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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 of Victoria, 2001 M . A . , Queen's University, 2002 A thesis submitted in partial fulfillment of the requirements for the degree of 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 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 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 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 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. The second paper develops and estimates an unobserved components model for purposes of 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 of unobserved state variables. 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 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. The primary contribution of this third paper is the development of a Bayesian procedure to estimate the levels of 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 of the procedure proposed in the third paper for the measurement of the stance of monetary policy in a small open economy with a i i i Monte Carlo experiment. This Bayesian estimation procedure is found to yield reasonably accurate and precise results in samples of currently available size. iv TABLE OF CONTENTS A B S T R A C T • u T A B L E OF C O N T E N T S - iv LIST OF T A B L E S : ix LIST OF F I G U R E S .' • x P R E F A C E •. x i i A C K N O W L E D G E M E N T S xvi i i C H A P T E R 1 1 Monetary Policy Analysis in a Small Open Economy: A Dynamic Stochastic General Equilibrium Approach 1.1. Introduction 1 1.2. Model Development 4 1.2.1. The Uti l i ty 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 rm 9 1.2.2.1. Employment and Investment Behaviour 9 1.2.2.2. Output Supply and Price Setting Behaviour 12 1.2.3. The Value Maximization Problem of the Representative Importer 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 18 1.2.4.1. The Monetary Authority 19 V 1.2.4.2. The F i s c a l Au tho r i t y 19 1.2.5. M a r k e t C l e a r i n g Condi t ions 20 1.2.6. The A p p r o x i m a t e L inea r M o d e l , 22 1.2.6.1. C y c l i c a l Components 22 1.2.6.2. T rend Components = 29 1.3. Estimation, Inference and Forecasting 31 1.3.1. Es t ima t ion ' 32 1.3.1.1. Es t ima t ion Procedure : 33 1.3.1.2. Es t ima t ion Resul ts 36 1.3.2. Inference 39 1.3.2.1. E m p i r i c a l Impulse Response A n a l y s i s 39 1.3.2.2. Theore t ica l Impulse Response A n a l y s i s 44 1.3.3. Forecast ing 45 1.4. Conclusion 46 Appendix l .A. Description of the Data Set 47 Appendix l .B. Tables and Figures 48 References • 69 C H A P T E R 2 • 72 A n Unobse rved Components M o d e l o f the Mone ta ry Transmiss ion 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 Components 75 2.2.2. Trend Components 78 2.3. Estimation of Unobserved State Variables 79 2.3.1. Unrestr ic ted Es t ima t ion o f Unobserved State Var iab les 80 2.3.2. Restr ic ted Es t ima t ion o f Unobserved State Var iab les 82 vi 2.4. Estimation, Inference and Forecasting 85 2.4.1. Estimation ; 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 95 2.5. Conclusion '. 97 Appendix 2.A. Description df the Data Set 98 Appendix 2.B. Tables and Figures 99 References 113 C H A P T E R 3 •' H 5 Measuring the Stance of 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 Uti l i ty 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 Fi rm 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 • 136 3.2.4.1. The Monetary Authority '.. 136 3.2.4.2. The Fiscal Authority 136 3.2.5. Market Clearing Conditions 137 3.2.6. The Approximate Linear Model 139 3.2.6.1. Cycl ical 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 151 3.3.1.2. Estimation Results '. 155 3.3.2. Inference 158 3.3.2.1. Quantifying the Stance of 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 ....193 C H A P T E R 4 196 Measuring the Stance of Monetary Policy in a Small Open Economy: A Monte Carlo Evaluation 4.1. Introduction 196 4.2. Model Development 199 4.2.1. The Uti l i ty Maximization Problem of the Representative Household 199 4.2.2. The Value Maximization Problem of the Representative Fi rm 201 4.2.2.1. Employment Behaviour ....202 4.2.2.2. Output Supply and Price Setting Behaviour 203 4.2.3. International Trade and Financial Linkages 205 4.2.3.1. International Trade Linkages 205 V l l l 4.2.3.2. International Financial Linkages 207 4.2.4. Monetary Pol icy ....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 , 211 4.3.2. Second Best Approximation : 212 4.3.2.1. Cycl ical Components : 212 4.3.2.2. Trend Components 214 4.4. Estimation and Inference 215 4.4.1. Estimation 216 4.4.2. Inference 220 4.5. Conclusion 227 References 228 LIST OF T A B L E S ix CHAPTER 1 Table 1.1. Deterministic steady state equilibrium values of great ratios 37 Table 1.2. Model selection criterion function values 40 Table 1.3. Results of tests of overidentifying restrictions 41 Table 1.4. Bayesian estimation results 48 CHAPTER 2 Table 2.1. Ful l information maximum likelihood estimation results, domestic economy 99 Table 2.2. Ful l information maximum likelihood estimation results, foreign economy 100 CHAPTER 3 Table 3.1. Deterministic steady state equilibrium values of great ratios 156 Table 3.2. Model selection criterion function values 163 Table 3.3. Results of tests of overidentifying restrictions 164 Table 3.4. Bayesian estimation results 171 CHAPTER 4 Table 4.1. True values of 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 of 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 O F F I G U R E S 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 of 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 of observed endogenous variables 54 Figure 1.8. Smoothed trend components of 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 of levels of 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 of observed endogenous variables 101 Figure 2.6. Filtered cyclical components of observed endogenous variables 102 Figure 2.7. Smoothed cyclical components of 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 of levels of observed endogenous variables 110 Figure 2.15. Dynamic forecasts of ordinary differences of observed endogenous variables 111 Figure 2.16. Dynamic forecasts of seasonal differences of observed endogenous variables 112 C H A P T E R 3 Figure 3.1. Predicted, fdtered and smoothed estimates of the natural rate of interest 160 Figure 3.2. Predicted, fdtered and smoothed estimates of 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 of observed endogenous variables 173 Figure 3.6. Filtered cyclical components of observed endogenous variables 174 Figure 3.7. Smoothed cyclical components of observed endogenous variables 175 Figure 3.8. Predicted trend components of observed endogenous variables 176 Figure 3.9. Filtered trend components of 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 : 187 Figure 3.20. Mean squared prediction error differentials for ordinary differences 188 Figure 3.21. Mean squared prediction error differentials for seasonal differences 189 Figure 3.22. Dynamic forecasts of levels of observed endogenous variables 190 Figure 3.23. Dynamic forecasts of ordinary differences of observed endogenous variables 191 Figure 3.24. Dynamic forecasts of seasonal differences of 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, DSGE 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 DSGE 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 DSGE 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. Closed economy versions of these papers have also been written, and are available at grad.econ.ubc.ca/fvitek. 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 M o n e t a r y P o l i c y A n a l y s i s i n a S m a l l O p e n E c o n o m y : A D y n a m i c S t o c h a s t i c G e n e r a l E q u i l i b r i u m A p p r o a c h 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. As 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 Yun (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 DSGE 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. As 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. As 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. As 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 DSGE 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 s and labour supply f s represented by intertemporal utility function where subjective discount factor B satisfies 0<B<1. The intratemporal utility function is additively separable and represents external habit formation preferences in consumption, CO (1) <C,s,L,s) = v's > c ( C „ - g C _ l ) 1 - | / g {Lj (2 ) 1 - 1 / t T 1 + 1/7 where 0 < a < 1. This intratemporal utility function is strictly increasing with respect to consumption i f and only i f vcs > 0 , and given this parameter restriction is strictly decreasing with 5 respect to labour supply i f and only i f vL > 0. Given these parameter restrictions, this 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 *4._,, and foreign currency denominated bonds Bf/ which yield interest at risk free rate i/. It also holds a diversified portfolio of shares {xjj s}lJ=0 in domestic intermediate good firms which pay dividends {TJf s}\=0, and a diversified portfolio of shares {xfk s}[=0 in domestic intermediate good importers which pay dividends {n/s}\=0. The representative household supplies differentiated intermediate labour service L. , earning labour income at nominal wage Wj s . Households pool their labour income, and the government levies a tax on pooled labour income at rate rs. These sources of private wealth are summed in household dynamic budget constraint: b:/+£X./- \vjXj^+ K C ^ ^ I H J C ^ O ^ , ) ^ (3) 7=0 k=0 1=0 According to this dynamic budget constraint, at the end of period s, the representative household purchases domestic bonds Bf/, and foreign bonds Bf/ at price £ s . It also purchases a diversified portfolio of shares {x]jS+X}XJ=0 in intermediate good firms at prices {y]tS}lJ=o> ar>d a diversified portfolio of shares {x/k J + 1 } [ = 0 in intermediate good importers at prices {VkMs}]k=0. Finally, the representative household purchases final consumption good Cls at price PSC- ' In period t, the representative household chooses state contingent sequences for consumption {C,.^}™,, domestic bond holdings {Bff+]}™=l, foreign bond holdings {Bf/}^,, share holdings in intermediate good firms {{xlJiS+l}lJ=0}™=t, and share holdings in intermediate good importers {{x^ks+]yk=0}^, to maximize intertemporal utility function (1) subject to dynamic budget constraint (3) and terminal nonnegativity constraints Bfjh+i > 0 , Bf/X > 0 , xj'. r + 1 > 0 and xfk r + 1 > 0 for T - > co . In equilibrium, selected necessary first order conditions associated with this utility maximization problem may be stated as uc(Cl,L.,) = PlcAn ( 4 ) 4=/?Cl + ' , ) E , A , + 1 , (5) £,A, = B(l + if)E,Sl+lAl+l, (6) 6 0=/rc,(/tf,+, + C.H+.. (8) where kis denotes the Lagrange multiplier associated with the period 5 household dynamic budget constraint. In equilibrium, necessary complementary slackness conditions associated with the terminal nonnegativity constraints may be stated as: l i m £ ^ / & + I = 0 , (9) hm £^£,+X;L=o, (10) hm £ ^ y ! i + T X r =0, (11) l i m yM U Q ( L 2 ) T ^ 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 uc ( C , L.t) = BE, (1 + /,) |1 uc (C, + 1 , L.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 W ^ j ) . £ , ^ ^ ( 14 ) 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 Li: supplied by households in a monopolistically competitive labour market to produce final labour service Lt according to constant elasticity of substitution production function \(L.,)e' di (15) where Of > 1. The representative final labour service firm maximizes profits derived from production of the final labour service n,L=w,L,- jW.,L.,di, (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~e'di (18) 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-eoL 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 W,. = f n C Y " f pC \ X ' r pC \rt-2 J w, i,t-\' (19) where 0 < yL < 1. Under this specification, although households adjust their wage every period, 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: w. uc(CnL;,,) ' Mc(C*>A>) ri-l Y ( pc } ri-\ i-r" w, (K) J pc w, ( pc ^ r ( pC*) ri-\ \-y-pC pc w, (20) 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 - coL of households that adjust their wage optimally in period t, and the average of the wages set by the remaining fraction coL of households that adjust their wage according to partial indexation rule (19): W. = (\-a>Lw;y-e- +coL rl-\ r rt-\ pc pC 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 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. 