is summarized by a multivariate normal prior distribution having mean vector 0, and covariance matrix Q: f{0) = (2x) \Q\2  exp\~(0-0A Q (0-0 )\.  2  J  1  (144)  t  Independent priors are represented by a diagonal covariance matrix, under which diffuse priors are represented by infinite variances. Inference on the parameters is based on an asymptotic normal approximation to the posterior distribution around its mode.  Under regularity conditions stated in Geweke (2005), posterior  mode 0 satisfies T  ylf(0 -6 ) T  where 0 e0 o  Hessian  X A/-(0,-?C),  0  (145)  denotes the pseudotrue parameter vector. Following Engle and Watson (1981), may be estimated by  *r = ^ £ E  M  [ v X m / 0 > , |J ^ ^ l +J - ^ X , [ v X m / ( H >  \I,_J j T  (146)  where  V.V] l n / ( j , | I _J )] t  T  =- V  0  y l ^ y  o  y ^  -I  fi{_ (fir . ,  V f  1  1  ® <#_,  W Q,^ , e  and  v,v>M) = -ir'. 1.3.1.2. Estimation Results  The set o f parameters associated with this D S G E model of a small open economy is partitioned into two subsets.  The first subset is calibrated to approximately match long run  averages of functions of observed endogenous variables where possible, and estimates derived from existing microeconometric studies where necessary.  The second subset is estimated with  the Bayesian procedure described above, conditional on prior information concerning the values of parameters and trend components. Subjective discount factor (3 is restricted to equal 0.99, implying an annualized deterministic steady state equilibrium real interest rate of approximately 0.04.  In deterministic steady state  37 equilibrium, the output price markup — - , import price markup -gr- , and wage markup —are restricted to equal 1.15. Depreciation rate parameter 8 is restricted to equal 0.015, implying an annualized deterministic steady state equilibrium depreciation rate o f approximately 0.06. In deterministic steady state equilibrium, the consumption import share \-<jf, investment import share 1 - ^ ' , and government consumption import share 1 - ^  G  are restricted to equal 0.30. The  deterministic steady state equilibrium ratio of consumption to output — is restricted to equal 0.60, while the deterministic steady state equilibrium ratio of domestic output to foreign output yr is restricted to equal 0.11. In deterministic steady state equilibrium, the foreign consumption import share l - ^  c  /  ,  foreign investment import share  , and foreign government  consumption import share \-<fi are restricted to equal 0.02. The deterministic steady state GJ  equilibrium income share of labour — is restricted to equal 0.50. In deterministic steady state py  ^  equilibrium, the ratio of government consumption to output — is restricted to equal 0.20, while the tax rate v is restricted to equal 0.22. Table 1.1. Deterministic steady state equilibrium values of great ratios R  a  t  i  o  CIY  Value  Ratio  Value  0.6000  WLIPY  0.5000  HY  0.1723  KIY  2.8710  GIY  0.2000  B IPY  -0.4950  X  Y  0.3194  BIPY  -0.6866  MIY  0.2917  I  G  Note: Deterministic steady state equilibrium values are reported at an annual frequency based on calibrated parameter values.  Bayesian estimation o f the remaining parameters of this D S G E model of a small open economy is based on the levels of twenty six observed endogenous variables for Canada and the United States described in Appendix l . A . Those parameters associated with the conditional mean function are estimated subject to cross-economy equality restrictions. Those parameters associated exclusively with the conditional variance function are estimated conditional on diffuse priors. Initial conditions for the cyclical components of exogenous variables are given by their unconditional means and variances, while the initial values of all other state variables are treated as parameters,  and are calibrated to match functions of preliminary estimates o f trend  components calculated with the linear filter described in Hodrick and Prescott (1997). The posterior mode is calculated by numerically maximizing the logarithm of the posterior density kernel with a modified steepest ascent algorithm. Estimation results pertaining to the period 1971Q3 through 2005Q1 are reported in Appendix l . B .  The sufficient condition for the  existence of a unique stationary rational expectations equilibrium due to Klein (2000) is satisfied in a neighbourhood around the posterior mode, while the estimator o f the Hessian is not nearly  38 singular at the posterior mode, suggesting that the approximate linear state space representation of this D S G E model of a small open economy is locally identified. The prior mean of indexation parameter y  Y  is 0.75, implying considerable output price  inflation inertia, while the prior mean of nominal rigidity parameter co implies an average Y  duration of output price contracts of two years.  The prior mean of capital utilization cost  parameter k is 0.10, while the prior mean of elasticity of substitution parameter 9 is 0.75, implying that utilized capital and effective labour are moderately close complements in production. The prior mean of habit persistence parameter a is 0.95, while the prior mean of intertemporal elasticity of substitution parameter a is 2.75, implying that consumption exhibits considerable persistence and moderate sensitivity to real interest rate changes. The prior mean of investment adjustment cost parameter x i 5.75, implying moderate sensitivity of investment to s  changes in the relative shadow price of capital. The prior mean of indexation parameter y  M  is  0.75, implying moderate import price inflation inertia, while the prior mean of nominal rigidity parameter co  M  implies an average duration of import price contracts of two years. The prior  mean of elasticity of substitution parameter y/ is 1.50, implying that domestic and foreign goods are moderately close substitutes in consumption, investment, and government consumption. The prior mean of indexation parameter y  L  is 0.75, implying considerable sensitivity of the real  wage to changes in consumption price inflation, while the prior mean of nominal rigidity parameter co implies an average duration of wage contracts of two years. The prior mean of L  elasticity of substitution parameter rj is 2.00, implying considerable insensitivity of the real wage to changes in employment. The prior mean of the consumption price inflation response coefficient E," in the monetary policy rule is 1.50, while the prior mean of the output response coefficient E, is 0.125, ensuring convergence of the level of consumption price inflation to its Y  target value.  The prior mean of the net foreign debt response coefficient £  G  in the fiscal  expenditure rule is -0.10, while the prior mean of the net government debt response coefficient <^ in the fiscal revenue rule is 1.00, ensuring convergence of the levels of the ratios of net r  foreign debt and net government debt to nominal output to their target values. A l l autoregressive parameters p have prior means of 0.85, implying considerable persistence of shocks driving variation in cyclical components. The posterior modes of these structural parameters are all close to their prior means, reflecting the imposition of tight independent priors to ensure the existence of a unique stationary rational expectations equilibrium.  The estimated variances of shocks driving variation in  cyclical components are all well within the range of estimates reported in the existing literature, after accounting for data rescaling. The estimated variances of shocks driving variation in trend components are relatively high, indicating that the majority of variation in the levels of observed endogenous variables is accounted for by variation in their trend components.  39 Prior information concerning the values of trend components is generated by fitting third order deterministic polynomial functions to the levels of all observed endogenous variables by ordinary least squares.  Stochastic restrictions on the trend components of all observed  endogenous variables are derived from the fitted values associated with these ordinary least squares regressions, with innovation variances set proportional to estimated prediction variances assuming known parameters.  A l l stochastic restrictions are independent, represented by a  diagonal covariance matrix, and are harmonized, represented  by a common factor of  proportionality. Reflecting little confidence in these preliminary trend component estimates, this common factor of proportionality is set equal to one. Predicted, filtered and smoothed estimates of the cyclical and trend components of observed endogenous variables are plotted together with confidence intervals in Appendix l . B . These confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters.  The predicted estimates are conditional on past  information, the filtered estimates are conditional on past and present information, and the smoothed estimates are conditional on past, present and future information. Visual inspection reveals close agreement with the conventional dating of business cycle expansions and recessions.  1.3.2. Inference Whether this estimated D S G E model approximately accounts for the empirical evidence concerning the monetary transmission mechanism in a small open economy is determined by comparing its impulse responses to domestic and foreign monetary policy shocks with impulse responses derived from an estimated S V A R model.  1.3.2.1. Empirical Impulse Response Analysis Consider the following S V A R model of the monetary transmission mechanism in a small open economy Ay^MiO  where p{t)  + zZ^-i  + Bs,,  (147)  denotes a third order deterministic polynomial function and e, ~ iid A/"(0, / ) .  Vector stochastic process y  t  consists of domestic output price inflation n], domestic output  40 lnF , (  domestic consumption price inflation  n],  domestic consumption  InC,, domestic  investment price inflation n[, domestic investment In / , , domestic import price inflation domestic exports In X , t  nf,  domestic imports In M , , domestic nominal interest rate /,, nominal  exchange rate l n £ , , foreign output price inflation  TI] , foreign output J  \nY/,  foreign  consumption In C] , foreign investment In / / , and foreign nominal interest rate / / .  The  diagonal elements of parameter matrix A are normalized to one, while the off diagonal elements Q  of positive definite parameter matrix B are restricted to equal zero, thus associating with each equation a unique endogenous variable, and with each endogenous variable a unique structural innovation. This S V A R model is identified by imposing restrictions on the timing of the effects of monetary policy shocks and on the information sets of the monetary authorities, both within and across the domestic and foreign economies. Within the domestic and foreign economies, prices and quantities are restricted to not respond instantaneously to monetary policy shocks, while the monetary authorities can respond instantaneously to changes in these variables.  Across the  domestic and foreign economies, the domestic monetary authority is restricted to not respond instantaneously to foreign monetary policy shocks, while foreign variables are restricted to not respond to domestic monetary policy shocks. This S V A R model of the monetary transmission mechanism in a small open economy is estimated by full information maximum likelihood over the period 1971Q3 through 2005Q1. As discussed in Hamilton (1994), in the absence of model misspecification, this full information maximum likelihood estimator is consistent and asymptotically normal, irrespective of the cointegration rank and validity of the conditional multivariate normality assumption. The lag order is selected to minimize multivariate extensions of the model selection criterion functions of Akaike (1974), Schwarz (1978), and Hannan and Quinn (1979) subject to an upper bound equal to the seasonal frequency. These model selection criterion functions generally prefer a lag order of one. Table 1.2. Model selection criterion function values p 1 2 3 4  AIC(p)  SC(p)  HQ(P)  -110.8778  -102.3386*  -107.4079*  -111.2069  -98.2780  -105.9532  -111.1651  -93.8465  -104.1276  -111.6281*  -89.9197  -102.8068  Note: M i n i m i z e d values o f model selection criterion functions are indicated by *.  Since this S V A R model is estimated to provide empirical evidence concerning the monetary transmission mechanism in a small open economy, it is imperative to examine the empirical  41 validity of its overidentifying restrictions prior to the conduct of impulse response analysis. On the basis of bootstrap likelihood ratio tests, these overidentifying restrictions are not rejected at conventional levels of statistical significance.  278.1389  Asymptotic  Parametric Bootstrap  Nonparametric Bootstrap  0.0000  0.9499  0.9990  Note: This likelihood ratio test statistic is asymptotically distributed as fa . Bootstrap distributions are based on 999 replications.  Theoretical impulse responses to a domestic monetary policy shock are plotted versus empirical impulse responses in Figure 1.1. Following a domestic monetary policy shock, the domestic nominal interest rate exhibits an immediate increase followed by a gradual decline. The domestic currency appreciates, with the nominal exchange rate exhibiting delayed overshooting. These nominal interest rate and nominal exchange rate dynamics induce persistent and generally statistically significant hump shaped negative responses of domestic output price inflation,  output,  consumption price inflation,  consumption, investment  price  inflation,  investment, import price inflation, exports and imports, with peak effects realized after approximately one year. These results are qualitatively consistent with those of S V A R analyses of the monetary transmission mechanism in open economies such as Eichenbaum and Evans (1995), Clarida and Gertler (1997), K i m and Roubini (1995), and Cushman and Zha (1997).  42 Figure 1.1, Theoretical versus empirical impulse responses to a domestic monetary policy shock DLPGDP(APR)  0 5  10  15  20  25  DLPCON (APR)  30  35  40  0  5  10  15  20  25  30  3S40  5  DLPINV (APR)  0  5  10  15  20  25  10  15  20  25  30  35  30  35  40  0  5  10  15  20  25  30  35  40  0  5  10  15  20  25  30  35  40  35  40  35  40  DLPIMP (APR)  30  35  5  10  15  20  25  30  0  5  10  t5  20  25  NINT(APR)  5  10  15  20  25  30  35  40  0  5  10  15  20  25  DLPGDPF (APR)  30  35  40  0  5  10  15  20  25  30  35  40  0  5  10  15  20  25  30  NINTF (APR)  5  10  IS  X  35  40  5  10  15  2 0 2 5 3 0 3 5 4 0  0 5  10  15  2 0 2 5 3 0 3 5 4 0  0 5  10  15  2 0 2 5 X  Note: Theoretical impulse responses to a 50 basis point monetary p o l i c y shock are represented by black lines, w h i l e blue lines depict empirical impulse responses to a 50 basis point monetary policy shock.  A s y m m e t r i c 9 5 % confidence intervals are calculated w i t h a nonparametric  bootstrap simulation w i t h 999 replications.  Theoretical impulse responses to a foreign monetary policy shock are plotted versus empirical impulse responses in Figure 1.2. Following a foreign monetary policy shock, the foreign nominal interest rate exhibits an immediate increase followed by a gradual decline. In response to these nominal interest rate dynamics, there arise persistent and generally statistically significant  hump  shaped  negative  responses of foreign output price inflation, output,  consumption and investment, with peak effects realized after approximately one to two years. Although domestic output, consumption, investment and imports decline, domestic consumption price inflation, investment price inflation and import price inflation rise due to domestic currency depreciation.  These results are qualitatively consistent with those of S V A R analyses of the  monetary transmission mechanism in closed economies such as Sims and Zha (1995), Gordon  43 and Leeper (1994), Leeper, Sims and Zha (1996), and Christiano, Eichenbaum and Evans (1998, 2005). Figure 1.2. Theoretical versus empirical impulse responses to a foreign monetary policy shock DLPGDP (APR)  0  5  10  15  20  25  30  DLPCON (APR)  35  40  0  5  10  15  20  25  30  35  40  5  10  DLPINV(APR)  5  10  IS  15  20  25  30  36  40  30  35  40  5  10  IS  20  S  10  15  20  DLPIMP(APR)  2D  0  5  10  15  20  25  30  15  20  25  0  NINT(APR)  5  10  15  20  25  30  5  10  15  20  25  25  30  35  40  30  35  40  30  35  40  DLPGDPF (APR)  30  35  *3  5  10  15  20  25  30  35  40  5  10  15  20  25  NINTF (APR)  0  5  10  15  30  25  30  35  40  5  10  15  20  25  30  35  40  0  5  10  15  20  25  30  35  40  5  10  15  20  25  Note: Theoretical impulse responses to a 50 basis point monetary policy shock are represented by black lines, w h i l e blue lines depict empirical impulse responses to a 50 basis point monetary policy shock.  Asymmetric 9 5 % confidence intervals are calculated w i t h a nonparametric  bootstrap simulation w i t h 999 replications.  Visual inspection reveals that the theoretical impulse responses to domestic and foreign monetary  policy  shocks  generally lie within confidence  intervals associated  corresponding empirical impulse responses, suggesting that this estimated  with the  D S G E model  approximately accounts for the empirical evidence concerning the monetary transmission mechanism in a small open economy.  However, these confidence intervals are rather wide,  indicating that considerable uncertainty surrounds this empirical evidence.  44  1.3.2.2. Theoretical Impulse Response Analysis  In an open economy, exchange rate adjustment contributes to both intratemporal and intertemporal equilibration, while business cycles are generated by interactions among a variety of nominal and real shocks originating both domestically and abroad.  Theoretical impulse  responses and forecast error variance decompositions to domestic and foreign preference, output technology, investment technology, import technology, output price markup, import price markup, wage markup, monetary policy, fiscal expenditure, and fiscal revenue shocks are plotted in Appendix l . B . Following a domestic output technology shock, there arise persistent hump shaped positive responses of domestic output,  consumption, investment,  and government  consumption.  Domestic output price inflation, consumption price inflation, investment price inflation, and government consumption price inflation exhibit persistent hump shaped declines in response to a reduction in real marginal cost. The domestic nominal and real interest rates exhibit persistent hump shaped declines in response to a reduction in consumption price inflation, mitigated by an increase in output. The domestic currency appreciates in nominal terms and depreciates in real terms, while the terms of trade deteriorate.  Since the increase in nominal output exceeds the  increase in domestic demand, the trade balance rises, facilitating an intertemporal resource transfer between the domestic and foreign economies. Following a domestic monetary policy shock, the domestic nominal and real interest rates exhibit immediate increases followed by gradual declines, inducing persistent hump shaped negative responses of domestic output, consumption, investment, and government consumption. The nominal and real exchange rates overshoot, with immediate appreciations followed by gradual depreciations. Domestic output price inflation, consumption price inflation, investment price inflation, and government consumption price inflation exhibit persistent hump shaped declines in response to a reduction in real marginal cost.  These declines in domestic  consumption price inflation, investment price inflation, and government consumption price inflation are amplified and accelerated by an improvement in the terms o f trade. This reduction in the price of imports in terms of exports induces intratemporal expenditure switching, with a decline in the trade balance reflecting a reduction in nominal output relative to domestic demand. Following a domestic fiscal expenditure shock, there arise immediate positive responses of domestic output and government consumption, together with persistent hump shaped negative responses of domestic consumption and investment. response to an increase in real marginal cost.  Domestic output price inflation rises in  The domestic nominal and real interest rates  exhibit immediate increases followed by gradual declines, causing the domestic currency to  45 appreciate in nominal and real terms, while the terms of trade improve. Domestic consumption price inflation, investment price inflation, and government consumption price inflation rise in response to an increase in real marginal cost, amplified and accelerated by an improvement in the terms of trade.  Since the increase in nominal output is less than the increase in domestic  demand, the trade balance declines, facilitating an intertemporal resource transfer between the domestic and foreign economies.  1.3.3. Forecasting While it is desirable that forecasts be unbiased and efficient, the practical value of any forecasting model depends on its relative predictive accuracy. In the absence of a well defined mapping between forecast errors and their costs, relative predictive accuracy is generally assessed with mean squared prediction error based measures.  A s discussed in Clements and  Hendry (1998), mean squared prediction error based measures are noninvariant to nonsingular, scale preserving linear transformations, even though linear models are.  It follows that mean  squared prediction error based comparisons may yield conflicting rankings across models, depending on the variable transformations examined. To compare the dynamic out of sample forecasting performance of the D S G E and S V A R models, forty quarters of observations are retained to evaluate forecasts one through eight quarters ahead, generated conditional on parameters estimated using information available at the forecast origin.  The models are compared on the basis of mean squared prediction errors in  levels, ordinary differences, and seasonal differences.  The D S G E model is not recursively  estimated as the forecast origin rolls forward due to the high computational cost of such a procedure, while the S V A R model is. Presumably, recursively estimating the D S G E model would improve its predictive accuracy. Mean squared prediction error differentials are plotted together with confidence intervals accounting for contemporaneous and serial correlation of forecast errors in Appendix l . B . If these mean squared prediction error differentials are negative then the forecasting performance of the D S G E model dominates that of the S V A R model, while i f positive then the D S G E model is dominated by the S V A R model in terms of predictive accuracy. The null hypothesis of equal squared prediction errors is rejected by the predictive accuracy test of Diebold and Mariano (1995) if and only i f these confidence intervals exclude zero. The asymptotic variance of the average loss differential is estimated by a weighted sum of the autocovariances of the loss differential, employing the weighting function proposed by Newey and West (1987).  Visual  inspection reveals that these mean squared prediction error differentials are generally negative,  46 suggesting that the D S G E model dominates the S V A R model in terms of forecasting performance, in spite of a considerable informational disadvantage.  However, these mean  squared prediction error differentials are rarely statistically significant at conventional levels, perhaps because the predictive accuracy test due to Diebold and Mariano (1995), which is univariate, typically lacks power to detect dominance in forecasting performance, as evidenced by Monte Carlo evaluations such as Ashley (2003) and McCracken (2000). Dynamic out of sample forecasts of levels, ordinary differences, and seasonal differences are plotted together with confidence intervals versus realized outcomes in Appendix l . B . These confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Visual inspection reveals that the realized outcomes generally lie within their associated confidence intervals, suggesting that forecast failure is absent.  However, these confidence intervals are rather wide, indicating that considerable  uncertainty surrounds the point forecasts.  1.4. Conclusion  This paper develops and estimates a D S G E model o f a small open economy which approximately accounts for the empirical evidence concerning the monetary transmission mechanism, as summarized by impulse response functions derived from an estimated S V A R model, while dominating that S V A R  model in terms of predictive accuracy.  Cyclical  components are modeled by linearizing equilibrium conditions around a stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path.  This estimated D S G E model consolidates much existing theoretical and empirical  knowledge concerning the monetary transmission mechanism in a small open economy, provides a framework for a progressive research strategy, and suggests partial explanations for its own deficiencies. Jointly modeling cyclical and trend components as unobserved components while imposing theoretical restrictions derived from the approximate multivariate linear rational expectations representation of a D S G E model confers a number of benefits o f particular importance to the conduct of monetary policy. A s discussed in Woodford (2003), the levels of the flexible price and wage equilibrium components of various observed and unobserved endogenous variables are important inputs into the optimal conduct of monetary policy, in particular the measurement of the stance of monetary policy. Jointly modeling cyclical and trend components as unobserved components facilitates estimation of the levels of the flexible price and wage equilibrium  47 components of endogenous variables while imposing relatively weak identifying restrictions on their trend components.  The analysis of optimal monetary policy under an inflation targeting  regime and the estimation of the levels of flexible price and wage equilibrium components within the framework of an extended and refined version of this D S G E model of a small open economy remains an objective for future research.  Appendix l.A. Description of the Data Set The data set consists of quarterly seasonally adjusted  observations  on twenty six  macroeconomic variables for Canada and the United States over the period 1971Q1 through 2005Q1.  A l l aggregate prices and quantities are expenditure based.  Model consistent  employment is derived from observed nominal labour income and a nominal wage index, while model consistent tax rates are derived from observed nominal output and disposable income. The nominal interest rate is measured by the three month Treasury bill rate expressed as a period average, while the nominal exchange rate is quoted as an end of period value. National accounts data for Canada was retrieved from the C A N S I M database maintained by Statistics Canada, national accounts data for the United States was obtained from the F R E D database maintained by the Federal Reserve Bank of Saint Louis, and other data was extracted from the IFS database maintained by the International Monetary Fund.  Appendix l.B. Tables and Figures Table 1.4, Bayesian estimation results Prior Distribution a X  n K  V a 3  r' r" r L  a,' m  M  co  L  4" 4' C PA  Py' P,» P„> P»«  P,f P,< P." P,-  Posterior Distribution  Mean  Standard Error  Mode  0.950000 5.750000 2.000000 0.100000 1.500000 2.750000 0.750000 0.750000 0.750000 0.750000 0.875000 0.875000 0.875000 1.500000 0.125000 -0.100000 1.000000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000  0.000950 0.005750 0.002000 0.000100 0.001500 0.002750 0.000750 0.000750 0.000750 0.000750 0.000875 0.000875 0.000875 0.001500 0.000125 0.000100 0.001000 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850  0.949020 5.745100 2.000100 0.099999 1.500000 2.751200 0.750000 0.749910 0.750060 0.750020 0.874010 0.874270 0.874950 1.499600 0.124950 -0.099998 0.999040 0.849250 0.850500 0.851580 0.850060 0.850060 0.850020 0.850030 0.852660 0.850300 0.850150 0.203690 0.056415 0.306120 0.110820 0.285680 0.253650 0.290040 0.015802 0.441830 0.242930 0.109320 0.102130 0.226170 0.193580 0.243750 0.243030 0.220930 0.073864 0.239540 0.394560  a],  -  00 00  a, 1  00  a  l-<  OO  <~  OC  <• °\  -  OO  a',  -  oo  a ,.  -  1  a, 2  ,' J  (YL  °\< cr  , " 2  •c  oo CO  -  oc  ...  OO  -  OO  -  OC  _  oo  -  OO  )i  -  CC  cr,,  -  OO  a  l 1  <T,, , 2  a ,, 1  OO -  X  Standard Error 0.000925 0.005750 0.002000 0.000100 0.001500 0.002750 0.000750 0.000750 0.000750 0.000750 0.000872 0.000874 0.000874 0.001500 0.000125 0.000100 0.001000 0.000850 0.000849 0.000849 0.000850 0.000850 0.000850 0.000850 0.000844 0.000849 0.000849 0.058358 0.013065 0.052463 0.014633 1.557700 2.182200 3.829100 0.002923 0.062389 0.030785 0.028552 0.016336 0.035216 0.502700 1.049700 13.116000 2.595100 0.009300 0.030234 0.038424  Parameter  4 4 \  Prior Distribution Mean -  Mode  Standard Error  oc  0.622800  0.078904  00  0.062492  0.008983  oc  0.577490  0.073005  00  0.096753  0.011677  -  X'  0.708920  0.088374  -  oc  0.449850  0.056239  -  a  4  Posterior Distribution  Standard Error  4  -  00  0.712530  0.090229  -  00  0.123730  0.026314  4 4 4 4  -  00  0.362830  0.040564  -  oc  1.460000  0.185300  _  oc  0.533020  0.062373  00  0.783770  0.098795  00  0.114950  0.020643  -  00  0.002181  0.000388  -  OC  0.178260  0.027004  00  0.448390  0.052147  00  0.303080  0.037457  00  0.021367  0.002802  -  00  0.026166  0.002760  -  00  0.497880  0.055539  00  0.086916  0.015808  00  2.147700  0.258730  00  0.352380  0.041770  00  0.080368  0.014333  -  OC  0.001500  0.000242  -  00  0.053588  0.011479  -  00  0.000136  0.000019  -  00  0.000004  0.000005  -  OC  0.000025  0.000009  00  0.000068  0.000012  -  00  0.000022  0.000011  -  CO  0.000029  0.000012  1  al  -I 4< 4 4 4 4 4, 44  a] a, a", a\,  -  -  -  -  1  Note: A l l observed endogenous variables are rescaled by a factor o f 100.  50 Figure 1.3. Predicted cyclical components of observed endogenous variables  Note: Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the Economic C y c l e Research Institute reference c y c l e .  51 Figure 1.4. Filtered cyclical components of observed endogenous variables  Note: Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated by the Economic C y c l e Research Institute reference cycle.  52 Figure 1.5. Smoothed cyclical components o f observed endogenous variables LPGDP  LRGDP  LPCON  LRCON  LPINV  LNEXCH  LPGDPF  LRGDPF  LRCONF  LRINVF  1975 1980 1995 1 990  1995 2000 3005  1 975 1980 1995  1990 1995 2109 2005  1975 I960 1995 1990 1995 2000 2005  1976  1980 1985 1990 1995 2000 2005  1975 1999 1995 1990 1995  2000 2005  LTAXRATEF  Note:  Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and  k n o w n parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  Note: Observed levels are represented by black lines, while blue lines depict estimated trend components. Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  54 Figure 1.7. Filtered trend components of observed endogenous variables L  P  G  0  P  L"GOF>  LPCON  LRCON  Note: Observed levels are represented by black lines, while blue lines depict estimated trend components. Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  55  Figure 1.8. Smoothed trend components of observed endogenous variables LPGDP  LRGDP  LPCON  LRCON  LPINV  Note: Observed levels are represented by black lines, while blue lines depict estimated trend components. Symmetric 95% confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  56  Note: Theoretical impulse responses to a unit standard deviation innovation are represented by blue lines.  Note: Theoretical impulse responses to a unit standard deviation innovation are represented by blue lines.  Figure 1.11. Theoretical impulse responses to a domestic fiscal expenditure shock  Note: Theoretical impulse responses to a unit standard deviation innovation are represented by blue lines.  59  Figure 1,12. Theoretical impulse responses to a foreign output technology shock  Note: Theoretical impulse responses to a unit standard deviation innovation are represented by blue lines.  Note: Theoretical impulse responses to a unit standard deviation innovation are represented by blue lines.  Note: Theoretical impulse responses to a unit standard deviation innovation are represented by blue lines.  63 Figure 1.16. Mean squared prediction error differentials for levels LPGDP  LRGDP  LPCON  LRCON 60  6040-  40  100-  0-  0.  20  0-  20-too-  40  20  20-  -20-  30  -40-601  2  4  3  5  7  6  -40-  -40-  •602  4  3  LP1NV  5  6  7  i  2  4  3  LRINV  5  7  6  2  4  3  LPIMP  5  6  7  {  6  7  8  6  7  I  6  7  LREXP  40400  80020-  400-  ft.  0.  -800.  4  3  5  2  r  6  -400-800-  -400-  40 2  0-  •Mi-  400-  -20-  800-  200  400-  0,  1200-  4  3  LRIMP  5  6  7  -12002  NINT  1000-  4  3  5  7  6  7  4  3  LNEXCH  5  LPGDPF  2-  20'"IOO  900.  10-  0-i  0-4 -ia  -sou500-  •2a  -2-  1000H 2  3  4  5  6  •  2  4  3  LRGDPF  5  6  7  2  4  3  LRCONF  5  7  6  2  4  3  LRINVF  5  NINTF  BO*.  4. 2000-  4a  3-  50-  20-  2-  IOUO-  0-  0,  20-  1000-  2-  -50-  40-  -3-  2000-,  -4.  <*]• 2  3  4  5  6  7  8  1  2  3  4  5  6  7  B  1  2  3  4  5  6  7  8  1  2  3  4  5  Note: M e a n squared prediction error differentials are defined as the mean squared prediction error for the D S G E model less that for the S V A R model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors.  64 Figure 1.17. Mean squared prediction error differentials for ordinary differences DLPGDP  DLRGDP  DLPCON  DLRCON 2  2 z  1  a  0  0  "~"  -2.  1  -2  -2 1  2  3  4  5  6  7  1  2  3  4  DLPINV  6  5  7  2 1  I  2  3  4  DLRINV  1.5-  6  5  7  1  B-  10  0.0-  0-  0 5-  •10-  10-  -1.0-  •20-  -20-  6  7  6  7  8  6  7  5  10-  -30-  -8-  -402  3  4  6  5  7  2  3  DNINT  20.  8  20.  DLRIMP  40.  7  0-  -305  6  30-  20  4  5  DLRE)0?  1.0-  3  4  40-  0.5.  2  3  DLPIMP  30  1 5-  2  4  6  5  7  2  3  DLNEXCH  08.  20.  04.  ia  00.  0-  4  5  DLPGDPF 1.2. 0 804.  (J. za  -0 4.  -.04.  -10-  .082  3  4  5  6  7  -0.8-12-  -20-  «t. 2  3  DLRGDPF  4  6  5  7  2  3  DLRCONF  4  6  5  7  2  DLRINVF  3  4  5  DNINTF  22. oe. sa 04.  ii.  a.  0-  .on 04.  -80.  -0822  3  4  5  6  7  5  1  2  3  4  5  6  7  Note: M e a n squared prediction error differentials are defined as the mean squared prediction error for the D S G E model less that for the S V A R model. Symmetric 95% confidence intervals account for contemporaneous and serial correlation o f forecast errors.  65 Figure 1,18. Mean squared prediction error differentials for seasonal differences  5  6  7  5  6  7  8  5  6  7  5  6  7  5  6  7  8  5  6  7  5  6  7  5  6  7  400.  5  6  7  5  6  7  B  8  8  Note: M e a n squared prediction error differentials are defined as the mean squared prediction error for the D S G E model less that for the S V A R model. Symmetric 95% confidence intervals account for contemporaneous and serial correlation o f forecast errors.  66 Figure 1.19. Dynamic forecasts o f levels o f observed endogenous variables LPGDP  LRGDP  LPCON  Note: Realized outcomes are represented by black lines, while blue lines depict point forecasts.  LRCON  LPIW  Symmetric 95% confidence intervals assume  multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters.  67  Figure 1.20. Dynamic forecasts o f ordinary differences of observed endogenous variables  Note: Realized outcomes are represented b y black lines, while blue lines depict point forecasts.  Symmetric 95% confidence intervals assume  multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters.  68 Figure 1.21. Dynamic forecasts o f seasonal differences o f observed endogenous variables  Note: Realized outcomes are represented by black lines, while blue lines depict point forecasts.  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Rogoff (1996), Foundations of International Macroeconomics, M I T Press.  71 Rotemberg, J. and M . Woodford (1995), Dynamic general equilibrium models with imperfectly competitive product markets, Frontiers of Business Cycle Research, Princeton University Press. Rotemberg, J. and M . Woodford (1997), A n optimization-based econometric framework for the evaluation o f monetary policy, NBER Macroeconomics Annual, M I T Press. Ruge-Murcia, F. (2003), Methods to estimate dynamic stochastic general equilibrium models, CIREQ Paper, 17.  Working  Schwarz, G . (1978), Estimating the dimension o f a model, Annals of Statistics, 6, 461-464. Sims, C . and T. Zha (1995), Does monetary policy generate recessions?, Unpublished  Manuscript.  Smets, F. and R. Wouters (2003), A n estimated dynamic stochastic general equilibrium model of the Euro area, Journal of the European Economic Association,  1,1123-1175.  Smets, F. and R. Wouters (2005). Comparing shocks and frictions in U S and Euro area business cycles: A Bayesian D S G E approach, Journal of Applied Econometrics, 20, 161-183. Vitek, F. (2006a), A n unobserved components model of the monetary transmission mechanism in a closed economy, Unpublished Manuscript. Vitek, F. (2006b), A n unobserved components model o f the monetary transmission mechanism in a small open economy, Unpublished  Manuscript.  Woodford, M . (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. Y u n , T. (1996), Nominal price rigidity, money supply endogeneity, and business cycles, Journal of Economics, 37, 345-370.  Monetary  72  CHAPTER 2  An Unobserved Components Model of the Monetary Transmission Mechanism in a Small Open Economy  2.1. Introduction  In  recent  years,  the c e n t r a l b a n k s  monetary policy regimes. three p r i m a r y elements.  o f many  economies  have  adopted  inflation  targeting  A n inflation targeting monetary p o l i c y r e g i m e is characterized b y F i r s t , there e x i s t s a n e x p l i c i t i n f l a t i o n target, w h i c h is t y p i c a l l y q u i t e  l o w a n d i s o f t e n s p e c i f i e d as a n i n t e r v a l . S e c o n d , a c h i e v i n g a n i n f l a t i o n c o n t r o l o b j e c t i v e , i n the form o f m i n i m i z i n g deviations o f inflation achieving  an  output  f r o m its target v a l u e , i s e m p h a s i z e d r e l a t i v e  stabilization objective.  Third,  the  conduct  of  monetary  policy  to is  characterized b y a h i g h degree o f transparency and accountability. A s t y l i z e d q u a l i t a t i v e d e s c r i p t i o n o f the m o n e t a r y t r a n s m i s s i o n m e c h a n i s m i n a s m a l l o p e n e c o n o m y d i s t i n g u i s h e s a m o n g i n s t r u m e n t s , i n d i c a t o r s , a n d targets.  U n d e r an inflation targeting  m o n e t a r y p o l i c y r e g i m e , the c e n t r a l b a n k p e r i o d i c a l l y adjusts a s h o r t t e r m n o m i n a l interest rate in response to inflationary pressure.  P r o v i d e d that t h i s r e s p o n s e  i s s u f f i c i e n t l y l a r g e , i n the  p r e s e n c e o f s h o r t r u n n o m i n a l r i g i d i t i e s o r i m p e r f e c t i n f o r m a t i o n , a n i n c r e a s e i n the s h o r t t e r m n o m i n a l interest rate c a u s e s a n i n c r e a s e i n the s h o r t t e r m r e a l interest rate, i n d u c i n g i n t e r t e m p o r a l reductions i n c o n s u m p t i o n and investment.  In a n o p e n e c o n o m y , a n i n c r e a s e i n the s h o r t t e r m  n o m i n a l interest rate c a u s e s a n o m i n a l a p p r e c i a t i o n , w h i l e a n i n c r e a s e i n the s h o r t t e r m r e a l interest rate c a u s e s a r e a l a p p r e c i a t i o n .  T h i s adjustment  o f the r e a l e x c h a n g e rate i n d u c e s a n  intratemporal r e d u c t i o n i n exports together w i t h an intratemporal increase i n imports. presence  o f s h o r t r u n n o m i n a l r i g i d i t i e s o r i m p e r f e c t i n f o r m a t i o n , the r e s u l t a n t  o u t p u t is a s s o c i a t e d w i t h a d e c l i n e i n o u t p u t p r i c e i n f l a t i o n .  I n the  reduction in  I n a n o p e n e c o n o m y , the  resultant  r e d u c t i o n i n c o n s u m p t i o n p r i c e i n f l a t i o n is a m p l i f i e d a n d a c c e l e r a t e d b y the a d j u s t m e n t  o f the  r e a l e x c h a n g e rate. D e s p i t e the r e m a r k a b l e s u c c e s s o f m a n y i n f l a t i o n t a r g e t i n g c e n t r a l b a n k s at a c h i e v i n g l o w a n d stable  i n f l a t i o n , the  development  o f a mutually consistent  A version o f this chapter has been accepted for publication.  set o f a c c u r a t e  and  precise  Vitek, F „ A n unobserved components model o f the monetary transmission  mechanism in a small open economy, Journal of World Economic Review.  73 indicators of inflationary pressure remains elusive.  Theoretically prominent indicators of  inflationary pressure such as the natural rate of interest and natural exchange rate are unobservable. A s discussed in Woodford (2003), the natural rate of interest provides a measure of the neutral stance o f monetary policy, with deviations of the real interest rate from the natural rate of interest generating inflationary pressure.  Within the framework o f an unobserved  components model of selected elements of the monetary transmission mechanism in a closed economy, Laubach and Williams (2003) find that estimates o f the natural rate of interest are relatively imprecise, as evidenced by relatively wide confidence intervals. Jointly estimating this and other indicators of inflationary pressure conditional on a larger information set may be expected to yield efficiency gains. Definitions o f indicators of inflationary pressure such as the natural rate o f interest and natural exchange rate vary. Following Laubach and Williams (2003), we define the natural rate of interest as that short term real interest rate consistent with achieving inflation control and output stabilization objectives in the absence of shocks having temporary effects.  In this long  run equilibrium, there does not exist a cyclical stabilization role for monetary policy generated by nominal rigidities or imperfect information. In contrast, Woodford (2003) defines the natural rate of interest as that short term real interest rate consistent with achieving inflation control and output stabilization objectives in the absence of nominal rigidities. In this short run equilibrium, although there does not exist a cyclical stabilization role for monetary policy, the natural rate of interest varies in response to shocks having both temporary and permanent effects.  Given an  interest rate smoothing objective derived from a concern with financial market stability, it may be optimal for a central bank to adjust the short term nominal interest rate primarily in response to variation in the natural rate of interest caused by shocks having permanent effects. Within the framework of a linear state space model, prior information concerning the values of unobserved state variables is often available in the form of deterministic or stochastic restrictions.  Within the framework of an unobserved components model, prior information  concerning the values of unobserved components is often available from alternative estimators. The primary methodological contribution of this paper is the development of a procedure to estimate a linear state space model conditional on prior information concerning the values o f unobserved state variables. This prior information assumes the form o f a set o f deterministic or stochastic restrictions on linear combinations of unobserved state variables.  In addition to  mitigating potential model misspecification and identification problems, exploiting such prior information may be expected to yield efficiency gains in estimation. This paper develops and estimates an unobserved components model for purposes of monetary policy analysis and inflation targeting in a small open economy. In an extension of the empirical framework developed by Laubach and Williams (2003), cyclical components are  74 modeled as a multivariate linear rational expectations model of the monetary transmission mechanism, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path. Although not derived from microeconomic foundations, this unobserved components model of the monetary transmission mechanism in a small open economy arguably provides a closer approximation to the data generating process than existing dynamic stochastic general equilibrium models, as fewer cross-coefficient restrictions are imposed.  Full information maximum likelihood estimation of this unobserved components  model, conditional on prior information concerning the values of trend components, provides a quantitative description of the monetary transmission mechanism in a small open economy, yields a mutually consistent set of indicators of inflationary pressure together with confidence intervals, and facilitates the generation of relatively accurate forecasts. The organization of this paper is as follows.  The next section develops an unobserved  components model of the monetary transmission mechanism in a small open economy. In section three, unrestricted and restricted estimators of unobserved state variables are derived within the framework of a linear state space model. Estimation, inference and forecasting within the framework of a linear state space representation of our unobserved components model are the subjects of section four.  Finally, section five offers conclusions and recommendations for  further research.  2.2. The Unobserved Components Model  Consider two structurally isomorphic economies which are asymmetric in size.  The  domestic economy is of negligible size relative to the foreign economy, and hence takes the rational expectations equilibrium of the foreign economy as exogenous. The central banks of the domestic and foreign economies pursue inflation control and output stabilization objectives. Cyclical components are modeled as a multivariate linear rational expectations model of the monetary transmission mechanism, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path. In what follows, E x (  l+s  denotes the rational expectation of variable x , conditional on t+s  information available at time t. Also, x, denotes the cyclical component of variable x,, while x, denotes the trend component of variable x . Cyclical and trend components are additively (  separable, that is x, - x + x . t  t  75  2.2.1. Cyclical Components  The c y c l i c a l component o f output price inflation K] depends on a linear combination o f past and expected future c y c l i c a l components o f output price inflation driven b y the contemporaneous c y c l i c a l component o f output according to output price Phillips curve  % = AA  + A ^ l x + 0 In Y, + ef, U  a\),  ef ~ i i d jV(0,  where the level o f output price inflation satisfies n) = A In P*.  (1)  T h e sensitivity o f the c y c l i c a l  component o f output price inflation to changes in the c y c l i c a l component o f output is increasing in 6  U  >0.  The c y c l i c a l component o f consumption price inflation nf  depends o n a linear combination  o f past and expected future c y c l i c a l components o f consumption price inflation driven b y the contemporaneous c y c l i c a l component o f output according to consumption price Phillips curve ^  =  +&, EA ;, +0 2  c  2l  InY  t  (2) - ^ A l n Q , ,  +0 A\nQ, 2a  -^ f5 E,AlnC} 2  22  where the level o f consumption price inflation satisfies n  c s  /+1  + ef , ef  - i i d Af^a .,  ),  2  = A In Pf . The c y c l i c a l component o f  consumption price inflation also depends o n past, contemporaneous, a n d expected  future  proportional changes i n the c y c l i c a l component o f the real exchange rate. T h e sensitivity o f the c y c l i c a l component o f consumption price inflation to changes i n the c y c l i c a l component o f output is increasing i n 9 , > 0 , and to changes in the c y c l i c a l component o f the real exchange 2  rate is increasing i n 0 < 6  2 2  < 1.  The c y c l i c a l component o f output In Y follows a stationary second order autoregressive t  process driven b y the contemporaneous c y c l i c a l component o f foreign output a n d a linear combination o f the past c y c l i c a l components o f the real interest rate and real exchange rate  In Y, = ^ In t , + ^ , 2 In t  2  + 0 In Yf + 9 r _ + 0„ In Q 3J  32  t  x  tA  where the level o f the real interest rate satisfies r =i -E nf , t  exchange rate satisfies l n g , = InS, + In Pf  -\nPf  l  l  l  + ef, ef ~ iid JV(0, <?]), (3) w h i l e the level o f the real  . T h e sensitivity o f the c y c l i c a l component  o f output to changes i n the c y c l i c a l component o f foreign output is increasing i n 0 , > 0 , to 3  changes i n the c y c l i c a l component o f the real interest rate is decreasing i n 6  3 2  changes i n the c y c l i c a l component o f the real exchange rate is increasing in 0  33  The  cyclical  component  o f consumption  I n C , follows  < 0 , and to  > 0.  a stationary  second  autoregressive process driven b y the past cyclical component o f the real interest rate:  order  76  InC, = ^ , InC,_, + £  2  InC,_ + f 7 r 4l  2  + ef, ef ~ i i d  M  (0,erf).  (4)  The sensitivity o f the c y c l i c a l component o f consumption to changes in the c y c l i c a l component o f the real interest rate is decreasing in <9, < 0. 4  The c y c l i c a l component o f investment I n / , follows a stationary second order autoregressive process driven by the contemporaneous c y c l i c a l component o f output:  In /, = </>, In /,_, + (j> In /,_ +fi.,In Y, + e\, e\ ~ iid jV(0, a)). 5  2  sl  (5)  The sensitivity o f the c y c l i c a l component o f investment to changes i n the c y c l i c a l component o f output is increasing i n 0 , > 0. 5  The c y c l i c a l component o f exports l n X , follows a stationary second order autoregressive process driven b y the contemporaneous c y c l i c a l component o f foreign output and the past c y c l i c a l component o f the real exchange rate: I n * , =4>b/nX,A+<Pb2\nXl_2+ebi\nYl<+ 0 \nQ b2  lA  + ef, ef - i i d J V ( 0 , < ^ ) .  (6)  The sensitivity o f the c y c l i c a l component o f exports to changes i n the c y c l i c a l component o f foreign output is increasing i n # exchange rate is increasing i n 9  b2  61  > 0 , and to changes i n the c y c l i c a l component o f the real  > 0.  The c y c l i c a l component o f imports In M  l  follows a stationary second order autoregressive  process driven b y the contemporaneous cyclical component o f output and the past c y c l i c a l component o f the real exchange rate:  In M , = <f> , In Af,_, + ^ In M , _ + 0 , In Y + 0 2  7  7  t  In Q,_ + ef , ef ~ i i d J V ( 0 , a\). 1  12  f  (7)  The sensitivity o f the c y c l i c a l component o f imports to changes i n the c y c l i c a l component o f output is increasing i n 6  1X  rate is decreasing in $  7 2  > 0 , and to changes in the c y c l i c a l component o f the real exchange  < 0.  The c y c l i c a l component o f wage inflation nf  depends on a linear combination o f past and  expected future c y c l i c a l components o f wage inflation driven b y the contemporaneous c y c l i c a l component o f the unemployment rate according to wage Phillips curve  (°)  77 where the level o f wage inflation satisfies nf = A In W .  The cyclical component of wage  t  inflation also depends on past, contemporaneous, and expected future cyclical components o f consumption price inflation.  The sensitivity of the cyclical component o f wage inflation to  changes in the cyclical component of the unemployment rate is decreasing in 9 < 0 , and to %x  changes in the cyclical component of consumption price inflation is increasing in 0 < 0 The cyclical  component  o f employment  InZ, follows  a stationary  g2  < 1.  second order  autoregressive process driven by the contemporaneous cyclical component of output: In L, =fa,In Z,_, + fa In Z,_ + 6 In Y + e , ef ~ iid  (0, CJ\ ).  L  2  2  9J  t  t  (9)  The sensitivity of the cyclical component of employment to changes in the cyclical component of output is increasing in 9 , > 0. 9  The cyclical component o f the unemployment rate «, follows a stationary second order autoregressive process driven by the contemporaneous cyclical component o f output: «r = d o . . " , - . + d o A - 2 + K\  ln  ?, + ef, ef - i i d Af(0, <7„ ).  (10)  2  The sensitivity o f the cyclical component of the unemployment rate to changes in the cyclical component of output is decreasing in 9 , < 0. W  The cyclical component o f the nominal interest rate /, follows a stationary first order autoregressive process driven by the contemporaneous cyclical components o f consumption price inflation and output: l=<l>nM+ m^  ei~MM(0,af).  d  (11)  The sensitivity of the cyclical component of the nominal interest rate to changes in the cyclical component of consumption price inflation is increasing in 9  XXX  component of output is increasing in 9  XX2  > 0 , and to changes in the cyclical  > 0.  The cyclical component o f the nominal exchange rate ln S , depends on a linear combination 1  of past and expected future cyclical components of the nominal exchange rate driven by the contemporaneous cyclical component of the nominal interest rate differential: In5, =4, ln5 _, +^ , E, lnS, +9 (i, -i/) + ef, ef - iid Af(0,a .). 2  1  (  12  2  +1  m  (12)  The sensitivity of the cyclical component of the nominal exchange rate to changes in the cyclical component of the nominal interest rate differential is decreasing in 9  X2l  < 0.  78 2.2.2. Trend Components  The trend components of the prices of output In Pf and consumption XnPf follow random walks with time varying drift n,: In Pf = n + In Pf + ef , ef  ~ iid  In Pf = n, + In Pf + ef , ef  ~ iid jV(0,  t  jV(0,a ), 2  (13)  p)  af)  (14)  It follows that the trend component of the relative price of consumption follows a random walk without drift. This implies that along a balanced growth path, the level of this relative price is time independent but state dependent. The trend components of output In Y , consumption In C , investment In I , exports In X , t  t  and imports In M, follow random walks with time varying drift In % = g, + n, + In Y  t  t  + ef, ef ~ iid jV(0,a ),  (15)  2  tA  9  In C, = g, + n, + In C,_, + ef, ef ~ iid M(0, a}.),  I = g, + n, + In l _  In  t  x  + ef,  I n M , =g +n +\nM _ +ef, l  l  l  (16)  + ej, ej ~ iid Af(0, erl),  In X, = g, + n, + In  (17)  ef ~ iid j\f(0,a\),  (18)  ef ~ iid ^ ( 0 , ^ ) .  1  t  g,+n :  (19)  It follows that the trend components of the ratios of consumption, investment, exports, and imports to output follow random walks without drifts. This implies that along a balanced growth path, the levels of these great ratios are time independent but state dependent. The trend component of the nominal wage In W follows a random walk with time varying t  drift 7r +g , r  t  while the trend component of employment In L follows a random walk with time t  varying drift n : t  In W,=K, + g,+ In W,_ + ef, x  ef ~ iid AT(0,of),  InL, = n, + InL,_ + ej, ej ~ iid j\f(0,ar\). x  (20) (21)  It follows that the trend component of the income share of labour follows a random walk without drift. This implies that along a balanced growth path, the level of the income share of labour is time independent but state dependent.  79 The trend components of the unemployment rate u,, nominal interest rate i,, and nominal exchange rate ln S, follow random walks without drifts: u, = u,_ + ef, ef - iid M(0, al),  (22)  T = T-i + e'i,  (23)  x  ln S, = ln  e] ~ iid Af(0, crl),  + ef, ef - iid A/"(0, cr|).  (24)  It follows that along a balanced growth path, the levels of the unemployment rate, nominal interest rate, and nominal exchange rate are time independent but state dependent. Long run balanced growth is driven by three common stochastic trends. Trend inflation it,, productivity growth g,, and population growth n, follow random walks without drifts: it,=it,_ +e;,  <~iid  S,=g +£f,  ef - iid A T ( 0 , C T ) ,  A  tA  jV{0,trl),  (25)  2  (26)  n, =»,_,+ e*, e* ~ iid jV(0, a]).  ( 2 7 )  It follows that along a balanced growth path, growth rates are time independent but state dependent. As an identifying restriction, all innovations are assumed to be independent, which combined with our distributional assumptions implies multivariate normality.  2.3. Estimation of Unobserved State Variables  State space models consist o f signal and state equations.  The signal equation specifies a  static relationship between a vector of observed signal variables and a vector of unobserved state variables, while the state equation specifies a dynamic relationship governing the evolution of this vector of unobserved state variables. The objective of state space analysis is to estimate the state vector, given the signal vector. Within the framework of a linear state space model, i f the signal and state innovation vectors are multivariate normally distributed and independent, then conditional on the parameters associated with the signal and state equations, mean squared error optimal estimates of the state vector may be calculated with the filter due to Kalman (1960). If the signal and state innovation vectors are not multivariate normally distributed, then these state vector estimates retain minimum mean squared error status within the class of linear estimators. Estimation, inference  80 and forecasting within the framework o f a linear state space model is discussed in Hamilton (1994), K i m and Nelson (1999), and Durbin and Koopman (2001). Within the framework of a linear state space model, prior information concerning the values of state variables is often available in the form o f deterministic or stochastic restrictions. This section derives unrestricted and restricted estimators of state variables within the framework of a linear state space model. The former approach is standard, while the latter is a contribution o f this paper.  In addition to mitigating potential model misspecification and identification  problems, exploiting prior information concerning the values of state variables may be expected to yield efficiency gains in estimation.  2.3.1. Unrestricted Estimation of Unobserved State Variables Let y  t  denote a vector stochastic process consisting of 7 V  observed nonpredetermined  endogenous variables, let x, denote a vector stochastic process consisting o f M observed exogenous or predetermined endogenous variables, and let z, denote a vector stochastic process consisting o f K unobserved state variables. Suppose that these vector stochastic processes have linear state space representation y,=A +A z,+A £ , lXl  2  3  (28)  u  z =^x,+B z,_ +B e , t  2  i  3  (29)  2j  where e,, ~ iid A/"(0,27,), e  2j  ~~ iid A/"(0,27,) and z ~ J\F(z ,P ). 0  0]0  0[0  The signal and state  innovation vectors are assumed to be independent, while the initial state vector is assumed to be independent from the signal and state innovation vectors, which combined with our distributional assumptions implies multivariate normality. Within  the framework  /»,., =Var(z | J , . , ) , y, l  of this  linear state space  =E(y, | X,_,) and Q,  M  M  model, define  z , = E(z, 1 ) , M  = Var(y, | T,_.), where J,_, = { { y  s  } £ „ } •  Conditional on the parameters associated with the signal and state equations, these conditional means and variances satisfy prediction equations: z,^ = B x, + B z,_ , t  ,\t-\ =  P  yt\t-\  -  B  2  2  P  A\X  t  (30)  [M  , \\, \ Z  + ^ 2  B  B  + A z ^_ , 2  /  ]  B  J ^  (  3 1  )  (32)  81  Q,i - =^A,-^ +A 2: Aj.  (33)  T  l  l  2  3  i  These predicted estimates of the means and variances of the signal and state vectors are conditional on past information. Given these predicted estimates, estimates of the state vector conditional on past and present information may be derived with Bayesian updating.  Define z , as that argument which l{  maximizes posterior density function:  /<«,  y , l , ^  f  l  y  i  ff\  (34)  Z  f(y, 127.,) Under the assumption of multivariate normally distributed signal and state innovation vectors, z,i, minimizes objective function  - J„,-.|) e,i'-,(j, - J , X  S(z,) = (z, -z ,^) P^iz,-: )-U  (35)  T  J  tl  )H  M  subject to signal equation (28). The necessary first order condition associated with the implied unconstrained minimization problem yields  where K = P^AjQf^ t  . This necessary first order condition is sufficient i f  positive definite. Define P as the mean squared error o f z , conditional on X l]t  l}l  -A Qf_. A i  2  M  ]  2  is  . Within the  framework of this linear state space model, this mean squared error matrix satisfies: P^P^-K^P^.  (37)  Under our distributional assumptions,  z , equals the mean o f posterior density  function  r]  f(z, | j , , X ) , and is therefore mean squared error optimal. Given initial conditions z M  0|0  and  P , recursive evaluation of equations (30), (31), (32), (33), (36) and (37) yields predicted and Q{0  filtered estimates of the state vector. Given these predicted and filtered estimates, estimates of the state vector conditional on past, present and future information may be derived with Bayesian updating.  Define z ,  |r  as that  argument which maximizes posterior density function: J(z,\z ,l )l+l  t  — — f(z, !I,)  •  (38)  +]  Under the assumption of a multivariate normally distributed state innovation vector, minimizes objective function  z,  ]r  82  S(z,) = («, -z, ) PJ(z, T  lt  -z )-(z lV  l+]  -  y  (- ,  Z)+Il  (39)  W  subject to state equation (29). The necessary first order condition associated with the implied unconstrained minimization problem yields (40) where  This necessary first order condition is sufficient if  J, - P^B]P ' \ . T + T  positive definite. Define P  T]T  as the mean squared error of  V  P  U  - B  , conditional o n J , .  J  P  is  B  X  WithVthe  framework of this linear state space model, this mean squared error matrix satisfies: (41) Under our distributional assumptions, f(z, | z ,T ), l+l  P  W  equals the mean of posterior density function  v  and is therefore mean squared error optimal. Given terminal conditions z  t  TT  and  obtained from the final evaluation of the prediction and updating equations, recursive  evaluation of equations (40) and (41) yields smoothed estimates of the state vector.  2.3.2. Restricted Estimation of Unobserved State Variables Let y  t  denote a vector stochastic process consisting of N observed nonpredetermined  endogenous variables, let x, denote a vector stochastic process consisting of M observed exogenous or predetermined endogenous variables, and let z, denote a vector stochastic process consisting of K unobserved state variables. Suppose that these vector stochastic processes have linear state space representation y, =A x, + A z +A e , ]  2  z, =B x,+B z,_, l  l  3  (42)  u  +B e ,  2  3  (43)  2j  where e,, ~ iid ^(0,27,), E , ~ iid Af(0,27 ) 2  2  and z ~ JV(Z ^,P ) 0  0  0[0  stochastic process consisting of J observed synthetic variables.  .  Let w, denote a vector Suppose that this vector  stochastic process satisfies ">,=C z,+C e , 1  2  3j  (44)  83 where « _, ~ iid JV(0,27 ). Conditional on known parameter values, this signal equation defines 3  3  a set of deterministic or stochastic restrictions on linear combinations of unobserved state variables.  The signal and state innovation vectors are assumed to be independent, while the  initial state vector is assumed to be independent from the signal and state innovation vectors, which combined with our distributional assumptions implies multivariate normality. Within the  framework  !»,_, = Var(z, | J ) ,  y,  M  J?,  M  of this  linear state space model, define  = E(y, | J ) ,  [lA  Q, ,_, = Var(y, | J,_,),  M  {  = {{J,K 1,{M'J': ,{JCJ' } .  = Var(w, | J,_,), where  I  1  =  s=i  z ,_, = E(z, | J,_,), (|  w,„_, = E(w, | J , )  and  Conditional on the parameters  associated with the signal and state equations, these conditional means and variances satisfy prediction equations: z,\,-\ -  + ^2 /-i|(-p  (45)  z  ^=B P,_^B +B £ Bl  (46)  J  2  •V-i  2  =  3  2  +4 V P  M  (  4 7  >  Q^-A^Al+A^Aj,  (48)  "Vi  ( >  = C  iVi'  49  ^.^C.^X+^Cj.  (50)  These predicted estimates of the means and variances of the signal and state vectors are conditional on past information. Given these predicted estimates, estimates of the state vector conditional on past and present information may be derived with Bayesian updating.  Define z , as that argument which A  maximizes posterior density function:  Az, 1 y„ „l,,)  =  W  f { y  i "»MH»' 1 Z  1  ^<- > '-).  z  )f{z  (51)  1J  Under the assumption of multivariate normally distributed signal and state innovation vectors, together with conditionally contemporaneously  uncorrelated signal vectors,  z,  |(  minimizes  objective function S(z,) - (z, - z , ) P,^_ (z, - z , ) T  M  x  H  ^  84 subject to signal equations (42) and (44). The necessary first order condition associated with the i m p l i e d unconstrained m i n i m i z a t i o n problem yields  h ~Z, where  H  +  K (y y<  -  t  J  2  sufficient i f Pf  x  + K (w,-  )  A  error o f z , conditional o n 2~_,.  This necessary first order condition is  x  is positive definite.  -CjR^C,  2  (53)  | M  - P^ C]Rf .  w  -A]Q;fA  H> ),  Wi  and K  K =P^A Q~f y  J/M  D e f i n e / » , as the mean squared  W i t h i n the framework o f this linear state space m o d e l , this  l]t  mean squared error matrix satisfies:  P ^ P ^ - K ^ P ^ - K ^ P ^ U n d e r o u r distributional f(z, \ y , , w l n  P , 0]0  t A  (54)  assumptions, z  equals the mean o f posterior density  t]l  function  ) , and is therefore mean squared error optimal. G i v e n initial conditions z  0 | 0  and  recursive evaluation o f equations (45), (46), (47), (48), (49), (50), (53) and (54) yields  predicted and filtered estimates o f the state vector. G i v e n these predicted and filtered estimates, estimates o f the state vector conditional on past, present and future information may be derived with Bayesian updating.  Define z,  as that  w  argument w h i c h m a x i m i z e s posterior density function:  / U,1 V P A )  77—r^r-  •  U n d e r the assumption o f a multivariate  (55)  normally distributed state innovation vector,  z,  w  m i n i m i z e s objective function S(z,) = (z, - z |,) P,\, (z, - z,\ )-(z, i - z, || ) ^ + i | , ( z , + i X (  t  subject to state equation (43).  +  t  (56)  (  The necessary first order condition associated w i t h the i m p l i e d  unconstrained m i n i m i z a t i o n problem yields I\T  Z  =  +  "^/(z,+i|r ~ /+i|r)'  where J = P^BJP'f,. t  positive definite.  (57)  z  T h i s necessary first order condition is sufficient i f Pf -B P l\ B 2  Define P  I]T  as the mean squared error o f z , conditional on l . l}T  t  l  l  2  is  W i t h i n the  framework o f this linear state space m o d e l , this mean squared error matrix satisfies:  P  l]T  = P, J,(P , -P, , )jJ. ll+  l+ T  + t  (58)  85 Under our distributional assumptions, f(z, | z, ,1,),  and is therefore mean squared error optimal. Given terminal conditions z  +]  P  w  equals the mean of posterior density function  v  and  obtamed from the final evaluation of the prediction and updating equations, recursive  evaluation of equations (57) and (58) yields smoothed estimates of the state vector  2.4. Estimation, Inference and Forecasting  Although  unobserved  components  models  feature  prominently  in  the  empirical  macroeconomics literature, an unobserved components model o f the monetary transmission mechanism has yet to be developed and estimated.  Given that the monetary transmission  mechanism is a cyclical phenomenon, it seems natural to model it within the framework of an unobserved components model.  2.4.1. Estimation  The traditional econometric interpretation of macroeconometric models regards them as representations of the joint probability distribution of the data.  Adopting this traditional  econometric interpretation, the parameters and trend components of our unobserved components model of the monetary transmission mechanism in a small open economy are jointly estimated by full information maximum likelihood, conditional on prior information concerning the values of trend components.  2.4.1.1. Estimation Procedure  Let JC, denote a vector stochastic process consisting of the levels of N nonpredetermined endogenous variables, of which M are observed.  The cyclical components of this vector  stochastic process satisfy third order stochastic linear difference equation  4)*/ = 4*M + 4*,- + 4 ,*, , + . E  2  where e  1(  +  (59)  ~ iid A/'(0,2' ). If there exists a unique stationary solution to this multivariate linear 1  rational expectations model, then it may be expressed as: x, =B x„ +B x,. l  l  2  2  + Be . 3  lJ  (60)  86  This unique stationary solution is calculated with the matrix decomposition based algorithm due to Klein (2000). The trend components of vector stochastic process x  t  satisfy first order stochastic linear  difference equation C x, = C,v, + C x,_ +£,_„ 0  2  (61)  1  where E2I - iid /V(0,27,). Vector stochastic process v, consists o f the levels o f L common stochastic trends, and satisfies first order stochastic linear difference equation /= /-i 3,/»  v  (62)  + £  v  where £ , - i i d A/"(0,27 ). 3  x=  3  Cyclical and trend components are additively separable, that is  x +x  t  t  Let  r  y  denote a vector stochastic process consisting o f the levels of M  t  nonpredetermined endogenous variables.  observed  Also, let z, denote a vector stochastic process  consisting of the contemporaneous levels of N - M unobserved nonpredetermined endogenous variables, the contemporaneous and lagged cyclical components of N  nonpredetermined  endogenous  nonpredetermined  variables, the contemporaneous  trend components  of  N  endogenous variables, and the levels of L common stochastic trends. Given unique stationary solution (60), these vector stochastic processes have linear state space representation y, = F&>  (63) 2 4,n  Z< = \Z,-\ G  where s  +G  () 64  £  - iid A/"(0,27 ) and z - A (z ,/ | ). r  4l  4  0  >  0[0  0  (l  Let w, denote a vector stochastic process  consisting of preliminary estimates of the trend components of M observed nonpredetermined endogenous variables. Suppose that this vector stochastic process satisfies w,=H z,+e x  where e  51  (65)  yn  - iid A/"(0,27 ). Conditional on known parameter values, this signal equation defines 5  a set of stochastic restrictions on selected unobserved state variables.  The signal and state  innovation vectors are assumed to be independent, while the initial state vector is assumed to be independent from the signal and state innovation vectors, which combined with our distributional assumptions implies multivariate normality.  87 Conditional on the parameters associated with these signal and state equations, estimates of unobserved state vector z, and its mean squared error matrix P may be calculated with the fdter t  derived previously. Given initial conditions z  0|0  and P , estimates conditional on information 0|Q  available at time / -1 satisfy prediction equations: t\t-\  z  p  (66) ^Zfi],  \ i-\\ -\ \~  =  G  p  G  t  (67)  y,\,-\= Fz  (68)  = FP  F  T  (69) (70)  =  ,\i-\  R  HP _H l  l{l  l  T  (71)  l  Given these predictions, under the assumption of multivariate normally distributed signal and state innovation vectors, together with conditionally contemporaneously  uncorrected signal  vectors, estimates conditional on information available at time t satisfy updating equations  U  V i y  Z,k =  + K  _  >Vi)  + K  ~ , ( > " " V i >'  (  w  P =P ^-K F P -K H^, t[l  t  y  x  lVA  where K = i > , H i  ?  y  (J  M  2  )  (73)  K  T l  7  and K = P^^HjR^[ . }  Given terminal conditions z  T{T  and P  T]T  obtained from the final evaluation o f these prediction and updating equations, estimates conditional on information available at time T satisfy smoothing equations  P =P J,(P, -P, „)J], 1]T  III+  +MT  +  (75)  where J = P^GjP'J^ . Under our distributional assumptions, these estimators o f the unobserved t  state vector are mean squared error optimal. Let 0 e0 cR  K  denote a K dimensional vector containing the parameters associated with  the signal and state equations o f this linear state space model. The full information maximum likelihood estimator 0 o f this parameter vector maximizes conditional loglikelihood function: T  88 (76)  Under the assumption of multivariate normally distributed signal and state innovation vectors, together with conditionally contemporaneously uncorrelated signal vectors, the contributions to this conditional loglikelihood function satisfy 1,(0) - f. (0) + C (0), where: w  ln(2^)-iln | Q,  (*) = - y  L, (0) =  [IA  -|ln I  ^ )  - y  \ - -(y,-y ,  -jy,),  {  l{l  | -^(M>,  (77)  - n y , ) /?„'_,(*, - n y , ).  (78)  T  Under regularity conditions stated in Watson (1989), full information maximum likelihood estimator 0 is consistent and asymptotically normal, T  (79)  where 0 e0 denotes the true parameter vector. consistent estimators of and B are given by  Following Engle and Watson (1981),  o  0  A^j^aXK),  (80)  A=^Y,b,(0 )b,(0 )\ T  (81)  T  where a (0 ) = a (9 ) + a (0 ) t  T  y  T  w  and b (0 ) = b (0 ) + b (0 ).  T  l  T  y  T  w  Under our distributional  T  assumptions,  w )=v^j-,a v^/k_, + i v,ej., T  « ,«?r) =  , V ^ , , _ ,+  w  ( r) = V tf  , ® a )v a , H  M  (  iv /?;,(/?, '  (0 ) and Z>„ (0 ) = V /  fl  k l  (^).  fl  OiJ-^V,^.,,  (82)  H  (83)  If the signal and state innovation vectors  are multivariate normally distributed, then the conditional information matrix equality holds and r  r  w  89 2.4.1.2. Estimation Results  Our unobserved components model of the monetary transmission mechanism in a small open economy is estimated by full information maximum likelihood, conditional on prior information concerning the values of trend components.  The data set consists of the levels of twenty  observed endogenous variables for Canada and the United States described in Appendix 2.A. The initial values of state variables are treated as parameters, and are calibrated to match functions of preliminary estimates of trend components calculated with the linear filter described in Hodrick and Prescott (1997).  The conditional loglikelihood function is maximized  numerically with a modified steepest ascent algorithm.  Estimation results pertaining to the  period 1972Q1 through 2005Q1 appear in Appendix 2.B, with robust t ratios reported in parentheses.  The sufficient condition for the existence of a unique stationary rational  expectations equilibrium due to Klein (2000) is satisfied in a neighbourhood around the full information maximum likelihood estimate, while the outer product o f the gradient estimator of the information matrix is not nearly singular at the full information maximum likelihood estimate, suggesting that the linear state space representation of this unobserved components model is locally identified. Prior information concerning the values of trend components is generated by fitting third order deterministic polynomial functions to the levels of all observed endogenous variables by ordinary least squares.  Stochastic restrictions on the trend components of all observed  endogenous variables are derived from the fitted values associated with these ordinary least squares regressions, with innovation variances set proportional to estimated prediction variances assuming known parameters.  