1.2.2.1. Employment and Investment Behaviour The representative intermediate good firm sells shares {xjV+1}J=0 to domestic households at price VY}f. Recursive forward substitution for Vjt+S with s > 0 in necessary first order 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 - (22) s=l+\ 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 (22) imposes transversality condition (11), which rules out self-fulfilling speculative asset price bubbles. Shares entitle households to dividend payments equal to net profits TlJs, defined as after tax earnings less investment expenditures: n], = a - *, )(pIyj, ~ wslj,s ) - p!i, • (24) Earnings are defined as revenues derived from sales of differentiated intermediate output good Yjs at price Pjs less expenditures on final labour service Lj s. The government levies a tax on earnings at rate r t , and negative dividend payments are a theoretical possibility. 10 The representative intermediate good firm utilizes capital Ks at rate ujs and rents final labour service Ljs given labour augmenting technology coefficient As to produce differentiated intermediate output good Yjs according to constant elasticity of substitution production function 9-\ 9-\ ((py(uhsKs)° +(\-(py(AsLJs)° 9-\ (25) where 0 < 1, & >0 and As > 0. This constant elasticity of substitution production function 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 s,Ks) denominated in terms of output: Y.s=F(u.sKs,AsL.s)-G{u.s,Ks). (26) Following Christiano, Eichenbaum and Evans (2005), this capital utilization cost is increasing in the rate of capital utilization at an increasing rate, G(u.s,Ks) = M [ e ^ - l ] K s (27) 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 Ks, which subsequently evolves according to accumulation function K^={\-S)KI+H{I„I,_X\ (28) where depreciation rate parameter 8 satisfies 0 < S < 1. Following Christiano, Eichenbaum and Evans (2005), effective investment function H(IS,IS_X) incorporates convex adjustment costs, n(is,is_l) = v!s f i - i v V *.v-l J (29) where ^ > 0 and v's > 0. In deterministic steady state equilibrium, these adjustment costs equal zero, and effective investment equals actual investment. ' Invoking [.'Hospital's rule yields l im InT(u h ,K s , AisLjs) = q>ln(i/^/(,) + (1 -<p)\n(A„L u)-<p]n<p-([-ip)\n(\-<p), which implies that Mm T(u j A , Aj = <p-*(\-py'^)(u^KJ(A,LjJ''. 11 In period t, the representative intermediate good firm chooses state contingent sequences for employment {Lis}™=l, capital utilization {ujs}™=l, investment a ° d the capital stock t 0 maximize pre-dividend stock market value (23) subject to net production function (26), capital accumulation function (28), and terminal nonnegativity constraint KT+l > 0 for T - > c o . In equilibrium, demand for the final labour service satisfies necessary first order condition ^Mj,tKt,AlLj,)0.^(\-Tl)^-, (30) where PsY<t>JS denotes the Lagrange multiplier associated with the period s production 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 + 1 (7 , + 1 , 1 , ) = P/, (32) which equates the expected present discounted value of an additional unit of investment to its price, where Qjs 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 + , > ) ] + ( ! - } > (33) 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 ^±LQi+TKl+T+l=0. (34) 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 conditions, this transversality condition is sufficient for the unique 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 Yjt supplied by intermediate good firms in a monopolistically competitive output market to produce final output good Yt according to constant elasticity of substitution production function Y = e,-\ (35) 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 6Y 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-coY of intermediate good firms adjust their price optimally. The remaining fraction coY of intermediate good firms adjust their price to account for past output price inflation according to partial indexation rule ( nr Y K = i - r r 1 j,t-\' (39) where 0 < yY < 1. 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 PY' given by necessary first order condition: E,I(*/) X, ' J's ( Py > ri-\ r ( W ^ rt-\ ( py*\ PY PY pY E,I(*/)J x, (0Y-W-T,) ( pY ^ rt-\ rt~\ f PY''} 1 PY pY p' PY PjY. (40) 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 - coY of intermediate good firms that adjust their price optimally in period t, and the average of the prices set by the remaining fraction coY of intermediate good firms that adjust their price according to partial indexation rule (39): PY = n - ^ x p r r - +coy f nY \ f nY V<-2 J V ' ' - 2 J (41) 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 {xfkl+l})=0 to domestic households at price Vkl . Recursive forward substitution for VkMl+s 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 TIks, defined as earnings less fixed costs: 15 K = PkM, MKs - £sPjJMktS - r , . (44) Earnings are defined as revenues derived from sales o f differentiated intermediate import good Mks at price PkMs less expenditures on foreign final output good Mk<s. The representative intermediate good importer purchases the foreign final output good at domestic currency price £sPj'f and differentiates it, generating zero gross profits on average. The law 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, £ PYJ Q = ( 4 5 > which measures the price o f foreign output in terms o f domestic output. A l s o define the terms o f trade, pM 7>-^F> (46) S which measures the price o f imports in terms o f exports. V i o l a t i o n o f the law o f one price drives a wedge *FS = £sPj'f IP/ between the real exchange rate and the terms o f trade, S > = ^ . (47) where the domestic currency price o f exports satisfies Pf = Pj . Under the law o f one price *FS=\, and the real exchange rate and terms o f trade coincide. There exist a large number o f perfectly competitive firms w h i c h combine a domestic intermediate good Zht e {Chl,IhJ,GhJ} and a foreign intermediate good ZfJ e {Cft,Ift,Gfl} to produce final good Z , e {C,, /,, G,} according to constant elasticity o f substitution production function Z , = (48) where 0<^ Z<1, \f/ >\ and v/ >0. The representative final good f irm maximizes profits derived from production o f the final good n?=P?Z,-P?Zh-PlMZfJ, (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: "h,l ( pY Y* I (50) Zft=(\-c/>z) f pM \ I MpZ \ I t J I ,M ' (51) Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final good firm earns zero profit, implying aggregate price index: Pz = f r>M\ \ t J (52) Combination of this aggregate price index with intermediate good demand functions (50) and (51) yields: \vt J (53) Zf=(\-<t>z) ( 1 - ^ Z ) + ^ Z f T Y~l \vi J ..M (54) 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 Mkl 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. = J ( ^ * , ) ~ Jit (55) where <9(w > 1. The representative final import good firm maximizes profits derived from production of the final import good n?=p,MM,- \pkMtMkldk, (56) with respect to inputs of intermediate import goods, subject to production function (55). The necessary first order conditions associated with this profit maximization problem yield intermediate import good demand functions: rk,l 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: \(P"V-dk k=0 1 (58) As the price elasticity of demand for intermediate import goods $ t M 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 - coM 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 1 * , / - ! » (59) where 0 < yM < 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 PtM' given by necessary first order condition pM,* I pM E,Z(^ ) s-l 'A QMy ( P M ^ r ( p M " ) Y pM ( pM* \ 1 pM V r i - i ) pM \ s-l ) pM 1 pM \ r> J PMM.. E,Z(^ ) f pM V rt-\ pM V J ( p M * \ I pM V r i J -, (60) PMM, where Ws = SsPsY'f I Pf measures real marginal cost. 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 \-coM of intermediate good importers that adjust their price optimally in period t, and the average of the prices set by the remaining fraction coM of intermediate good importers that adjust their price according to partial indexation rule (59): pM _ (\-a)M)(P,M')1-0' +o)" f r>M\ 1-0," 1_ (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 where > 1 and %Y > 0 . As specified, the deviation of the nominal interest rate from its deterministic steady state equilibrium value is a linear increasing function of the contemporaneous 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 where C,G <0. 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 i,-i,= r - Jcf) + ? (In Y- In + V, (62) (63) 20 In r, - In t, = In 5, a \ v - I n (64) where £"r >0. As 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 of 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'h which yield interest at risk free rate /,_,, and foreign currency denominated bonds Bf'f which yield interest at risk free rate if_{. It also levies taxes on the pooled labour income of households and the earnings of intermediate good firms at rate r,. These sources of public wealth are summed in government dynamic budget constraint: + Erf'* = (1 + / M )Bf* + £, (1 + ) B ? - ' \ 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 at price £ t . It also purchases final government consumption good G, at price . 1.2.5. Market Clearing Conditions A rational expectations equilibrium in this DSGE 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 = PCJ = P.J = po,f = pfJ and £ = Mf_ = ^ = Q *t I 1 t Clearing of the final output good market requires that exports Xt equal production of the domestic final output good less the cumulative demands of domestic households, firms, and the government, 21 X,=*-Cn-ln-GhJ, (66) where Xt = MJ . Clearing of the final import good market requires that imports Mt satisfy the cumulative demands of domestic households, firms, and the government for the foreign final output good, M,=CfJ+IfJ + GfJ, (67) where Mt = Xf. In equilibrium, combination of these final output and import good market clearing conditions yields aggregate resource constraint: PiyYi = tfc, +P,'I,+ F?Gt +P,XX,- PtMMr (68) The trade balance equals export revenues less import expenditures, or equivalently nominal output less domestic demand. Let Bl+] 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 B^x = + StB^{ and public sector bond holdings Bf+l = Bf+f + £,B°'/ , since domestic bond holdings cancel out when the private and public sectors are consolidated: *L+1=2£1+*,C+1. (69) 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 M - *,') = E,_, HI-t,)P/Y, -Ifq-^I,]. (70) Al~\ At-\ 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 ^ ( ^ i - ^ C ) = E l . 1 ^ ( i f . I f l f + r f ^ i ; - / f G f ) . (71) V i 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,XX, - P/M,). (72) 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 of 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, xl+s denotes the rational expectation of variable xl+s, conditional on information available at time t. Also, jc, denotes the cyclical component of variable x,, while xt denotes the trend component of variable x ( . Cyclical and trend components are additively separable, that is xt - xt + xt. 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 T , . I l + y'fi \ + yYB ' '+l coY(\ + yYp) XnO, + l n f , - — \nOY ' l - r ' 0Y-I ' (73) 0 —l where 0 = (l - t)-^- . The persistence of the cyclical component of output price inflation is increasing in indexation parameter yY, while the sensitivity of the cyclical component of output 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 yY. 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 InK = 1 6Y WL GY-\PY 6Y WL Hu,K,) + 7^7—j—HA,L,), (74) where y = j ^ J ^ ~ 'pj)' a P P r o x i m a t e linear net production function is subject to 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: lnw, = — 01 WL 0Y-\PY k9 + - 9Y WL 6Y -\PY Xnt: ( 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: /lI , . Y n ni-\ + . Y i-t \ + yYB \ + rrB ' M coY{\ + rYp) ln<P,+ — l n f , ^—\nOY ' 1 - r ' 6Y-\ ' (76) 7 T 7 ^ A 1 » t + < , ^ ) A 1 " # - T W E ' A l 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 of 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 , lnC, + l — cr——— l + a \ + a r,c + E, l n - ^ v. (77) 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 l n /^ ,+ J3 1 + /? E , l n / , + 1 + -In ( 6 ^ P>' V ri J (78) 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) eY 3 0 - 1 PY (79) 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 m t f , + 1 = ( l - ^ l n £ , + £ t a ( i ? , 7 , ) , (80) where ~ = 5. 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, pYy ••cG In B, pry (81) 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: lnr, =^ rln 5, pry 11 Ii J + V . (82) 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, , +• P \ + rMB (\-mM)(\-coMB) coM(l + yM{3) £ PYJ l n ^ — 1 6>A - lnf l A .(83) The persistence of the cyclical component of import price inflation is increasing in indexation parameter yM, 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>M and indexation parameter yM. This import price Phillips curve is subject to 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 C C If Y Y F I n J r f = ( l - ^ ) ^ t a ^ + ( l - ^ ) ± t a ^ + 0 - ^ l n ^ -v 4>CJ{\ - + - *'J)U f ' d - <t>GJ) G In I L (84) where x u > G G Y </> y . The sensitivity of the cyclical component of exports to Y Y Y 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, investment, government consumption, and the terms of trade according to approximate linear import demand function -V In (85) where y = ( l - ^ c ) y + ( l - ^ ' ) y + ( l - ^ G ) y . 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 of 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 yL, to 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 coL , and to changes in the cyclical component of employment is decreasing in elasticity of 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 of labour according to approximate linear implicit labour demand function: 1 ^ , w , r . . 1 ln<2> = l n ^ lnr, PYA, 1 - r '3 ( eY wl^ dY-\PY , ii.K. l n T f - (87) AL, it 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 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 ^ C + ^ l n ^ + i ? ; . (88) 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 = it - E ( ky+] , while the cyclical component of the consumption based real interest rate satisfies rtc = it - E ( ;r, c + 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: l n 5 , = E , l n £ , + 1 - ( / , - / / ) . (89) 28 The cyclical component of the real exchange rate satisfies InQ = In £t + \nPfJ -\nPf, while the cyclical component of the terms of trade satisfies In Tt = In Pf - In Pf , where In Pf = In Pj . 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(PfYl) = j\n(PfCl) + Un(P/ll) + ^  (90) 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 of nominal government consumption according to approximate linear government dynamic budget constraint E , - , ln(-5 , G + l ) : + r B o \ PY tH^P/YJ-jH^G,) (91) where py \-p\t~~y)' ^ s approximate 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 of export revenues, and the contemporaneous cyclical component of import expenditures according to approximate linear national dynamic budget constraint _l_r H-B,) + l + B PY | l n ( ^ i , ) - ^ l n ( ^ M , ) (92) where py ~~^z^{y~y)' ^ s a P P r o x U T i a t e linear national dynamic budget constraint is well defined only i f 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?,c =p Inift + < , ef ~ iid Af(0,(T2,), (93) 29 In A, = pA ln4_, + ef ef ~ iid Af(0,a]), (94) In v\ = P y l \nvf + < ' , < ' ~ iid A/"(0, cr2,), (95) In vf = In v,w, + < " , ef ~ iid JV(0, <M), (96) \n9j=PfjV Indf+ef, ef - i i d Af(0,a2gt), (97) \n0f=p9Uln0f+ef,ef~M M(Q,af), (98) l n # = / v l n 4 i , + , ef ~ iid jV (0 ,oJ ), (99) v;=Pv,v;_l+£;\ < ~ i i d A/-(O,<T2,), (100) vf = PyGvf + ef, ef ~ iid Af{Q,af,), (101) vT,=pvJU+ef ,ef ~M 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. In deterministic steady state equilibrium, vc =v' =vM =1 and cr2c=a2=cr2 =a2 =<j2.= <V = = = °"2- = cr2r = 0 . U V v V 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 ~ iid A/"(0,a2p,), (103) In Pf = n, + In Pf + ef , ef ~ i id a2, ), (104) In Pf = n, + In Pf + ef , ef ~ iid M(0,a2pl), (105) In Pf = n, + In Pf + ef, ef ~ iid M(0,af ), (106) 30 In PtM = n, + In P™ + ef, ef ~ iid Af(0,a2ptl ). (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, +nl: lnF, = g, + n, + In l 7 + ef, ef ~ iid JV(0, a\), (108) InC, = g, + n, + In C,_, + ef, ef ~ iid M(0, a\), (109) inT; =  g, + n, + In + ef, ef ~ iid AA(0, a]), (110) InG, = g, + n, + In G,_, + ef, ef ~ iid A^ (0,4), (111) l n X , = g, + «, + In + ^ , ef ~ iid W(0,4), (112) InM, = g , + « , + l n M , _ , + ^ , ff*~iid jV(0,4). (113) It follows that the trend components of the ratios of consumption, investment, government 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,+ gt, while the trend component of employment follows a random walk with time varying drift nt: In W,=nt+gt+ In Wt_x + ef, ef ~ iid JV(0 ,a£), (114) InL,=n,+ I n I M + ef, ef ~ iid A T ( 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 Qt = In Pf , while the trend component of the capital stock satisfies In ^ p- = In % . 31 The trend components of the nominal interest rate, tax rate, and nominal exchange rate follow random walks without drifts: Tl=J_l+ ej, ej ~ iid A/"(0, a]), (116) In t, = In t,_{ + ej, ej ~ iid Af(0, o\), (117) In £ = In St_x + ef, ef ~ iid JV(0, o £ ) . (118) 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. The trend component of the output based real interest rate satisfies rt Y = /' - E ; nf, while the trend component of the consumption based real interest rate satisfies rf =it-Et xf+i. The trend component of the real exchange rate satisfies InQ = l n £ , +lnPlrj - InFf, while the trend component of the terms of trade satisfies In7^ = In Pf1 -\nP,x , where lnPtx = In Pf. The trend component of the net government debt satisfies l n l - ^ - l = lnl 1, while the trend component of the net foreign debt satisfies In I - 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,_x+e% e~ iid M(0, a]). (121) 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 DSGE 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 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 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. 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 xt 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 second order stochastic linear difference equation AqX, =A\x,_x +A2E,xl+l +A3v,, (122) where vector stochastic process v, consists of the cyclical components of K exogenous variables. This vector stochastic process satisfies stationary first order stochastic linear difference equation v, (123) where su — iid A/"(0,27,). The trend components of vector stochastic process x, satisfy first order stochastic linear difference equation CQx, = C, + C2u, + C3*,_, + e2 „ (124) where e 2, - iid A/X0,27 2). Vector stochastic process ut consists of the levels of L common stochastic trends, and satisfies nonstationary first order stochastic linear difference equation = w / - i + £xn (125) where £ 3 ( ~ i id A/"(0,273). Cyclical and trend components are additively separable, that is JC, = Jc, + X, . If there exists a unique stationary solution to multivariate linear rational expectations model (122), then it may be expressed as: x,=D,xtX+D2vr (126) This unique stationary solution is calculated with the matrix decomposition based algorithm due to Klein (2000). Let y, denote a vector stochastic process consisting of the levels of M observed nonpredetermined endogenous variables. 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 z, = (7, + G2z,_l +G3e4j, (127) (128) where s4j ~ iid Af(0,Z4) and z0 ~ Af(za]0,PQlQ). 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 (129) where sSj ~ iid J\f(0,Z5). 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 Pt 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 z0[Q and P 0 | 0 , estimates conditional on information available at time t -1 satisfy prediction equations: zt\t~\ -Gx + G2z,_\\i_x, Pl\i-\ =G2P,-\\l-\Gl ^G^fi], R, = FP F1 1 \ rt\i-\l i ' (130) (131) (132) (133) (134) (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 + Ky (y> - JV-i) + K», (w/ - "V-i )' (136) P ^ P ^ - K ^ P ^ - K ^ P ^ , (137) where K yi = Pl]/AFf Q^_x and K^ = P ^ H j . Given terminal conditions zT]T and P7 obtained from the final evaluation of these prediction and updating equations, estimates conditional on information available at time T satisfy smoothing equations Zt\T ~ Zt\t +Jt(zi+\\T -<v+l |r)' (138) P,T=P!v+JtiP^T-Pl^)Jf (139) 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\lT)Kf(IT\0)f(9), (140) where I , = {{j t}'=1,{H>v}',=1}. Under the assumption of multivariate normally distributed signal and state innovation vectors, together with conditionally contemporaneously uncorrelated signal vectors, conditional density function / ( 1 T 10) satisfies: /(iT 1 0 ) = n r u i z;-.. * ) • n I z,-> m ( H O t=\ i=\ Under our distributional assumptions, conditional density functions f(yAXlA,0) and / (w, | 2 ~ M , 0 ) satisfy: f(y,\l^,0) = (2n) 2 i a | M | 2 e x p | - - U - ^ . 1 ) T f i f V . 1 U - J ' , H ) [ . (142) f(w,\I,_x,e) = {27v) 2 | J ? , H | 2 exp \ - - (yv , -w^ x f 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) 2\Q\2 exp\~(0-0AJQ1(0-0t)\. (144) 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 0T satisfies ylf(0T-60) X A / - ( 0 , - ? C ) , (145) where 0oe0 denotes the pseudotrue parameter vector. Following Engle and Watson (1981), Hessian may be estimated by * r = ^ £ E M [ v X m / 0 > , | J ^ ^ l + J - ^ X , [ v X m / ( H > \I,_JTj (146) where V.V] l n / ( j , | It_JT)] = - V 0 y l ^ y o y ^ - I V f f i { _ 1 ( f i r , . 1 ® <#_, WeQ,^ , v,v>M) = - i r ' . and 1.3.1.2. 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 (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 of 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 \-<fiGJ are restricted to equal 0.02. The deterministic steady state 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 Value Ratio Value CIY 0.6000 WLIPY 0.5000 HY 0.1723 KIY 2.8710 GIY 0.2000 BGIPY -0.4950 X I Y 0.3194 BIPY -0.6866 MIY 0.2917 Note: Deterministic steady state equi l ibr ium 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 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 of 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 of 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 yY is 0.75, implying considerable output price inflation inertia, while the prior mean of nominal rigidity parameter coY implies an average 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 s 5.75, implying moderate sensitivity of investment to changes in the relative shadow price of capital. The prior mean of indexation parameter yM is 0.75, implying moderate import price inflation inertia, while the prior mean of nominal rigidity parameter coM 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 yL is 0.75, implying considerable sensitivity of the real wage to changes in consumption price inflation, while the prior mean of nominal rigidity parameter coL implies an average duration of wage contracts of two years. The prior mean of 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,Y is 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 <^r 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. 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 + zZ^-i + Bs,, ( 1 4 7 ) where p{t) denotes a third order deterministic polynomial function and e, ~ iid A/"(0, / ) . Vector stochastic process yt 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 nf, domestic exports In Xt, domestic imports In M , , domestic nominal interest rate /,, nominal exchange rate ln£, , foreign output price inflation TI]J , foreign output \nY/, foreign consumption In C] , foreign investment In / / , and foreign nominal interest rate / / . 