A l l stochastic restrictions are independent, represented by a  diagonal covariance matrix, and are harmonized, represented by a common factor of proportionality. Reflecting little confidence in these preliminary trend component estimates, this common factor of proportionality is set equal to one. The signs of all parameter estimates are consistent with our priors, while most are statistically significant at conventional levels.  Estimates of the variances of innovations  associated with both cyclical and trend components are often statistically significant at conventional levels, suggesting that the levels of the observed endogenous variables under consideration are subject to shocks having both temporary and permanent effects. Predicted, filtered and smoothed estimates of the cyclical and trend components of observed endogenous variables are plotted together with confidence intervals in Appendix 2.B. These confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters.  The predicted estimates are conditional on past  information, the filtered estimates are conditional on past and present information, and the  90 smoothed estimates are conditional on past, present and future information. Visual inspection reveals close agreement with the conventional dating of^business cycle expansions and recessions. In order to examine whether our unobserved components model of the monetary transmission mechanism in a small open economy is dynamically complete in mean and variance, we subject the levels and squares of the predicted standardized residuals to the autocorrelation test of Ljung and Box (1978). We also examine whether there exist significant departures from conditional normality with the test of Jarque and Bera (1980). The predicted standardized residual vector is related to the predicted ordinary residual vector %t\t-\ ~ yt ~ y \t-\ • t  |(  = Qff$ _ ,  _, by  l[t  {  where  The inverse square root of predicted conditional covariance matrix (? ,_, is (|  calculated with a spectral decomposition as Qff = X^^A'ffxJ^^,  where A" ,_, denotes a  square matrix containing distinct orthonormal eigenvectors, while /1,  (|  M  denotes a diagonal  matrix containing the corresponding positive eigenvalues. We find moderate evidence of autocorrelation in the predicted standardized residuals, suggesting that the conditional mean function is dynamically incomplete. Furthermore, we find strong evidence of autoregressive conditional heteroskedasticity in the predicted standardized residuals, suggesting that the conditional variance function is dynamically incomplete. Finally, we find strong evidence of departures from normality in the predicted standardized residuals, in part attributable to the existence of excess kurtosis. These residual diagnostic test results suggest that our full  information maximum likelihood estimation  results  are consistent and  asymptotically normal, but are asymptotically inefficient.  2.4.2. Inference Achieving low and stable inflation calls for accurate and precise indicators of inflationary pressure, together with an accurate and precise quantitative description of the monetary transmission mechanism.  Our unobserved components model of the monetary transmission  mechanism in a small open economy addresses both of these challenges within a unified empirical framework.  2.4.2.1. Quantifying Inflationary Pressure Theoretically prominent indicators of inflationary pressure such as the natural rate of interest and natural exchange rate are unobservable. As discussed in Woodford (2003), the natural rate  91  of interest provides a measure of the neutral stance of monetary policy, with deviations of the real interest rate from the natural rate of interest generating inflationary pressure. It follows that the key to achieving low and stable inflation is the conduct of a monetary policy under which the short term nominal interest rate tracks variation in the natural rate of interest as closely as possible. Predicted, fdtered and smoothed estimates of the natural rate of interest are plotted together with confidence intervals versus corresponding estimates of the real interest rate in Figure 2.1. This concept of the natural rate of interest represents that short term real interest rate consistent with achieving inflation control and output stabilization objectives in the absence of shocks having temporary effects.  Visual inspection reveals that our estimates of the natural rate of  interest exhibit persistent low frequency variation and are relatively precise, as evidenced by relatively narrow confidence intervals. Periods during which the estimated real interest rate exceeds the estimated natural rate o f interest are closely aligned with the conventional dating of recessions, suggesting that tight monetary policy was to varying degrees a contributing factor. Figure 2.1. Predicted, filtered and smoothed estimates o f the natural rate o f interest RINT_P (APR)  RINT_F (APR)  RINTS (APR)  Note: Estimated levels are represented by black lines, while blue lines depict estimated trend components. Symmetric 95% confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters.  Shaded regions indicate  recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  Predicted, filtered and smoothed estimates of the natural exchange rate are plotted together with confidence intervals versus the observed real exchange rate in Figure 2.2. This concept of the natural exchange rate represents that real exchange rate consistent with achieving inflation control and output stabilization objectives in the absence of shocks having temporary effects. Visual inspection reveals that our estimates of the natural exchange rate exhibit persistent low frequency variation and are relatively precise, as evidenced by relatively narrow confidence intervals.  92 Figure 2.2. Predicted, filtered and smoothed estimates o f the natural exchange rate LREXCH_P  LREXCHF  LREXCH_S  Note: Observed levels are represented b y black lines, while blue lines depict estimated trend components. Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters.  Shaded regions indicate  recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  2.4.2.2. Quantifying the Monetary Transmission Mechanism The monetary transmission mechanism describes the dynamic effects of unsystematic variation in the instrument of monetary policy on indicators and targets.  In a small open  economy, the monetary transmission mechanism features both interest rate and exchange rate channels, while an inflation targeting central bank must react to shocks originating both domestically and abroad. Estimated impulse responses to domestic and foreign monetary policy shocks are plotted in Figure 2.3 and Figure 2.4, providing a quantitative description of the monetary transmission mechanism in a small open economy. In response to a domestic monetary policy shock, the domestic nominal and real interest rates exhibit immediate increases followed by gradual declines. The domestic currency appreciates in nominal and real terms, with the nominal exchange rate exhibiting delayed overshooting. These real interest rate and real exchange rate dynamics induce persistent hump shaped reductions in domestic output, consumption, investment, exports and imports, together with persistent hump shaped declines in domestic output price inflation and consumption price inflation, with peak effects realized after one to two years. These output dynamics are associated with a persistent hump shaped reduction in domestic employment, together with a persistent hump shaped increase in the domestic unemployment rate, inducing a persistent hump shaped decline in domestic wage inflation, with peak effects realized after one to two years.  These results are  qualitatively consistent with those of structural vector autoregressive analyses of the monetary transmission mechanism in open economies such as Eichenbaum and Evans (1995), Clarida and Gertler (1997), K i m and Roubini (1995), and Cushman and Zha (1997).  93  Figure 2.3. Estimated impulse responses to a domestic monetary policy shock D L P G D P (APR)  LPGDP  D L P C O N (APR)  LPCON  Note: Estimated impulse responses to a 50 basis point monetary policy shock are depicted.  In response to a foreign monetary policy shock, the foreign nominal and real interest rates exhibit immediate increases followed by gradual declines. These real interest rate dynamics induce persistent hump shaped reductions in foreign output, consumption and investment, together with a persistent hump shaped decline in foreign inflation, with peak effects realized after one to two years. These output dynamics are associated with a persistent hump shaped reduction in foreign employment, together with a persistent hump shaped increase in the foreign unemployment rate, inducing a persistent hump shaped decline in foreign wage inflation, with  94 peak effects realized after one to two years. The domestic currency depreciates in nominal and real terms, with the nominal exchange rate exhibiting delayed overshooting. These real interest rate and real exchange rate dynamics induce persistent hump shaped reductions in domestic output, exports and imports, together with persistent hump shaped declines in domestic output price inflation and consumption price inflation.  These output dynamics are associated with a  persistent hump shaped reduction in domestic employment, together with a persistent hump shaped increase in the domestic unemployment rate, inducing a persistent hump shaped decline in domestic wage inflation.  These results are qualitatively consistent with those of structural  vector autoregressive analyses of the monetary transmission mechanism in closed economies such as Sims and Zha (1995), Gordon and Leeper (1994), Leeper, Sims and Zha (1996), and Christiano, Eichenbaum and Evans (1998, 2005).  95 Figure 2.4. Estimated impulse responses to a foreign monetary policy shock  DLNWAGE (APR)  DLPGDPF (APR)  DLNWAGEF (APR)  Note: Estimated impulse responses to a 50 basis point monetary policy shock are depicted.  2.4.3. Forecasting While it is desirable that forecasts be unbiased and efficient, the practical value of any forecasting model depends on its relative predictive accuracy. As a benchmark against which to evaluate the predictive accuracy of our unobserved components  model of the  monetary  transmission mechanism in a small open economy, we consider the autoregressive integrated  96 moving average or A R I M A class of models. In particular, we consider A R I M A models for the levels of observed endogenous variables y  it  of the form (84)  where s  it  ~ iid j\f(0,crf).  Theoretical support for this univariate forecasting framework is  provided by the decomposition theorem due to Wold (1938), which states that any covariance stationary purely linearly indeterministic scalar stochastic process has an infinite order moving average representation. A s discussed in Clements and Hendry (1998), any infinite order moving average process can be approximated to any required degree of accuracy by an autoregressive moving average process, with the required autoregressive and moving average orders typically being relatively low. The A R I M A models are estimated by maximum likelihood over the period 1972Q3 through 2005Q1. The autoregressive, ordinary difference, and moving average orders are jointly selected to minimize the model selection criterion function proposed by Schwarz (1978). Those A R I M A 1  model specifications deemed optimal are employed throughout our forecast  performance  evaluation exercise. In the absence of a well defined mapping between forecast errors and their costs, relative predictive accuracy is generally assessed with mean squared prediction error based measures. A s discussed in Clements and Hendry (1998), mean squared prediction error based measures are noninvariant to nonsingular, scale preserving linear transformations, even though linear models are.  It follows that mean squared prediction error based comparisons may yield conflicting  rankings across models, depending on the variable transformations examined. To evaluate the dynamic out of sample forecasting performance o f our unobserved components model of the monetary transmission mechanism in a small open economy, we retain forty quarters of observations to evaluate forecasts one through eight quarters ahead, generated conditional on parameters estimated using information available at the forecast origin.  The  models are compared on the basis of mean squared prediction errors in levels, ordinary differences, and seasonal differences.  The unobserved components model is not recursively  estimated as the forecast origin rolls forward due to the high computational cost of such a procedure, while the A R I M A models are. Presumably, recursively estimating the unobserved components model would improve its predictive accuracy.  The autoregressive order p , , ordinary difference order d , i  two and two, respectively.  and m o v i n g average order q are j o i n t l y selected subject to upper bounds o f four, t  97 Mean squared prediction error differentials are plotted together with confidence intervals accounting for contemporaneous and serial correlation of forecast errors in Appendix 2.B. If these mean squared prediction error differentials are negative then the forecasting performance of the unobserved components model dominates that of the A R I M A models, while i f positive then the unobserved components model is dominated by the A R I M A models in terms of predictive accuracy. The null hypothesis of equal squared prediction errors is rejected by the predictive accuracy test of Diebold and Mariano (1995) i f and only i f these confidence intervals exclude zero. The asymptotic variance of the average loss differential is estimated by a weighted sum of the autocovariances of the loss differential, employing the weighting function proposed by Newey and West (1987). Visual inspection reveals that these mean squared prediction error differentials are of variable sign, suggesting that the unobserved components model matches the A R I M A models in terms of forecasting performance, in spite of a considerable informational disadvantage. However, these mean squared prediction error differentials are rarely statistically significant at conventional levels, perhaps because the predictive accuracy test due to Diebold and Mariano (1995), which is univariate, typically lacks power to detect dominance in forecasting performance, as evidenced by Monte Carlo evaluations such as Ashley (2003) and McCracken (2000). Dynamic out of sample forecasts of levels, ordinary differences, and seasonal differences are plotted together with confidence intervals versus realized outcomes in Appendix 2.B. These confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters.  Visual inspection reveals that the realized outcomes  generally lie within their associated confidence intervals, suggesting that forecast failure is absent.  However, these confidence intervals are rather wide, indicating that considerable  uncertainty surrounds the point forecasts.  2.5. Conclusion This paper develops and estimates an unobserved components model of the monetary transmission mechanism in a small open economy for purposes of monetary policy analysis and inflation targeting.  This estimated unobserved components model provides a quantitative  description of the monetary transmission mechanism in a small open economy, yields a mutually consistent set of indicators of inflationary pressure together with confidence intervals, and facilitates the generation of relatively accurate forecasts. Definitions of indicators of inflationary pressure such as the natural rate of interest and natural exchange rate vary, while estimates are typically sensitive to identifying restrictions. It  98 follows that combinations of estimates of indicators of inflationary pressure derived under alternative definitions from dissimilar models may be more useful for purposes of monetary policy analysis and inflation targeting in a small open economy than any of the constituents: A n examination of the inflation control and output stabilization benefits conferred by combining alternative estimates remains an objective for future research.  Appendix 2.A. Description of the Data Set The data set consists of quarterly seasonally adjusted observations on twenty macroeconomic variables for Canada and the United States over the period 1971Q1 through 2005Q1. A l l aggregate prices and quantities are expenditure based.  Employment is derived from observed  nominal labour income and a nominal wage index, while the unemployment rate is expressed as a period average.  The nominal interest rate is measured by the three month Treasury bill rate  expressed as a period average, while the nominal exchange rate is quoted as an end of period value. National accounts data for Canada was retrieved from the C A N S I M database maintained by Statistics Canada, national accounts data for the United States was obtained from the F R E D database maintained by the Federal Reserve Bank of Saint Louis, and other data was extracted from the IFS database maintained by the International Monetary Fund.  99 Appendix 2.B. Tables and Figures  Table 2.1. Full information maximum likelihood estimation results, domestic economy 0u 0.385  0.2  02,1  02.2  03.1  03,2  04,1  04,2  05.1  05.2  06.1  06,2  0.277  0.392  0.213  0.976  -0.190  0.879  0.011  0.973  -0.323  0.299  -0.014  (2.472)  (0.325)  (1.957)  (0.156)  (6.861)  (-1.331)  (5.843)  (0.077)  (7.536)  (-3.028)  (2.798)  (-0.157)  07,1  07.2  08.1  08,2  09,1  09,2  00,1  00.2  011,1  012,1  012,2  0.651  -0.313  0.445  0.184  0.572  0.092  0.830  -0.237  0.725  0.571  0.372  (3.912)  (-2.819)  (0.866)  (0.073)  (3.069)  (0.589)  (5.886)  (-2.186)  (9.827)  (0.236)  (0.104)  #,,  # 2 ,  #2,2  #3,2  #3.3  #4,1  #5.1  *«..  #6,2  #7,1  #7,2  0.048  0.031  0.010  0.199  -0.185  0.007  -0.336  0.871  1.614  0.089  1.735  -0.051  (0.549)  (0.395)  (0.465)  (4.374)  (-0.917)  (0.621)  (-1.859)  (4.242)  (6.654)  (1.533)  (4.875)  (-0.793)  <?8,i  ^8,2  #9.1  0.570  #n,i 0.120  #12,1  0.164  #io,i -0.237  #11,2  -0.046  0.081  -0.048  (-0.205)  (0.751)  (5.916)  (-5.289)  (1.545)  (5.491)  (-0.064)  <4  -I  0.132  0.072  0.231  0.291  2.533 '  2.433  2.184  0.129  0.200  0.025  0.037  1.752  (0.970)  (0.633)  (4.688)  (4.651)  (5.394)  (3.920)  (3.838)  (0.328)  (1.806)  (2.625)  (5.072)  (0.117)  <t  al  4-  °l  °l  °)  4  al  -I  0.112  0.088  0.196  0.270  2.022  2.141  2.921  0.165  0.357  0:041  i 0.010  1.919  (2.757)  (3.071)  (4.469)  (4.321)  (3.992)  (4.050)  (4.097)  (3.714)  (3.221)  (3.334)  (3.806)  (4.360)  ~ 2 *  „ g  2 n  3.49xl0" 5.24xl0" 5.36xl0~ 3  (2.228)  6  (0.605)  5  (1.276)  7-M  S,t,-i  h,\,-\  hi\,-\  £"iii-i  ht\t-\  5.458*  1.328  2(2)  0.058  1.566  2.228  4.417  3.931-  0.675  12.263"'  2.583  0.779  0.338  2(4)  5.489  2.093  14.461"  10.258"  4.819  4.099  12.759"  3.010  2.481  1.377  8.737'  8.569'  2  e (2) e (4)  38.452"  26.663"  47.865"  20.507*"  47.034"'  47.410"  48.900"'  20.631'"  43.456"*  46.364**'  71.222"*  34.705*"  J  78.884"  56.533"  84.406"  58.357"'  75.306""  74.097"  61.142'*'  66.940"'  155.382"*  Skewness  -0.272  -0.180  -0.258  -0.509"  0.137  0.253  0.438"  Kurtosis  3.291  3.379  3.226  4.286*"  2.953  2.979  2.910  3.893"  4.711*"  3.702*  4.690**'  3.670  JB  2.110  1.510  1.756  14.906"*  0.428  1.423  4.289  6.108"  16.232"'  10.445"'  19.354"'  2.716  116.574"' 42.569"* 0.276  -0.004  Note: Rejection o f the null hypothesis at the 1%, 5 % and 10% levels is indicated by * * * , * * and * , respectively.  0.590"'  0.399'  69.776'** -0.102  100 Table 2.2. Full information maximum likelihood estimation results, foreign economy 01,1  <t>\,2  03.2  04.1  04.2  05.1  05.2  0.493  0.290  1.177  -0.310  1.092  -0.190  0.300  -0.121  (0.489)  (0.098)  (12.784)  (-3.060)  (9.550)  (-1.770)  (3.607)  (-1.682)  $.1  0s.i  0U  09,i  09.2  010.1  010,2  011,1  0.105  0.142  0.935  -0.194  0.746  -0.192  0.747  (0.393)  (0.073)  (6.322)  (-1.536)  (8.632)  (-2.714)  (9.210)  ^3.2  *4,1  09,1  0io,i  -0.632  -0.789  0 , 4.061  ^8,2  0.007  -0.041  0.413  0.306  -0.224  (0.144)  (-1.626)  (-3.030)  (8.986)  (-0.414)  (3.483)  (5.836)  (-6.891)  5  0|i,i  ^11.2  0.208  0.056  (1.683)  (4.800)  0.025  0.407  0.235  2.777  0.037  0.104  0.012  0.025  (0.223)  (5.272)  (5.705)  (3.312)  (1.470)  (2.665)  (2.056)  (3.315)  <4  °i  °1  al  0.037  0.096  0.079  1.944  0.080  0.212  0.023  0.007  (3.046)  (3.024)  (3.386)  (2.920)  (4.239)  (3.948)  (3.589)  (2.869)  °\  a)  -I  °>T  i  °l 2.14xl0" (2.495)  3  6.19xl0" (2.129)  5  1.40X10"  4  (1.701)  SiK-l  2(2)  3.334  C<i>-\  1.281  8,821"  1.816  7.823*  7.363  8.704'  10.755*  16.416"*'  12.753"  1.441  20.437"'  12.939""  11.448*  69.938""  55.532"'  24.154"'  28.954"'  23.797""  23.569'  122.141""  108.846"'  40.219"*  58.337"*  -0.305  -0.283  -0.452"  0.063  0.991"'  1.170"*  2.474  3.218  4.927*"  10.565'*'  6.059"  0.352  42.354"'  347.477*"  G(4) g (2)  29.357*"  12.149*"  G (4) 2  Cii-i  6.285"  15.198"'  2  h,\t-\  46.188*"  18.945*"  Skewness  0.209  0.509"  Kurtosis  4.029"  6.060""  4.606"*  6.040*  JB  6.841"  57.641'"  16.360"'  52.977'  11.126""  7.377"'  C(0 ) = -6605.462 T  Note: Rejection o f the null hypothesis at the 1%, 5 % and 1 0 % levels is indicated by * * * , * * and * , respectively.  101 Fi gure 2.5. Predicted cyclical components of observed endogenous variables  Note: Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the Economic C y c l e Research Institute reference c y c l e .  102 Fi gure 2.6. Filtered cyclical components of observed endogenous variables  Note: Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  103 Figure 2.7. Smoothed cyclical components o f observed endogenous variables L P G D P  1975  1930  1965  1990  L P C O N  1995  2000  2006  1975  1960  1985  1990  L R G D P  1995  20O0  2005  1975  0  1975  1980  1985  1990  19B5  2000  2005  1975  1930  19B5  1990  1995  2000  2005  I960  1985  1990  1995  2000  2005  1975  1960  1985  1990  1995  2030  2005  1975  1980  1985  1900  1995  2000  2005  •y 1  18751980  1985  NIN T  1990  (APR)  ft  fl \  'Ii  A '•4'%'  if 1975  1980  1985  1990  1995  2000  2006  1975  1990  1985  1990  1985  2000  1975  19B0  1985  1990  1995  2000  2005  1975  I960  1985  1990  1995  2000  2005  1975  19B0  1985  1990  1995  2003  2005  1975' ' 1980 ' 1935' ' 1990  NINTF  1975  1980  1985  1990  1995  1975  1980  1985  1990  1995  2000  2005  1975  1980  1985  1990  (APR)  1995  Note; Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated by the Economic C y c l e Research Institute reference c y c l e .  104 Figure 2.8. Predicted trend components of observed endogenous variables LPGDP  LPCON  LRGDP  LRCON  Note: Observed levels are represented by black lines, while blue lines depict estimated trend components. Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  105 Fi gure 2.9. Filtered trend components of observed endogenous variables  Note: Observed levels are represented by black lines, while blue lines depict estimated trend components. Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  106 Figure 2,10. Smoothed trend components of observed endogenous variables  1975  1980  1985  1975  1980  1985  1990  1995  2000  2005  2000  2005  2000  2005  2000  2005  NINT(APR)  1990  1995  NINTF(APR)  1975  1980  1965  1990  1965  1990  1996  2000  2005  1975  1980  1985  1990  1995  2000  2005  1975  1980  1985  1990  1995  Note: Observed levels are represented by black lines, while blue lines depict estimated trend components. S y m m e t r i c 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  107 Figure 2 . 1 1 . Mean squared prediction error differentials for levels LPGDP  LPCON  LRGDP  LRC01N  3 B  20  2  i  1  10.  0  0  0  -4.  -  -10  •fl-  -2  %  •2  -20-  -4  -31  2  3  4  5  6  7  l  1  2  3  LRINV  .  s  e  ;  1  1 2  3  LREXP  4  5  6  7  1  1  2  3  LRIUP  4  5  6  7  8  6  7  8  6  7  8  6  7  8  6  7  8  LNWAGE 20  iCO  400  400  400  XO  10-  200  200  0-  0  200  •200-  4(1)  -200  10  -400-400-  800 6002  3  4  5  6  7  2  3  LEMP  4  5  6  -20  7  2  3  RUNENP  5  6  7  1  3  4  5  LNEXCH  0 6-  20-  100-  0.4  0-  2  NINT  3.  4(1  4  0  0.0.  20-  0  -0 4. 100H  2. 40-  -0 8-  -3. 2  3  4  5  6  7  2  3  LPGDPF  4  5  6  7  2  3  LRGDPF  4  5  6  7  2  3  LRCONF  4  5  LRINVF  20-  B-  500.  10.  4.  202-  4.  -10-4.  -500.  fl•6-  •202  3  4  5  6  7  I  2  3  LNWAGEF  4  5  6  7  2  3  LEMPF  IC  4  5  6  7  2  3  RUNEMPF  60.  3.  40-  2-  4  5  NINTF 2.  20i-  0  0-  -2010-  -2-  -W-602  3  4  5  6  7  8  -32  3  4  5  6  7 ;  2  3  4  5  6  7  (  2  3  4  5  Note: M e a n squared prediction error differentials are defined as the mean squared prediction error for the unobserved components model less that for the AR1MA model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors.  108 Figure 2.12. Mean squared prediction error differentials for ordinary differences DLPGDP  DLPCON  DLRGDP  .« 2  .3  05 1 X  •2  •.«  - 05  -2  -5  4  -.10 1  3  2  4  5  6  7  -.3  1  1  2  3  DLRINV  4  5  8  7  8  1 2  3  5  6  7  1  2  3  DLRIfvF  DLREXP  12.  4  4  5  6  7  8  6  7  8  6  7  8  6  7  8  6  7  8  DLNWAGE  10  84  s.  o-  0-  •4  %  2  *  -5  -8-  -2  -10-  -12-  \  •10 '*  2  3  4  5  6  7  8  1  2  3  4  5  6  7  1  1.2. 0 8.  7  1  •  2  3  "  -.01-  0 8-  4  5  DLNEX;H 10-  02-  00-  -.05.  6  5-  01-  rm  0.4.  5  03-  05-  ^—~~  4  DNINT  10-  ——  3  DRUNErVP  DLEMP  0 0-  2  _____  \_  -  -5.  -.02. - ia  •1 2-  -102  3  4  5  5  7  2  3  DLPGDPF  4  5  6  7  2  3  DLRGDPF  1-  4  5  6  7  2  3  DLRCONF  2.  5  DLRINVF  J.  0.8-  4  20.  i0.4-  ii.  10-  o.a  i.  •0.4.  2.  10-  -OB. 2  3  4  5  6  7  20-  6  2  3  DLNWAGEF  4  5  6  7  l  3  •  DLEMPF  ;  •  •  •  '  i  2  3  DRUNENPF  4  5  DNINTF 06-  2-  u2.1.  08-  04.  '•  04.  0-  00  nn  i.  0*.  -.02.  -?-  OB-  - 04-  02-  08. 1  2  3  »  5  6  7  8  1  2  3  4  5  6  7  2  3  4  5  6  7  8  2  3  4  5  Note: M e a n squared prediction error differentials are defined as the mean squared prediction error for the unobserved components model less that for the A R I M A model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors.  109 Figure 2.13. Mean squared prediction error differentials for seasonal differences SDLPCON  SDLRGDP  e  3 0.8  4  0.4  2 '  2  9  0.0  -2  -0.4  -4  -5 -2  •0.8.  •ft  -3 1  2  3  4  5  6  7  1  1  2  3  4  SDLRINV  5  6  7  t  1 2  3  SDLREXP  4  5  6  ?  1  0  0  5  6  6  7  8  -4-  «-  -150 4  7  -2  -100  -200. 3  6  2.  -50-  2  5  100  -100  1  4  ft.  50  101)  3  SDLNWAGE  150 100  2  SDLRINP  20U  100  1  7  2  3  SDLEMP  4  5  6  7  1  2  3  SDRUNEIvP  4  5  6  7  1  2  3  SDNINT  4  5  B  SDLNEXCH  15. 1.0-  105-  0 5-  0 -  0.0.  50-  .2.  —  ...  0.  — .  -5-  •  "—-—....  -0.5-  -10.  2  3  4  5  6  •  -50-  -1.0-  •15.  — —  •2.  7  2  3  SDLPGDPF  4  5  6  7  2  3  SDLRGDPF  4  5  6  7  2  3  SOLRCONF  4  5  6  7  8  6  7  8  6  7  8  SDLRINVF 300-  4  102005-  100-  < U-  0-  fc  -5-  4-  -100.  >-3-  -10-  -2O0-  -4. 2  3  4  5  6  7  8  2  3  SDLNWAGEF  4  5  6  7  -300-  6  2  3  SDLEMPF  4  5  6  7  8  2  3  SDRUNEMPF  4  5  SDNINTF  15. 201.0-  2.  5.  0.5-  J-  0 """"""  z.  —  —  4.  0.0. I I 5-  10-  1 D-  -20-  6.  -.5-  l E 1  2  3  4  5  B  7  8  1  2  3  4  5  6  7  2  3  4  5  6  7  8  2  3  4  5  Note: M e a n squared prediction error differentials are defined as the mean squared prediction error for the unobserved components model less that for the A R I M A model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors.  110 Figure 2.14. Dynamic forecasts of levels of observed endogenous variables  Note: Realized outcomes are represented by black lines, while blue lines depict point forecasts.  Symmetric 9 5 % confidence intervals assume  multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters.  Note: Realized outcomes are represented by black lines, while blue lines depict point forecasts.  Symmetric 95% confidence intervals assume  multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters.  112 Figure 2.16. Dynamic forecasts of seasonal differences of observed endogenous variables  Note: Realized outcomes are represented by black lines, while blue lines depict point forecasts.  Symmetric 95% confidence intervals assume  multivariate normally distributed and independent signal and state innovation vectors and known parameters.  113  References Ashley, R. (2003), Statistically significant forecasting improvements: H o w much out-of-sample data is likely necessary?, International Journal of Forecasting, 19, 229-239. Christiano, L . , M . Eichenbaum and C . Evans (1998), Monetary policy shocks: What have we learned and to what end?, NBER Working Paper, 6400. Christiano, L . , M . Eichenbaum and C . Evans (2005), Nominal rigidities and the dynamic effects o f a shock to monetary policy, Journal of Political Economy, 113, 1-45. Clarida, R. and M . Gertler (1997), H o w the Bundesbank conducts monetary policy, Reducing Inflation: Motivation and Strategy, University o f Chicago Press. Clements, M . and D . Hendry (1998), Forecasting Economic Time Series, Cambridge University Press. Cushman, D . and T. Zha (1997), Identifying monetary policy in a small open economy under flexible exchange rates, Journal of Monetary Economics, 39, 433-448. Diebold, F . and R. Mariano (1995), Comparing predictive accuracy, Journal of Business and Economic Statistics, 13,253-263. Durbin, J. and S. Koopman (2001), Time Series Analysis by State Space Methods, Oxford University Press. Eichenbaum, M . and C Evans (1995), Some empirical evidence on the effects o f shocks to monetary policy on exchange rates, Quarterly Journal of Economics, 110, 1975-1010. Engle, R. and M . Watson (1981), A one-factor multivariate time series model o f metropolitan wage rates, Journal of the American Statistical Association, 76, 774-781. Gordon, D . and E . Leeper (1994), The dynamic impacts o f monetary policy: A n exercise in tentative identification, Journal of Political Economy, 102, 1228-1247. ' Hamilton, J. (1994), Time Series Analysis, Princeton University Press. Hodrick, R. and E. Prescott (1997), Post-war U . S . business cycles: A descriptive empirical investigation, Journal of Money, Credit, and Banking, 29, 1-16. Jarque, C and A . Bera (1980), Efficient tests for normality, homoskedasticity and serial dependence o f regression residuals, Economics Letters, 6, 255-259. Kalman, R. (1960), A new approach to linear filtering and prediction problems, Transactions ASME Journal of Basic Engineering, 82, 35-45. K i m , C . and C . Nelson (1999), State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications, M I T Press. K i m , S. and N . Roubini (1995), Liquidity and exchange rates: A structural V A R approach, Unpublished Manuscript. Klein, P. (2000), Using the generalized Schur form to solve a multivariate linear rational expectations model, Journal of Economic Dynamics and Control, 24, 1405-1423. Laubach, T. and J. Williams (2003), Measuring the natural rate o f interest, Review of Economics and Statistics, 85, 1063-1070. Leeper, E . , C . Sims and T. Zha (1996), What does monetary policy do?, Brookings Papers on Economic Activity, 2, 1-63. Ljung, G . and G . B o x (1978), On a measure o f lack o f fit in time series models, Biometrika, 65, 297-303. McCracken, M . (2000), Robust out-of-sample inference, Journal of Econometrics, 99, 195-223. Newey, W . and K . West (1987), A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica, 55, 703-708. Schwarz, G . (1978), Estimating the dimension o f a model, Annals of Statistics, 6,461-464.  114 Sims, C . and T. Zha (1995), Does monetary policy generate recessions?, Unpublished  Manuscript.  Watson, M . (1989), Recursive solution methods for dynamic linear rational expectations models, Journal  of  Econometrics, 41, 65-89. Wold, H . (1938), A Study in the Analysis of Stationary Time Series, Almqvist and Wiksell. Woodford, M . (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press.  115  CHAPTER 3  Measuring the Stance of Monetary Policy in a Small Open Economy: A Dynamic Stochastic General Equilibrium Approach  3.1. Introduction Estimated dynamic stochastic general equilibrium or D S G E models have recently emerged as quantitative monetary policy analysis and inflation targeting tools.  A s extensions of real  business cycle models, D S G E models explicitly specify the objectives and constraints faced by optimizing households and firms, which interact in an uncertain environment to determine equilibrium prices and quantities. The existence of short run nominal price and wage rigidities generated by monopolistic competition and staggered reoptimization in output and labour markets permits a cyclical stabilization role for monetary policy, which is generally implemented through control of the short term nominal interest rate according to a monetary policy rule. The persistence of the effects of monetary policy shocks on output and inflation is often enhanced with other features such as habit persistence in consumption, adjustment costs in investment, and variable capital utilization.  Early examples of closed economy D S G E models incorporating  some of these features include those of Y u n (1996), Goodfriend and King (1997), Rotemberg and Woodford (1995, 1997), and McCallum and Nelson (1999), while recent examples of closed economy D S G E models incorporating all of these features include those of Christiano, Eichenbaum and Evans (2005), Altig, Christiano, Eichenbaum and Linde (2005), Smets and Wouters (2003, 2005), and Vitek (2006c). Open economy D S G E models extend their closed economy counterparts to allow for international trade and financial linkages, implying that the monetary transmission mechanism features both interest rate and exchange rate channels. Building on the seminal work of Obstfeld and Rogoff (1995, 1996), these open economy D S G E models determine trade and current account balances through both intratemporal and intertemporal optimization, while the nominal  A version o f this chapter has been submitted for publication.  V i t e k , F., Measuring the stance o f monetary p o l i c y in a closed economy: A  dynamic stochastic general equilibrium approach, International Journal of Central Banking.  116 exchange rate is determined by an uncovered interest parity condition. Existing open economy D S G E models differ primarily with respect to the degree of exchange rate pass through. Models in which exchange rate pass through is complete include those of Benigno and Benigno (2002), McCallum and Nelson (2000), Clarida, Gali and Gertler (2001, 2002), and Gertler, Gilchrist and Natalucci (2001), while models in which exchange rate pass through is incomplete include those of Adolfson (2001), Betts and Devereux (2000), Kollman (2001), Corsetti and Pesenti (2002), Monacelli (2005), and Vitek (2006d). Recent research has emphasized the implications o f developments in the housing market for the conduct of monetary policy. Existing D S G E models incorporating a housing market include those of A o k i , Proudman and Vlieghe (2004) and Iacoviello (2005), both of which focus on the implications of financial market frictions for the monetary transmission mechanism. In addition to abstracting from open economy elements of the monetary transmission mechanism, these papers do not consider the implications of developments in the housing market for the measurement of the stance of monetary policy. Existing D S G E models featuring long run balanced growth driven by trend inflation, productivity growth, and population growth generally predict the existence of common deterministic or stochastic trends. Estimated D S G E models incorporating common deterministic trends include those of Ireland (1997) and Smets and Wouters (2005), while estimated D S G E models incorporating common stochastic trends include those of Altig, Christiano, Eichenbaum and Linde (2005) and Del Negro, Schorfheide, Smets and Wouters (2005).  However, as  discussed in Clements and Hendry (1999) and Maddala and K i m (1998), intermittent structural breaks  render  such common deterministic or stochastic  trends empirically inadequate  representations o f low frequency variation in observed macroeconomic variables.  For this  reason, it is common to remove trend components from observed macroeconomic variables with deterministic polynomial functions or linear filters, such as the difference filter or the low pass filter described in Hodrick and Prescott (1997), prior to the conduct of estimation, inference and forecasting.  A s an alternative, Vitek (2006c, 2006d) proposes jointly modeling cyclical and  trend components as unobserved components while imposing theoretical restrictions derived from the approximate multivariate linear rational expectations representation of a D S G E model. This  merging of modeling paradigms  drawn  from the  theoretical  and empirical  macroeconomics literatures confers a number of important benefits. First, the joint estimation of parameters and trend components ensures their mutual consistency, as estimates of parameters appropriately reflect estimates of trend components, and vice versa. As shown by Nelson and Kang (1981) and Harvey and Jaeger (1993), decomposing integrated observed endogenous variables into cyclical and trend components with atheoretic deterministic polynomial functions or low pass filters may induce spurious cyclical dynamics, invalidating subsequent estimation,  117 inference and forecasting. Second, basing estimation on the levels as opposed to differences of observed endogenous variables may be expected to yield efficiency gains. A central result of the voluminous cointegration literature surveyed by Maddala and K i m (1998) is that, i f there exist cointegrating relationships, then differencing all integrated observed endogenous variables prior to the conduct of estimation, inference and forecasting results in the loss of information. Third, the proposed unobserved components framework ensures stochastic nonsingularity of the resulting approximate linear state space representation of the D S G E model, as associated with each observed endogenous variable is at least one exogenous stochastic process. As discussed in Ruge-Murcia (2003), stochastic nonsingularity requires that the number of observed endogenous variables used to construct the loglikelihood function associated with the approximate linear state space representation of a D S G E model not exceed the number of exogenous stochastic processes, with efficiency losses incurred i f this constraint binds.  