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 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 AIC(p) SC(p) HQ(P) 1 -110.8778 -102.3386* -107.4079* 2 -111.2069 -98.2780 -105.9532 3 -111.1651 -93.8465 -104.1276 4 -111.6281* -89.9197 -102.8068 Note: M in im ized 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. Asymptotic Parametric Bootstrap Nonparametric Bootstrap 278.1389 0.0000 0.9499 0.9990 Note: This l ikel ihood 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) DLPCON (APR) 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 3 S 4 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 DLPINV (APR) 0 5 10 15 20 25 30 35 5 10 15 20 25 30 DLPIMP (APR) 0 5 10 t5 20 25 30 35 0 5 10 15 20 25 30 35 40 NINT(APR) DLPGDPF (APR) 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 5 10 IS X 35 40 NINTF (APR) 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 35 40 Note: Theoretical impulse responses to a 50 basis point monetary pol icy shock are represented by black lines, whi le blue lines depict empir ical impulse responses to a 50 basis point monetary pol icy shock. Asymmetr ic 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 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 35 40 DLPINV(APR) 5 10 IS 2D 5 10 15 20 25 30 0 5 10 15 30 25 30 35 40 DLPCON (APR) 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 36 40 DLPIMP(APR) 0 5 10 15 20 25 30 15 20 25 30 35 40 NINT(APR) 5 10 15 20 25 30 35 *3 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 5 10 IS 20 0 S 10 15 20 25 30 35 40 DLPGDPF (APR) 5 10 15 20 25 30 35 40 NINTF (APR) 5 10 15 20 25 30 35 40 Note: Theoretical impulse responses to a 50 basis point monetary pol icy shock are represented by black lines, whi le blue lines depict empir ical impulse responses to a 50 basis point monetary pol icy shock. Asymmetr ic 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 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 of 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. Domestic output price inflation rises in response to an increase in real marginal cost. 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. As 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 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 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 of particular importance to the conduct of monetary policy. 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 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 DSGE 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 Posterior Distribution Mean Standard Error Mode Standard Error a 0.950000 0.000950 0.949020 0.000925 X 5.750000 0.005750 5.745100 0.005750 n 2.000000 0.002000 2.000100 0.002000 K 0.100000 0.000100 0.099999 0.000100 V 1.500000 0.001500 1.500000 0.001500 a 2.750000 0.002750 2.751200 0.002750 3 0.750000 0.000750 0.750000 0.000750 r' 0.750000 0.000750 0.749910 0.000750 r" 0.750000 0.000750 0.750060 0.000750 rL 0.750000 0.000750 0.750020 0.000750 a,' mM 0.875000 0.000875 0.874010 0.000872 0.875000 0.000875 0.874270 0.000874 coL 0.875000 0.000875 0.874950 0.000874 4" 1.500000 0.001500 1.499600 0.001500 4' 0.125000 0.000125 0.124950 0.000125 -0.100000 0.000100 -0.099998 0.000100 C 1.000000 0.001000 0.999040 0.001000 0.850000 0.000850 0.849250 0.000850 PA 0.850000 0.000850 0.850500 0.000849 Py' 0.850000 0.000850 0.851580 0.000849 P,» 0.850000 0.000850 0.850060 0.000850 P„> 0.850000 0.000850 0.850060 0.000850 P»« 0.850000 0.000850 0.850020 0.000850 P,f 0.850000 0.000850 0.850030 0.000850 P,< 0.850000 0.000850 0.852660 0.000844 P." 0.850000 0.000850 0.850300 0.000849 P,- 0.850000 0.000850 0.850150 0.000849 a], - 00 0.203690 0.058358 00 0.056415 0.013065 a1, 00 0.306120 0.052463 al-< OO 0.110820 0.014633 <~ OC 0.285680 1.557700 <• - OO 0.253650 2.182200 °\ - •c 0.290040 3.829100 a', - oo 0.015802 0.002923 a1,. - oo 0.441830 0.062389 a2, CO 0.242930 0.030785 (YL,' J - oc 0.109320 0.028552 °\< ... OO 0.102130 0.016336 c r , 2 " - OO 0.226170 0.035216 - OC 0.193580 0.502700 _ oo 0.243750 1.049700 - OO 0.243030 13.116000 a)il 1 - CC 0.220930 2.595100 cr,, - OO 0.073864 0.009300 <T2,, , OO 0.239540 0.030234 a1,, - X 0.394560 0.038424 Parameter Prior Distribution Posterior Distribution Mean Standard Error Mode Standard Error 4 - oc 0.622800 0.078904 00 0.062492 0.008983 4 - oc 0.577490 0.073005 a\ 00 0.096753 0.011677 4 - X ' 0.708920 0.088374 - oc 0.449850 0.056239 4 - 00 0.712530 0.090229 - 00 0.123730 0.026314 4 - 00 0.362830 0.040564 4 - oc 1.460000 0.185300 4 _ oc 0.533020 0.062373 4 - 00 0.783770 0.098795 1 - 00 0.114950 0.020643 - 00 0.002181 0.000388 al - OC 0.178260 0.027004 -I 00 0.448390 0.052147 4< - 00 0.303080 0.037457 4 00 0.021367 0.002802 4 - 00 0.026166 0.002760 4 - 00 0.497880 0.055539 4 - 00 0.086916 0.015808 4 , - 00 2.147700 0.258730 4- - 00 0.352380 0.041770 4 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 a] - OC 0.000025 0.000009 a1, 00 0.000068 0.000012 a", - 00 0.000022 0.000011 a\, - CO 0.000029 0.000012 Note: A l l observed endogenous variables are rescaled by a factor o f 100. Figure 1.3. Predicted cyclical components of observed endogenous variables 50 Note: 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 Economic Cyc le Research Institute reference cycle. Figure 1.4. Filtered cyclical components of observed endogenous variables 51 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 Cyc le Research Institute reference cycle. 52 Figure 1.5. Smoothed cyclical components of 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 known parameters. Shaded regions indicate recessions as dated by the Economic Cyc le Research Institute reference cycle. Note: Observed levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 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 Cyc le 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, whi le blue lines depict estimated trend components. Symmetr ic 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 Cyc le Research Institute reference cycle. 5 5 Figure 1.8. Smoothed trend components of observed endogenous variables LPGDP LRGDP LPCON LRCON LPINV Note: Observed levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 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 Economic Cyc le 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. Figure 1,12. Theoretical impulse responses to a foreign output technology shock 59 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 100-60-40 20-40-20 60 40 20 0- 0. 0--too-20--40--60-30 -40--20--40-•60-1 2 3 4 5 6 7 2 3 4 5 6 7 i 2 3 4 5 6 7 2 3 4 5 6 7 { LP1NV LRINV LPIMP LREXP 40-800- 400 1200-20-400- 200 800-400-0, 0. ft. 0--20-40 400--800. •Mi--400--400--800--1200-2 3 4 5 6 r 2 3 4 5 6 7 2 3 4 5 6 7 7 3 4 5 6 7 8 LRIMP NINT LNEXCH LPGDPF 1000-900. 0-i 2-' " I O O 0-4 20-10--sou-1000H -2-500--ia •2a 2 3 4 5 6 • 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 I LRGDPF LRCONF LRINVF NINTF BO*. 4. 4a 50-2000- 3-2-20- IOUO-0- 0, 20-40-<*]• -50-1000-2000-, 2--3--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 6 7 Note: Mean 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 D L P G D P D L R G D P D L P C O N D L R C O N z 1 2 2 a 0 0 -2 . "~" 1 -2 -2 2 1 2 3 4 5 6 7 1 2 3 4 5 6 7 I 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 D L P I N V D L R I N V D L P I M P D L R E ) 0 ? 1.5-1.0-0.5. 0.0-30 20 10 0-B-0-40-30-2 0 . 10-0 5--1 .0-1 5-•10-•20--30-- 8 -10--20-- 3 0 -- 4 0 -2 3 4 5 6 7 D L R I M P 2 3 4 5 6 7 D N I N T 2 3 4 5 6 7 D L N E X C H 2 3 4 5 6 7 8 D L P G D P F 4 0 . 20. (J. 08. 04. 00. 20. i a 0-1.2. 0 8-0 4 . z a « t . - .04. .08--10--20--0 4. -0.8-- 1 2 -2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 5 2-D L R G D P F D L R C O N F D L R I N V F D N I N T F ii. 2 . a . s a 0-oe . 04. .on 2--80. 04 . - 0 8 -2 3 4 5 6 7 5 1 2 3 4 5 6 7 Note: Mean 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. Figure 1,18. Mean squared prediction error differentials for seasonal differences 65 5 6 7 8 5 6 7 8 5 6 7 5 6 7 4 0 0 . 5 6 7 5 6 7 8 5 6 7 B 5 6 7 8 5 6 7 5 6 7 Note: Mean 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 of levels of observed endogenous variables LPGDP LRGDP L P C O N L R C O N L P I W Note: Real ized outcomes are represented by black lines, whi le blue lines depict point forecasts. Symmetric 95% confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Figure 1.20. Dynamic forecasts of ordinary differences of observed endogenous variables 67 Note: Real ized outcomes are represented by black lines, while blue lines depict point forecasts. Symmetr ic 95% confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Figure 1.21. Dynamic forecasts of seasonal differences of observed endogenous variables 68 Note: Real ized outcomes are represented by black lines, whi le blue lines depict point forecasts. Symmetr ic 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. 69 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. Al t ig , D. , L . Christiano, M . Eichenbaum and J. 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(1978), Estimating the dimension of 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 of 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. Yun, T. (1996), Nominal price rigidity, money supply endogeneity, and business cycles, Journal of Monetary Economics, 37, 345-370. 72 CHAPTER 2 An Unobserved Components Model of the Monetary Transmission Mechanism in a Small Open Economy 2.1. Introduction In recent years , the cen t r a l b a n k s o f m a n y e c o n o m i e s h a v e a d o p t e d i n f l a t i o n t a rge t ing mone t a ry p o l i c y r eg imes . A n i n f l a t i o n ta rge t ing m o n e t a r y p o l i c y r e g i m e is cha r ac t e r i z ed b y three p r i m a r y e lements . F i r s t , there exis ts an 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 qu i t e l o w a n d is of ten s p e c i f i e d as an i n t e r v a l . S e c o n d , a c h i e v i n g an i n f l a t i o n c o n t r o l ob j ec t i ve , i n the f o r m o f m i n i m i z i n g d e v i a t i o n s o f i n f l a t i o n f rom its target v a l u e , i s e m p h a s i z e d r e l a t ive to a c h i e v i n g an output s t a b i l i z a t i o n ob jec t ive . T h i r d , the c o n d u c t o f m o n e t a r y p o l i c y is cha rac t e r i zed b y a h i g h degree o f t r ansparency a n d a c c o u n t a b i l i t y . A s t y l i z e d qua l i t a t i ve d e s c r i p t i o n o f the mone ta ry 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 ngu i shes a m o n g ins t ruments , i nd ica to r s , a n d targets. U n d e r an i n f l a t i o n t a rge t ing m o n e t a r y p o l i c y r e g i m e , the cen t ra l bank p e r i o d i c a l l y adjusts a shor t t e r m n o m i n a l interest rate i n response to i n f l a t i o n a r y pressure . P r o v i d e d that th i s r esponse is s u f f i c i e n t l y la rge , i n the presence o f short r u n n o m i n a l r i g i d i t i e s o r imper fec t i n f o r m a t i o n , an inc rease i n the short t e r m n o m i n a l interest rate causes an increase i n the short t e r m rea l interest rate, i n d u c i n g i n t e r t empora l r educ t ions i n c o n s u m p t i o n a n d inves tment . In an o p e n e c o n o m y , an increase i n the short t e r m n o m i n a l interest rate causes a n o m i n a l app rec i a t i on , w h i l e an inc rease i n the short t e r m rea l interest rate causes a r ea l a p p r e c i a t i o n . T h i s adjus tment o f the r ea l e x c h a n g e rate i n d u c e s a n in t ra tempora l r e d u c t i o n i n expor t s together w i t h an in t r a t empora l inc rease i n impor t s . In the presence o f short r u n n o m i n a l r i g i d i t i e s o r imper fec t i n f o r m a t i o n , the resul tant r e d u c t i o n i n output is a s soc ia t ed w i t h a d e c l i n e i n output p r i ce i n f l a t i o n . In an o p e n e c o n o m y , the resul tant 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 acce le ra t ed b y the adjus tment o f the rea 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 success o f m a n y i n f l a t i o n ta rge t ing cen t ra 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 d e v e l o p m e n t o f a m u t u a l l y cons i s ten t set o f accura te a n d p rec i s e A version of this chapter has been accepted for publication. 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. As discussed in Woodford (2003), 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. Within the framework of an unobserved components model of selected elements of the monetary transmission mechanism in a closed economy, Laubach and Williams (2003) find that estimates of 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 of indicators of inflationary pressure such as the natural rate of 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 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. 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 ( xl+s denotes the rational expectation of variable xt+s, conditional on 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, - xt + xt. 75 2.2.1. Cyclical Components The cyc l ica l component o f output price inflation K] depends on a linear combination o f past and expected future cycl ica l components o f output price inflation driven by the contemporaneous cycl ica l component o f output according to output price Phi l l ips curve % = A A + A ^ l x + 0U In Y, + ef, ef ~ i i d jV(0, a\), (1) where the level o f output price inflation satisfies n) = A In P*. The sensitivity o f the cyc l ica l component o f output price inflation to changes in the cycl ical component o f output is increasing in 6U > 0 . The cycl ica l component o f consumption price inflation nf depends on a linear combination o f past and expected future cyc l ica l components o f consumption price inflation driven by the contemporaneous cyc l ica l component o f output according to consumption price Phi l l ips curve ^ = + & , 2 E A c ; , + 02l InYt (2) - ^ A l n Q , , +02aA\nQ, -^2f52 2E,AlnC}/ + 1 + ef , ef - i i d Af^a2., ), where the level o f consumption price inflation satisfies ncs = A In Pf . The cyc l ica l component o f consumption price inflation also depends on past, contemporaneous, and expected future proportional changes i n the cyc l ica l component o f the real exchange rate. The sensitivity o f the cycl ica l component o f consumption price inflation to changes in the cyc l ica l component o f output is increasing in 92, > 0 , and to changes in the cycl ical component o f the real exchange rate is increasing in 0 < 62 2 < 1. The cycl ica l component o f output In Yt follows a stationary second order autoregressive process driven by the contemporaneous cycl ica l component o f foreign output and a linear combination o f the past cyc l ica l components o f the real interest rate and real exchange rate In Y, = ^  In t , + ^ , 2 In t 2 + 03J In Yf + 932rt_x + 0„ In QtA + ef, ef ~ i i d JV(0, <?]), (3) where the level o f the real interest rate satisfies rt =il-Elnfl, whi le the level o f the real exchange rate satisfies l n g , = InS, + In Pf -\nPf . The sensitivity o f the cyc l ica l component o f output to changes in the cycl ica l component o f foreign output is increasing in 03, > 0 , to changes i n the cyc l ica l component o f the real interest rate is decreasing in 63 2 < 0 , and to changes in the cyc l ica l component o f the real exchange rate is increasing in 03 3 > 0 . The cyc l ica l component o f consumption InC, follows a stationary second order autoregressive process driven by the past cycl ical component o f the real interest rate: 76 InC, = ^ , InC,_, + £ 2 InC,_2 + f7 4 l r M + ef, ef ~ i i d (0,erf). (4) The sensitivity o f the cyc l ica l component o f consumption to changes in the cyc l ica l component o f the real interest rate is decreasing in <94, < 0. The cycl ica l component o f investment In/, follows a stationary second order autoregressive process driven by the contemporaneous cyc l ica l component o f output: In /, = </>5, In /,_, + (j>sl In /,_2 + fi., In Y, + e\, e\ ~ i id jV(0, a)). (5) The sensitivity o f the cycl ica l component o f investment to changes in the cyc l ica l component o f output is increasing i n 05, > 0. The cyc l ica l component o f exports l n X , follows a stationary second order autoregressive process driven by the contemporaneous cycl ical component o f foreign output and the past cyc l ica l component o f the real exchange rate: I n * , =4>b/nX,A+<Pb2\nXl_2+ebi\nYl<+ 0b2\nQlA + ef, ef - i i d J V ( 0 , < ^ ) . (6) The sensitivity o f the cycl ica l component o f exports to changes i n the cyc l ica l component o f foreign output is increasing in # 6 1 >0, and to changes in the cyc l ica l component o f the real exchange rate is increasing in 9b 2 > 0. The cycl ica l component o f imports In Ml fol lows a stationary second order autoregressive process driven by the contemporaneous cycl ical component o f output and the past cyc l ica l component o f the real exchange rate: In M , = <f>7, In Af,_, + ^  In M , _ 2 + 07, In Yt + 012 In Q,_f + ef1, ef ~ i i d J V ( 0 , a\). (7) The sensitivity o f the cyc l ica l component o f imports to changes i n the cycl ica l component o f output is increasing in 61X > 0 , and to changes in the cycl ica l component o f the real exchange rate is decreasing in $7 2 < 0 . The cycl ica l component o f wage inflation nf depends on a linear combination o f past and expected future cycl ica l components o f wage inflation driven by the contemporaneous cyc l ica l component o f the unemployment rate according to wage Phi l l ips curve (°) 77 where the level of wage inflation satisfies nf = A In Wt. The cyclical component of wage inflation also depends on past, contemporaneous, and expected future cyclical components of consumption price inflation. The sensitivity of the cyclical component of wage inflation to changes in the cyclical component of the unemployment rate is decreasing in 9% x < 0, and to changes in the cyclical component of consumption price inflation is increasing in 0 < 0g 2 < 1. The cyclical component of employment InZ, follows a stationary second order autoregressive process driven by the contemporaneous cyclical component of output: In L, = fa, In Z,_, + fa2 In Z,_2 + 69J In Yt + eLt , ef ~ i id (0, CJ\ ). (9) The sensitivity of the cyclical component of employment to changes in the cyclical component of output is increasing in 99, > 0. The cyclical component of the unemployment rate «, follows a stationary second order autoregressive process driven by the contemporaneous cyclical component of output: «r = do. . " , - . + doA-2 + K\ln ?, + ef, ef - i id Af(0, <7„2). (10) The sensitivity of the cyclical component of the unemployment rate to changes in the cyclical component of output is decreasing in 9W, < 0. The cyclical component of the nominal interest rate /, follows a stationary first order autoregressive process driven by the contemporaneous cyclical components of consumption price inflation and output: l=<l>nM+dm^ ei~MM(0,af). (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 9XXX > 0 , and to changes in the cyclical component of output is increasing in 9XX 2 > 0. The cyclical component of the nominal exchange rate ln S1, depends on a linear combination 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, =4,1ln5(_, +^ 1 2, 2E, lnS, + 1 +9m(i, -i/) + ef, ef - iid Af(0,a2.). (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 9X2l < 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 = nt + In Pf + ef , ef ~ iid jV(0,a2p)), In Pf = n, + In Pf + ef , ef ~ iid jV(0, af) 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 Yt, consumption In Ct, investment In It, exports In Xt, and imports In M, follow random walks with time varying drift g,+nt: In % = g, + n, + In YtA + ef, ef ~ iid jV(0,a29), (15) In C, = g, + n, + In C,_, + ef, ef ~ iid M(0, a}.), (16) In I = g, + n, + In lt_x + ej, ej ~ iid Af(0, erl), (17) In X, = g, + n, + In + ef, ef ~ i id j\f(0,a\), (18) InM, =gl+nl+\nMl_1+ef, ef ~ iid ^ ( 0 , ^ ) . (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 Wt follows a random walk with time varying drift 7rr+gt, while the trend component of employment In Lt follows a random walk with time varying drift nt: In W,=K, + g,+ In W,_x + ef, ef ~ iid AT(0,of), (20) InL, = n, + InL,_x + ej, ej ~ iid j\f(0,ar\). (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. (13) (14) 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,_x + ef, ef - i id M(0, al), (22) T = T-i + e'i, e] ~ i id Af(0, crl), (23) ln S, = ln + 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,_A+e;, < ~ i i d jV{0,trl), (25) (26) n, =»,_,+ e*, e* ~ iid jV(0, a]). ( 2 7 ) S,=gtA+£f, ef - iid A T ( 0 , C T 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. 2.3. Estimation of Unobserved State Variables State space models consist of 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 of a linear state space model is discussed in Hamilton (1994), K im 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 of 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 of 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 yt denote a vector stochastic process consisting of 7 V 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,=AlXl+A2z,+A3£u, (28) zt=^x,+B2z,_i+B3e2j, (29) where e,, ~ iid A/"(0,27,), e2j ~~ iid A/"(0,27,) and z0 ~ J\F(z0]0,P0[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 of this linear state space model, define z , M = E(z, 1 ) , /»,., =Var(z l | J , . , ) , y,M =E(y, | X,_,) and Q,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,^ = Btx, + B2z,_[M, (30) P,\t-\ =  B 2 P , \\, \BZ +  B ^ 2 B J ^ ( 3 1 ) yt\t-\ - A\Xt + A2z/^_], (32) 81 Q,il-l=^A,-^T2+A32:iAj. (33) 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 zl{, as that argument which maximizes posterior density function: /<«, y , l , ^ f l y i Zff\ (34) f(y, 127.,) Under the assumption of multivariate normally distributed signal and state innovation vectors, z,i, minimizes objective function S(z,) = (z, -ztl,^)JP^iz,-:)H)-U - J„,-.|)Te,i'-,(j, - J , M X (35) subject to signal equation (28). The necessary first order condition associated with the implied unconstrained minimization problem yields where Kt = P^AjQf^ . This necessary first order condition is sufficient i f -A2iQf_.]A2 is positive definite. Define Pl]t as the mean squared error of zl}l, conditional on X M . Within the framework of this linear state space model, this mean squared error matrix satisfies: P^P^-K^P^. (37) Under our distributional assumptions, zr], equals the mean of posterior density function f(z, | j , , X M ) , and is therefore mean squared error optimal. Given initial conditions z 0 | 0 and PQ{0, recursive evaluation of equations (30), (31), (32), (33), (36) and (37) yields predicted and 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,\zl+l,lt)- — — • (38) f(z,+]!I,) Under the assumption of a multivariate normally distributed state innovation vector, z,]r minimizes objective function 82 S(z,) = («, -z,lt)TPJ(z, -z l V )-(z l + ] - Z)+Ily ( - W , (39) subject to state equation (29). The necessary first order condition associated with the implied unconstrained minimization problem yields (40) where J, - P^B]PT'+\T. This necessary first order condition is sufficient if P U - B J P X B is positive definite. Define PT]T as the mean squared error of V , conditional o n J , . WithVthe framework of this linear state space model, this mean squared error matrix satisfies: (41) Under our distributional assumptions, v equals the mean of posterior density function f(z, | zl+l,Tt), and is therefore mean squared error optimal. Given terminal conditions zTT and PW 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 yt 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, + A2zl+A3eu, (42) z, =Blx,+B2z,_, +B3e2j, (43) where e,, ~ iid ^(0,27,), E 2 , ~ iid Af(0,272) and z0 ~ JV(Z0^,P0[0) . Let w, denote a vector stochastic process consisting of J observed synthetic variables. Suppose that this vector stochastic process satisfies ">,=C1z,+C2e3j, (44) 83 where «3_, ~ iid JV(0,273). Conditional on known parameter values, this signal equation defines 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 of this linear state space