Fourth, the proposed unobserved  components framework facilitates the direct generation of forecasts of the levels of endogenous variables as opposed to their cyclical components together with confidence intervals, while ensuring that these forecasts satisfy the stability restrictions associated with balanced growth. These stability restrictions are necessary but not sufficient for full cointegration, as along a balanced growth path, great ratios and trend growth rates are time independent but state dependent, robustifying forecasts to intermittent structural breaks that occur within sample. The primary contribution of this paper is the development of a procedure to estimate the levels of the flexible price and wage equilibrium components of endogenous variables while imposing relatively weak, and hence relatively credible, identifying restrictions on their trend components. Based on an extension and refinement of the unobserved components framework proposed by Vitek (2006c, 2006d), this estimation procedure confers a number of benefits of particular importance to the conduct of monetary policy. First, as discussed in Woodford (2003), the levels of the flexible price and wage equilibrium components of various observed and unobserved endogenous variables are important inputs into the optimal conduct of monetary policy.  In particular, the level of the natural rate of interest, defined as that short term real  interest rate consistent with price and wage flexibility, provides a measure of the neutral stance of monetary policy, with deviations of the real interest rate from the natural rate of interest generating inflationary pressure.  The proposed unobserved components framework facilitates  estimation of the levels as opposed to cyclical components of the flexible price and wage equilibrium components of endogenous variables, while ensuring that they satisfy the stability restrictions associated with balanced growth. Second, given an interest rate smoothing objective derived from a concern with financial market stability, variation in the natural rate of interest caused by shocks having permanent effects may call for larger monetary policy responses than variation caused by shocks having temporary effects.  The proposed unobserved components  118 framework yields a decomposition of the levels of the flexible price and wage equilibrium components of endogenous variables into cyclical and trend components, together with confidence intervals which account for uncertainty associated with the detrending procedure. Third, as discussed in Clements and Hendry (1999) and Maddala and K i m (1998), accommodating the existence of intermittent structural breaks requires flexible trend component specifications.  However, the joint derivation of empirically adequate cyclical and trend  component specifications from microeconomic foundations is a formidable task. The proposed unobserved components framework facilitates estimation of the levels of the flexible price and wage equilibrium components of endogenous variables while allowing for the possibility that the determinants of their trend components are unknown but persistent. The secondary contribution of this paper is the estimation of the levels of the flexible price and wage equilibrium components of various observed and unobserved endogenous variables while imposing relatively weak identifying restrictions on their trend components, with an emphasis on the levels of the natural rate of interest and natural exchange rate. Definitions of indicators of inflationary pressure such as the natural rate of interest and natural exchange rate vary, and numerous alternative procedures for estimating the natural rate of interest have been proposed, several of which are discussed in a survey paper by Giammarioli and Valla (2004). Within the framework of a calibrated D S G E model of a closed economy, Neiss and Nelson (2003) find that estimates of the deviation of the real interest rate from the natural rate of interest exhibit economically significant high frequency variation. Within the framework of an estimated D S G E model o f a closed economy, Smets and Wouters (2003) find that estimates of the deviation of the real interest rate from the natural rate of interest exhibit economically significant high frequency variation and are relatively imprecise, as evidenced by relatively wide confidence intervals. In addition to abstracting from open economy elements of the monetary transmission mechanism, these papers abstract from the trend component of the natural rate of interest, as they employ estimation procedures which involve the preliminary removal o f trend components from observed macroeconomic variables with atheoretic deterministic polynomial functions. This paper develops and estimates a D S G E model of a small open economy for purposes of monetary policy analysis and inflation targeting.  This estimated D S G E model provides a  quantitative description of the monetary transmission mechanism in a small open economy, yields a mutually consistent set of indicators of inflationary pressure together with confidence intervals, and facilitates the generation of relatively accurate forecasts. The model features short run nominal price and wage rigidities generated by monopolistic competition and staggered reoptimization in output and labour markets. The resultant inertia in inflation and persistence in output is enhanced with other features such as habit persistence in consumption and labour supply, adjustment costs in housing and capital investment, and variable capital utilization.  119 Incomplete exchange rate pass through is generated by short run nominal price rigidities in the import market, with monopolistically competitive importers setting the domestic currency prices of differentiated intermediate import goods subject to randomly arriving reoptimization opportunities. Cyclical components are modeled by linearizing equilibrium conditions around a stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path. Parameters and unobserved components are jointly estimated with a novel Bayesian procedure, conditional on prior information concerning the values of parameters and trend components. The organization of this paper is as follows. The next section develops a DSGE model of a small open economy.  Estimation, inference and forecasting within the framework of a linear  state space representation of an approximate unobserved components representation of this DSGE model are the subjects of section three.  Finally, section four offers conclusions and  recommendations for further research.  3.2. Model Development Consider two open economies which are asymmetric in size, but are otherwise identical. The domestic economy is of negligible size relative to the foreign economy.  3.2.1. The Utility Maximization Problem of the Representative Household There exists a continuum of households indexed by z'e[0,l].  Households supply  differentiated intermediate labour services, but are otherwise identical.  3.2.1.1. Consumption, Saving and Investment Behaviour The representative infinitely lived household has preferences defined over consumption C . , (  housing H , js  and labour supply L represented by intertemporal utility function i s  U„=E f P -u{C, ,H ,L. ), s  l j  s  s=l  is  (1)  s  1  120  where subjective discount factor /? satisfies 0 < /3 < 1. The intratemporal utility function is additively separable and represents external habit formation preferences in consumption, housing, and labour supply, (C,, -a'C^r*  (H -a H^r"  H  is  l-i/o-  y L  l-i/o-  where 0 < « < 1 , 0 < « < 1 and 0 < a < l . C  (L  H  | y  L  M/n  sl  (2)  \+\in  v  This intratemporal utility function is strictly  t  W  —a L )  js  increasing with respect to consumption if and only if vf > 0 , and given this parameter restriction is strictly increasing with respect to housing if and only if v" > 0 , and is strictly decreasing with respect to labour supply if and only if v > 0 .  Given these parameter restrictions, this  L  intratemporal utility function is strictly concave if a > 0 and r/ > 0 . The representative household enters period 5 in possession of previously purchased domestic currency denominated bonds Bff which yield interest at risk free rate i _ , and foreign currency s  denominated bonds Bf/ which yield interest at risk free rate if. portfolio of shares {xf y js  j=0  in domestic intermediate goodfirmswhich pay dividends  and a diversified portfolio of shares {xf }[ k  pay dividends {I7 }\ . M  k s  =0  x  It also holds a diversified  s  {I7f }f , s  0  in domestic intermediate good importers which  =0  The representative household supplies differentiated intermediate  labour service L , earning labour income at nominal wage W . Households pool their labour s (  is  income, and the government levies a tax on pooled labour income at rate T . These sources of S  private wealth are summed in household dynamic budget constraint:  )KA,,^n;=o  + J (K j=0  + K>KA  j  «  ,  (iH- )C^a+e,)^  ^=  /  1  *=o  ( ) 3  + (1 -1,) ) W L dl - PfC -  + K sK,A  + J«  M  u  ls  Pflf.  Ls  1=0  k=0  According to this dynamic budget constraint, at the end of period s, the representative household purchases domestic bonds Bff , and foreign bonds Bf/ +]  purchases a diversified portfolio of shares [x]'. Y S+]  {Vf Yj=o>  diversified portfolio of shares {xf  a n Q a  s  {V  M k  s  }[ . =0  k  v  +  =  It also  s  in intermediate good firms at prices  J=0  i}[  at price £ .  0  in intermediate good importers at prices  Finally, the representative household purchases final consumption good C, . at price s  Pf, andfinalhousing investment good lf at price Pf" . s  The representative household enters period 5 in possession of previously accumulated housing stock H , which subsequently evolves according to accumulation function js  % r ( l - ^ ) ^ ^ " ( C C ) .  (4)  121 where depreciation rate parameter 8  satisfies 0 < 5" < 1.  H  Effective housing investment  function H '(//l>^"-i) incorporates convex adjustment costs, H  1- X  (^.Vi)- '.  n  1  where % > 0 and v'  > 0.  H  s  f  jH 1  _ j H  i,s  1  V  i,s-\  I"  i"  (5)  In deterministic steady state equilibrium, these adjustment costs  equal zero, and effective investment equals actual investment. In period  t,  the representative  consumption {C }^,, is  household chooses  investment in housing {l",}™ ,, the stock of housing {H }™ , =  iiS+l  bond holdings {Bff }™ , foreign bond holdings +X  firms  {{xjj } }%,, [  s+l  j=0  state contingent sequences  for  domestic  =l  , share holdings in intermediate good  =l  and share holdings in intermediate good importers {{xf  } }™ , ]  k  s+x  k=0  =  to  maximize intertemporal utility function (1) subject to dynamic budget constraint (3), housing accumulation function (4), and terminal nonnegativity ^constraints Bfj  +X  > 0 , xjj  > 0 and xf  j+l  kT+]  H.  T+l  > 0,  , > 0,  > 0 for T —» oo . In equilibrium, selected necessary first order  conditions associated with this utility maximization problem may be stated as u (C„H„L. ) c  l  = PfA ,  (6)  l  (7)  + Q.-S  )Q  M  A,=/J(1 + /,)E,A, ,  (9)  +1  S A, = B(l + l  (8)  if)E £ A , !  l+l  l+l  (10) (11) (12)  where A  ( >  denotes the Lagrange multiplier associated with the period s household dynamic  budget constraint, and A Q" denotes the Lagrange multiplier associated with the period s Ls  housing accumulation function. In equilibrium, necessary complementary slackness conditions associated with the terminal nonnegativity constraints may be stated as:  122  (13)  A.  lim  B,;  B  lim PL A±L£ T  (14)  = 0,  T+X  P  =0  /  (15)  1  T-Ko  Hm P *>+T yr r r"i X j,t+T j,,+T+l > T  (16)  = f J  y  r->co  X  U  (17)  1  Provided that the intertemporal utility function is bounded and strictly concave, together with all necessary first order conditions, these transversality conditions are sufficient for the unique utility maximizing state contingent intertemporal household allocation. Combination of necessary first order conditions (6) and (9) yields intertemporal optimality condition  Mc  (C,,/7,,L,,) = ^E,(l + / , ) ^  W c  (C  ( + l  ,/Y, ,L,, ), + 1  (18)  + 1  which ensures that at a utility maximum, the representative household cannot benefit from feasible intertemporal consumption reallocations.  Combination of necessary  first  order  conditions (6) and (7) yields intertemporal optimality condition n T-/H(T U, « l (A H  H  f " U C  ^ c(^i+\'^t+\'^i,t+]) U  ,~  +  I  T  T  u (c H„h<) c  s  n  Pf  /-,/7  ~^C~^ 2 H  i_/H / jH ( V l '  r 7  /  H>. >  D =  P  l"  >  (19)  '  :  p  which equates the expected present discounted value of an additional unit of investment in housing to its price.  Combination of necessary first order conditions (6) and (8) yields  intertemporal optimality condition Bu {C ,H ,L ) c  M  l+]  u (C„H,,Lu) c  pC  il+x  tf.  "H(Q|^I+I'A.H-I)  _  rH  ^H n  (20)  123 which equates the shadow price of housing to the expected present discounted value of the sum of the future marginal cost of housing, and the future shadow price of housing net of depreciation.  Finally, combination of necessary first order conditions (6), (9) and (10) yields  intratemporal optimality condition  u (C„H„L,,)  J*'  c  u {C„H„L ) c  u  ftS,  ''  which equates the expected present discounted values of the gross real returns on domestic and foreign bonds.  3.2.1.2. Labour Supply and Wage Setting Behaviour There exist a large number of perfectly competitive firms which combine differentiated intermediate labour services L  it  supplied by households in a monopolistically competitive  labour market to produce final labour service L according to constant elasticity of substitution t  production function  9,-1  L. =  \(k,) ' 9  where df > 1.  di  (22)  The representative final labour service firm maximizes profits derived from  production of the final labour service nf=W Lt  \W L.,di, it  (23)  with respect to inputs of intermediate labour services, subject to production function (22). The necessary first order conditions associated with this profit maximization problem yield intermediate labour service demand functions:  L... =  A-  (24)  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final labour service firm earns zero profit, implying aggregate wage index:  124  \(wJ~Ui  W.  (25)  As the wage elasticity of demand for intermediate labour services Of increases, they become closer substitutes, and individual households have less market power. In an extension of the model of nominal wage rigidity proposed by Erceg, Henderson and Levin (2000) along the lines of Smets and Wouters (2003, 2005), each period a randomly selected fraction 1 - m of households adjust their wage optimally. The remaining fraction co L  L  of households adjust their wage to account for past consumption price inflation according to partial indexation rule r  P  t-\  c\  r  pC  W,.  pC  (26)  where 0 < y < 1. Under this specification, although households adjust their wage every period, L  they infrequently adjust their wage optimally, and the interval between optimal wage adjustments is a random variable. If the representative household can adjust its wage optimally in period t, then it does so to maximize intertemporal utility function (1) subject to dynamic budget constraint (3), housing accumulation function (4), intermediate labour service demand function (24), and the assumed form of nominal wage rigidity. Since all households that adjust their wage optimally in period t solve an identical utility maximization problem, in equilibrium they all choose a common wage Wf given by necessary first order condition: '( pC '  u (C ,H ,L ) L  s  s  is  r-l pC r  1"'' 1  P'-'u {C ,H ,L ) c  s  s  is  w  ( P l-\ c  \  pc  u  w  .  'pc\ -r  L  r  u (C„H„L ) c  BC  V s-\ r  w.  1-1  pc  1  w,  4  (27)  (wf -oi  This necessary first order condition equates the expected present discounted value of the consumption benefit generated by an additional unit of labour supply to the expected present discounted value of its leisure cost. Aggregate wage index (25) equals an average of the wage set by the fraction 1 - co of households that adjust their wage optimally in period t, and the L  average of the wages set by the remaining fraction a> of households that adjust their wage L  according to partial indexation rule (26):  125  ( P ^ c  i\-co w;t ' L  +co  e  L  t-\ pC r  V i-2 r  )  y-  ( P  \  c  i-i pC r  w.  (28)  Since those households able to adjust their wage optimally in period / are selected randomly from among all households, the average wage set by the remaining households equals the value of the aggregate wage index that prevailed during period t -1, rescaled to account for past consumption price inflation. If all households were able to adjust their wage optimally every period, then co = 0 and L  necessary first order condition (27) would reduce to:  ( 1  f  )  f , _  u {C,,H,,L,) ef-\u {C H ,L^  (29)  L  c  n  t  In flexible price and wage equilibrium, each household sets its after tax real wage equal to a time varying markup over the marginal rate of substitution between leisure and consumption, and labour supply is inefficiently low.  3.2.2. The Value Maximization Problem of the Representative Firm There exists a continuum of intermediate good firms indexed by j e [0,1]. Intermediate good firms supply differentiated intermediate output goods, but are otherwise identical. Entry into and exit from the monopolistically competitive intermediate output good sector is prohibited.  3.2.2.1. Employment and Investment Behaviour The representative intermediate good firm sells shares {xf }) jl+l  price Vj . t  Recursive forward substitution for Vj  l+s  =0  to domestic households at  with s > 0 in necessary first order condition  (11) applying the law of iterated expectations reveals that the post-dividend stock market value of the representative intermediate good firm equals the expected present discounted value of future dividend payments:  .9=/+!  A  t  126  Acting in the interests of its shareholders, the representative intermediate good firm maximizes its pre-dividend stock market value, equal to the expected present discounted value of current and future dividend payments:  (31)  The derivation of result (30) imposes transversality condition (16), which rules out self-fulfilling speculative asset price bubbles. Shares entitle households to dividend payments equal to net profits IJ , defined as after tax Y  js  earnings less expenditures on investment in capital: //;. = ( l - r j ( - ^ ^ . J - P ' v  /«.  (32)  Earnings are defined as revenues derived from sales of differentiated intermediate output good Y  at price Pj  js  less expenditures on final labour service Z,. . The government levies a tax on  s  s  earnings at rate t , and negative dividend payments are a theoretical possibility. s  The representative intermediate good firm utilizes capital K  s  labour service L  at rate u  js  and rents final  given labour augmenting technology coefficient A to produce differentiated  j s  s  intermediate output good Yj according to constant elasticity of substitution production function s  9-1  Huj, K ,A L. )s  s  s  (<pY(u K,)>  s  JtS  9-\  +{\-(pY{A L ) s  9-1  js  (33)  where 0 < <p < 1, 9 > 0 and A > 0. This constant elasticity of substitution production function s  exhibits constant returns to scale, and nests the production function proposed by Cobb and Douglas (1928) under constant returns to scale for 9 = 1.  1  In utilizing capital to produce output, the representative intermediate good firm incurs a cost G(u ,K ) Js  denominated in terms of output:  s  Y. =Hu K ,A L )-Q{u ,K ). s  hs  s  s  js  hs  (34)  s  Following Christiano, Eichenbaum and Evans (2005), this capital utilization cost is increasing in the rate of capital utilization at an increasing rate,  ' Invoking L ' H o s p i t a l ' s  rule yields  l i m I n F ( u K „ A , L , . J = (oln(» __KJ + (1 -<p)\n{A,LjJ-q>\n(p-(\-<p)\n(\-(p), j J  i  i ™ •^("/..^.^^j=^^i-pr'"'H»r^)'K;)'"'J  l  i  J  y  w h i c h implies that  127 (35) where p > 0 and K> 0. In deterministic steady state equilibrium, the rate of capital utilization is normalized to one, and the cost of utilizing capital equals zero. Capital is endogenous but not firm-specific, and the representative intermediate good firm enters period 5 with access to previously accumulated capital stock K ^ , which subsequently evolves according to accumulation function K =(l-S )K +H (lf,d K  (36)  K  s+l  s  where depreciation rate parameter S  satisfies 0 < S  K  K  < 1. Following Christiano, Eichenbaum  and Evans (2005), effective capital investment function H (I*,I*_ ) K  }  incorporates convex  adjustment costs,  I  (37)  K  2  I  K  s-\  1  where  >0  a n  d ' v  >0-  s  I  n  deterministic steady state equilibrium,'these adjustment costs  equal zero, and effective investment equals actual investment. In period / , the representative intermediate good firm chooses state contingent sequences for employment stock  JS  }" , investment in capital {I*}™ , and the capital =/  =l  to maximize pre-dividend stock market value (31) subject to net production  {K }™ S+I  capital utilization  {L }^,, =L  function (34), capital accumulation function (36), and terminal nonnegativity constraint K  T + L  >0  for T —> oo . In equilibrium, demand for the final labour service satisfies necessary first order condition (38)  F (U.,K ,A L.,)0. =(\-T,) AL  where  I  Pj<P  JS  1  i  denotes the Lagrange multiplier associated with the period s  production  technology constraint. This necessary first order condition equates real marginal cost <£,, to the ratio of the after tax real wage to the marginal product of labour. In equilibrium, the rate of capital utilization satisfies necessary first order condition r (u K„A,L ) uK  JJ  jtl  =  K,  (39)  128  w h i c h equates the marginal product o f utilized capital to its marginal cost.  In e q u i l i b r i u m ,  demand for the final capital investment good satisfies necessary first order condition  QfK  df, / * , ) + E , ^  Qf X +  (/*, ,lf) = if,  (40)  w h i c h equates the expected present discounted value o f an additional unit o f investment i n capital to its price, where Q*  denotes the Lagrange multiplier associated w i t h the period s  capital accumulation function.  In equilibrium, this shadow price o f capital satisfies necessary  t  first order condition  ^  =  E  ' ^  {^'  ± L  < Z >  y - — C >-r .^(«y,, i^ ,. ^ .^y,, i) ~ ^ M  +  +  +  +  +  (« -, , y  /+1  ) ] + (1 ~  }, ( 4 1 )  w h i c h equates it to the expected present discounted value o f the sum o f the future marginal cost o f capital, and the future shadow price o f capital net o f depreciation. necessary  complementary  slackness  condition associated  In e q u i l i b r i u m , the  w i t h the terminal  nonnegativity  constraint m a y be stated as:  Hm l^JLQ^ K =Q. T  (42)  l+T+{  P r o v i d e d that the pre-dividend stock market value o f the representative intermediate good firm is bounded  and strictly concave,  transversality  together  condition is sufficient  with  a l l necessary  for the unique  value  first  order  conditions,  m a x i m i z i n g state  this  contingent  intertemporal firm allocation.  3.2.2.2. Output Supply and Price Setting Behaviour There exist a large number o f perfectly competitive firms w h i c h combine differentiated intermediate output goods  Y  jt  supplied b y intermediate good  firms  i n a monopolistically  competitive output market to produce final output good Y according to constant elasticity o f t  substitution production function  Y=\  (43)  129 where dj > 1.  The representative final output good firm maximizes profits derived from  production of the final output good n =P!Y-  (44)  Y  with respect to inputs of intermediate output goods, subject to production function (43).  The  necessary first order conditions associated with this profit maximization problem yield intermediate output good demand functions:  Y.  (45)  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final output good firm earns zero profit, implying aggregate output price index:  (46) .7=0  As the price elasticity of demand for intermediate output goods 6] increases, they become closer substitutes, and individual intermediate good firms have less market power. In an extension of the model of nominal output price rigidity proposed by Calvo (1983) along the lines of Smets and Wouters (2003, 2005), each period a randomly selected fraction 1 - of of intermediate good firms adjust their price optimally. The remaining fraction co of intermediate Y  good firms adjust their price to account for past output price inflation according to partial indexation rule P  Y  f  r  7,'  !-\  P V i-2  ^r  where 0 < y < 1. Y  pY  i-\  r  P  Y  r  \  P  Y  Y  )  \-r  r  (  \ l-2 J  (47)  i,t-\'  r  J  Under this specification, optimal price adjustment opportunities arrive  randomly, and the interval between optimal price adjustments is a random variable. If the representative intermediate good firm can adjust its price optimally in period t, then it does so to maximize to maximize pre-dividend stock market value (31) subject to net production function (34), capital accumulation function (36), intermediate output good demand function (45), and the assumed form of nominal output price rigidity. Since all intermediate good firms  130 that adjust their price optimally in period t solve an identical value maximization problem, in equilibrium they all choose a common price Pj'* given by necessary first order condition:  ( P ^ Y  A,  '  s  P  J s  Y  p  ( P" \ Y  '-''Pi]  i-\  r  t-\  r  P'  Y  P,%  P  Y  -• (48) Y  ( P OX i-\  r  (0;-i)(i-r,) p  P  P  Y  PJ  \ s-\ )  V's-I )  ( P '* \  PI  r  Y  Y  Y  r  PlY.  This necessary first order condition equates the expected present discounted value of the after tax revenue benefit generated by an additional unit o f output supply to the expected present discounted value of its production cost. Aggregate output price index (46) equals an average of the price set by the fraction 1 - co of intermediate good firms that adjust their price optimally in Y  period t, and the average of the prices set by the remaining fraction a> of intermediate good Y  firms that adjust their price according to partial indexation rule (47): 1  f r>r ^ Y  f  pY  t-\  r  P = (l-tf/)(7f0'~*' +a  v  P  Y  x-ej  \-r  r  \  P  Y  i-\  (49)  r  Since those intermediate good firms able to adjust their price optimally in period t are selected randomly from among all intermediate good firms, the average price set by the remaining intermediate good firms equals the value o f the aggregate output price index that prevailed during period t - l , rescaled to account for past output price inflation. If all intermediate good firms were able to adjust their price optimally every period, then a? = 0 and necessary first order condition (48) would reduce to (l-f)P '=  °< P 0  Y  I'  Y  I  QY  ]  (50)  '  where PJ* = Pj. In flexible price and wage equilibrium, each intermediate good firm sets its after tax price equal to a time varying markup over nominal marginal cost, and output supply is inefficiently low.  131 3.2.3. The Value Maximization Problem of the Representative Importer There exists a continuum of intermediate good importers indexed by k e [0,1]. Intermediate good importers supply differentiated intermediate import goods, but are otherwise identical. Entry into and exit from the monopolistically competitive intermediate import good sector is prohibited.  3.2.3.1. The Real Exchange Rate and the Terms of Trade The representative intermediate good importer sells shares {x" }]= kl+l  at price V  M k  t  .  Recursive forward substitution for V  M k  l+s  to domestic households  0  with s > 0 in necessary first order  condition (12) applying the law of iterated expectations reveals that the post-dividend stock market value of the representative intermediate good importer equals the expected present discounted value of future dividend payments: (51)  Acting in the interests of its shareholders, the representative intermediate good importer maximizes its pre-dividend stock market value, equal to the expected present discounted value of current and future dividend payments:  K^^t^Ks=l  (52)  t  A  The derivation of result (51) imposes transversality condition (17), which rules out self-fulfilling speculative asset price bubbles. Shares entitle households to dividend payments equal to gross profits TI , ks  defined as  earnings less fixed costs: K= "M ,-£ Pj M ,-r . p  (53)  J  k  s  k  s  Earnings are defined as revenues derived from sales of differentiated intermediate import good M  k  s  at price P  M k s  less expenditures on foreign final output good M . k s  The representative  intermediate good importer purchases the foreign final output good at domestic currency price £,P, '  Y f  and differentiates it, generating zero gross profits on average.  132 The law of one price asserts that arbitrage transactions equalize the domestic currency prices of domestic imports and foreign exports. Define the real exchange rate, (54)  which measures the price of foreign output in terms of domestic output. Also define the terms of trade,  T  (55)  =-  which measures the price of imports in terms of exports. Violation of the law of one price drives a wedge *F = £ Pj'  f  S  s  IP  between the real exchange rate and the terms of trade,  M S  (56) where the domestic currency price of exports satisfies Pf - Pj.  Under the law of one price  F = 1, and the real exchange rate and terms of trade coincide.  t  s  There exist a large number of perfectly competitive firms which combine a domestic intermediate Z  fJ  good  e {C j,l"„If ,G j} f  tt  f  Z  hl  e {C , if, if, G } hl  ht  and  a  foreign  to produce final good Z, e{C ,l"',lf t  ,G,}  intermediate  good  according to constant  elasticity of substitution production function  Z. =  (57)  where 0 < (/> < 1, y/ > 1 and vf > 0 . The representative final good firm maximizes profits z  derived from production of the final good n^=P^7-P 7 ,  -P 7 M  (58)  with respect to inputs of domestic and foreign intermediate goods, subject to production function (57).  The necessary first order conditions associated with this profit maximization problem  imply intermediate good demand functions:  (59)  133  z =(\-<t> ) z  v  M  p  (60)  M "  Z  fJ  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final good firm earns zero profit, implying aggregate price index:  (61) \l  J  v  Combination of this aggregate price index with intermediate good demand functions (59) and (60) yields:  <t> +(WZ)  (62)  z  z ,,=<l>  z  h  Z =(\-0 ) Z  (63)  fl  K: v  J  These demand functions for domestic and foreign intermediate goods are directly proportional to final good demand, with a proportionality coefficient that varies with the terms of trade.  3.2.3.2. Import Supply and Price Setting Behaviour There exist a large number of perfectly competitive firms which combine differentiated intermediate import goods M  kl  supplied by intermediate good importers in a monopolistically  competitive import market to produce final import good M, according to constant elasticity of substitution production function  i  M,=  \(M ) ' 6  kl  dk  (64)  k=0  where Of > 1.  The representative final import good firm maximizes profits derived from  production of the final import good  134  n?=P?M-  (65)  \P ,M dk, M  k  kJ  k=0  with respect to inputs of intermediate import goods, subject to production function (64).  The  necessary first order conditions associated with this profit maximization problem yield intermediate import good demand functions:  k,l  M,.  (66)  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final import good firm earns zero profit, implying aggregate import price index:  j CO"  _  pM  dk  (67)  MO  As the price elasticity of demand for intermediate import goods 9f  increases, they become  closer substitutes, and individual intermediate good importers have less market power. In an extension of the model of nominal import price rigidity proposed by Monacelli (2005) along the lines of Smets and Wouters (2003, 2005), each period a randomly selected fraction 1 - co  M  of intermediate good importers adjust their price optimally. The remaining fraction co  M  of intermediate good importers adjust their price to account for past import price inflation according to partial indexation rule f  pM  i-\  r  k,t  where 0 < y  ~  M  Y  f pM  t-\  r  A "' 1  1  k,<-\>  (68)  < 1. Under this specification, the probability that an intermediate good importer  has adjusted its price optimally is time dependent but state independent. If the representative intermediate good importer can adjust its price optimally in period t, then it does so to maximize to maximize pre-dividend stock market value (52) subject to intermediate import good demand function (66), and the assumed form of nominal import price rigidity. Since all intermediate good importers that adjust their price optimally in period t solve an identical value maximization problem, in equilibrium they all choose a common price P * M  t  given by necessary first order condition  135 \-J»  -10. pM  s  ~t-\ j/W,*  P M. M  pM  V t  J  r  ( P  M  (6^-1)  pM  r  where P = £ Pj' l  IP  f  s  M S  1  ^r  \ s-\ )  s  p M * \  (  pM  l  "t-\  ~  pM  pM  s'  (  pM*  r  1  \  1  pM  \ s-\ )  measures real marginal cost.  (69)  ..M r  P M, M  pM  \  i  r  J  T h i s necessary first order condition  equates the expected present discounted value o f the revenue benefit generated b y an additional unit o f import supply to the expected present discounted value o f its production cost. Aggregate import price index (67) equals an average o f the price set b y the fraction 1 - co  M  o f intermediate  good importers that adjust their price optimally i n period t, and the average o f the prices set b y the remaining fraction a> o f intermediate good importers that adjust their price according to M  partial indexation rule (59):  (  pM  l-\  r  K 1-2 1  Y"  (70) J  \ i-2. A  Since those intermediate good importers able to adjust their price optimally i n period t are selected randomly f r o m among a l l intermediate good importers, the average price set b y the remaining intermediate good importers equals the value o f the aggregate import price index that prevailed during period t-\,  rescaled to account for past import price inflation.  If all intermediate good importers were able to adjust their price optimally every period, then co - 0 and necessary first order condition (69) w o u l d reduce to M  e. 6T - 1 M  pM,~ _  pM  uj  (71)  where P * = P] . In flexible price and wage equilibrium, each intermediate g o o d importer sets M  M  t  its price equal to a time varying markup over nominal marginal cost, and import supply is inefficiently low.  136 3.2.4. Monetary and Fiscal Policy The government consists of a monetary authority and a fiscal authority.  The monetary  authority implements monetary policy, while the fiscal authority implements fiscal policy.  3.2.4. L The Monetary Authority The monetary authority implements monetary policy through control of the short term nominal interest rate according to monetary policy rule (72) where  > 1 and % > 0 .  As specified, the deviation of the nominal interest rate from its  Y  flexible price and wage equilibrium value is a linear increasing function of the contemporaneous deviation of consumption price inflation from its target value nf = Jrf, and the contemporaneous proportional deviation of output from its flexible price and wage equilibrium value.  Persistent  departures from this monetary policy rule are captured by serially correlated monetary policy shock v\ .  3.2.4.2. The Fiscal A uthority The fiscal authority implements fiscal policy through control of nominal government consumption and the tax rate applicable to the pooled labour income of households and the earnings of intermediate good firms. In equilibrium, this distortionary tax collection framework corresponds to proportional output taxation. The ratio of nominal government consumption to nominal output satisfies fiscal expenditure rule P G.  P G, i—In P Y, PY G  ln  Y  where £  G  < 0.  *  In  B.  -In  + v;  (73)  As specified, the proportional deviation of the ratio of nominal government  consumption to nominal output from its deterministic steady state equilibrium value is a linear decreasing function of the contemporaneous proportional deviation of the ratio of net foreign debt to nominal output from its target value. This fiscal expenditure rule is well defined only i f  137 the net foreign debt is positive.  Persistent departures from this fiscal expenditure rule are  captured b y serially correlated fiscal expenditure shock vf . The tax rate applicable to the pooled labour income o f households and the earnings o f intermediate good firms satisfies fiscal revenue rule  l n r , - l n r , =<^  where ^ steady  > 0.  B.G  In  r  \  B,G  •In  \  (74)  +v  PY.  A s specified, the proportional deviation o f the tax rate from its deterministic  state equilibrium value  is a linear increasing  function  o f the  contemporaneous  proportional deviation o f the ratio o f net government debt to n o m i n a l output from its target value.  This fiscal revenue rule is w e l l defined only i f the net government debt is positive.  