model, define z(|,_, = E(z, | J,_,), !»,_, = Var(z, | J M ) , y,[lA = E(y, | J M ) , Q,{,_, = Var(y, | J,_,), w,„_, = E(w, | J , ) and J?,M = Var(w, | J,_,), where = {{J,K=1,{M'J':I1,{JCJ's=i} . Conditional on the parameters associated with the signal and state equations, these conditional means and variances satisfy prediction equations: (45) z,\,-\ - + ^ 2 z / - i | ( - p ^=B2P,_^BJ2+B3£2Bl (46) • V - i = M + 4 V P ( 4 7 > Q^-A^Al+A^Aj, (48) " V i = C i V i ' (49> ^ . ^ C . ^ X + ^ C j . (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 zA, as that argument which maximizes posterior density function: Az, 1 y„W„l,,) = f { y i 1 Z"»MH»'1 z^<-)f{z>1J'-). (51) 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 , M ) T P,^_x (z, - z , H ) ^ 84 subject to signal equations (42) and (44). The necessary first order condit ion associated with the impl ied unconstrained minimizat ion problem yields h ~ Z , H + Ky<(yt - J / M) + KWi(w,- H> | M ) , (53) where Ky=P^AJ2Q~f and Kw - P^AC]Rfx. This necessary first order condit ion is sufficient i f Pfx -A]Q;fA2 -CjR^C, is positive definite. Def ine /», as the mean squared error o f zl]t, condit ional on 2~_,. Wi th in the framework o f this l inear state space model, this mean squared error matrix satisfies: P ^ P ^ - K ^ P ^ - K ^ P ^ (54) Under our distributional assumptions, zt]l equals the mean o f posterior density function f(z, \ y , , w n l t A ) , and is therefore mean squared error optimal. G i ven init ial conditions z 0 | 0 and P0]0, 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 iven these predicted and filtered estimates, estimates o f the state vector condit ional on past, present and future information may be derived with Bayesian updating. Def ine z,w as that argument which maximizes posterior density function: / U,1 V P A ) 77—r^r- • (55) Under the assumption o f a multivariate normally distributed state innovation vector, z,w minimizes objective function S(z,) = (z, - z(|,) P,\, (z, - z,\t)-(z,+i - z,t||() ^ + i | , ( z , + i X (56) subject to state equation (43). The necessary first order condit ion associated wi th the impl ied unconstrained minimizat ion problem yields ZI\T = +"^/(z,+i|r ~ z/+i|r)' (57) where Jt = P^BJP'f,. This necessary first order condit ion is sufficient i f Pf -B2Pll\lB2 is positive definite. Def ine PI]T as the mean squared error o f zl}T, condit ional on lt. Wi th in the framework o f this linear state space model , this mean squared error matrix satisfies: Pl]T = P,ll+J,(Pl+,T-P,+,t)jJ. (58) 85 Under our distributional assumptions, v equals the mean of posterior density function f(z, | z,+],1,), and is therefore mean squared error optimal. Given terminal conditions z and Pw 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 of 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*,-2 + 4E,*,+, + . (59) where e 1 ( ~ iid A/'(0,2'1). If there exists a unique stationary solution to this multivariate linear rational expectations model, then it may be expressed as: x, =Blx„l+B2x,.2 + B3elJ. (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 xt satisfy first order stochastic linear difference equation C 0x, = C,v, + C2x,_1 +£,_„ (61) where E2I - iid /V(0,27,). Vector stochastic process v, consists of the levels of L common stochastic trends, and satisfies first order stochastic linear difference equation v / = v / - i + £3,/» (62) where £ 3 , - i i d A/"(0,273). Cyclical and trend components are additively separable, that is xt= xt+xr Let yt denote a vector stochastic process consisting of the levels of M observed nonpredetermined endogenous variables. 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 variables, the contemporaneous trend components of N nonpredetermined 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) Z< =G\Z,-\  +G2£4,n (64) where s4l - iid A/"(0,274) and z0 - A r(z 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,=Hxz,+eyn (65) where e51 - i id A/"(0,275). 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. 87 Conditional on the parameters associated with these signal and state equations, estimates of unobserved state vector z, and its mean squared error matrix Pt may be calculated with the fdter derived previously. Given initial conditions z0 |0 and P 0 | Q, estimates conditional on information available at time / -1 satisfy prediction equations: (66) ^Zfi], (67) (68) (69) (70) (71) 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 Z,k = V i + Ky U _ >Vi) + K ~, (w> " " V i >' ( 7 2 ) Pt[l=Pt^-KyFxPlVA-KKH^, (73) zt\t-\ p = G\pi-\\t-\G\~ y,\,-\ = Fz = FP FT R,\i-\ = HlPl{l_lHlT where Ky = i > , H i ? l T ( J M and K = P^^HjR^[}. Given terminal conditions zT{T and PT]T obtained from the final evaluation of these prediction and updating equations, estimates conditional on information available at time T satisfy smoothing equations P1]T=PIII+J,(P,+MT-P,+„)J], (75) where Jt = P^GjP'J^ . Under our distributional assumptions, these estimators of the unobserved state vector are mean squared error optimal. Let 0 e0 cRK denote a K dimensional vector containing the parameters associated with the signal and state equations of this linear state space model. The full information maximum likelihood estimator 0T of this parameter vector maximizes conditional loglikelihood function: 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) + Cw (0), where: (*) = - y ln(2^)-iln | Q,[IA \ -{-(y,-yl{l, - j y , ) , (77) L, (0) = - y ^ ) - | l n I | - ^ ( M > , - ny , ) T /?„'_,(*, - ny , ). (78) Under regularity conditions stated in Watson (1989), full information maximum likelihood estimator 0T is consistent and asymptotically normal, (79) where 0oe0 denotes the true parameter vector. Following Engle and Watson (1981), consistent estimators of and B0 are given by A^j^aXK), (80) A=^Y,b,(0T)b,(0T)\ (81) where at(0T) = ay(9T) + aw(0T) and bl(0T) = by (0T) + bw (0T). Under our distributional assumptions, wT)=v^j-,aM v /^k_, + i v,ej., , ® a H )vflaH, (82) « w ,«?r) = , V ^ , ( , _ , +ivfl/?;,(/?,k'l O i J - ^ V , ^ . , , (83) are (t f r ) = V (0r) and Z>„ (0 r) = V / w (^). If the signal and state innovation vectors multivariate normally distributed, then the conditional information matrix equality holds and 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 of 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 |(_, by = Qff$l[t_{, where %t\t-\ ~ yt ~ yt\t-\ • 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 of 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 of the natural rate of interest RINT_P (APR) RINT_F (APR) RINTS (APR) Note: Estimated levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 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 Economic Cyc le 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 of the natural exchange rate LREXCH_P L R E X C H F LREXCH_S Note: Observed levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 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 Cyc le 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) L P G D P DLPCON (APR) L P C O N Note: Estimated impulse responses to a 50 basis point monetary pol icy 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). Figure 2.4. Estimated impulse responses to a foreign monetary policy shock 95 DLNWAGE (APR) DLPGDPF (APR) DLNWAGEF (APR) Note: Estimated impulse responses to a 50 basis point monetary pol icy 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 where sit ~ 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. As 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).1 Those A R I M A 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. As 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 of 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 di, and moving average order qt are jo int ly selected subject to upper bounds o f four, two and two, respectively. levels of observed endogenous variables yit of the form (84) 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. Ful l information maximum likelihood estimation results, domestic economy 0u 0 . 2 02,1 02.2 03.1 03,2 04,1 04,2 05.1 05.2 06.1 06,2 0.385 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 #io,i #n,i # 1 1 , 2 #12,1 -0.046 0.164 0.570 -0.237 0.120 0.081 -0.048 (-0.205) (0.751) (5.916) (-5.289) (1.545) (5.491) (-0.064) 4- <4 -I °) °l °l 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) al 4 <t al i -I 0.112 0.088 0.196 0.270 2.022 2.141 2.921 0.165 0.357 0:041 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 „ 2 * g n 3.49xl0" 3 5.24xl0" 6 5.36xl0~ 5 (2.228) (0.605) (1.276) h,\,-\ S,t,-i hi\,-\ 7-M £"iii-i ht\t-\ 2(2) 0.058 1.566 2.228 4.417 3.931- 0.675 12 .263" ' 2.583 0.779 0.338 5.458* 1.328 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' e2(2) 3 8 . 4 5 2 " 2 6 . 6 6 3 " 4 7 . 8 6 5 " 20 .507*" 4 7 . 0 3 4 " ' 4 7 . 4 1 0 " 4 8 . 9 0 0 " ' 2 0 . 6 3 1 ' " 43 .456"* 46.364**' 71 .222"* 34 .705*" eJ(4) 78 .884 " 56 .533 " 84 .406" 58 .357 " ' 75 .306" " 74 .097" 116.574" ' 42 .569"* 61.142'* ' 6 6 . 9 4 0 " ' 155.382"* 69.776'** Skewness -0 .272 -0 .180 -0 .258 - 0 . 5 0 9 " 0.137 0.253 0 .438 " 0.276 -0 .004 0 . 5 9 0 " ' 0.399' -0 .102 Kurtosis 3.291 3.379 3.226 4 .286* " 2.953 2.979 2.910 3 .893 " 4 . 7 1 1 * " 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 Note: Rejection o f the nul l hypothesis at the 1%, 5 % and 10% levels is indicated by * * * , * * and * , respectively. 100 Table 2.2. Full information maximum likelihood estimation results, foreign economy 01,1 <t>\,2 $.1 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) 0s.i 0 U 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 0 5 , ^8 ,2 09,1 0io,i 0.007 -0.632 -0.789 4.061 -0.041 0.413 0.306 -0.224 (0.144) (-1.626) (-3.030) (8.986) (-0.414) (3.483) (5.836) (-6.891) 0|i,i ^11.2 0.208 0.056 (1.683) (4.800) °\ -I a) 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) °>T <4 °i °1 al i 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) °l 2.14xl0" 3 6.19xl0" 5 1 .40X10" 4 (2.495) (2.129) (1.701) S i K - l h,\t-\ C i i - i C<i>-\ 2(2) 3.334 6 .285" 1.816 7.823* 11.126"" 7.377"' 1.281 8 , 8 2 1 " G(4) 15.198" ' 7.363 8.704' 10.755* 16.416"*' 12 .753" 1.441 2 0 . 4 3 7 " ' g 2 ( 2 ) 29.357*" 12.149*" 12.939"" 11.448* 69 .938" " 5 5 . 5 3 2 " ' 2 4 . 1 5 4 " ' 2 8 . 9 5 4 " ' G 2 ( 4 ) 46.188* " 18.945*" 23 .797" " 23.569' 122.141"" 108 .846" ' 40 .219"* 58 .337"* Skewness 0.209 0 .509 " -0 .305 -0 .283 - 0 . 4 5 2 " 0.063 0 . 9 9 1 " ' 1.170"* Kurtosis 4 . 0 2 9 " 6 .060"" 4 .606"* 6.040* 2.474 3.218 4 .927* " 10.565'* ' JB 6 . 8 4 1 " 5 7 . 6 4 1 ' " 16 .360" ' 52.977' 6 .059" 0.352 4 2 . 3 5 4 " ' 347 .477*" C(0T) = -6605.462 Note: Rejection o f the nul l hypothesis at the 1%, 5 % and 10% levels is indicated by * * * , * * and * , respectively. Fi gure 2.5. Predicted cyclical components of observed endogenous variables 101 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 Cyc le Research Institute reference cycle. Fi gure 2.6. Filtered cyclical components of observed endogenous variables 102 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 Cyc le Research Institute reference cycle. 103 Figure 2.7. Smoothed cyclical components of observed endogenous variables L P G D P L P C O N L R G D P 1975 1930 1965 1990 1995 2000 2006 1975 1960 1985 1990 1995 20O0 2005 1975 I960 1985 1990 1995 2000 2005 1975 1960 1985 1990 1995 2030 2005 0 •y 1 1975 1980 1985 1990 19B5 2000 2005 1975 1930 19B5 1990 1995 2000 2005 18751980 1985 1990 N I N T ( A P R ) ft \ A '•4'%' if fl 'Ii 1975 1980 1985 1900 1995 2000 2005 1975 1980 1985 1990 1995 2000 2006 1975 1990 1985 1990 1985 2000 2005 1975 19B0 1985 1990 1995 2003 2005 1975' ' 1980 ' 1935' ' 1990 1975 19B0 1985 1990 1995 2000 2005 1975 I960 1985 1990 1995 2000 N I N T F ( A P R ) 1975 1980 1985 1990 1995 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 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 Cyc le Research Institute reference cycle. 