Persistent departures from this fiscal revenue rule are captured b y serially correlated fiscal revenue shock The  v\.  fiscal authority enters period  denominated bonds  Bf  h  denominated bonds Bf  J  t  holding previously purchased domestic  w h i c h y i e l d interest at risk free rate  , and foreign  currency currency  w h i c h y i e l d interest at risk free rate /,{,. It also levies taxes on the  pooled labour income o f households and the earnings o f intermediate good firms at rate r , . These sources o f public wealth are summed i n government dynamic budget constraint:  B% + Erf*  = (1 + /,_, )Bf  +£,{1 + if  h  i  i  )Bf  f  i  +*•, j jW^dldi  + r,  i=0 /=o  (75)  l(PfY -W,L. )dj-P, G G  jJ  l  r  y=o  A c c o r d i n g to this dynamic budget constraint, at the end o f period t, the fiscal authority purchases domestic bonds B, \ , and foreign bonds B°f G  h  +  government consumption good G at price t  at price £ . t  It also purchases  final  Pf.  3.2.5. Market Clearing Conditions A rational expectations e q u i l i b r i u m i n this D S G E model o f a small open economy consists o f state contingent intertemporal allocations for domestic and foreign households and firms w h i c h solve their constrained optimization problems given prices and p o l i c y , together w i t h state contingent intertemporal allocations for domestic and foreign governments w h i c h satisfy their p o l i c y rules and constraints given prices, w i t h supporting prices such that a l l markets clear.  138 Since the domestic economy is of negligible size relative to the foreign economy, in equilibrium rj c,f »J ^f o.f pxj and ^ = ^ = = 0. P  =  P  =  P;  =  P  =  p  =  Clearing of the final output good market requires that exports X  t  equal production of the  domestic final output good less the cumulative demands of domestic households, firms, and the government, X^Y-C^-I^-Il-G^ where X = Mf.  (76)  Clearing of the final import good market requires that imports M  t  t  satisfy the  cumulative demands of domestic households, firms, and the government for the foreign final output good, M,=C +I^+I^  +G,  ftl  where M = Xf.  (77)  fJ  In equilibrium, combination of these final output and import good market  t  clearing conditions yields aggregate resource constraint: PjY = P, C, + P " If + Pflf C  t  + PfG, + PfX, - P M .  l  (78)  M  t  t  t  The trade balance equals export revenues less import expenditures, or equivalently nominal output less domestic demand. Let  denote the net foreign asset position of the economy, which in equilibrium equals  the sum of the domestic currency values of private sector bond holdings Bf public sector bond holdings Bf = Bf +]  h x  + £,Bf(,  = Bf\ + £ Bf{ t  and  since domestic bond holdings cancel out when  the private and public sectors are consolidated: B, =Bf Bf +x  l+  (79)  v  The imposition of equilibrium conditions on household dynamic budget constraint (3) reveals that the expected present discounted value of the net increase in private sector asset holdings equals the expected present discounted value of private saving less domestic investment: E,_, ^(Bf A  ,-\  -Bf) = E _ , 4%,*; V i  H1-T,)P?Y,-PfC-Pflf  L  -P'flfX  (80) J  The imposition of equilibrium conditions on government dynamic budget constraint (75) reveals that the expected present discounted value of the net increase in public sector asset holdings equals the expected present discounted value of public saving:  139 E,_, &L{B?«  - Bf) = E,_, f^(/,_,5,  t-\  c  +  T,Pf  Yl  - FfG,).  (81)  \ - \  A  Combination of these household and government dynamic budget constraints with aggregate resource constraint (78) reveals that the expected present discounted value of the net increase in foreign asset holdings equals the expected present discounted value of the sum of net international investment income and the trade balance, or equivalently the expected present discounted value of national saving less domestic investment: E , - *, ^ ^ (+i , -B,) —i' = E,_Ai,_ — / - i * B, ]  <  +P/X, - / - - /  -Pf*Mf). -< - - I / -  (82)  t-\  . - V i  A  In equilibrium, the current  account balance  is determined by both intratemporal and  intertemporal optimization.  3.2.6. The Approximate Linear Model Estimation, inference and forecasting are based on a linear state space representation of an approximate unobserved components representation of this D S G E model o f a small open economy.  Cyclical components are modeled by linearizing equilibrium conditions around a  stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path. In what follows, E, x  l+s  denotes the rational expectation of variable x  information available at time t.  m  , conditional on  Also, x, denotes the cyclical component of variable x,, x,  denotes the flexible price and wage equilibrium component of variable x,, and x, denotes the trend component of variable x,. Cyclical and trend components are additively separable, which implies that x, = x, + x, and x, =x +x , t  t  where x, = x,.  3.2.6.1. Cyclical Components  The cyclical component of output price inflation depends on a linear combination of past and expected future cyclical components of output price inflation driven by the contemporaneous cyclical components of real marginal cost and the tax rate according to output price Phillips curve  140  _ .  r  E  \+  ( i - * / ) ( ! - « / / ? ) ln 0. + -lnf, 1-r  Y  fp  e -\ Y  -\ne , (83)  where 0 = (\-z)^—f-. The persistence of the cyclical component of output price inflation is increasing in indexation parameter y , while the sensitivity of the cyclical component of output Y  price inflation to changes in the cyclical components of real marginal cost and the tax rate is decreasing in nominal rigidity parameter of and indexation parameter y . Y  This output price  Phillips curve is subject to output price markup shocks. The cyclical component of output depends on the contemporaneous cyclical components of utilized capital and effective labour according to approximate linear net production function 9  WL  Y  I n Z = 1-  e  -\PY J  Y  = ^  where /  \n(u K ) + l  l  e  Y  WL  e -IPY  HAL,),  (84)  Y  WL  \ [ ^ - y j - - ^ 1. This approximate linear net production function is subject to  T  \—p{\—o) \ 0  PY J  output technology shocks. The cyclical component of the rate of capital utilization depends on the contemporaneous cyclical component of the ratio of capital to effective labour according to approximate linear implicit capital utilization function: 6  Y  lnw, = —  WL  6  WL  Y  9 -IPY  6 -\PY Y  Y  (85)  A,L,  The sensitivity of the cyclical component of the rate of capital utilization to changes in the cyclical component of the ratio of capital to effective labour is decreasing in capital utilization cost parameter K and elasticity of substitution parameter &. This approximate linear implicit capital utilization function is subject to output technology shocks. The cyclical component of consumption, housing investment, capital investment or government consumption price inflation depends on a linear combination of past and expected future cyclical components of consumption, housing investment, capital investment or government consumption price inflation driven by the contemporaneous cyclical components of real marginal cost and the tax rate according to Phillips curves:  r  P 1+  l + yB Y  \  p  Y +  Y  (l-co )(l-co fi) Y  z  Y  af(\ + y p)  Aln^L + O - ^ A l n ^ -  Y  ln<Z> ,  1-r  • ln x, - -  E,Aln%.  e -\ Y  (86)  141 Reflecting the entry of the price of imports into the aggregate consumption, housing investment, capital investment or government consumption price index, the cyclical component of consumption, housing investment, capital investment or government consumption price inflation also depends on past, contemporaneous, and expected future proportional changes in the cyclical component of the terms of trade. These Phillips curves are subject to output price markup and import technology shocks. The cyclical component of consumption depends on a linear combination of past and expected future cyclical components of consumption driven by the contemporaneous cyclical component of the consumption based real interest rate according to approximate linear consumption Euler equation: l-a  L  InC, =•  c  l+a  lnC,_,+  C  l+a  E , lnC, , +  Z;  c  l+ a  L  +  E,ln^  (87)  v.  The persistence of the cyclical component of consumption is increasing in habit persistence parameter a , c  while the sensitivity of the cyclical component of consumption to changes in the  cyclical component of the consumption based real interest rate is increasing in intertemporal elasticity of substitution parameter cr and decreasing in habit persistence parameter a . c  This  approximate linear consumption Euler equation is subject to preference shocks. The cyclical component of investment in housing depends on a linear combination of past and  expected future  cyclical components  of  investment  in housing  driven by the  contemporaneous cyclical component of the relative shadow price of housing according to approximate linear housing investment demand function:  ln/"=-  1  -In/.",  1+0  +•  i+0-  EMI/  1  +  •In  ra+/5)  p  (88)  The sensitivity of the cyclical component of investment in housing to changes in the cyclical component of the relative shadow price of housing is decreasing in housing investment adjustment cost parameter x" • This approximate linear housing investment demand function is subject to housing investment technology shocks. The cyclical component of the relative shadow price of housing depends on the expected future cyclical component of the relative shadow price of housing, the contemporaneous cyclical component of the consumption based real interest rate, and the expected future cyclical component of the marginal rate of substitution between housing and consumption according to approximate linear housing investment Euler equation:  142  6"  O" t+i  1  \-B{\-S")  \x\H^-a \nH  lnC, , - a  H  ]  t  \-a  (89)  l-a  l  a  InC,  c  +  L  The sensitivity of the cyclical component of the relative shadow price of housing to changes in the cyclical component of the ratio of adjusted housing to adjusted consumption is decreasing in intertemporal elasticity of substitution parameter  a.  The cyclical component of the stock of housing depends on the past cyclical component of the stock of housing and the contemporaneous cyclical component of investment in housing according to approximate linear housing accumulation function lnH, =(\-S )lnH S H  where -ff  =  d"-  InO?,'"/,")  H  +l  l+  (90)  ;  This approximate linear housing accumulation function is subject to housing  investment technology shocks. The cyclical component of investment in capital depends on a linear combination of past and expected future cyclical components of investment in capital driven by the contemporaneous cyclical component of the relative shadow price of capital according to approximate linear capital investment demand function: ln/,*=-  1+ 0  1 E , l n / f , +1+ 0 Z <\ + K  f  -In  P) V  AK  '  ^  (91)  r>I  K  "  J  The sensitivity of the cyclical component of investment in capital to changes in the cyclical component of the relative shadow price of capital is decreasing in capital investment adjustment cost parameter % . This approximate linear capital investment demand function is subject to K  capital investment technology shocks. The cyclical component of the relative shadow price of capital depends on the expected future cyclical component of the relative shadow price of capital, the contemporaneous cyclical component of the output based real interest rate, the expected future cyclical component of real marginal cost, and the expected future cyclical component of the marginal product of capital according to approximate linear capital investment Euler equation:  143  ln^- = A l - ^ ) E , l n % ^ (92)  The sensitivity of the cyclical component of the relative shadow price of capital to changes in the cyclical component of the ratio of utilized capital to effective labour is decreasing in elasticity of substitution parameter 9.  This approximate linear capital investment Euler equation is subject  to output technology shocks. The cyclical component of the stock of capital depends on the past cyclical component of the stock of capital and the contemporaneous cyclical component of investment in capital according to approximate linear capital accumulation function lnK  =(l-S )lnK  8  K  l+l  K  l+  where ^ - = S .  Inff/*/*),  (93)  This approximate linear capital accumulation function is subject to capital  K  investment technology shocks. The cyclical component of the ratio of nominal government consumption to nominal output depends on the contemporaneous cyclical component of the ratio of net foreign debt to nominal output according to fiscal expenditure rule: P G, „„. ( l n ^ = C In P.Y. G  B  A l+X  + vf  (94)  This fiscal expenditure rule ensures convergence of the level of the ratio of net foreign debt to nominal output to its target value in deterministic steady state equilibrium, and is subject to fiscal expenditure shocks. The cyclical component of the tax rate depends on the contemporaneous cyclical component of the ratio of net government debt to nominal output according to fiscal revenue rule:  ln£, =^ ln  ,\  B  r  +  V  Pftj  (95)  This fiscal revenue rule ensures convergence of the level of the ratio of net government debt to nominal output to its target value in deterministic steady state equilibrium, and is subject to fiscal revenue shocks. The cyclical component of import price inflation depends on a linear combination of past and expected future cyclical components of import price inflation driven by the contemporaneous  144 cyclical component of the deviation of the domestic currency price of foreign output from the price of imports according to import price Phillips curve: M  n, = - Y ^ 1+Y P  - 1  +  P i+r p  ,  E  CpYJ  {\-a) ){\-co B) M  M  1  co (i+ P)  M  M  M  -In 0,M  (96)  r  The persistence of the cyclical component of import.price inflation is increasing in indexation parameter y ,  while the sensitivity of the cyclical component of import price inflation to  M  changes in the cyclical component of real marginal cost is decreasing in nominal rigidity parameter co  and indexation parameter y .  M  This import price Phillips curve is subject to  M  import price markup shocks. The cyclical component of exports depends on the contemporaneous cyclical components of foreign consumption, housing investment, capital investment, government consumption, and the terms of trade according to approximate linear export demand function  Y  -v  y  *  (97)  C /  f  c  Y  v:  f/ 4.C d - 0 ) f In •MJ  Y  where — = 1 - <b — - <f>'" ~-d> Y  Y  v.  Y  Y  aG  —.  The sensitivity of the cyclical component of  exports to changes in the cyclical component of the foreign terms of trade is increasing in elasticity of substitution parameter y/.  This approximate linear export demand function is  subject to foreign import technology shocks. The cyclical component of imports depends on the contemporaneous cyclical components of consumption, housing investment, capital investment, government consumption, and the terms of trade according to approximate linear import demand function M  InM, (98) ^(1-^)|  where  = (l-f)^+  ([-/'  +  /'(1  )!l + (i-0'  ' ) ^  K  +  ^ ( l - ^ ) ^  )Il + (i-0 )G G  .  +  ^ ( l - ^ ) |  The sensitivity of the cyclical  component of imports to changes in the cyclical component of the terms of trade is increasing in elasticity of substitution parameter y/. subject to import technology shocks.  This approximate linear import demand function is  145 The cyclical component of the real wage depends on a linear combination of past and expected future cyclical components of the real wage driven by the contemporaneous cyclical component of the deviation of the marginal rate of substitution between leisure and consumption from the after tax real wage according to wage Phillips curve: l n  4=_L,n%+^_E,,n%+J^^ 1+  pf  pf  {\-G) )(\-6> p) 1 In L, - a L  +  \+p  pf L  rj  I  L  -  '-'  ln Z,_, +  a  htP_f _P_ -c \ p E/  +  1+  1+  (99)  +  1 ln C, - a ° ln C,_, . T . „ , W. '• - —'— + Inf, - l n ^ r f \ - a 1-r  a  r  -"—  Intf  Reflecting the existence of partial wage indexation, the cyclical component of the real wage also depends on past, contemporaneous, and expected future cyclical components of consumption price inflation.  The sensitivity of the cyclical component of the real wage to changes in the  cyclical component of consumption price inflation is increasing in indexation parameter y , to L  changes in the cyclical component of the deviation of the marginal rate of substitution between leisure and consumption from the after tax real wage is decreasing in nominal rigidity parameter co , and to changes in the cyclical component of adjusted employment is decreasing in elasticity L  of substitution parameter rj. This wage Phillips curve is subject to wage markup shocks. The cyclical component of real marginal cost depends on the contemporaneous cyclical component of the deviation of the after tax real wage from the marginal product of labour according to approximate linear implicit labour demand function: W  r  ln&, = ln-rr-^  R A,  1 lnf,  1-r  '  $  e -\PY Y  In 4 ^ . AL. ••f"t  (100)  The sensitivity of the cyclical component of real marginal cost to changes in the cyclical component of the ratio of utilized capital to effective labour is decreasing in elasticity of substitution parameter 3. This approximate linear implicit labour demand function is subject to output technology shocks. The  adjusted  cyclical  component  of the  nominal interest  rate  depends  on  the  contemporaneous adjusted cyclical. components of consumption price inflation and output according to monetary policy rule:  This monetary policy rule ensures convergence of the level of consumption price inflation to its target value in flexible price and wage equilibrium, and is subject to monetary policy shocks.  146 The cyclical component of the output based real interest rate satisfies rf = i, - E, fcf, while the cyclical component of the consumption based real interest rate satisfies rf = /, - E, fcf . x  The cyclical component of the nominal exchange rate depends on the expected future cyclical component of the nominal exchange rate and the contemporaneous cyclical component of the nominal interest rate differential according to approximate linear uncovered interest parity condition: ln£, = E , l n £  / + 1  -(/,  -if).  (102)  The cyclical component of the real exchange rate satisfies ln Q, = In E + ln Pf' - ln Pf, while the f  t  cyclical component of the terms of trade satisfies ln T = ln Pf - In Pf, where ln Pf - ln Pf . t  The cyclical component o f nominal output depends on the contemporaneous cyclical components o f nominal consumption, housing investment, capital investment, government consumption, exports, and imports according to approximate linear aggregate  resource  constraint: ta(«)  = £ l i i ( ^ < ^  Iff  If_  Y  Y  Y  (103)  ^ln(PfX )-y\n(P, M ). M  +  l  l  In equilibrium, the cyclical component of output is determined by the cumulative demands of domestic and foreign households, firms, and governments. The cyclical component of the net government debt depends on the past cyclical component of the net government debt, the past cyclical component of the nominal interest rate, the contemporaneous cyclical component of tax revenues, and the contemporaneous  cyclical  component o f nominal government consumption according to approximate linear government dynamic budget constraint  V , ln(-73, ) = ^[ln(-73, ) + G  G  +l  where — = ~~ffpy~Y]'  ^  m  s a  PP  ]+  r o x  i  m a t e  ^  rW PfY )--\n(PfG ) t  l  t  (104)  linear government dynamic budget constraint is  well defined only i f the level of the net government debt is positive. The cyclical component of the net foreign debt depends on the past cyclical component of the net foreign debt, the past cyclical component of the nominal interest rate, the contemporaneous cyclical component o f export revenues, and the contemporaneous cyclical component of import expenditures according to approximate linear national dynamic budget constraint  147  r  P where  = T^(y  B  Xj  '  _  ( 1  H-B,) + i,_ +  a  PP  r o x  i  -1  li  m a t e  ^ln(yflj-^ln(^M,)  n e a r  (105)  national dynamic budget constraint is well  defined only i f the level of the net foreign debt is positive. Variation in cyclical components is driven by eleven exogenous stochastic processes. The cyclical components of the preference, output technology, housing investment technology, capital investment technology, import technology, output price markup, import price markup, wage markup, monetary policy, fiscal expenditure, and fiscal revenue shocks follow stationary first order autoregressive processes: ln vf = p  vC  ln vf + ef,  ef  ~ iid /V(0,  In 4 = p l n i , _ , + ef, sf ~ iid  of),  (106)  Af(0,o ),  (107)  2  A  A  lni?/" =p ,„ Invf +ef",  ef  -iid  jV(0,of),  (108)  \nvf  ef  ~ iid  /V(0,of),  (109)  v  =p ,< \nvf v  ln vf = p  vM  +ef,  In vf + ef,  \n0f=p \ndf+ef,  ef  er  lnff = ^  l n t % + sf,  ln^ =^  ef  l n ^ _ , + ef,  ~ iid A^(0,  v  l  v;=p^;_  ef  (111)  ~ iid AA(0,CT^ ),  (112)  ef ~ iid A ^ O , ^ ),  ef ~M  i  i  d  (113) (114)  ^(0,CT ), 2  AT(0,a ),  (115)  2  c  < ' , < ~ i i d AA(0,cr ). r  1 +  (110)  ~ iid A ^ O , ^ ) ,  ^ = / V t > + < ' , < ' ~ y?=P cvf. +ef,  of),  2  (116)  r  The innovations driving these exogenous stochastic processes are assumed to be independent, which combined with our distributional assumptions implies multivariate normality. In flexible price and wage equilibrium, co = co = co = 0 and o , = 0 . Y  M  L  2  In deterministic steady state  148  equilibrium, o- , = a 2  2 G  v = v'  =v'  c  = erf  =v  =1  M  and  of  = a = <r „ = a 2  2  A  2  ;  /K  = af  = cr = of 2  gY  = a] = L  = 0.  3.2.6.2. Trend Components The trend components of the prices of output, consumption, housing investment, capital investment, government consumption, and imports follow random walks with time varying drift  \nPf =  K +\nP/+ef, l  \nPf = K, + In Pf + ef,  Mid  M(0,a ),  (117)  -iid  M(0,cr ),  (118)  2  r  2  pc  l n / f = x, + ]nPf"+ef"  -iid  /V(0,o-J,„),  (119)  In i f = n + \n.P '_ + ef  - i i d AT(0,(71,.),  (120)  t  t  / h\P, =  K,+]nPf+sf,  G  \nPf  =  x  -iid pM  +In  Af(0,crl ),  (121)  c  ~iid M(0,CT ). 2  (122)  PM  It follows that the trend components of the relative prices of consumption, housing investment, capital investment, government consumption, and imports follow random walks without drifts. This implies that along a balanced growth path, the levels of these relative prices are time independent but state dependent. The trend components of output, consumption, housing investment, capital investment, government consumption, exports, and imports follow random walks with time varying drift  In Y, = g, + n, + In Y,_ + ef x  ef  InC, =g,+n, + \nC,_ +ef, {  In I" = g, +  + In If + ef,  In 7? =g, +n, +\nJf  + ef,  ~ iid / V ( 0 , ^ ) ,  (123)  ef - iid Af(0,a ),  (124)  2  £  ef ef  ~ iid / V ( 0 , a ), 2  H  - iid Af(0,cr ), 2  K  (125)  (126)  149 InG, =g,+n + l  InG,_ + ?,e?~M  JV(0,a\),  £  x  (127)  In x, = g, + n, + In X _ + ef,  ef - iid Af(0, a\),  I n M , =g, + «, + l n M , +  ~ iid .A/(0,e4).  t  x  ,  (  (128) (129)  It follows that the trend components of the ratios of consumption, housing investment, capital investment, government consumption, exports, and imports to output follow random walks without drifts. This implies that along a balanced growth path, the levels o f these great ratios are time independent but state dependent.  The trend component of the shadow price of housing  satisfies In Qf = \nPf" , while the trend component of the housing stock satisfies l n - ^ - = In y . The trend component of the nominal wage follows a random walk with time varying drift n, + g,, while the trend component of employment follows a random walk with time varying drift n : t  \nW, = n +g, + \nW,_ + ef, t  x  l n l , = n, +lnl,_, +ef,  ef - iid N{Q,a ),  (130)  2  w  ef ~ iid Af(0,a ).  (131)  2  z  It follows that the trend component of the income share of labour follows a random walk without drift. This implies that along a balanced growth path, the level of the income share of labour is time independent but state dependent.  The trend component of real marginal cost satisfies  In 0 = In 0, while the trend component of the rate of capital utilization satisfies In w, = 0. The t  trend component of the shadow price of capital satisfies In Qf = In Pf , while the trend component of the capital stock satisfies In  K  K  = In —-.  i,  i  The trend components of the nominal interest rate, tax rate, and nominal exchange rate follow random walks without drifts: ef - iid N(0,<T J),  (132)  2  lnr; = l n v , +ef, ef - iid A/"(0,c^),  (133)  In£, = In S _ + ef, ef - iid /V(0,a\).  (134)  t  x  It follows that along a balanced growth path, the levels of the nominal interest rate, tax rate, and nominal exchange rate are time independent but state dependent. output based real interest rate satisfies 7 = {-E, t  consumption based real interest rate satisfies r  c t  exchange rate satisfies InQ = l n £ + l n y ^ (  Ky  The trend component of the  nf , while the trend component of the  Y  +x  = i - E, 7rf . The trend component of the real t  +x  - kiPf, while the trend component of the terms of  )  150 trade satisfies ln7^ = InPf  -]nP*,  where l n P  x t  = InP/.  The trend component of the net  government debt satisfies l n | - ^ ~ - | = l n | - - ^ y j , while the trend component of the net foreign debt satisfies I  n  =  In ( ~ )  '•  Long run balanced growth is driven by three common stochastic trends.  Trend inflation,  productivity growth, and population growth follow random walks without drifts: n, = n,_ + < , e' ~ iid A f ( 0 , a \ \  (135)  g,  g  (136)  « , = « , _ , + < , < ~ i i d Af(0,cr ).  (137)  x  t  e?~M  jV(0,o- ), 2  2  It follows that along a balanced growth path, growth rates are time independent but state dependent.  A l l innovations driving variation in trend components  are assumed to be  independent,  which  implies multivariate  combined with  our distributional assumptions  normality.  3.3. Estimation, Inference and Forecasting Quantitative monetary policy analysis and inflation targeting should be based on empirically adequate models of the economy, ones which approximately account for the existing empirical evidence in all measurable respects, at all frequencies. The monetary transmission mechanism is a cyclical phenomenon, involving dynamic interrelationships among deviations of the levels of various observed and unobserved endogenous variables from the levels of their flexible price and wage equilibrium components.  Measurement of the stance of monetary policy involves  estimation of the levels of the flexible price and wage equilibrium components of particular unobserved endogenous variables, while inflation targeting involves the generation of forecasts of the levels of particular observed endogenous variables. Within a D S G E framework, a first best approach to the conduct of quantitative monetary policy analysis and inflation targeting entails the joint derivation of empirically adequate cyclical and trend component specifications from microeconomic foundations.  This approach, which  should promote invariance to monetary policy regime shifts for reasons identified by Lucas (1976), is complicated by the existence of intermittent structural breaks, accounting for which requires flexible trend component specifications, as discussed in Clements and Hendry (1999) and Maddala and K i m (1998).  Within a D S G E framework, a second best approach to the  conduct of quantitative monetary policy analysis and inflation targeting entails the derivation of  151 empirically adequate cyclical component specifications from microeconomic foundations, augmented with flexible trend component specifications. This approach, proposed by Vitek (2006c, 2006d), is based on the presumption that the determinants of trend components are unknown but persistent, and is extended and refined in this paper.  3.3.1. Estimation The traditional econometric interpretation of macroeconometric models regards them as representations of the joint probability distribution of the data.  Adopting this traditional  econometric interpretation, the parameters and unobserved components of a linear state space representation of an approximate unobserved components representation of this D S G E model of a small open economy are jointly estimated with a Bayesian procedure, conditional on prior information concerning the values of parameters and trend components.  3.3.1.1. Estimation Procedure Let x denote a vector stochastic process consisting of the levels of N nonpredetermined t  endogenous variables, of which M are observed.  The cyclical components of this vector  stochastic process satisfy second order stochastic linear difference equation (138) where vector stochastic process x,  consists of the flexible price and wage equilibrium  components of /V nonpredetermined endogenous variables. The cyclical components of this vector stochastic process satisfy second order stochastic linear difference equation B x, = B x,_ + B E,x 0  x  x  2  l+x  + B,v  n  where vector stochastic process v, consists of the cyclical components of K variables.  (139) exogenous  This vector stochastic process satisfies stationary first order stochastic linear  difference equation (140) where s , ~ iid Af(0,27,). x  The trend components of vector stochastic process x, satisfy first  order stochastic linear difference equation  152  D x, =D +D u,+ Z> *,_, + e , 0  {  where e  3  2  (141)  2j  ~ iid A/"(0,27 ). Vector stochastic process «, consists of the levels of L common  2l  2  stochastic trends, and satisfies nonstationary first order stochastic linear difference equation =",-i+ 3,,>  (142)  £  where £ , ~ iid J\f(0,£ ). 3  Cyclical and trend components are additively separable, which  3  x, and x = x + J t , , where i t , = x,.  implies that x,=x + t  t  t  If there exists a unique stationary solution to multivariate linear rational expectations model (138) , then it may be expressed as: x, =S x _ +S x _,+S v . l  t  l  2  t  i  (143)  l  If there exists a unique stationary solution to multivariate linear rational expectations model (139) , then it may be expressed as: +r v.  i, =  2  (144)  (  These solutions are calculated simultaneously with the matrix decomposition based algorithm due to Klein (2000). Let  y  t  denote a vector stochastic process consisting of the levels of M  nonpredetermined endogenous variables. consisting of the levels of N-M  observed  Also, let z, denote a vector stochastic process  unobserved nonpredetermined endogenous variables, the  cyclical components of TV nonpredetermined endogenous variables, the cyclical components of the flexible price and wage equilibrium components of N  nonpredetermined  endogenous  variables, the trend components of N nonpredetermined endogenous variables, the cyclical components of K exogenous variables, and the levels of L common stochastic trends.  Given  unique stationary solutions (143) and (144), these vector stochastic processes have linear state space representation (145)  y, = F,z  n  z,=Gi+  G z _t + 2  t  G,e ,,  (146)  4>  where « , ~ iid A/"(0,27 ) and z ~ N(z \ ,P \ ). 4  4  0  0 0  0 0  Let w denote a vector stochastic process t  consisting of preliminary estimates of the trend components of M observed nonpredetermined endogenous variables. Suppose that this vector stochastic process satisfies  153  w, =  H z,+s „ x  (147)  5<  where e , ~ iid Af(0,£ ) 5  . Conditional on known parameter values, this signal equation defines  5  a set of stochastic restrictions on selected unobserved state variables.  The signal and state  innovation vectors are assumed to be independent, while the initial state vector is assumed to be independent from the signal and state innovation vectors, which combined with our distributional assumptions implies multivariate normality. Conditional on the parameters associated with these signal and state equations, estimates of unobserved state vector z, and its mean squared error matrix P, may be calculated with the filter proposed by Vitek (2006a, 2006b), which adapts the filter due to Kalman (1960) to incorporate prior information. Given initial conditions z  0l0  and P , estimates conditional on information 0|0  available at time t - 1 satisfy prediction equations: Z,\t-\  \  G  l\,-\  P  (148)  2 t-\\l-\'  +  G  Z  = 2 ,-l\,-\ 2 G  P  +  G  G  J ^ 4  G  (149)  i  (150)  y,\,=FP F  (151)  1  \ ty-\* i '  1  r  (152) - \ ,\,-\ \ H P  (153)  Z-  H  +  5  Given these predictions, under the assumption of multivariate normally distributed signal and state innovation vectors, together with conditionally contemporaneously uncorrelated signal vectors, estimates conditional on information available at time t satisfy updating equations (154) t\t  -  P  t\,-\ - y , \ , \ - \  P  K  F  P  l  K  P  where K = P, _ F Q~_ and K y  V  X  X  (155)  - w,H\ t\,-\,  x  w<  = P __ H R~f. l[t  x  x  Given terminal conditions z  T}T  and P  T]T  obtained from the final evaluation of these prediction and updating equations, estimates conditional on information available at time T satisfy smoothing equations I\T  Z  + J,(Z, ,  +X T  Z, u,). +  (156)  154  P, =P, J,(P -P ,)JJ, LT  LL+  L+]LT  where / , = P^G  Under our distributional assumptions, these estimators of the unobserved  P~ \ .  7  (157)  L+N  + LT  state vector are mean squared error optimal. Let 0 e0cz~R  J  denote a J dimensional vector containing the parameters associated with  the signal and state equations o f this linear state space model. The Bayesian estimator o f this parameter vector has posterior density function f(0\l )<xf(l \0)f(0), T  where J = {{y }' ^,{w }' ^}. t  (158)  T  s  s  s  s  Under the assumption o f multivariate normally distributed signal  and state innovation vectors, together with conditionally contemporaneously uncorrelated signal vectors, conditional density function / ( 1 \ 0) satisfies: T  f(T \0)  = flf(y,\l^,0)-flf(w,\l,_ ,0).  T  Under  our distributional assumptions,  f{w,\l,_ ,0) x  (159)  ]  conditional density  functions  f(y  t  |2~_ 0) p  and  satisfy: p e x p | - i ( j , - y , ^ ) Q ^ i y , - J,,,-,)},  f(y, | J _„0) = ( 2 / r ) ^ |  7  f  f(w, | Z,_„0) = (2n)^  | /?,„., r exp j-ito - »v, _,) Rffw, T  |(  -  (160)  (161)  Prior information concerning parameter vector 0 is summarized by a multivariate normal prior distribution having mean vector 0 and covariance matrix Q: X  m - ^ r W ' ^ - ^ v - ^ } .  (.62)  Independent priors are represented by a diagonal covariance matrix, under which diffuse priors are represented by infinite variances. Inference on the parameters is based on an asymptotic normal approximation to the posterior distribution around its mode. mode 0 satisfies T  Under regularity conditions stated in Geweke (2005), posterior  155 (163) where 0 <E0 denotes the pseudotrue parameter vector. Following Engle and Watson (1981), o  Hessian  may be estimated by 1 ^  1 (164)  1 + -V,V>/(<U where  E,_, [v V  E,.,  lnf(w,  [ V  T  g  S  V J  g  In f(y,  | J,_.,0 )] r  |J ,0 )] M  =  -V  r  J_,J?- _ V 1  F L W  -{v.fij..^, ®  = - V ^ . ^ V ^ , , , . , 1  F L W ( | (  .  I  -{V,*J_,(*-'_.  ®  g ^ , ) V , e ,  ,  M  ,  and  3.3.1.2. Estimation Results The set of parameters associated with this D S G E model of a small open economy is partitioned into two subsets.  The first subset is calibrated to approximately match long run  averages of functions of observed endogenous variables where possible, and estimates derived from existing microeconometric studies where necessary.  The second subset is estimated with  the Bayesian procedure described above, conditional on prior information concerning the values of parameters and trend components. Subjective discount factor f3 is restricted to equal 0.99, implying an annualized deterministic steady state equilibrium real interest rate of approximately 0.04.  In deterministic steady state  equilibrium, the output price markup - p - j - , import price markup -^-j , and wage markup ^—are restricted to equal 1.15. Depreciation rate parameter S  H  is restricted to equal 0.01, implying  an annualized deterministic steady state equilibrium depreciation rate of approximately 0.04, while depreciation rate parameter 8  K  is restricted to equal 0.02, implying an annualized  deterministic steady state equilibrium depreciation rate of approximately 0.08. In deterministic steady state equilibrium, the consumption import share \-(/> , housing investment import share c  \-(f>' , capital investment import share X — jr  1  \-(f>  G  are restricted to equal 0.30.  , and government consumption import share  The deterministic steady state equilibrium ratio of  consumption to output — is restricted to equal 0.60, while the deterministic steady state equilibrium ratio of domestic output to foreign output yr  * restricted to equal 0.11. s  In  deterministic steady state equilibrium, the foreign consumption import share 1 - (j> , foreign CJ  housing investment import share 1 - <fi' , foreign capital investment import share 1 - </>' , and J  J  156 foreign government consumption import share \-<fP'  are restricted to equal 0.02.  f  The  deterministic steady state equilibrium income share of labour — is restricted to equal 0.65, while the deterministic steady state equilibrium ratio of housing to output — is restricted to equal 6.00. In deterministic steady state equilibrium, the ratio of government consumption to output — is restricted to equal 0.20, while the tax rate r is restricted to equal 0.22. Table 3.1. Deterministic steady state equilibrium values o f great ratios Ratio  Value  Ratio  Value  CIY  0.6000  WLIPY  0.6500  I"  0.0600  HIY  1.5000  IY  JK.  