104 Figure 2.8. Predicted trend components of observed endogenous variables L P G D P L P C O N L R G D P L R C O N Note: Observed levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 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 Cyc le Research Institute reference cycle. Fi gure 2.9. Filtered trend components of observed endogenous variables 105 Note: Observed levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 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 Cyc le 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 NINT(APR) 1990 1995 2000 2005 N I N T F ( A P R ) 1975 1980 1965 1990 1965 1990 1996 2000 2005 1975 1980 1985 1990 1995 2000 2005 1975 1980 1985 1990 1995 2000 2005 Note: Observed levels are represented by black lines, whi le blue lines depict estimated trend components. Symmetr ic 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 Cyc le Research Institute reference cycle. 107 Figure 2 . 1 1 . Mean squared prediction error differentials for levels LPGDP LPCON LRGDP L R C 0 1 N B 3 2 20 i 0 1 0 10. 0 % -4. - -10 •2 •fl- -2 -3--20--4 1 2 3 4 5 6 7 l 1 2 3 . s e ; 1 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 8 LRINV LREXP LRIUP LNWAGE iCO 400 XO 0-400 200 0 400 200 20 10-200 4(1) 800 •200--400-600--200 -400-10 -20 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 1 2 3 4 5 6 7 8 LEMP RUNENP NINT LNEXCH 4(1 3. 0 6-20- 0.4 100-0- 0 0.0. 0 20-2. -0 4. 100H 40- -3. -0 8-2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 LPGDPF 20-10. LRGDPF LRCONF LRINVF B-4. 2-500. 4. fl--10-•20-2--4. •6-0--500. 2 3 4 5 6 7 I 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 LNWAGEF LEMPF RUNEMPF NINTF IC i-60. 40-20-0 3. 2-0-2. 10--20--W--60--2--3-2 3 4 5 6 7 8 2 3 4 5 6 7 ; 2 3 4 5 6 7 ( 2 3 4 5 6 7 8 Note: Mean 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 D L P G D P D L P C O N D L R G D P . « .3 2 05 X 1 •2 - 05 •.« 4 - 5 - 2 -.10 -.3 1 2 3 4 5 6 7 1 1 2 3 4 5 8 7 8 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 D L R I N V D L R E X P DLRIfvF D L N W A G E 12. 10 8-4 o-s. 0-% 2 •4 * -5 -8- -2 -12- -10- •10 \ '* 2 3 4 5 6 7 8 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 D L E M P DRUNErVP D N I N T D L N E X ; H 1.2. 10- 03- 10-0 8. 02-05- 5-01-0 0- — — ^—~~ rm 00-0.4. -.05. • " -.01- \ _ _____ -0 8- -.02. -5. •1 2- - ia -10-2 3 4 5 5 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 D L P G D P F D L R G D P F D L R C O N F DLRINVF 1-0.8- J. 20. 2. i-ii. 0.4-o.a 10-i. •0.4. 2. 10--OB. 20-2 3 4 5 6 7 6 2 3 4 5 6 7 l 3 • • • • ' i 2 3 4 5 6 7 8 D L N W A G E F D L E M P F D R U N E N P F DNINTF ; 2- 08-06-04. '• 04. 02-u- 0- 00 nn 2- i. 0*. -.02. .1. -?- OB- - 04-08. 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 6 7 8 Note: Mean 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. Figure 2.13. Mean squared prediction error differentials for seasonal differences 109 S D L P C O N S D L R G D P e 4 2 ' 0.8 0.4 0.0 9 3 2 -2 -4 •ft -0.4 •0.8. -5 -2 -3 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 t 1 2 3 4 5 6 ? 1 1 2 3 4 5 6 7 8 20U 100 0 S D L R I N V S D L R E X P S D L R I N P S D L N W A G E 100 0 150 100 50 ft. 2. 101) -100 -50--100 -2 -4--200. -150 «-1 2 3 4 5 6 7 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 B S D L E M P S D R U N E I v P S D N I N T S D L N E X C H 15. 10-5-1.0-0 5- .2. 50-0 - 0.0. -5--10. •15. . . . — "—-—.... • — . 0. —— • -0.5--1.0-•2. -50-2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 8 S D L P G D P F S D L R G D P F S O L R C O N F S D L R I N V F 10-4 300-200-U-5-0-< 100-fc -5- >- -100. 4--10- -3--4. -2O0--300-2 3 4 5 6 7 8 2 3 4 5 6 7 6 2 3 4 5 6 7 8 2 3 4 5 6 7 8 S D L N W A G E F S D L E M P F S D R U N E M P F S D N I N T F 2. J-20-0 -15. 1.0-0.5-0.0. 5. z. " " " " " " — — 10- I I 5-4. 6. -20-1 D-l E -.5-1 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 6 7 8 Note: Mean 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. Figure 2.14. Dynamic forecasts of levels of observed endogenous variables 110 Note: Real ized outcomes are represented by black lines, whi le blue lines depict point forecasts. Symmetr ic 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Note: Real ized outcomes are represented by black lines, whi le blue lines depict point forecasts. Symmetr ic 95% confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. Figure 2.16. Dynamic forecasts of seasonal differences of observed endogenous variables 112 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: How 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 of a shock to monetary policy, Journal of Political Economy, 113, 1-45. Clarida, R. and M . Gertler (1997), How the Bundesbank conducts monetary policy, Reducing Inflation: Motivation and Strategy, University o f Chicago Press. 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(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 C H A P T E R 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. As 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 Yun (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 tek, F., Measuring the stance of monetary pol icy in a closed economy: A dynamic stochastic general equi l ibr ium approach, International Journal of Central Banking. 116 exchange rate is determined by an uncovered interest parity condition. Existing open economy DSGE 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 of developments in the housing market for the conduct of monetary policy. Existing DSGE models incorporating a housing market include those of Aoki , 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 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 filter described in Hodrick and Prescott (1997), prior to the conduct of estimation, inference and forecasting. As 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 DSGE 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 of 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 of 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 Hjs, and labour supply Li s represented by intertemporal utility function U„=ElfjPs-u{C,s,His,L.s), s=l 1 (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* | y H (His -aHH^r" y L (Ljs —aLLsl)M/n l-i/o- l-i/o- v \+\in (2) where 0 < « C < 1 , 0 < « W < 1 and 0 < a t < l . This intratemporal utility function is strictly 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 vL > 0 . Given these parameter restrictions, this 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 is_x, and foreign currency denominated bonds Bf/ which yield interest at risk free rate if. It also holds a diversified portfolio of shares {xfjsyj=0 in domestic intermediate good firms which pay dividends {I7fs}f0, and a diversified portfolio of shares {xfk s}[=0 in domestic intermediate good importers which pay dividends {I7kMs}\=0. The representative household supplies differentiated intermediate labour service Ls ( , earning labour income at nominal wage Wis. Households pool their labour income, and the government levies a tax on pooled labour income at rate TS . These sources of private wealth are summed in household dynamic budget constraint: )KA,,^n- j « , ^ = ( i H - 1 ) C ^ a + e , ) ^ / ;=o *=o ( 3 ) + J (K + K>KA + J « + KMsK,A + (1 -1 , ) ) WuLlsdl - PfCLs - Pflf. j=0 k=0 1=0 According to this dynamic budget constraint, at the end of period s, the representative household purchases domestic bonds Bff+], and foreign bonds Bf/ at price £ s . It also purchases a diversified portfolio of shares [x]'.S+]YJ=0 in intermediate good firms at prices {VfsYj=o> a n Q a diversified portfolio of shares {xfk v + i } [ = 0 in intermediate good importers at prices {VkMs }[=0. Finally, the representative household purchases final consumption good C, s. at price Pf, and final housing investment good lfs at price Pf" . The representative household enters period 5 in possession of previously accumulated housing stock Hj s, which subsequently evolves according to accumulation function % r ( l - ^ ) ^ ^ " ( C C ) . (4) 121 where depreciation rate parameter 8H satisfies 0 < 5" < 1. Effective housing investment function HH'(//l>^"-i) incorporates convex adjustment costs, n ( ^ . V i ) - 1 ' . 1-X f jH _ j H 1 i,s  1 i,s-\ I" V i" (5) where %H > 0 and v's > 0. In deterministic steady state equilibrium, these adjustment costs equal zero, and effective investment equals actual investment. In period t, the representative household chooses state contingent sequences for consumption {Cis}^,, investment in housing {l",}™=,, the stock of housing {HiiS+l}™=l, domestic bond holdings {Bff+X}™=l, foreign bond holdings , share holdings in intermediate good firms {{xjjs+l}[j=0}%,, and share holdings in intermediate good importers {{xfk s+x}]k=0}™=, to maximize intertemporal utility function (1) subject to dynamic budget constraint (3), housing accumulation function (4), and terminal nonnegativity ^constraints H. T+l > 0, , > 0 , Bfj+X > 0 , xjjj+l > 0 and xfkT+] > 0 for T —» oo . In equilibrium, selected necessary first order conditions associated with this utility maximization problem may be stated as uc(C„H„L.l) = PfAl, (6) (7) + Q.-S )QM A,=/J(1 + /,)E,A, + 1 , SlA, = B(l + if)E!£l+lAl+l, (8) (9) (10) (11) (12) where A ( > denotes the Lagrange multiplier associated with the period s household dynamic budget constraint, and ALsQ" denotes the Lagrange multiplier associated with the period s housing accumulation function. In equilibrium, necessary complementary slackness conditions associated with the terminal nonnegativity constraints may be stated as: 122 A. lim B,;T+X = 0, lim PLTA±L£ B P / =0 T-Ko 1 Hm PT*>+T yr r = f J r " i X yj,t+TXj,,+T+l U> r->co 1 (13) (14) (15) (16) (17) 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 M c ( C , , / 7 , , L , , ) = ^ E , ( l + / , ) ^ W c ( C ( + l , / Y , + 1 , L , , + 1 ) , (18) 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 nHT-/H(TH f " U C ^Uc(^i+\'^t+\'^i,t+]) Pf / - , / 7 i_/H / jH rH>. Dl" U, « l (A + ,~ I T T s ~^C~^H2 ( V l ' 7 / > = P > ' uc(cnH„h<) p: (19) 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 Buc{CM,Hl+],Lil+x) uc(C„H,,Lu) tf. pC " H ( Q | ^ I + I ' A . H - I ) _ rH ^nH (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 uc(C„H„L,,) J * ' uc{C„H„Lu) 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 Lit supplied by households in a monopolistically competitive labour market to produce final labour service Lt according to constant elasticity of substitution production function L. = \(k,)9' di 9,-1 (22) where df > 1. The representative final labour service firm maximizes profits derived from production of the final labour service nf=WtL- \WitL.,di, (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 W. \(wJ~Ui (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 - mL of households adjust their wage optimally. The remaining fraction coL of households adjust their wage to account for past consumption price inflation according to partial indexation rule r P c \ rt-\ pC pC W,. (26) where 0 < yL < 1. Under this specification, although households adjust their wage every period, 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: w. P'-'uc{Cs,Hs,Lis) uc(C„H„Lu) uL(Cs,Hs,Lis) '( pC ' r r - l 11-1 w 1"'' 4 pC BC V rs-\ . w ( Pc \ rl-\ pc 'pc\ pc -rL 1 w, (wf -oi (27) 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 - coL of households that adjust their wage optimally in period t, and the average of the wages set by the remaining fraction a>L of households that adjust their wage according to partial indexation rule (26): 125 i\-coLw;te' +coL ( Pc ^ rt-\ y- ( Pc \ ri-i pC V ri-2 ) pC 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 coL = 0 and necessary first order condition (27) would reduce to: ( 1 f ) f , _ uL{C,,H,,L,) ef-\uc{CnHt,L^ (29) 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 {xfjl+l})=0 to domestic households at price Vjt. Recursive forward substitution for Vjl+s 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=/+! At 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 IJYjs, defined as after tax earnings less expenditures on investment in capital: / / ; . v = ( l - r j ( - ^ ^ . J - P ' / « . (32) Earnings are defined as revenues derived from sales of differentiated intermediate output good Yjs at price Pjs less expenditures on final labour service Z,. s . The government levies a tax on earnings at rate ts, and negative dividend payments are a theoretical possibility. The representative intermediate good firm utilizes capital Ks at rate ujs and rents final labour service Lj s given labour augmenting technology coefficient As to produce differentiated intermediate output good Yj s according to constant elasticity of substitution production function Huj,sKs,AsL.s)-9-1 9-\ (<pY(uJtSK,)> +{\-(pY{AsLjs) 9-1 (33) where 0 < <p < 1, 9 > 0 and As > 0. This constant elasticity of substitution production function 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(uJs,Ks) denominated in terms of output: Y.s=HuhsKs,AsLjs)-Q{uhs,Ks). (34) Following Christiano, Eichenbaum and Evans (2005), this capital utilization cost is increasing in the rate of capital utilization at an increasing rate, ' Invoking L 'Hospi ta l 's rule yields l im I n F ( u j J K „ A i , L , . J = (oln(»y__KJ + (1 -<p)\n{A,LjJ-q>\n(p-(\-<p)\n(\-(p), which implies that i ™ •^ ("/..