IY  0.1138  GIY  0.2000  XIY  0.3183  MIY  0.2921  K/Y B  G  1.4224  IPY  -0.4950  BIPY  -0.6487  Note: Deterministic steady state equilibrium values are reported at an annual frequency based on calibrated parameter values.  Bayesian estimation of the remaining parameters of this D S G E model of a small open economy is based on the levels of twenty nine observed endogenous variables for Canada and the United States described in Appendix 3 .A. Those parameters associated with the conditional mean function are estimated subject to cross-economy equality restrictions. Those parameters associated exclusively with the conditional variance function are estimated conditional on diffuse priors. Initial conditions for the cyclical components of exogenous variables are given by their unconditional means and variances, while the initial values of all other state variables are treated as parameters, and are calibrated to match functions of preliminary estimates o f trend components calculated with the linear filter described in Hodrick and Prescott (1997).  The  posterior mode is calculated by numerically maximizing the logarithm of the posterior density kernel with a modified steepest ascent algorithm. Estimation results pertaining to the period 1971Q3 through 2005Q3 are reported in Appendix 3.B.  The sufficient condition for the  existence of a unique stationary rational expectations equilibrium due to Klein (2000) is satisfied in a neighbourhood around the posterior mode, while the estimator of the Hessian is not nearly singular at the posterior mode, suggesting that the approximate linear state space representation of this D S G E model of a small open economy is locally identified. The prior mean of indexation parameter y  Y  is 0.75, implying considerable output price  inflation inertia, while the prior mean of nominal rigidity parameter a> implies an average Y  duration of output price contracts of two years.  The prior mean of capital utilization cost  parameter K is 0.10, while the prior mean of elasticity of substitution parameter 9 is 0.75, implying that utilized capital and effective labour are moderately close complements in  157 production. The prior mean of habit persistence parameter a  c  is 0.95, while the prior mean of  intertemporal elasticity of substitution parameter a is 2.75, implying that consumption exhibits considerable persistence and moderate sensitivity to real interest rate changes. The prior mean of habit persistence parameter a  H  cost parameter x"  is 0.95, while the prior mean of housing investment adjustment  is 1-25, implying considerable sensitivity of housing investment to changes  in the relative shadow price of housing. The prior mean of capital investment adjustment cost parameter x  K  is 5.75, implying moderate sensitivity of capital investment to changes in the  relative shadow price of capital. The prior mean of indexation parameter y  M  is 0.75, implying  moderate import price inflation inertia, while the prior mean of nominal rigidity parameter eo  M  implies an average duration of import price contracts of two years. The prior mean of elasticity of substitution parameter y/ is 1.50, implying that domestic and foreign goods are moderately close substitutes in consumption, housing investment, capital investment, and government consumption.  The prior mean of indexation parameter y  L  is 0.75, implying considerable  sensitivity of the real wage to changes in consumption price inflation, while the prior mean of nominal rigidity parameter m  L  implies an average duration of wage contracts of two years. The  prior mean of habit persistence parameter a  L  is 0.95, while the prior mean of elasticity of  substitution parameter 77 is 0.75, implying considerable insensitivity of the real wage to changes in employment. The prior mean of the consumption price inflation response coefficient E>" in the monetary policy rule is 1.50, while the prior mean of the output response coefficient % is Y  0.125, ensuring convergence of the level of consumption price inflation to its target value. The prior mean of the net foreign debt response coefficient ^  G  in the fiscal expenditure rule is -0.10,  while the prior mean of the net government debt response coefficient £  T  in the fiscal revenue  rule is 1.00, ensuring convergence of the levels of the ratios of net foreign debt and net government debt to nominal output to their target values. A l l autoregressive parameters p have prior means of 0.85, implying considerable persistence of shocks driving variation in cyclical components. The posterior modes of these structural parameters are all close to their prior means, reflecting the imposition of tight independent priors to ensure the existence of a unique stationary rational expectations equilibrium.  The estimated variances of shocks driving variation in  cyclical components are all well within the range of estimates reported in the existing literature, after accounting for data rescaling. The estimated variances of shocks driving variation in trend components are relatively high, indicating that the majority of variation in the levels of observed endogenous variables is accounted for by variation in their trend components. Prior information concerning the values of trend components is generated by fitting third order deterministic polynomial functions to the levels of all observed endogenous variables by ordinary least squares.  Stochastic restrictions on the trend components of all observed  158 endogenous variables are derived from the fitted values associated with these ordinary least squares regressions, with innovation variances set proportional to estimated prediction variances assuming known parameters.  A l l stochastic restrictions are independent, represented by a  diagonal covariance matrix, and are harmonized, represented by a common factor of proportionality. Reflecting little confidence in these preliminary trend component estimates, this common factor of proportionality is set equal to one. Predicted, filtered and smoothed estimates of the cyclical and trend components of observed endogenous variables are plotted together with confidence intervals in Appendix 3.B. These confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters.  The predicted estimates are conditional on past  information, the filtered estimates are conditional on past and present information, and the smoothed estimates are conditional on past, present and future information. Visual inspection reveals close agreement with the conventional dating of business cycle expansions and recessions. Predicted, filtered and smoothed estimates of deviations of the levels of observed endogenous variables from their flexible price and wage equilibrium components, in addition to the levels of these flexible price and wage equilibrium components, are plotted together with confidence intervals in Appendix 3.B. Visual inspection reveals that a relatively low proportion of variation in the cyclical components of observed endogenous variables is accounted for by variation in the cyclical components of their flexible price and wage equilibrium components. This result suggests that a relatively high proportion of business cycle variation is accounted for by short run nominal price and wage rigidities, which amplify and propagate the effects of a variety of nominal and real shocks having temporary effects.  3.3.2. Inference Achieving low and stable inflation calls for accurate and precise indicators of inflationary pressure, together with an accurate and precise quantitative description of the monetary transmission mechanism. This estimated D S G E model of a small open economy addresses both of these challenges within a unified framework.  159 3.3.2.1. Quantifying the Stance of Monetary Policy Theoretically prominent indicators of inflationary pressure such as the natural rate of interest and natural exchange rate are unobservable. A s discussed in Woodford (2003), the level of the natural rate of interest provides a measure of the neutral stance of monetary policy, with deviations of the real interest rate from the natural rate of interest generating inflationary pressure.  It follows that the key to achieving low and stable inflation is the conduct of a  monetary policy under which the short term nominal interest rate tracks variation in the level of the natural rate of interest as closely as possible, although also achieving an interest rate smoothing objective derived from a concern with financial market stability may call for larger monetary policy responses to variation in the natural rate of interest caused by shocks having permanent effects than to variation caused by shocks having temporary effects. Definitions of indicators of inflationary pressure such as the natural rate of interest and natural exchange rate vary. Following Neiss and Nelson (2003), we define the natural rate of interest as that short term real interest rate consistent with past, present and future price and wage flexibility.  Under this definition, the natural rate of interest is a function only of exogenous  variables. In contrast, Woodford (2003) defines the natural rate of interest as that short term real interest rate consistent with current and future price and wage flexibility, conditional on the state of the economy. Under this definition, the natural rate of interest is a function of both exogenous and predetermined endogenous variables. A s argued by Neiss and Nelson (2003), it seems odd to define the natural rate of interest such that it depends on predetermined endogenous variables, and by implication past monetary policy shocks given short run nominal price and wage rigidities. Predicted, filtered and smoothed estimates of the level and trend component of the consumption based natural rate of interest are plotted together with confidence intervals versus corresponding estimates of the consumption based real interest rate in Figure 3.1.  Visual  inspection reveals that predicted estimates of the level of the natural rate of interest exhibit economically significant low frequency variation and are relatively imprecise, as evidenced by relatively wide confidence intervals, while filtered and smoothed estimates exhibit economically and statistically significant high frequency variation and are relatively precise, as evidenced by relatively narrow confidence intervals. Visual inspection also reveals that predicted, filtered and smoothed estimates of the trend component of the natural rate of interest exhibit economically and statistically significant low frequency variation and are relatively precise, as evidenced by relatively narrow confidence intervals. Given delays in data availability, these results suggest that accurate and precise measurement of the neutral stance of monetary policy on the basis of the level of the natural rate of interest can occur only retrospectively in practice, while inaccurate  160 but precise measurement of the neutral stance of monetary policy on the basis of the trend component of the natural rate of interest can take place contemporaneously in practice. This is problematic, as periods during which the estimated real interest rate exceeds the estimated natural rate of interest are closely aligned with the conventional dating of recessions, suggesting that tight monetary policy was to varying degrees a contributing factor. Figure 3.1. Predicted, fdtered and smoothed estimates of the natural rate o f interest RINTC_P (APR)  RINTC_F (APR)  RINTC_S (APR)  Note: Estimated levels are represented b y black lines, w h i l e blue and red lines depict estimated flexible price and wage equilibrium components and trend components, respectively. Symmetric 95% confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated b y the E c o n o m i c C y c l e Research Institute reference cycle.  In an open economy, the level of the consumption based natural rate of interest should fluctuate in response to a variety of shocks having both temporary and permanent effects, originating both domestically and abroad. In particular, the cyclical component of the natural rate of interest should fluctuate in response to a variety of real shocks having temporary effects, while the trend component of the natural rate of interest should fluctuate in response to a variety of nominal and real shocks having permanent effects. A s noted by Woodford (2003), it is not obvious that the level of the natural rate of interest should be expected to evolve smoothly, given its dependence on such a diverse set of shocks. The dynamic effects of a variety of real shocks having temporary effects on the level of the consumption based natural rate of interest, and the relative contributions of these real shocks to variation in its cyclical component, may be analyzed with theoretical impulse responses and forecast error variance decompositions.  Visual inspection of theoretical impulse responses  plotted in Appendix 3.B reveals that the level of the natural rate of interest declines in response to a temporary foreign output technology shock, and rises in response to a temporary foreign fiscal expenditure shock. Visual inspection of theoretical forecast error variance decompositions plotted in Appendix 3.B reveals that approximately 89% of variation in the cyclical component of the natural rate of interest is accounted for by the foreign output technology shock at all  161 horizons, while approximately 7% of this variation is accounted for by the foreign fiscal expenditure shock at all horizons. Predicted, filtered and smoothed estimates of the level and trend component of the natural exchange rate are plotted together with confidence intervals versus the observed real exchange rate in Figure 3.2. This concept of the natural exchange rate represents that real exchange rate consistent with past, present and future price and wage flexibility. Visual inspection reveals that predicted, filtered and smoothed estimates of both the level and trend component of the natural exchange rate exhibit economically and statistically significant high frequency variation and are relatively precise, as evidenced by relatively narrow confidence intervals. Visual inspection also reveals that a relatively high proportion of variation in the observed real exchange rate is accounted for by variation in the level of the natural exchange rate, while a relatively high proportion of variation in the level of the natural exchange rate is accounted for by variation in its trend component. These results suggest that a relatively high proportion of variation in the observed real exchange rate is accounted for by nominal and real shocks having permanent effects, while a relatively high proportion of variation in the cyclical component of the real exchange rate is accounted for by real shocks having temporary effects.  It follows that a  relatively low proportion of cyclical real exchange rate variation is accounted for by short run nominal price and wage rigidities. Figure 3.2. Predicted, fdtered and smoothed estimates of the natural exchange rate LREXCH_P  LREXCH_F  LREXCH_S  Note: Observed levels are represented by black lines, w h i l e blue and red lines depict estimated flexible price and wage e q u i l i b r i u m components and trend components, respectively. Symmetric 95% confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated b y the E c o n o m i c C y c l e Research Institute reference cycle.  The dynamic effects of a variety of real shocks having temporary effects on the level of the natural exchange rate, and the relative contributions of these real shocks to variation in its cyclical component, may be analyzed with theoretical impulse responses and forecast error variance decompositions. Visual inspection of theoretical impulse responses plotted in Appendix 3.B reveals that the level of the natural exchange rate rises in response to a temporary domestic  162 output technology shock, corresponding to a real depreciation, and declines in response to a temporary foreign output technology shock, corresponding to a real appreciation.  Visual  inspection of theoretical forecast error variance decompositions plotted in Appendix 3.B reveals that approximately 53% of variation in the cyclical component of the natural exchange rate is accounted for by the domestic output technology shock at all horizons, while approximately 21% of this variation is accounted for by the foreign output technology shock at all horizons. The finite sample properties of the estimation procedure proposed in this paper are evaluated with a Monte Carlo experiment in Vitek (2006e), with an emphasis on the levels of the natural rate of interest and natural exchange rate. Joint estimation of the parameters and unobserved components of a linear state space representation of an approximate unobserved components representation of a relatively parsimonious D S G E model of a small open economy with this Bayesian procedure is found to yield reasonably accurate and precise results in samples of approximately the size considered in this paper.  In particular, estimates of the levels of the  natural rate of interest and natural exchange rate conditional on alternative information sets are approximately unbiased, while analytical root mean squared errors appropriately account for uncertainty surrounding them, irrespective of whether the data generating process features common deterministic or stochastic trends.  3.3.2.2. Quantifying the Monetary Transmission Mechanism Whether this estimated D S G E model provides an accurate quantitative description of the monetary transmission mechanism in a small open economy is determined by comparing its impulse responses to domestic and foreign monetary policy shocks with impulse responses derived from an estimated structural vector autoregressive or S V A R model. Consider the following S V A R model of the monetary transmission mechanism in a small open economy  1=1  where fi(t)  denotes a third order deterministic polynomial function and s ~ iid JV(0, I). t  Vector stochastic process y  t  consists of domestic output price inflation nf, domestic output  ln Y , domestic consumption price inflation nf, domestic consumption ln C , , domestic housing t  investment price inflation n\ price inflation nf,  , domestic housing investment ln if,  domestic capital investment ln if,  domestic exports ln X , t  domestic capital investment  domestic import price inflation  nf,  domestic imports ln M,, domestic nominal interest rate z,, nominal  163 exchange rate In £ , foreign output price inflation t  nf' , f  foreign output  In Y/,  foreign  consumption I n C / , foreign housing investment l n / , , foreign capital investment \nlf , w /  foreign nominal interest rate if.  f  and  The diagonal elements of parameter matrix AQ are normalized  to one, while the off diagonal elements of positive definite parameter matrix B are restricted to equal zero, thus associating with each equation a unique endogenous variable, and with each endogenous variable a unique structural innovation. This S V A R model is identified by imposing restrictions on the timing of the effects of monetary policy shocks and on the information sets of the monetary authorities, both within and across the domestic and foreign economies. Within the domestic and foreign economies, prices and quantities are restricted to not respond instantaneously to monetary policy shocks, while the monetary authorities can respond instantaneously to changes in these variables.  Across the  domestic and foreign economies, the domestic monetary authority is restricted to not respond instantaneously to foreign monetary policy shocks, while foreign variables are restricted to not respond to domestic monetary policy shocks. This S V A R model of the monetary transmission mechanism in a small open economy is estimated by full information maximum likelihood over the period 1971Q3 through 2005Q3. As discussed in Hamilton (1994), in the absence of model misspecification, this full information maximum likelihood estimator is consistent and asymptotically normal, irrespective of the cointegration rank and validity of the conditional multivariate normality assumption. The lag order is selected to minimize multivariate extensions of the model selection criterion functions of Akaike (1974), Schwarz (1978), and Hannan and Quinn (1979) subject to an upper bound equal to the seasonal frequency. These model selection criterion functions generally prefer a lag order of one. Table 3.2. Model selection criterion function values  p  AIC(p)  SC(p)  HQ{ ) P  1  -125.9962  -114.3833*  -121.2771*  2  -126.5847  -108.8516  -119.3785  3  -127.8420*  -103.9889  -118.1489  4  -126.7401  -98.8863  -115.4212  Note: M i n i m i z e d values o f model selection criterion functions are indicated b y * .  Since this S V A R model is estimated to provide empirical evidence concerning the monetary transmission mechanism in a small open economy, it is imperative to examine the empirical validity of its overidentifying restrictions prior to the conduct of impulse response analysis. On the basis of bootstrap likelihood ratio tests, these overidentifying restrictions are not rejected at conventional levels of statistical significance.  164  Table 3,3. Results o f tests o f overidentifying restrictions Test Statistic  359.2387  P Values Asymptotic  Parametric Bootstrap  Nonparametric Bootstrap  0.0000  1.0000  1.0000  Note: T h i s l i k e l i h o o d ratio test statistic is asymptotically distributed as xlr, • Bootstrap distributions are based on 999 replications.  Theoretical impulse responses to a domestic monetary policy shock are plotted versus empirical impulse responses in Figure 3.3. Following a domestic monetary policy shock, the domestic nominal interest rate exhibits an immediate increase followed by a gradual decline. The domestic currency appreciates, with the nominal exchange rate exhibiting delayed overshooting. These nominal interest rate and nominal exchange rate dynamics induce persistent and generally statistically significant hump shaped negative responses of domestic output price inflation, output, consumption price inflation, consumption, housing investment price inflation, housing investment, capital investment price inflation, capital investment, import price inflation, exports and imports, with peak effects realized after approximately one year. These results are qualitatively consistent with those of S V A R analyses of the monetary transmission mechanism in open economies such as Eichenbaum and Evans (1995), Clarida and Gertler (1997), K i m and Roubini (1995), Cushman and Zha (1997), and Vitek (2006d).  165 Figure 3.3. Theoretical versus empirical impulse responses to a domestic monetary policy shock D L P C O N (APR)  D L P G D P (APR)  5  10  15  20  25  30  35  40  0  5  10  15  2 0 2 5 3 0 3 5 4 0  0  5  10  15  20  25  15  20  25  30  35  40  5  10  15  20  25  30  35  40  30  35  40  5  10  15  20  25  30  35  40  DLPKINV(APR)  DLPHINV (APR)  5  10  30  35  40  0  5  10  15  5  2 0 2 5 3 0 3 5 4 0  10  15  20  25  NINT(APR)  DLPIMP (APR)  0  5  10  15  2 0 2 5 3 0 3 5 4 0  0  5  10  15  2 0 2 5 3 0 3 5 4 0  0  5  10  15  5  10  15  2 0 2 5 3 0 3 5 4 0  0  5  10  15  5  10  15  2 0 2 5 3 0 3 5 4 0  D L P G D P F (APR)  15  20  25  30  35  40  0  5  10  15  20  25  30  35  40  20  25  30  35  40  30  35  40  20  25  30  35  40  NINTF (APR)  5  10  15  20  25  30  35  40  5  10  15  20  25  30  35  40  0  5  10  15  20  25  Note: Theoretical impulse responses to a 50 basis point monetary policy shock are represented by black lines, while blue lines depict empirical impulse responses to a 50 basis point monetary policy shock.  A s y m m e t r i c 9 5 % confidence intervals are calculated with a nonparametric  bootstrap simulation with 999 replications.  Theoretical impulse responses to a foreign monetary policy shock are plotted versus empirical impulse responses in Figure 3.4.  Following a foreign monetary policy shock, the  foreign nominal interest rate exhibits an immediate increase followed by a gradual decline. In response to these nominal interest rate dynamics, there arise persistent and generally statistically significant  hump shaped negative responses of foreign  output price inflation,  output,  166 consumption, housing investment and capital investment, with peak effects realized after approximately one to two years. Although domestic output, consumption, housing investment, capital investment  and imports decline, domestic  consumption price inflation, housing  investment price inflation, capital investment price inflation and import price inflation rise due to domestic currency depreciation. These results are qualitatively consistent with those of S V A R analyses of the monetary transmission mechanism in closed economies such as Sims and Zha (1995), Gordon and Leeper (1994), Leeper, Sims and Zha (1996), Christiano, Eichenbaum and Evans (1998, 2005), and Vitek (2006c, 2006d).  167 Figure 3.4. Theoretical versus empirical impulse responses to a foreign monetary policy shock DLPGDP (APR)  0 5  10  15  20  25  LRGDP  30  35  40  0  5  10  DLPHINV(APR)  0 5  10  15  10  15  2 0 2 5 3 0 3 5 4 0  20  25  0 5  10  10  15  30  35  40  0  5  10  10  15  30  35  40  0  15  5  10  15  2 0 2 5 3 0 3 5 4 0  2O  2 0 2 5 3 0 3 5 4 0  0 5  10  15  0 5  10  0  5  10  15  25  30  35  4O  25  X  35  40  O S 1 0  1S  15  0 5  10  25  15  0 5  10  15  20  25  30  35  40  LRKINV  2 0 2 5 3 3 3 5 4 0  0 5  10  15  2 0 2 5 3 0 3 5 4 0  NINT(APR)  3  2 0 2 5 3 0 3 5 4 0  0 5  10  LRGDPF  2 0 2 5 3 0 3 5 4 0  20  20  LRIM  LRKINVF  2 0 2 5 3 0 3 5 4 0  15  LRCON  DLPKINV(APR)  DLPGDPF (APR)  LRHINVF  0 5  25  LREXP  LNEXCH  0 5  20  LRHINV  DLPIMP(APR)  0 5  15  DLPCON (APR)  15  20  25  303540  LRCONF  2 0 2 5 3 0 3 5 4 0  0 5  10  15  20  25  303640  NINTF (APR)  30  35  40  0 5  10  15  20  25  333540  Note: Theoretical impulse responses to a 50 basis point monetary policy shock are represented b y black lines, w h i l e blue lines depict empirical impulse responses to a 50 basis point monetary policy shock.  Asymmetric 9 5 % confidence intervals are calculated with a nonparametric  bootstrap simulation with 999 replications.  Visual inspection reveals that the theoretical impulse responses to domestic and foreign monetary  policy  shocks generally  lie within confidence  intervals  associated  with the  corresponding empirical impulse responses, suggesting that this estimated D S G E model provides an accurate quantitative description of the monetary transmission mechanism in a small open  168 economy.  However, these confidence intervals are rather wide, indicating that considerable  uncertainty surrounds this empirical evidence.  3.3.3. Forecasting While it is desirable that forecasts be unbiased and efficient, the practical value of any forecasting model depends on its relative predictive accuracy. To compare the dynamic out of sample forecasting performance of the D S G E and S V A R models, forty quarters of observations are retained to evaluate forecasts one through eight quarters ahead, generated conditional on parameters estimated using information available at the forecast origin.  The models are  compared on the basis of mean squared prediction errors in levels, ordinary differences, and seasonal differences. The D S G E model is not recursively estimated as the forecast origin rolls forward due to the high computational cost of such a procedure, while the S V A R model is. Presumably, recursively estimating the D S G E model would improve its predictive accuracy. Mean squared prediction error differentials are plotted together with confidence intervals accounting for contemporaneous and serial correlation of forecast errors in Appendix 3.B. If these mean squared prediction error differentials are negative then the forecasting performance of the D S G E model dominates that of the S V A R model, while i f positive then the D S G E model is dominated by the S V A R model in terms of predictive accuracy. The null hypothesis of equal squared prediction errors is rejected by the predictive accuracy test of Diebold and Mariano (1995) i f and only i f these confidence intervals exclude zero. The asymptotic variance of the average loss differential is estimated by a weighted sum of the autocovariances of the loss differential, employing the weighting function proposed by Newey and West (1987).  Visual  inspection reveals that these mean squared prediction error differentials are generally negative, suggesting that the D S G E model dominates the S V A R model in terms of forecasting performance, in spite of a considerable informational disadvantage.  However, these mean  squared prediction error differentials are rarely statistically significant at conventional levels, perhaps because the predictive accuracy test due to Diebold and Mariano (1995), which is univariate, typically lacks power to detect dominance in forecasting performance, as evidenced by Monte Carlo evaluations such as Ashley (2003) and McCracken (2000). Dynamic out of sample forecasts of levels, ordinary differences, and seasonal differences are plotted together with confidence intervals versus realized outcomes in Appendix 3.B.  These  confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Visual inspection reveals that the realized outcomes generally lie within their associated confidence intervals, suggesting that forecast failure is  169 absent.  However, these confidence intervals are rather wide, indicating that considerable  uncertainty surrounds the point forecasts.  3.4. Conclusion This paper develops and estimates a D S G E model of a small open economy for purposes of monetary policy analysis and inflation targeting which provides a quantitative description of the monetary transmission mechanism, yields a mutually consistent set of indicators of inflationary pressure together with confidence intervals, and facilitates the generation of relatively accurate forecasts.  Cyclical components are modeled by linearizing equilibrium conditions around a  stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path.  Parameters and unobserved components are jointly estimated  with a Bayesian procedure, conditional on prior information concerning the values of parameters and trend components. Definitions of indicators of inflationary pressure such as the natural rate of interest and natural exchange rate vary, while estimates are typically sensitive to identifying restrictions. It follows that combinations of estimates of indicators of inflationary pressure derived under alternative definitions from dissimilar models may be more useful for purposes of monetary policy analysis and inflation targeting in a small open economy than any of the constituents. A n examination of the inflation control and output stabilization benefits conferred by combining alternative estimates remains an objective for future research.  Appendix 3.A. Description of the Data Set The data set consists of quarterly seasonally adjusted  observations on twenty nine  macroeconomic variables for Canada and the United States over the period 1971Q1 through 2005Q3.  A l l aggregate prices and quantities are expenditure based.  Model consistent  employment is derived from observed nominal labour income and a nominal wage index, while model consistent tax rates are derived from observed nominal output and disposable income. The nominal interest rate is measured by the three month Treasury bill rate expressed as a period average, while the nominal exchange rate is quoted as an end of period value. National accounts data for Canada was retrieved from the C A N S 1 M database maintained by Statistics Canada, national accounts data for the United States was obtained from the F R E D database maintained by  170 the Federal Reserve Bank of Saint Louis, and other data was extracted from the IFS database maintained by the International Monetary Fund.  171 Appendix 3.B. Tables and Figures Table 3.4, Bayesian estimation results Parameter  a a" a c  L  x" x K  1 K  V  a 9  f  r r  M L  w co M  L  <T  ?  i  a  <T  P,.c  PA P ,~ V  />„,-  P„« Po<  p„* p>>  a ,. <r\ o-\, 1  4  a, *{, °\< -I,, 1  °\,  -V, -I  °\„ 4  Prior Distribution  Posterior Distribution  Mean 0.950000 0.950000 0.950000 1.250000 5.750000 0.750000 0.100000 1.500000 2.750000 0.750000 0.750000 0.750000 0.750000 0.875000 0.875000 0.875000 1.500000 0.125000 -0.100000 1.000000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000 0.850000  Standard Error 0.000950 0.000950 0.000950 0.001250 0.005750 0.000750 0.000100 0.001500 0.002750 0.000750 0.000750 0.000750 0.000750 0.000875 0.000875 0.000875 0.001500 0.000125 0.000100 0.001000 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850  --  00  -  --  00  00 00 00 00 00 00 00  1  CO  oo  00 00 00 00  --  OO  -  CO  --  OO  -  OO  00  oo oo oo  00 OO  Mode 0.942380 0.947530 0.940960 1.249300 5.746900 0.750400 0.099995 1.500100 2.751400 0.750000 0.750100 0.750140 0.750080 0.876850 0.874030 0.879820 1.499300 0.124940 -0.099998 0.999100 0.850280 0.851190 0.850130 0.851230 0.849990 0.850140 0.850020 0.850020 0.853240 0.849960 0.849970 0.204890 0.319060 0.270540 0.245970 0.237440 0.232260 0.258260 0.251000 0.208960 0.260570 0.254010 0.273060 0.097028 0.242180 0.241410 0.259430 0.092639 0.227650 0.248370 0.003574 0.116640 0.268170 0.617930 0.059436 0.582020  Standard Error 0.000880 0.000651 0.000905 0.001250 0.005750 0.000750 0.000100 0.001500 0.002749 0.000750 0.000750 0.000750 0.000750 0.000864 0.000873 0.000868 0.001500 0.000125 0.000100 0.001000 0.000850 0.000848 0.000850 0.000850 0.000850 0.000850 0.000850 0.000850 0.000840 0.000849 0.000849 0.062569 0.050146 0.058117 0.051460 0.035607 1.361800 2.058700 12.298000 0.028450 0.038933 0.031108 0.066943 0.012105 0.051732 0.052788 0.537140 0.106830 13.374000 2.473000 0.000491 0.015587 0.029474 0.077548 0.008994 0.073046  172  Parameter  Prior Distribution Mean  Standard Error  Mode  Standard Error  -  00  0.091754  0.011002  --  00  0.701000  0.083695  CO  0.971080  0.183350  00  0.794680  0.097908  CO  0.468040  0.056322  CO  0.705270  0.088417  CO  0.120910  0.023816  CO  0.344410  0.038142  CO  1.418000  0.180230  00  0.454340  0.052121  CO  0.787960  0.096018  00  0.111000  0.024453  00  0.002109  0.000411  00  0.168410  0.024267  00  0.492930  0.060998  00  0.301330  0.036485  00  0.018243  0.002367  00  0.023448  0.002263  00  0.755180  0.088658  00  0.407700  0.041494  00  0,082961  0.013603  00  2.071800  0.250180  00  0.352310  0.042287  00  0.074382  0.009045  00  0.001339  0.000334  00  0.057857  0.011437  CO  0.000138  0.000019  00  0.000004  0.000006  00  0.000025  0.000009  00  0.000068  0.000012  00  0.000023  0.000011  00  0.000029  0.000010  -  -  44  ----  --  a. 2  °l  o-] <>  Posterior Distribution  -  --  Note: A l l observed endogenous variables are rescaled b y a factor o f 100.  173  Figure 3.5. Predicted cyclical components of observed endogenous variables  1 z***^  1975  198 0 1985 1990  1990  1995  1000 20C  198S 2000 2005  Note: Estimated cyclical components are represented by blue lines, while red lines depict estimated deviations f r o m flexible price and wage equilibrium components.  Symmetric 95% confidence intervals assume multivariate normally distributed and independent signal and state  innovation vectors and known parameters. cycle.  Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference  174  Figure 3.6. Filtered cyclical components o f observed endogenous variables  ft  LPKINV  f \  ^Li  r\  A /  wm  (Mit  mm  u  •  f  |  Note: Estimated c y c l i c a l components are represented by blue lines, while red lines depict estimated deviations f r o m flexible price and wage equilibrium components.  Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state  innovation vectors and known parameters. cycle.  Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference  175  Figure 3.7. Smoothed cyclical components of observed endogenous variables  (V  A  LPKINV  A  L  .  i\  A.  Afote: Estimated c y c l i c a l components are represented by blue lines, while red lines depict estimated deviations f r o m flexible price and wage equilibrium components.  Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state  innovation vectors and known parameters. cycle.  Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference  176  Figure 3.8. Predicted trend components of observed endogenous variables LPGDP  LRGDP  LPCON  LRCON  ...  z.y  -20.  LRHINV  11J0  LPKINV  LRKINV  LREXP  LPHINv  LPIMP  LPGOV  LRGOV  LNWAGE  LEW  war  /  ...  LRIIvP  I1M  NINT (APR)  LTAXRATE  LNEXGH  LPGDPF  LRGDPF  LRGOVF  LPIWF  / LRCONF  LRHINVF  LRKINVF  LE W F  NINTF (APR)  MO. Jt*S lll.ll.,  , S  LNWAGEF  1*75 19*0 1965 1MO 1»5 2000 200  LTAXRATEF  US  Note: Observed levels are represented by black lines, while blue and red lines depict estimated flexible price and wage e q u i l i b r i u m components and trend components, respectively. S y m m e t r i c 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  177  Figure 3.9. Filtered trend components of observed endogenous variables  '  1240  E 1280.  J  Note: Observed levels are represented by black lines, while blue and red lines depict estimated flexible price and wage equilibrium components and trend components, respectively. Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  178  Figure 3.10. Smoothed trend components of observed endogenous variables  J  .40.  J  E /  13.0  A  J  / if'  Note: Observed levels are represented by black lines, w h i l e blue and red lines depict estimated flexible price and wage equilibrium components and trend components, respectively. Symmetric 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters. Shaded regions indicate recessions as dated by the E c o n o m i c C y c l e Research Institute reference cycle.  Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equilibrium are represented by blue lines, while green lines depict theoretical impulse responses to a unit standard deviation innovation under flexible price and wage equilibrium.  Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equilibrium are represented by blue lines, while green lines depict theoretical impulse responses to a unit standard deviation innovation under flexible price and wage equilibrium.  Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equilibrium are represented by blue lines, while green lines depict theoretical impulse responses to a unit standard deviation innovation under flexible price and wage equilibrium.  182  Figure 3.14. Theoretical impulse responses to a foreign output technology shock  Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equilibrium are represented by blue lines, while green lines depict theoretical impulse responses to a unit standard deviation innovation under flexible price and wage equilibrium.  183  Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equilibrium are represented by blue lines, while green lines depict theoretical impulse responses to a unit standard deviation innovation under flexible price and wage equilibrium.  184  Figure 3.16. Theoretical impulse responses to a foreign fiscal expenditure shock DLPGDPIAre)  LPQDP  LRGOP  0LPCON|*PR|  LPCON  LRCON  DLPHWV(*PR|  LKSTOCKIH)  NfjriAPR)  RWIY(APR)  RWTC(APR)  LTAXRATE  LNfJCH  LREXCH  Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equilibrium are represented by blue lines, while green lines depict theoretical impulse responses to a unit standard deviation innovation under flexible price and wage equilibrium.  185 Figure 3.17. Theoretical forecast error variance decompositions under sticky price and wage equilibrium  •NO TAIJ  BUM! er  •OMRMLff •lTWETAt_MT •LTMCTAi.U sNU a 'i' TAUT  7  mVUJT  187 Figure 3.19. Mean squared prediction error differentials for levels  2  3  4  5  6  7  LRKINV  .l« ' Ni-| • WOO.  5  6  7  2  3  4  5  6  7  NINT  800-  5  6  7  2  3  4  5  6  7  6  7  LRGDPF  5  1(100. 5  6  7  5  6  7  8  Note: M e a n squared prediction error differentials are defined as the mean squared prediction error for the D S G E model less that for the S V A R model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors.  188 Figure 3.20. Mean squared prediction error differentials for ordinary differences  5  1  2  3  5  6  6  7  5  7  2  3  4  5  6  6  7  5  7  2  3  DLRIMP  6  7  8  2  3  4  5  4  6  5  7  8  6  7  8  6  7  8  6  7  DNINT  6  7  8  2  3  4  5  DLRCONF  -0 4  -o.a. 5  5  6  6  7  7  5  6  7  1 2  3  4  5  8  8  Note: M e a n squared predietion error differentials are defined as the mean squared prediction error for the D S G E model less that for the S V A R model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors.  189 Figure 3.21. Mean squared prediction error differentials for seasonal differences SDLPCON  2  3  2  3  4  4  5  5  6  6  7  7  8  8  SDLPIMP  111!.  2  SDLNEXCH  3  4  5  6  7  2  3  4  5  6  7  2  2  4  5  6  T  6  7  SDLRCONF  2  1  3  SDLPGDPF  3  4  S  3  Note: M e a n squared prediction error differentials are defined as the mean squared prediction error for the D S G E model less that for the S V A R model. Symmetric 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors.  190 Figure 3.22. Dynamic forecasts of levels of observed endogenous variables  Note: Realized outcomes are represented by black lines, while blue lines depict point forecasts.  Symmetric 9 5 % confidence intervals assume  multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters.  191 Figure 3.23. Dynamic forecasts o f ordinary differences of observed endogenous variables  N—  '  Note: Realized outcomes are represented by black lines, while blue lines depict point forecasts.  Symmetric 9 5 % confidence intervals assume  multivariate normally distributed and independent signal and state innovation vectors and known parameters.  Fi gure 3.24. Dynamic forecasts of seasonal differences of observed endogenous variables  Note: Realized outcomes are represented by black lines, while blue lines depict point forecasts.  Symmetric 9 5 % confidence intervals  multivariate normally distributed and independent signal and state innovation vectors and k n o w n parameters.  193 References Adolfson, M . (2001), Monetary policy with incomplete exchange rate pass-through, Stockholm School of Economics Working Paper, 476. Akaike, H . (1974), A new look at the statistical model identification, IEEE Transactions on Automatic Control, 19, 716-723. A l t i g , D . , L . Christiano, M . 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Nelson (2000), Monetary policy for an open economy: A n alternative framework with optimizing agents and sticky prices, Oxford Review of Economic Policy, 16, 74-91. McCracken, M . (2000), Robust out-of-sample inference, Journal of Econometrics, 99, 195-223. Monacelli, T. (2005), Monetary policy in a low pass-through environment, Journal of Money, Credit, and Banking, 37, 1047-1066. Neiss, K . and E. Nelson (2003), The real-interest-rate gap as an inflation indicator, Macroeconomic Dynamics, 7, 239-262. Nelson, C . and H . K a n g (1981), Spurious periodicity in inappropriately detrended time series, Econometrica, 49, 741-751.  195 Newey, W . and K . West (1987), A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica, 55, 703-708. Obstfeld, M . and K . Rogoff (1995), Exchange rate dynamics redux, Journal of Political Economy, 103, 624-660. Obstfeld, M . and K . Rogoff (1996), Foundations of International Macroeconomics, M I T Press. Rotemberg, J. and M . Woodford (1995), Dynamic general equilibrium models with imperfectly competitive product markets, Frontiers of Business Cycle Research, Princeton University Press: Rotemberg, J. and M . Woodford (1997), A n optimization-based econometric framework for the evaluation o f monetary policy, NBER Macroeconomics Annual, M I T Press. Ruge-Murcia, F . (2003), Methods to estimate dynamic stochastic general equilibrium models, CIREQ Working Paper, 17. Schwarz, G . (1978), Estimating the dimension o f a model, Annals of Statistics, 6, 461-464. Sims, C . and T. Zha (1995), Does monetary policy generate recessions?, Unpublished Manuscript. Smets, F. and R. Wouters (2003), A n estimated dynamic stochastic general equilibrium model o f the Euro area, Journal of the European Economic Association, 1, 1123-1175. Smets, F. and R. Wouters (2005), Comparing shocks and frictions in U S and Euro area business cycles: A Bayesian D S G E approach, Journal ofApplied Econometrics, 20, 161-183. Vitek, F. (2006a), A n unobserved components model o f the monetary transmission mechanism in a closed economy, Unpublished Manuscript. Vitek, F. (2006b), A n unobserved components model o f the monetary transmission mechanism in a small open economy, Unpublished Manuscript. Vitek, F. (2006c),' Monetary policy analysis in a closed economy: A dynamic stochastic general equilibrium approach, Unpublished Manuscript. Vitek, F. (2006d), Monetary policy analysis in a small open economy: A dynamic stochastic general equilibrium approach, Unpublished Manuscript. Vitek, F. (2006e), Measuring the stance o f monetary policy in a small open economy: A Monte Carlo evaluation, Unpublished Manuscript. Woodford, M . (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. Y u n , T. (1996), Nominal price rigidity, money supply endogeneity, and business cycles, Journal of Monetary Economics, 37, 345-370.  196  CHAPTER 4  Measuring the Stance of Monetary Policy in a Small Open Economy: A Monte Carlo Evaluation  4.1. Introduction  The real business cycle or R B C class of models introduced by Kydland and Prescott (1982) and Long and Plosser (1983) was originally intended to provide a unified theoretical and empirical framework for the joint analysis of business cycle dynamics and long run growth. A s extensions of the neoclassical growth model due to Ramsey (1928) and Solow (1956), in the absence of shocks which generate business cycle fluctuations, R B C models generally converge to well defined balanced growth paths along which great ratios and trend growth rates are time and state independent.  However, evaluations o f the business cycle predictions of this class of  models based on comparisons between theoretical and empirical unconditional second moments typically abstract from their predictions for long run growth, as the comparisons are conditional on atheoretic decompositions of the levels of endogenous variables into cyclical and trend components. The dynamic stochastic general equilibrium or D S G E class of models has recently emerged as the dominant theoretical and empirical framework for the analysis of the monetary transmission mechanism and the optimal conduct of monetary policy. A s extensions of R B C models, the subclass of D S G E models generally employed in contemporary monetary policy analyses features short run nominal price rigidities generated by monopolistic competition and staggered reoptimization in output markets. Early examples of closed economy D S G E models within this subclass include those of Yun (1996), Goodfriend and King (1997), Rotemberg and Woodford (1995, 1997), and McCallum and Nelson (1999), while early examples of open economy D S G E models within this subclass include those of McCallum and Nelson (2000), Clarida, Gali and Gertler (2001, 2002), and Gertler, Gilchrist and Natalucci (2001). In parallel with the R B C methodology, evaluations of the predictions of these D S G E models with regards to the monetary transmission mechanism based on comparisons between theoretical and empirical impulse response functions typically abstract from their predictions for long run growth, as do measurements of the stance of monetary policy based on flexible price equilibrium concepts,  197 again being conditional on atheoretic decompositions of the levels of endogenous variables into cyclical and trend components. The existence of a well defined balanced growth path along which great ratios and trend growth rates are time independent is desirable, as it ensures the mutual stability of long horizon forecasts of the levels of endogenous variables. However, within a D S G E framework, ensuring the existence of a well defined balanced growth path requires the imposition of restrictions which potentially limit the empirical adequacy of cyclical and trend component specifications. A s discussed in King, Plosser and Rebelo (1988), ensuring the existence of a well defined balanced growth path restricts the classes of functions representing preferences and technologies which may be considered, potentially limiting the empirical adequacy of cyclical  component  specifications. Ensuring the existence of a well defined balanced growth path also restricts the types of exogenous stochastic processes responsible for driving both business cycle dynamics and long run growth which may be considered, potentially limiting the empirical adequacy of trend component specifications. As discussed in Canova, Finn and Pagan (1994), the prediction of D S G E models featuring long run balanced growth driven by trend inflation, productivity growth, and population growth that the levels of observed endogenous variables should fluctuate around common deterministic or stochastic trends is often rejected empirically. A central theme of the voluminous cointegration literature surveyed by Maddala and K i m (1998) is that, while empirical support for the existence of cointegrating relationships of a form consistent with the existence of a well defined balanced growth path often arises, it typically does so conditional on intermittent structural breaks.  As discussed in Clements and Hendry  (1999) , failure to allow for the existence of such intermittent structural breaks is a dominant source of forecast failure in macroeconometric models.  These observations suggest that  allowing the balanced growth paths towards which D S G E models converge in the absence of shocks to be state dependent should robustify estimation, inference and forecasting to intermittent structural breaks that occur within sample. Due to the curse o f dimensionality, D S G E models are generally solved with perturbation methods, which require the existence of a stationary deterministic steady state equilibrium around which to approximate equilibrium conditions. In cases where D S G E models feature long run balanced growth driven by trend inflation, productivity growth and population growth, a stationary deterministic steady state equilibrium may be obtained by appropriately deflating endogenous variables by common deterministic or stochastic trends. However, i f the balanced growth paths towards which D S G E models converge in the absence of shocks exhibit a flexible form of state dependence, then existing perturbation methods are not applicable. Quantitative monetary policy analysis and inflation targeting should be based on empirically adequate models of the economy, ones which approximately account for the existing empirical  198 evidence in all measurable respects, at all frequencies. As discussed in Woodford (2003), the monetary  transmission  mechanism  is  a  cyclical  phenomenon,  involving  dynamic  interrelationships among deviations of the levels of various observed and unobserved endogenous  variables from the levels of their flexible price equilibrium  components.  Measurement of the stance of monetary policy involves estimation of the level of the natural rate of interest, defined as that short term real interest rate consistent with flexible prices, while inflation targeting involves the generation of forecasts of the levels of particular observed endogenous variables. Within a D S G E framework, a first best approach to the conduct of quantitative monetary policy analysis and inflation targeting entails the joint derivation of empirically adequate cyclical and trend component specifications from microeconomic foundations.  This approach, which  should promote invariance to monetary policy regime shifts for reasons identified by Lucas (1976), is complicated by the existence o f intermittent structural breaks, accounting for which requires flexible trend component specifications, as discussed in Clements and Hendry (1999) and Maddala and K i m (1998).  Within a D S G E framework, a second best approach to the  conduct o f quantitative monetary policy analysis and inflation targeting entails the derivation of empirically adequate cyclical component specifications from microeconomic foundations, augmented with flexible trend component specifications. This approach, proposed by Vitek (2006c, 2006d), is based on the presumption that the determinants of trend components are unknown but persistent, and is extended and refined in Vitek (2006e, 2006f). The primary objective of this paper is to evaluate the finite sample properties of the procedure proposed by Vitek (2006f) for the measurement of the stance of monetary policy in a small open economy under alternative trend component specifications. Towards this end, the accuracy and precision of the Bayesian procedure proposed for the estimation of the levels of the flexible price equilibrium components of various observed and unobserved endogenous variables is analyzed with a Monte Carlo experiment, with an emphasis on the levels of the natural rate of interest and natural exchange rate. The secondary objective of this paper is to describe in a pedagogical manner the application of this procedure to the estimation of a simple but economically interesting D S G E model of a small open economy.  Joint estimation of the  parameters and unobserved components of a linear state space representation of an approximate unobserved components representation of this D S G E model with this Bayesian procedure, conditional on prior information concerning the values of parameters and trend components, is found to yield reasonably accurate and precise results in samples of currently available size. In particular, estimates of the levels of the natural rate of interest and natural exchange rate conditional on alternative information sets are approximately unbiased, while root mean squared errors are relatively small, irrespective of whether the data generating process features common  199 deterministic or stochastic trends. Moreover, analytical root mean squared errors appropriately account for uncertainty surrounding estimates of the levels of the natural rate of interest and natural exchange rate. The organization of this paper is as follows. The next section develops a D S G E model of a small open economy. Alternative approximate unobserved components representations of this D S G E model are described in section three. The design and results of a Monte Carlo experiment for analyzing the accuracy and precision of the procedure proposed for the measurement of the stance of monetary policy in a small open economy are discussed in section four.  Finally,  section five offers conclusions and recommendations for further research.  4.2. Model Development Consider two open economies which are asymmetric in size, but are otherwise identical. The domestic economy is of negligible size relative to the foreign economy.  4.2.1. The Utility Maximization Problem of the Representative Household There exists a continuum of identical households indexed by / e [0,1]. The representative infinitely lived household has preferences defined over consumption C . and labour supply f ( v  s  represented by intertemporal utility function  (1) where subjective discount factor B satisfies 0 < B < 1. The representative household consists of N identical members, and has intratemporal utility function: s  u(C ,L. ) Ks  s  = Nv s  (2)  The intratemporal utility function of the representative household member is multiplicatively separable: i l-l/o-  1-l/cT  exp  l-l/o-  -x I + I/77  (3)  200  In order to ensure the existence of a well defined balanced growth path, the marginal utility o f consumption is homogeneous  in consumption, while the marginal utility of leisure is  homogenous of one higher degree in consumption. The representative household enters period s in possession of a previously purchased diversified portfolio of internationally traded domestic currency denominated bonds 5  which  s  completely spans all relevant uncertainty. It also holds a diversified portfolio of shares {x  tJ  in domestic intermediate good firms which pay dividends {17'. }\ . s  The  =0  }^  s  =0  representative  household supplies final labour service L , earning labour income at nominal wage W . {s  s  These  sources o f wealth are summed in household dynamic budget constraint:  E , il.^B,,,  + | y,^,,,,di  =B  + J (77  u  + V )x . dj  ;v  js  L s  + WL t  - P C. .  (4)  c  u  s  s  According to this dynamic budget constraint, at the end of period 5 , the household purchases a diversified portfolio of state contingent bonds B  t J + 1  representative  , where Q  s s+[  denotes  the price of a bond which pays one unit of the domestic currency in a particular state in the following period, divided by the conditional probability of occurrence of that state. purchases a diversified portfolio of shares {JC,. .  }^.  v+1  at prices  =0  representative household purchases final consumption good C In period  t,  the  representative  household  chooses  ;s  at price  {V } . . ]  jv  is  {{ i,j,s+\) j=tM=t x  X  =l  state contingent is+X  Finally, the  Pf.  consumption { Q J ™ , , labour supply {L }™ , bond holdings {B }™ , t 0  =0  It also  =l  sequences for  and share holdings  maximize intertemporal utility function (1) subject to dynamic budget  constraint (4) and terminal nonnegativity constraints B  : T+j  > 0 and x  t J J + l  > 0 for T - » oo . In  equilibrium, selected necessary first order conditions associated with this utility maximization problem may be stated as u (C„L,)^PfA„  (5)  c  -u (C„L ) L  t  = WX t  (6)  n  fi, i4=M , +  (7)  +1  F /l,=^E,(77., , F., H , yV  where A  js  +  +  +1  +1  (8)  denotes the Lagrange multiplier associated with the period s household dynamic  budget constraint.  In equilibrium, necessary complementary slackness conditions associated  with the terminal nonnegativity constraints may be stated as:  201  lim £ A ^ Q  l  i  B  ff^Ly•  m  0  (9)  ,  =0.  X  (10)  Provided that the intertemporal utility function is bounded and strictly concave, together with all necessary first order conditions, these transversality conditions are sufficient for the unique utility maximizing state contingent'intertemporal household allocation. The absence of arbitrage opportunities requires that short term nominal interest rate ^ - = E, Q .  satisfy  Combination of this equilibrium asset pricing relationship with necessary first  LL+]  order conditions (5) and (7) yields intertemporal optimality condition u (C ,L ) c  l  l  = p^,(\ + i )^u (C ,L ), l  c  l+l  l+i  (11)  'i+i  which ensures that at a utility maximum, the representative household cannot benefit from feasible intertemporal consumption reallocations. Finally, combination of necessary first order conditions (5) and (6) yields intratemporal optimality condition  '~u (c„Ly c  r  P  { U )  which equates the marginal rate of substitution between leisure and consumption to the real wage.  4.2.2. The Value Maximization Problem of the Representative Firm There exists a continuum of intermediate good firms indexed by j e [0,1]. Intermediate good firms supply differentiated intermediate output goods, but are otherwise identical. Entry into and exit from the monopolistically competitive intermediate output good sector is prohibited.  202 4.2.2.1. Employment Behaviour The representative intermediate good firm sells shares {x,. . , }! +1  price Vjj. Recursive forward substitution for V  jJ+s  =0  to domestic households at  with s > 0 in necessary first order condition  (8) applying the law of iterated expectations reveals that the post-dividend stock market value of the representative intermediate good firm equals the expected present discounted value of future dividend payments:  Acting in the interests of its shareholders, the representative intermediate good firm maximizes its pre-dividend stock market value, equal to the expected present discounted value of current and future dividend payments:  The derivation of result (13) imposes transversality condition (10), which rules out self-fulfilling speculative asset price bubbles. Shares entitle households to dividend payments equal to profits 77. , defined as revenues s  derived from sales of differentiated intermediate output good Y. at price Pj s  less expenditures  s  on final labour service L,, nj,s  = Pj, Yj, -W L . s  s  s  (15)  Us  The representative intermediate good firm rents final labour service L  js  given labour  augmenting productivity coefficient A to produce differentiated intermediate output good y. s  5  according to production function Y , , - A ^ where A > 0. s  (16)  In order to ensure the existence of a well defined balanced growth path, this  production function is homogeneous of degree one. In period t, the representative intermediate good firm chooses a state contingent sequence for employment  {L }™ is  =l  to maximize pre-dividend stock market value (14) subject to production  function (16). In equilibrium, demand for the final labour service satisfies necessary first order condition  203  (17) where  P@  denotes the Lagrange multiplier associated with the period s production  Y  s  JiS  technology constraint. This necessary first order condition equates real marginal cost 0 to the t  ratio of the real wage to the marginal product of labour.  4.2.2.2. Output Supply and Price Setting Behaviour There exist a large number of perfectly competitive firms which combine differentiated intermediate output goods Y  jt  supplied by intermediate good firms in a monopolistically  competitive output market to produce final output good Y according to constant elasticity o f t  substitution production function  (18)  where 6>>1. The representative final output good firm maximizes profits derived from production of the final output good  n]=pjY-  \plY dj, j<t  (19)  with respect to inputs of intermediate output goods, subject to production function (18). The necessary first order conditions associated with this profit maximization problem yield intermediate output good demand functions: (P \ Y  P>  Y.  (20)  V ' J  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final output good firm earns zero profit, implying aggregate output price index:  Pj =  \(Pl,Vdj  (21)  204 As the price elasticity of demand for intermediate output goods 9 increases, they become closer substitutes, and individual intermediate good firms have less market power. In an adaptation of the model of nominal output price rigidity proposed by Calvo (1983), each period a randomly selected fraction \-oo  of intermediate good firms adjust their price  optimally. The remaining fraction co o f intermediate good firms adjust their price to account for past steady state output price inflation according to indexation rule:  p.\  p' i f  -  (22)  v  Under this specification, optimal price adjustment opportunities arrive randomly, and the interval between optimal price adjustments is a random variable. If the representative intermediate good firm can adjust its price optimally in period / , then it does so to maximize to maximize pre-dividend stock market value (14) subject to production function (16), intermediate output good demand function (20), and the assumed form of nominal price rigidity.  Since all intermediate good firms that adjust their price optimally in period t  solve an identical value maximization problem, in equilibrium they all choose a common price Pf  given by necessary first order condition: pY  f  CO  e  0-1  p Y \  pY  ( p Y  s  t-\  r  E,5>  pYy s  pY  r  (23)  p Y \  PY Y  s  s=t  s  s  This necessary first order condition equates the expected present discounted value of the revenue benefit generated by an additional unit of output supply to the expected present discounted value of its production cost. Aggregate output price index (21) equals an average of the price set by the fraction 1 - co of intermediate good firms that adjust their price optimally in period t, and the average of the prices set by the remaining fraction co of intermediate good firms that adjust their price according to indexation rule (22):  (  pY  t-\  r  P = (i~a>)(pry- +co Y  d  pY  \ t-2 r  \  pY  \-0  (24)  t-\  r  J  Since those intermediate good firms able to adjust their price optimally in period t are selected randomly from among all intermediate good firms, the average price set by the remaining  205 intermediate good firms equals the value of the aggregate output price index that prevailed during period t -1, rescaled to account for past steady state output price inflation. If all intermediate good firms were able to adjust their price optimally every period, then co = 0 and necessary first order condition (23) would reduce to \  ~h '^  Pr=  p  (25)  where Pf* = PJ. In flexible price equilibrium, each intermediate good firm sets its price equal to a constant markup over nominal marginal cost, and output supply is inefficiently low.  4.2.3. International Trade and Financial Linkages In an open economy, exchange rate adjustment contributes to both intratemporal and intertemporal equilibration, while business cycles are generated by interactions among a variety of nominal and real shocks originating both domestically and abroad.  4.2.3.1. International Trade Linkages The law of one price asserts that arbitrage transactions equalize the domestic currency prices of domestic imports and foreign exports.  Let £  s  denote the nominal exchange rate, which  measures the price of foreign currency in terms of domestic currency, and define the real exchange rate, £ P Qs=^y-,  YJ  (26)  which measures the price of foreign output in terms of domestic output. Under the law of one price, the real exchange rate coincides with the terms of trade, which measures the price of imports in terms of exports. There exist a large number of perfectly competitive firms which combine a domestic intermediate consumption good Q , and a foreign intermediate consumption good C  fl  to  produce final consumption good C, according to constant elasticity of substitution production function  206 _!_  if/-\  \_  C. =  (27)  where 0 < <> / < 1 and ^ > 1. The representative final consumption good firm maximizes profits derived from production of the final consumption good nf  =PfC,~PjC -E PjfC , Kl  t  (28)  f!  with respect to inputs of domestic and foreign intermediate consumption goods, subject to production function (27).  The necessary first order conditions associated with this profit  maximization problem imply intermediate consumption good demand functions:  (29)  pC  V~  J  (30)  Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final consumption good firm earns zero profit, implying aggregate consumption price index: P  c t  =[<P(Pft* + ( 1 - 0 ( 5 , / * / ) H K ] ^  (31)  Combination of this aggregate consumption price index with intermediate consumption good demand functions (29) and (30) yields:  hj  c  (32)  =  C / , , = ( i - ^ ) [ ( i - ^ ^ r ' f  c,.  (33)  These demand functions for domestic and foreign intermediate consumption goods are directly proportional to final consumption good demand, with a proportionality coefficient that varies with the real exchange rate.  207 4.2.3.2. International Financial Linkages Under the assumption of complete international financial markets, utility maximization by domestic and foreign households implies intertemporal optimality conditions /3u (C „L )  Pf  u (C„ )  Pff  c  Q  '  M  ~  t+  c  M  Ll  _/3u (Cf ,L{ ) c  +l  ( 3 4 )  Pf  £,  f  +i  u cf,L{)  pffe f  ci  ( 3 5 )  l+  respectively. Combination of these intertemporal optimality conditions with real exchange rate definition (26) yields international risk sharing condition:  u (C„L,) c  P,'  Under the assumption that the domestic economy is of negligible size relative to the foreign economy, this international risk sharing condition induces stationarity of consumption per unit of effective labour, and of the real net foreign asset position per unit of effective labour, which equals zero in deterministic steady state equilibrium.  4.2.4. Monetary Policy The government consists of a monetary authority which implements monetary policy through control of the short term nominal interest rate according to monetary policy rule i,-i, = ^nf-nf)  + a^Y,-lnY,)  + v,  (37)  l  where £, > 1 and ^ > 0. A s specified, the deviation of the nominal interest rate from its flexible price equilibrium value is a linear increasing function of the contemporaneous deviation of consumption price inflation from its target value  nf - nf,  and the  contemporaneous  proportional deviation of output from its flexible price equilibrium value. Persistent departures from this monetary policy rule are captured by serially correlated monetary policy shock v,.  208 4.2.5. Market Clearing Conditions A rational expectations equilibrium in this D S G E model of a small open economy consists of state contingent intertemporal allocations for domestic and foreign households and firms which solve their constrained optimization problems given prices and policy, together with state contingent intertemporal allocations for domestic and foreign governments which satisfy their policy rules, with supporting prices such that all markets clear. Clearing of the final output good market requires that production of the final output good equal the cumulative demands of domestic and foreign households: Y,=C  +C .  (38)  f  hJ  fJ  The assumption that the domestic economy is of negligible size relative to the foreign economy is represented by parameter restriction <f> = 1, under which Pj f  J  =P  CJ I  in equilibrium.  4.3. The Approximate Linear Model Estimation and inference are based on a linear state space representation of an approximate unobserved components representation of this D S G E model of a small open economy. A first best approximation is considered in which cyclical and trend component specifications are jointly derived from microeconomic foundations.  Under this approach, along the balanced  growth path towards which the economy converges in the absence of shocks, great ratios and trend growth rates are time and state independent.  A second best approximation is also  considered in which cyclical component specifications are derived from microeconomic foundations, and are combined with more flexible trend component specifications. Under this approach, along the balanced growth path towards which the economy converges in the absence of shocks, great ratios and trend growth rates are time independent but state dependent. In what follows, E, x  l+s  denotes the rational expectation of variable x  information available at time t.  ( + !  , conditional on  Also, x, denotes the cyclical component o f variable x , t  x  t  denotes the flexible price equilibrium component of variable x,, and x, denotes the trend component of variable x,.  Cyclical and trend components are additively separable, which  implies that x, - x, + x, and x, = x, + x,, where x, = x,.  209 4.3.1. First Best Approximation Cyclical components are modeled by applying stationarity inducing transformations consistent with the existence of a well defined balanced growth path along which all variables are constant or grow at constant rates to equilibrium conditions, then linearizing them around the resultant stationary deterministic steady state equilibrium, while trend components are modeled by imposing the cointegrating relationships implied by this balanced growth path.  4.3.1.1. Cyclical Components The cyclical component of output price inflation depends on the expected future cyclical component of output price inflation and the contemporaneous cyclical component of real marginal cost according to output price Phillips curve f  - + -  co  where /?' = /?(l + «)(l + g )  M / C T  .  A,N, </>  yf ^ l n ^ + O/zQ + ^ - ^ l n Q ,(39) AN.  Reflecting the existence of international trade linkages, the  cyclical component of real marginal cost depends not only on the contemporaneous cyclical component of domestic output per unit of effective labour, but also on the contemporaneous cyclical components of foreign output per unit of effective labour and the real exchange rate. The cyclical component of consumption price inflation depends on the expected future cyclical component of consumption price inflation and the contemporaneous cyclical component of real marginal cost according to consumption price Phillips curve:  c  , (l-a>)(l-«/?') co  r  \ 1^ — + — ln0 n A,N,  1-0 (j>  Y l n ^ ^ + O/zO + ^ - ^ l n g ; f  (40)  +(1 - (/>) In - P - - p\\ - 0)E, In %  Reflecting the entry of the price of imports into the aggregate consumption price index, the cyclical component of consumption price inflation also depends on contemporaneous and expected future proportional changes in the cyclical component of the real exchange rate. The cyclical component of output per unit of effective labour depends on the expected future cyclical component of output per unit of effective labour and the contemporaneous cyclical component of the real interest rate according to approximate linear consumption Euler equation:  210  Y Y l n ^ r - = E, l n ^ 4 ,N, A, N,  or,  0-1 \ + (/>(o-\) 9  A  +X  i-<  E, A ln  0-1 \ + (/>(o~\) 0  \  (41) + y/(\ + 0)E, ln ^  ,+ A  -E,ln-P-  +  Reflecting the existence o f international trade linkages, the cyclical component of output per unit of effective labour also depends on expected future proportional changes in the cyclical components of foreign output per unit of effective labour and the real exchange rate. The  adjusted  contemporaneous  cyclical  component  of  the  nominal  interest  rate  depends  on  the  adjusted cyclical components of consumption price inflation and output  according to monetary policy rule: i,-i,=  t;{fcf -tf)  + a m Y, - ln ?,) + v,.  (42)  This monetary policy rule ensures convergence of the level of consumption price inflation to its target value in flexible price equilibrium. The cyclical component of the real exchange rate depends on the contemporaneous cyclical components of the domestic and foreign marginal utilities of consumption according to approximate linear international risk sharing condition:  I+ ^CT-I)  0-1  ^ CT + K 1 + M 1 - < * ) ) 2  ln A,N,  A!  Nf  + ln^r + (l-^)ln-^Af  .(43)  N  f  The cyclical component of the real interest rate satisfies r, = i - E 7rf , while the cyclical t  component of the real exchange rate satisfies InQ = l n £ + \nPf  J  (  (  +x  - XnPf.  Variation in cyclical components is driven by three exogenous stochastic processes. cyclical components  The  of the productivity, population, and monetary policy shocks follow  stationary first order autoregressive processes: In A, = p. ln 4_, + ef, ef ~ iid /V(0, o\),  (44)  lnN =p.]nN,_ ef,  (45)  A  t  l+  v,=p v,-i v  +<> < - iid  ef ~M  M(0,o]\  M(Q,o ). 2  v  (46)  211 The innovations driving these exogenous stochastic processes are assumed to be independent, which combined with our distributional assumptions implies multivariate normality. In flexible price equilibrium, co = 0 and <j = 0. 2  4.3.1.2. Trend Components The trend components of the prices of output In Pf and consumption In P  c t  are driven by  common deterministic or stochastic trend In P , while the trend component of output In Y is t  t  driven by common deterministic or stochastic trends In A, and In N : t  In Pf = 7i + \nP f+ef,  (47)  l n T f =/r + ln/>5 +<?f,  (48)  \nY,=g + n + \nY,_ +ef+ef.  (49)  t  1  x  It follows that along a balanced growth path, the level of the relative price of consumption is time and state independent. The trend components of the nominal interest rate i and nominal exchange rate ln£, are t  time and state independent: W - P  (50)  l n ^ = ln^_,.  (51)  The trend component of the real interest rate satisfies T = i - E, nf , while the trend component t  of the real exchange rate satisfies ln(j> = ln£, + \nP '  Y f  l  t  +x  - InPf.  Long run balanced growth is driven by three common deterministic or stochastic trends. The trend components of the price level In P , productivity In A , and population In TV, follow t  T  random walks with constant drifts: XnP = n + ln/j_, + ef, ef ~ iid N(0,<T\), t  In A, - g + In 4_, + ef,  ef  ~ iid M(Q,CJ\),  In N, = n + In /Y,_, + ef, ef ~ iid AA(0, a\).  (52) (53)  (54)  212 If <jp = a\ - <J\ = 0 then these common trends are deterministic, and are otherwise stochastic. As an identifying restriction, all innovations are assumed to be independent, which combined with our distributional assumptions implies multivariate normality.  4.3.2. Second Best Approximation Cyclical components are modeled by linearizing equilibrium conditions around a stationary deterministic steady state equilibrium which abstracts from long run balanced growth, while trend components are modeled as random walks while ensuring the existence of a well defined balanced growth path.  4.3.2.1. Cyclical Components The cyclical component of output price inflation depends on the expected future cyclical component of output price inflation and the contemporaneous  cyclical component of real  marginal cost according to output price Phillips curve:  CO  — + — ln-  1-0 l n ^ ^ + a/(l + 0 - 0 1 n Q  A,N, </>  A,N,  .(55)  Reflecting the existence of international trade linkages, the cyclical component of real marginal cost depends not only on the contemporaneous cyclical component of domestic output per unit of effective labour, but also on the contemporaneous cyclical components of foreign output per unit of effective labour and the real exchange rate. The cyclical component of consumption price inflation depends on the expected future cyclical component of consumption price inflation and the contemporaneous cyclical component of real marginal cost according to consumption price Phillips curve:  CO  — + — ln-  A,N,  \4> vj  +(l-^)ln-S—/?(1-^)E, In%-  <f>  l n ^ r - + GKl + 0-0)ln£> (56)  213 Reflecting the entry of the price of imports into the aggregate consumption price index, the cyclical component of consumption price inflation also depends on contemporaneous and expected future proportional changes in the cyclical component of the real exchange rate. The cyclical component of output per unit of effective labour depends on the expected future cyclical component of output per unit of effective labour and the contemporaneous cyclical component of the real interest rate according to approximate linear consumption Euler equation: Y Y l n ^ r V = E, In ^ 4 A,N,  err - E , In-  i-< l + ^(o--l)  E A In .  0-1  (  h  Y  4  e  (57)  (\ + 0)E, In % •  +¥  9  Reflecting the existence of international trade linkages, the cyclical component of output per unit of effective labour also depends on expected future proportional changes in the cyclical components of foreign output per unit of effective labour and the real exchange rate. The  adjusted  cyclical  component  of the  nominal interest  rate  depends  on  the  contemporaneous adjusted cyclical components o f consumption price inflation and output according to monetary policy rule: 1,-1= £ 0 r - kf) + CQn Y — \nY ) + v c  t  l  (58)  r  This monetary policy rule ensures convergence of the level of consumption price inflation to its target value in flexible price equilibrium. The cyclical component of the real exchange rate depends on the contemporaneous cyclical components of the domestic and foreign marginal utilities of consumption according to approximate linear international risk sharing condition: 0-1  1 + ^(cr - 1 )  Y  Y  f  +  A,N,  l  A!N{< J  n  The cyclical component of the real interest rate satisfies r = i - E, nf, r  component of the real exchange rate satisfies InQ = ln£, + \aPj  J  t  A  A!  +  (  1  _ ^  )  l  n  A  (59)  N  F  while the cyclical  - \a.Pj.  Variation in cyclical components is driven by three exogenous stochastic processes.  The  cyclical components of the productivity, population, and monetary policy shocks follow stationary first order autoregressive processes:  214 In A = P In  + ^ , sf ~ iid /V(0,CT?),  A  lnA> = / > . t o A r _ < , 1 +  K  e  (60)  *~iid/v-(0,t7?),  (61)  + < , < ~ i i d /V(0,a ).  (62)  2  = PvK-i  v  The innovations driving these exogenous stochastic processes are assumed to be independent, which combined with our distributional assumptions implies multivariate normality. In flexible price equilibrium, co = 0 and cr* = 0.  4.3.2.2. Trend Components The trend components of the prices of output \n.Pf and consumption In Pf follow random walks with time varying drift n,, while the trend component of output ln Y follows a random t  walk with time varying drift g, +n : t  ln  = n + ln Pf + sf,  sf  ~ iid /V(0, a -, \  (63)  ln 7 f = x, + ln i f + sf,  sf  ~ iid Af(0, a ),  (64)  ~ iid /V(0, cr^).  (65)  Pf  ln Y, =  t  + n, + ln  g l  + sf,  1  2  pc  It follows that the trend component of the relative price of consumption follows a random walk without drift. This implies that along a balanced growth path, the level of this relative price is time independent but state dependent. The trend components of the nominal interest rate i, and nominal exchange rate Inc?, follow random walks without drifts: J=U+ef,  sf ~ iid JV(0, al),  IncT = l n ^ _ , + ef, sf ~ iid /V(0,<r|).  (66) (67)  It follows that along a balanced growth path, the levels of the nominal interest rate and nominal exchange rate are time independent but state dependent. The trend component of the real interest rate satisfies 7 = i - E nf, t  InQ = l n ^ + l n ^  t  r /  while the trend component of the real exchange rate satisfies  r  -ln^ . y  215 Long run balanced growth is driven by three common stochastic trends. Trend inflation  n, t  productivity growth g , , and population growth n follow random walks without drifts: t  n = t  + < , < ~ iid Af(0, a]),  (68)  g, =&_,+*,*, ef~x\AM^a\\  (69)  n, = «,_,+ s:, < ~ iid /V(0, o-„ ).  (70)  2  It follows that along a balanced growth path, growth rates are time independent but state dependent. As an identifying restriction, all innovations are assumed to be independent, which combined with our distributional assumptions implies multivariate normality.  4.4. Estimation and Inference The finite sample properties of the procedure proposed by Vitek (2006f) for the measurement of the stance of monetary policy in a small open economy are analyzed within the framework of these alternative approximate unobserved components representations of this D S G E model with a Monte Carlo experiment. Each replication of this Monte Carlo experiment consists of two steps. In the first step, a linear state space representation of the first best approximation to this D S G E model of a small open economy is simulated conditional on calibrated parameter values and initial conditions. In generating artificial data sets, both deterministic and stochastic trend component specifications are employed. In the second step, the parameters and unobserved components of a linear state space representation of the second best approximation to this D S G E model are jointly estimated with a Bayesian procedure, conditional on prior information concerning the values of parameters and trend components.  Averaging the differences and  squared differences between estimated and simulated levels of the flexible price equilibrium components of various observed and unobserved endogenous variables across replications of this Monte Carlo experiment, with an emphasis on the levels of the natural rate of interest and natural exchange rate, facilitates measurement of accuracy and precision in terms of bias and root mean squared error. The linear state space representation of the second best approximation to this D S G E model of a small open economy approximately nests the linear state space representation of the first best approximation, irrespective of whether common trends are deterministic or stochastic.  To  elaborate, the cyclical component specifications differ only with respect to the discount factor entering into the coefficients of the Phillips curves, while the trend component specifications are  216 fully nested under restrictions on variance parameters i f common trends are deterministic, and are approximately nested under restrictions on variance parameters i f common trends are stochastic.  Furthermore, under the first best approximation to this D S G E model, the exogenous  1  stochastic processes associated with the stochastic trend component specification nest commonly employed types of exogenous stochastic processes under parameter restrictions. It follows that 2  the estimated model associated with this Monte Carlo experiment is approximately correctly specified for the data generating process, in the sense that it approximately nests the data generating process, while the exogenous stochastic processes associated with the data generating process nest commonly employed types of exogenous stochastic processes.  4.4.1. Estimation  Let x denote a vector stochastic process consisting of the levels of /V nonpredetermined t  endogenous variables, of which M  are observed.  The cyclical components of this vector  stochastic process satisfy second order stochastic linear difference equation A x, = A ^+A E,x Q  lX  2  +A x, + A v„  l+l  3  where vector stochastic process  (71)  4  consists of the flexible price equilibrium components of N  nonpredetermined endogenous variables.  The cyclical components of this vector stochastic  process satisfy second order stochastic linear difference equation B x =B x _:+B E x +B v„ 0  l  l  l  2  l  l+l  (72)  3  where vector stochastic process v, consists o f the cyclical components of K variables.  exogenous  This vector stochastic process satisfies stationary first order stochastic linear  difference equation v,=C v,_ +e , x  '  Under  O p . , = al,  the  first  = al,  t£=o\= l=  = a,  (73)  u  best  approximation  = a,  2  ol  a  2  x  2  =o- =0, 2  =a, 2  o i  with  common  deterministic  trends  al,  = al  = 0 . Under the first best approximation with common stochastic trends al,  ,=<£,, cr -, 1  =  a\, + a -, 2  and  al, = a , = a . =a , 1  2  2  exogenous stochastic process has structural form representation ln A, = ln A, + ln A, , where ln A, = IA  with sf  + ef  M(0,al).  Under the assumption that sf  A l n A, = (1 - p- )g + p A l n A,_ + ef + 9 ef_, with sf — i i d A/"(0, a ), 2  A  t  t  A  A  where 0  and ef A  a n d a\  = al  = a„ =0 2  and  = al , a - =O 1- JKJ 1- , 1  f  =0 .  Consider the exogenous stochastic process governing the evolution o f the level o f productivity.  ln A, = g + ln A  = a~ = al = al = a\ = a  2  c  U n d e r the first best approximation, this ln A,_, +  with  ~ i i d JV(0, a?), and  are independent, it has reduced form representation are functions o f p- , <7? a n d A  al.  217 where e,, ~ iid A/"(0,27,). The parameter matrices differ depending on whether the first best or second best approximation is employed. The trend components of vector stochastic process JC, satisfy first order stochastic linear difference equation D x, 0  where e  = Z>, + Z) u, + 2  Z) x 3  + e ,,  (74)  2  — iid A/"(0,27 ). Vector stochastic process u consists of the levels of L common  2l  2  t  stochastic trends, and satisfies nonstationary first order stochastic linear difference equation ",=«,_,  +EXI,  (75)  where e , ~ iid A/"(0,27 ). The parameter matrices differ depending on whether the first best or 3  3  second best approximation is employed, in addition to whether common trends are deterministic or stochastic.  Cyclical and trend components are additively separable, which implies that  3  JC, = JC + JC, and JC, = JC, + JC, , where (  x,=x,.  If there exists a unique stationary solution to multivariate linear rational expectations model (71), then it may be expressed as: x,=S x,_ +S x _ +S v . i  l  2  t  }  3  (76)  l  If there exists a unique stationary solution to multivariate linear rational expectations model (72), then it may be expressed as: (77) These solutions are calculated simultaneously with the matrix decomposition based algorithm due to Klein (2000). Let  y  t  denote a vector stochastic process consisting of the levels of M  nonpredetermined endogenous variables. consisting of the levels of N-M  observed  Also, let z, denote a vector stochastic process  unobserved nonpredetermined endogenous variables, the  cyclical components of 7Y nonpredetermined endogenous variables, the cyclical components of the flexible price equilibrium components of /V nonpredetermined endogenous variables, the trend components of N nonpredetermined endogenous variables, the cyclical components of K exogenous variables, and the levels of L common stochastic trends. Given unique stationary solutions (76) and (77), these vector stochastic processes have linear state space representation  3  Under the first best approximation with c o m m o n deterministic trends D = 0 , Z = 0 and I, = 0 . Under the first best approximation with 2  c o m m o n stochastic trends D = 0 and X , = 0 . 2  2  218  y,=F z , x  (78)  t  z, = G + G z _ + C e „ 1  2  /  I  3  (79)  4 i  where «s , ~ iid /V(0,27 ) and z ~ N(z ,P ). 4  4  0  0l0  Let w, denote a vector stochastic process  0]0  consisting of preliminary estimates of the trend components of M observed nonpredetermined endogenous variables. Suppose that this vector stochastic process satisfies w>, = / / , z , + e „  (80)  5  where e  5j  ~ iid A/"(0,27 ). Conditional on known parameter values, this signal equation defines 5  a set of stochastic restrictions on selected unobserved state variables.  The signal and state  innovation vectors are assumed to be independent, while the initial state vector is assumed to be independent from the signal and state innovation vectors, which combined with our distributional assumptions implies multivariate normality. Conditional on the parameters associated with these signal and state equations, estimates of unobserved state vector z, and its mean squared error matrix P may be calculated with the filter t  proposed by Vitek (2006a, 2006b), which adapts the filter due to Kalman (1960) to incorporate prior information. Given initial conditions z available at time t-l  0|0  and P , estimates conditional on information Q{0  satisfy prediction equations: (81)  ~ \ + 2Vl|,-l> G  p 1  t\t-\  =  G  2 ,-\\t-\ l  G  P  +  G  G  3 ^ 4  G  (82)  l i  •V,|.-i = FT  (83)  P  = F  F  T  (84) (85)  ^1,-1  = H P ^Hj x  t  +27 .  (86)  5  Given these predictions, under the assumption of multivariate normally distributed signal and state innovation vectors, together with conditionally contemporaneously uncorrelated signal vectors, estimates conditional on information available at time t satisfy updating equations  =Vi  +  K  y,  U ~ ->Vi)  + K  », ( > ~ « k - i ) ' w  w  ( > 87  219 ,\t  =  P  where K  P  ,\,-\~  y  K  = P^Ff  y<  F l  P l  ,\,-\-  K  w,  H  Qf_ and K  \  P  t\,-\'  ( ) 88  = P _ H R^_ .  Given terminal conditions z  r  x  lXl  x  x  x  w  and P  T[r  obtained from the final evaluation of these prediction and updating equations, estimates conditional on information available at time T satisfy smoothing equations I\T  Z  <\t  =  z  (Zi+\\r ~ i+\\t)'  +  ,T= „+J,( ,  P  P  (89)  z  ,T- „)J,\  P  (90)  P  +  I+  where / , = P^Gj P ~ \ . Under our distributional assumptions, these estimators of the unobserved t  +  tl  state vector are mean squared error optimal. Let ^ € 0 c R  denote a J dimensional vector containing the parameters associated with  J  the signal and state equations of this linear state space model. The Bayesian estimator of this parameter vector has posterior density function f(O\l )Kf(T \0)f(6), T  (91)  T  where 2~ = {{y }' ,{ }' =\} • Under the assumption of multivariate normally distributed signal w  s  s={  s  s  and state innovation vectors, together with conditionally contemporaneously uncorrected signal vectors, conditional density function / ( 1 19) satisfies: T  f(i \o)-Ylf{y,\i^e).f[f( T  | j,_„0),  Wl  Under our distributional assumptions, f(w, \1,_ ,G) satisfy:  (92)  conditional density  functions  f(y,\l,_ ,0) x  and  X  M  f(y,\X _ ,e) t  x  —  / ( , | Z , _ „ 0 ) = (2;r) W  2  r  1  = {2K) ' | g ,  -  | ^  H  j  e x p | - ^ U - y^f  M  f  1  | ^  e x p  1  ^-( ,w  W l l  QfM, - J>„_,)j,  ,JRf ( x  W  ,~  W l l  ,_ )[. x  1  (93)  (94)  Prior information concerning parameter vector 0 is summarized by a multivariate normal prior distribution having mean vector 0 and covariance matrix Q: X  f{0) = (2ar)~* | Q f e x p { - | ( 0 - 0 ) Q ~ \ 0 - 0 , ) j . J  X  (95)  220 Independent priors are represented by a diagonal covariance matrix, under which diffuse priors are represented by infinite variances. Inference on the parameters is based on an asymptotic normal approximation to the posterior distribution around its mode. mode 0  Under regularity conditions stated in Geweke (2005), posterior  satisfies  T  (96) where 0 e0  denotes the pseudotrue parameter vector. Following Engle and Watson (1981),  o  Hessian ^  may be estimated by  *r =7ZE  M  [ v V > / ( ^ J ^ , t f ) ] +l j ; E f  I  r  1  t=i  1  M  [v V;in/( f  W |  i=i  |  I,_J ) T  (97)  +^v>M), where ^ V \nf(0 ) T  e  g  T  =  -Q- . I  4.4.2. Inference  The design of this Monte Carlo experiment is realist in the sense that the true parameter values are all well within the range of estimates reported in the existing literature, after accounting for data rescaling, while the sample size is consistent with the span and frequency of real data sets typically employed in the estimation of D S G E models.  The true values of  parameters are reported in Table 4.1. Under both deterministic and stochastic trend component specifications, artificial data sets consist of 200 simulated observations on the levels of eight observed endogenous variables, namely domestic and foreign price levels, outputs, nominal interest rates, and the nominal exchange rate. quarterly observations.  This sample size corresponds to 50 years of  221 Table 4.1. True values o f parameters Parameter  Value  P n  0.9900  (O  0.8000  P,  <t> V  0.7000  a  1.5000  a  1.0000  °*  0  7.6667  1.0000  Parameter  Value  P\  0.5000  Pi,  0.5000  2  A  a  2  v  A  1  Parameter  Value  g  0.2500  0.5000  n  0.5000  0.2500  °?  0.5000  0.2500 0.2500  4  0.2500  $  1.5000  0.2500  i  0.1250  0.2500  0.2500 0.1250 0.1250 0.2500 0.1250  4  0.1250  Note: T h e data generating process is calibrated at a quarterly frequency under the assumption that all observed endogenous variables are rescaled by a factor o f 100.  The set of parameters associated with this D S G E model of a small open economy is partitioned into two subsets. The first subset is calibrated to equal true values, while the second subset is estimated with the Bayesian procedure described above, conditional on prior information concerning the values of parameters and trend components.  Those parameters  associated with the conditional mean function are estimated conditional on cross-economy equality restrictions and informative independent priors, while those parameters associated exclusively with the conditional variance function are estimated conditional on diffuse priors. The means of informative marginal prior distributions equal true values, penalizing deviations from them. Initial conditions for the cyclical components of exogenous variables are given by their unconditional means and variances, while the initial values of all other state variables are treated as parameters, and are calibrated to equal true values. The posterior mode is calculated by numerically maximizing the logarithm of the posterior density kernel with a modified steepest ascent algorithm. Prior information concerning the values of trend components is generated by fitting first order deterministic polynomial functions to the levels of all observed endogenous variables by ordinary least squares.  Stochastic restrictions on the trend components o f all observed  endogenous variables are derived from the fitted values associated with these ordinary least squares regressions, with innovation variances set proportional to estimated prediction variances assuming known parameters.  A l l stochastic restrictions are independent, represented by a  diagonal covariance matrix, and are harmonized, represented  by a common factor of  proportionality. Reflecting little confidence in these preliminary trend component estimates, this common factor of proportionality is set equal to one. This Monte Carlo experiment indicates that joint estimation of parameters and unobserved components with the Bayesian procedure under consideration yields reasonably accurate and precise results. Parameter estimation results under deterministic and stochastic trend component  222 specifications are reported in Table 4.2 and Table 4.3, respectively. Examination of these results reveals that, under both deterministic and stochastic trend component specifications, the modes of the marginal posterior distributions of the parameters exhibit statistically insignificant differences from true values at conventional levels, while posterior standard errors are relatively small.  However, posterior standard errors based on asymptotic distribution theory tend to  overstate uncertainty surrounding estimates of the parameters, implying that inference on them based on an asymptotic normal approximation to the posterior distribution around its mode tends to be conservative. That estimates of those parameters associated with the conditional mean function are approximately unbiased is in part attributable to the design of this Monte Carlo experiment, under which the means of informative marginal prior distributions equal true values, penalizing deviations from them. Nevertheless, the data remain informative with respect to these parameters, as prior standard errors are larger than posterior standard errors.  Examination of  these results also reveals that the modes of the marginal posterior distributions of parameters tend to exhibit smaller deviations from true values under a deterministic trend component specification than under a stochastic trend component specification, while posterior standard errors are generally smaller. These results are to be expected, as prior information concerning the values of trend components represents the belief that common trends are deterministic as opposed to stochastic.  223  Table 4.2. Experimental results under deterministic trend specification, parameters rameter  True Value  Prior Distribution  Posterior Distribution  Mean  SE  Mode  SE  ASE  0.100000  1.001700  0.002682  0.098380  i  1.000000  1.000000  Q)  0.800000  0.800000  0.080000  0.817700  0.019118  0.010779  V  1.500000  1.500000  0.150000  1.529300  0.036579  0.053809  a  1.000000  1.000000  0.100000  1.055300  0.056936  0.056935  $  1.500000  1.500000  0.150000  1.486400  0.022647  0.089290  Q  0.125000  0.125000  0.012500  0.125280  0.000466  0.012484  Pi  0.500000  0.500000  0.050000  0.515920  0.016593  0.026879  P«  0.500000  0.500000  0.050000  0.493610  0.007825  0.036776  Py  0.500000  0.500000  0.050000  0.501490  0.006337  0.018997  a  0.250000  oo  0.247010  0.003462  0.042837  < a]  0.250000  -  oo  0.248150  0.002833  0.032829  oo  0.249110  0.001892  0.046709  oo  0.245890  0.004309  0.050068  oo  0.247250  0.003572  0.032892  oo  0.249250  0.002178  0.037266  00 OO OO OO 00  0.009145  0.009146  0.001130  0.009100  0.009101  0.001106  0.009897  0.009897  0.012641  0.009766  0.009766  0.005081  0.009765  0.009766  0.003443  oo  0.009404  0.009404  0.001445  00 OO OO OO OO CO OO  0.009900  0.009900  0.012376  0.009650  0.009650  0.002895  5.00X10  oo  5.00x10"'  2  A  0.250000 0.250000 0.250000 0.250000  4  0.000000 0.000000 0.000000  a2  0.000000  4  0.000000 0.000000 0.000000  of/  0.000000  al a\ al a], a, s al.  0.000000  2  0.000000 0.000000 0.000000 0.000000 0.000000  -. -  l.OOxlO"  6  1.00x10"*  3.36xl0"  6  5.00xl0"  7  5.00x10"'  7.27x|0"  6  5.00x10"'  7.27x10"  6  l.OOxlO"  6  2.87xl0"  6  5.00x10"'  6.66x10"  6  5.00x10"'  6.66x10"  6  5.00* 1(T l.OOxlO"  7  6  -7  Note: T h e ensemble modes, standard errors, and asymptotic standard errors o f the marginal posterior distributions o f parameters are calculated by averaging posterior modes, squared deviations o f posterior modes from true values, and asymptotic standard errors across 100 replications, respectively. The parameters are estimated subject to identifying restrictions a  2 g  = al  and cr*, = a , . 2  224 Table 4.3. Experimental results under stochastic trend specification, parameters Parameter  True Value  Prior Distribution  Posterior Distribution  Mean  SE  Mode  SE  ASE  0.100000  1.000700  0.005004  0.099044  1  1.000000  1.000000  0)  0.800000  0.800000  0.080000  0.787060  0.019327  0.019622  ¥  1.500000  1.500000  0.150000  1.573000  0.082897  0.077776  a  1.000000  1.000000  0.100000  1.047700  0.053657  0.069255  1.500000  1.500000  0.150000  1.471100  0.046440  0.105880  0.125000  0.125000  0.012500  0.124640  0.000718  0.012489  Pi P/,  0.500000  0.500000  0.050000  0.527120  0.029400  0.034489  0.500000  0.500000  0.050000  0.533860  0.037286  0.040902  A.  0.500000  0.500000  0.050000  0.489050  0.015958  0.029729  °\  0.250000  —  CO  0.252050  0.004991  0.054323  0.250000  -  00  0.257040  0.009317  0.050935  0.250000  —  00  0.252440  0.004238  0.071539  0.250000  —  CO  0.251070  0.003634  0.071831  0.250000  —  00  0.253530  0.005909  0.051491  0.250000  —  CO  0.251380  0.004521  0.054079  4 4 4  0.250000  —  CO  0.251700  0.006655  0.026205  0.250000  -  CO  0.252430  0.008012  0.026170  0.250000  —  CO  0.243260  0.008393  0.055033  a  2  0.000000  —  00  0.009734  0.009734  0.006039  4  0.000000  —  00  0.010058  0.010061  0.005022  0.250000  —  00  0.250430  0.006991  0.026737  0.250000  —  00  0.244940  0.007746  0.047782  0.000000  -  00  0.009455  0.009457  0.003662  0.000000  —  CO  l.OOxlO"  6  1.00x10-*  2.04x10"  0.000000  —  CO  5.00xl0"  7  5.00x10"'  2.05*10"  5  al  0.000000  —  CO  5.00x10"'  5.00x10"'  2.05xl0"  5  °\  0.000000  -  00  l.OOxlO'  1.00x10"*  4.03xl0"  5  00  5.00x10"'  5.00x10"' •  1.72xl0"  5  00  5.00x10-'  5.00x10"'  1.72xl0"  5  i  4  al  4 4  4 4  a, 2  4 4  0.000000 0.000000  -  6  5  Note: T h e ensemble modes, standard errors, and asymptotic standard errors o f the marginal posterior distributions o f parameters are calculated b y averaging posterior modes, squared deviations o f posterior modes from true values, and asymptotic standard errors across 100 replications, respectively. The parameters are estimated subject to identifying restrictions a\ = al  and a , 2  = a, 2  .  Theoretically prominent indicators of inflationary pressure such as the natural rate of interest and natural exchange rate are unobservable. As discussed in Woodford (2003), the level of the natural rate of interest provides a measure of the neutral stance of monetary policy, with deviations of the real interest rate from the natural rate of interest generating inflationary pressure.  It follows that the key to achieving low and stable inflation is the conduct of a  monetary policy under which the short term nominal interest rate tracks variation in the level of the natural rate of interest as closely as possible. This Monte Carlo experiment indicates that joint estimation of parameters and unobserved components with the Bayesian estimation procedure under consideration yields reasonably accurate and precise estimates of the level, cyclical component and trend component of the natural rate of interest conditional on alternative information sets, irrespective of whether the  225 data generating process features common deterministic or stochastic trends.  The results of  estimating the level, cyclical component and trend component of the natural rate of interest under deterministic and stochastic trend component specifications are reported in Table 4.4 and Table 4.5, respectively. This concept of the natural rate of interest represents that real interest rate consistent with past, present and future price flexibility. The predicted estimates are conditional on past information, the filtered estimates are conditional on past and present information, and the smoothed estimates are conditional on past, present and future information. Examination of these results reveals that, under both deterministic and stochastic trend component specifications, estimates of the level, cyclical component and trend component of the natural rate of interest conditional on alternative information sets are approximately unbiased, while root mean squared errors are relatively small.  That estimates of the natural rate of interest are generally more  accurate and precise under a deterministic trend component specification than under a stochastic trend component specification, as evidenced by smaller biases and root mean squared errors, reflects the design of this Monte Carlo experiment, under which prior information concerning the values of trend components represents the belief that common trends are deterministic as opposed to stochastic.  Examination of these results also reveals that analytical root mean  squared errors appropriately account for uncertainty surrounding estimates of the natural rate of interest, with size distortions being small under both deterministic and stochastic trend component specifications. These size distortions may be partially attributed to the fact that analytical root mean squared errors do not account for parameter uncertainty, whereas simulated root mean squared errors do, inflating them to some extent.  Table 4.4. Experimental results under deterministic trend specification, natural rate of interest Estimate  Level Bias  RMSE  Cyclical Component ARMSE  ARMSE  Trend Component  Bias  RMSE  Bias  RMSE  Predicted  0.002016  1.978920  1.977000  -0.000881  2.005960  1.921080  0.002897  0.517920  ARMSE 0.798320  Filtered  0.001975  0.967920  1.177400  -0.001352  1.142480  1.316880  0.003327  0.520640  0.695640  Smoothed  0.001120  0.867760  1.008440  -0.008835  0.970120  1.121880  0.009955  0.486760  0.545920  Note: T h e ensemble biases, root mean squared errors, and analytical root mean squared errors o f the marginal posterior distributions o f state variables are calculated b y averaging deviations o f posterior means from simulated values, squared deviations o f posterior means f r o m simulated values, and analytical mean squared errors across 100 replications, respectively. U n d e r the data generating process, the unconditional mean o f the natural rate o f interest is 4 % , expressed at an annual percentage rate. A l l results are reported at an annual percentage rate.  226 Table 4.5. Experimental results under stochastic trend specification, natural rate o f interest Estimate  Level  Cyclical Component  Trend Component  Bias  RMSE  Bias  RMSE  Bias  RMSE  Predicted  0.006736  2.092360  2.033000  -0.010660  2.094880  1.965280  0.017396  0.643840  0.856360  Filtered  0.006227  1.362800  1.333400  -0.012104  1.455360  1.441680  0.018331  0.647920  0.762360  Smoothed  0.016802  1.396280  1.197800  -0.001545  1.439960  1.294040  0.018347  0.600920  0.587240  ARMSE  ARMSE  ARMSE  Note: The ensemble biases, root mean squared errors, and analytical root mean squared errors o f the marginal posterior distributions o f state variables are calculated by averaging deviations o f posterior means from simulated values, squared deviations o f posterior means from simulated values, and analytical mean squared errors across 100 replications, respectively. Under the data generating process, the unconditional mean o f the natural rate o f interest is 4 % , expressed at an annual percentage rate. A l l results are reported at an annual percentage rate.  This Monte Carlo experiment indicates that joint estimation of parameters and unobserved components with the Bayesian procedure under consideration also yields reasonably accurate and precise estimates of the level, cyclical component and trend component of the natural exchange rate conditional on alternative information sets, irrespective of whether the data generating process features common deterministic or stochastic trends. The results of estimating the level, cyclical component and trend component of the natural exchange rate under deterministic and stochastic trend component specifications are reported in Table 4.6 and Table 4.7, respectively. This concept of the natural exchange rate represents that real exchange rate consistent with past, present and future price flexibility.  Examination of these results reveals  numerous parallels with the results of estimating the natural rate of interest.  Under both  deterministic and stochastic trend component specifications, estimates of the level, cyclical component and trend component of the natural exchange rate conditional on alternative information sets are approximately unbiased, while root mean squared errors are relatively small. Furthermore, estimates of the natural exchange rate are generally more accurate and precise under a deterministic trend component specification than under a stochastic trend component specification, as evidenced by smaller biases and root mean squared errors. Finally, analytical root mean squared errors appropriately account for uncertainty surrounding estimates of the natural exchange rate, with size distortions being small under both deterministic and stochastic trend component specifications.  227 Table 4.6, Experimental results under deterministic trend specification, natural exchange rate Estimate  Level  Cyclical Component  Trend Component  Bias  RMSE  ARMSE  Bias  RMSE  ARMSE  Bias  RMSE  Predicted  -0.001269  0.633730  0.656290  0.005225  0.646410  0.645090  -0.006495  0.146420  0.252810  Filtered  -0.000612  0.236010  0.319740  0.006317  0.313410  0.402800  -0.006930  0.148620  0.188510  0.000476  0.237530  0.293560  0.009998  0.280010  0.349460  -0.009522  0.151880  0.145340  f  Smoothed  ARMSE  Note: The ensemble biases, root mean squared errors, and analytical root mean squared errors o f the marginal posterior distributions o f state variables are calculated by averaging deviations o f posterior means from simulated values, squared deviations o f posterior means from simulated values, and analytical mean squared errors across 100 replications, respectively. U n d e r the data generating process, the unconditional mean o f the natural exchange rate is normalized to one.  Table 4.7. Experimental results under stochastic trend specification, natural exchange rate Estimate  Level  Cyclical Component  Trend Component  Bias  RMSE  ARMSE  Bias  RMSE  ARMSE  Bias  RMSE  ARMSE  -0.000822  0.965340  0.943340  0.008526  0.681870  0.667470  -0.009348  0.754540  0.735260  Filtered  0.002469  0.405620  0.305880  0.008835  0.513620  0.498300  -0.006366  0.347450  0.169120  Smoothed  0.004117  0.396570  0.263190  0.006391  0.511590  0.427850  -0.002274  0.381920  0.069611  Predicted  Note: The ensemble biases, root mean squared errors, and analytical root mean squared errors o f the marginal posterior distributions o f state variables are calculated by averaging deviations o f posterior means from simulated values, squared deviations o f posterior means from simulated values, and analytical mean squared errors across 100 replications, respectively. Under the data generating process, the unconditional mean o f the natural exchange rate is normalized to one.  4.5. Conclusion This paper evaluates the finite sample properties of a novel procedure proposed by Vitek (2006f) for the measurement of the stance of monetary policy in a small open economy with a Monte Carlo experiment. Joint estimation of the parameters and unobserved components of a linear state space representation of an approximate unobserved components representation of a D S G E model of a small open economy with this Bayesian procedure, conditional on prior information concerning the values of parameters and trend components, is found to yield reasonably accurate and precise results in samples of currently available size.  In particular,  estimates of the levels of the natural rate of interest and natural exchange rate conditional on alternative information sets are approximately unbiased, while root mean squared errors are relatively small,  irrespective of whether the  data generating process features common  deterministic or stochastic trends. Moreover, analytical root mean squared errors appropriately account for uncertainty surrounding estimates of the levels of the natural rate of interest and natural exchange rate. The design of this Monte Carlo experiment could be extended or refined along numerous dimensions. The data generating process could be estimated rather than calibrated, potentially enhancing its empirical realism.  However, the trend component specification of the data  \ 228  generating process under consideration is too restrictive to accommodate the existence o f intermittent structural breaks in real data sets spanning a reasonably long period, irrespective o f whether common trends are deterministic or stochastic. In order to analyze the robustness o f the finite sample properties of the estimation procedure under consideration to forms o f model misspecification not associated with approximation error, alternative data generating processes could be considered, perhaps driven by different types of exogenous stochastic processes. However,  the  set  of potential forms  of such model  misspecification  is large, and  the  computational cost o f evaluating the implications o f individual forms o f model misspecification is high, while the exogenous stochastic processes associated with the estimated model nest commonly employed types o f exogenous stochastic processes.  References  Calvo, G . 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