^ .^ J^ j=^ i^-prl'"'H»r^ )'Ki;J)'"'-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 Ks+l=(l-SK)Ks+HK(lf,d (36) where depreciation rate parameter SK satisfies 0 < SK < 1. Following Christiano, Eichenbaum and Evans (2005), effective capital investment function HK(I*,I*_}) incorporates convex adjustment costs, 2 IK 1s-\ IK (37) where >0 a n d v's >0- 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 {LJS}^,, capital utilization }"=/, investment in capital {I*}™=l, and the capital stock {KS+I}™=L to maximize pre-dividend stock market value (31) subject to net production 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 FAL(U.,KI,A1L.,)0.i=(\-T,) (38) where Pj<PJS 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 ruK(uJJK„A,Ljtl) = K, (39) 128 which equates the marginal product o f uti l ized capital to its marginal cost. In equil ibrium, demand for the final capital investment good satisfies necessary first order condition QfK df, / * , ) + E , ^ Qf+X (/*, ,lf) = if, ( 4 0 ) which equates the expected present discounted value o f an additional unit o f investment i n capital to its price, where Q*t denotes the Lagrange multiplier associated with the period s capital accumulation function. In equil ibrium, this shadow price o f capital satisfies necessary first order condition ^ = E ' ^ ± L { ^ ' < Z > y - — C M > - r + . ^ ( « y , , + i ^ + , . ^ + . ^ y , , + i ) ~ ^ ( « y - , / + 1 , ) ] + (1 ~ }, (41) which 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. In equil ibrium, the necessary complementary slackness condition associated with the terminal nonnegativity constraint may be stated as: Hm l^JLQ^TKl+T+{=Q. ( 42 ) Provided that the pre-dividend stock market value o f the representative intermediate good firm is bounded and strictly concave, together with a l l necessary first order conditions, this transversality condition is sufficient for the unique value maximiz ing state contingent intertemporal firm allocation. 3.2.2.2. Output Supply and Price Setting Behaviour There exist a large number o f perfectly competitive firms which combine differentiated intermediate output goods Yjt supplied by intermediate good firms in a monopolist ical ly competitive output market to produce final output good Yt according to constant elasticity o f substitution production function Y=\ ( 4 3 ) 129 where dj > 1. The representative final output good firm maximizes profits derived from production of the final output good nY=P!Y- (44) 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: .7=0 (46) 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 coY of intermediate good firms adjust their price to account for past output price inflation according to partial indexation rule 7,' f PY ^ r ! - \ r ( pY \ ri-\ PY V ri-2 ) PY \Jl-2 J \-rr PY ri,t-\' (47) where 0 < yY < 1. 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: A, s J's ( PY ^ rt-\ ri-\ '-''Pi] ( PY" \ PY p Y P' PY P,% (0; - i ) ( i -r , ) Y ( P O ri-\ X r PI ( PY'* \ p Y V ' s - I ) PY \ rs-\ ) PJ PY -• (48) PlY. 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 (46) equals an average of the price set by the fraction 1 - coY of intermediate good firms that adjust their price optimally in period t, and the average of the prices set by the remaining fraction a>Y of intermediate good firms that adjust their price according to partial indexation rule (47): PY = (l-tf/)(7f0'~*' +av f r>r ^ PY f pY \ rt-\ \-rr PY ri-\ 1 x-ej (49) 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 - 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)PY'= °< PY0 I' 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}]=0 to domestic households at price VkMt . Recursive forward substitution for VkMl+s 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^K- (52) s=l At 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 TIks, defined as earnings less fixed costs: K=p"Mk,-£sPjJMk,-rs. (53) Earnings are defined as revenues derived from sales of differentiated intermediate import good Mk s at price PkMs less expenditures on foreign final output good Mk 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 *FS = £sPj'f IPSM between the real exchange rate and the terms of trade, (56) where the domestic currency price of exports satisfies Pf - Pj. Under the law of one price tFs = 1, and the real exchange rate and terms of trade coincide. There exist a large number of perfectly competitive firms which combine a domestic intermediate good Zhl e {Chl, if, if, Ght} and a foreign intermediate good ZfJ e {Cfj,l"„Iftt,Gfj} to produce final good Z, e{Ct,l"',lf ,G,} according to constant elasticity of substitution production function Z. = (57) where 0 < (/>z < 1, y/ > 1 and vf > 0 . The representative final good firm maximizes profits derived from production of the final good n^=P^7-P,7 -PM7 (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 zfJ=(\-<t>z) v M p Z M " (60) Since the production function exhibits constant returns to scale, in competitive equilibrium the representative final good firm earns zero profit, implying aggregate price index: \vl J (61) Combination of this aggregate price index with intermediate good demand functions (59) and (60) yields: zh,,=<l>z <t>z+(WZ) (62) Zfl=(\-0Z) Kv: J (63) 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 Mkl 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 M,= i \(Mkl)6' dk k=0 (64) where Of > 1. The representative final import good firm maximizes profits derived from production of the final import good 134 n?=P?M- \PkM,MkJdk, (65) 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: pM _ j CO" MO dk (67) 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 - coM of intermediate good importers adjust their price optimally. The remaining fraction coM of intermediate good importers adjust their price to account for past import price inflation according to partial indexation rule k,t ~ f pM Y ri-\ f pM A 1"' rt-\ 1 k,<-\> (68) where 0 < yM < 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 PtM* given by necessary first order condition 135 j/W,* ~t-\ \-J» -10. pM s pM ( p M * \ pM V rt J PMM. ( 6 ^ - 1 ) ( PM ^ 1 ..M r "t-\ l ~ r pM s' ( pM* \ 1 pM \  rs-\ ) pM \  rs-\ ) pM 1 pM \ ri J (69) PMM, where lPs = £sPj'f IPSM measures real marginal cost. This necessary first order condit ion equates the expected present discounted value o f the revenue benefit generated by 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 by the fraction 1 - coM o f intermediate good importers that adjust their price optimally in period t, and the average o f the prices set by the remaining fraction a>M o f intermediate good importers that adjust their price according to partial indexation rule (59): ( pM Y " rl-\ K11-2 J \Ai-2. (70) Since those intermediate good importers able to adjust their price optimally in period t are selected randomly from among a l l intermediate good importers, the average price set by 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 a l l intermediate good importers were able to adjust their price optimally every period, then coM - 0 and necessary first order condit ion (69) wou ld reduce to pM,~ _ e. M 6T -1 pM uj (71) where PtM* = P]M . In f lexible price and wage equi l ibr ium, each intermediate good importer sets 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 %Y > 0 . As specified, the deviation of the nominal interest rate from its 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 ln PGG. P Y, i—In P G, PYY * In B. - I n + v; (73) where £G < 0. 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 by 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 , =<^r In B. G \ •In B, G \ PY. + v (74) where ^ > 0 . A s specified, the proportional deviation o f the tax rate from its deterministic steady state equil ibrium value is a linear increasing function o f the contemporaneous proportional deviation o f the ratio o f net government debt to nominal 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 by serially correlated fiscal revenue shock v\. The fiscal authority enters period t holding previously purchased domestic currency denominated bonds Bfh which y i e l d interest at risk free rate , and foreign currency denominated bonds BfJ which y ie ld 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 in government dynamic budget constraint: B% + Erf* = (1 + /,_, )Bfh +£,{1 + if )Bff i i i +*•, j jW^dldi + r, l(PfYjJ-W,L.l)dj-P,GGr (75) i=0 /=o 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,G+\h, and foreign bonds B°f at price £ t . It also purchases final government consumption good Gt at price Pf. 3.2.5. Market Clearing Conditions A rational expectations equil ibrium in 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 pol icy, together with state contingent intertemporal allocations for domestic and foreign governments which satisfy their pol icy rules and constraints given prices, with 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 Prj = Pc,f = P;»J = P^f = po.f = pxj and ^ = ^ = = 0. Clearing of the final output good market requires that exports Xt 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^ (76) where Xt = Mf. Clearing of the final import good market requires that imports Mt satisfy the cumulative demands of domestic households, firms, and the government for the foreign final output good, M,=Cftl+I^+I^ + GfJ, (77) where Mt = Xf. In equilibrium, combination of these final output and import good market clearing conditions yields aggregate resource constraint: PjYt = P,CC, + Plt" If + Pflf + PfG, + PfX, - PtMMt. (78) 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 = Bf\ + £tBf{ and public sector bond holdings Bf+] = Bfxh + £,Bf(, since domestic bond holdings cancel out when the private and public sectors are consolidated: B,+x=Bfl+Bfv (79) 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 -Bf) = E_ , 4%,*; H1-T,)P?Y,-PfC-Pflf -P'flfX (80) A , - \ V i L 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,c + T,PfYl - FfG,). (81) At-\ \ - \ 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 , - , ^ ^ , -B,) = E,_Ai,_]B, +P/X, -Pf*Mf). * (+i —i' — / - i * < - / - - / - < - - I / - (82) . - V i At-\ 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 of 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, xl+s denotes the rational expectation of variable x m , conditional on information available at time t. 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, =xt+xt, 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 \ + fp _ E . Y ( i - * / ) ( ! - « / / ? ) ln 0. + -1-r -lnf, eY-\ -\ne , (83) where 0 = (\-z)^—f-. The persistence of the cyclical component of output price inflation is increasing in indexation parameter yY, while the sensitivity of the cyclical component of output 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 yY. 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 I nZ = 1- 9Y WL eY -\PY \n(ulKl) + J WL eY WL eY-IPY HAL,), (84) where = ^ T\ [ ^ - y j - - ^ 1. This approximate linear net production function is subject to / \—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: lnw, = — 6Y WL 9Y-IPY 6Y WL 6Y-\PY A,L, (85) 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 l + yYB P 1 + z (l-coY)(l-coYfi) ln<Z> \ + YYp A l n ^ L + O - ^ A l n ^ -af(\ + yYp) 1 - r • ln x, - -eY-\ , E , A l n % . (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: InC, =• l + a clnC,_,+ l + a C E , lnC,+, l-aL l + aL Z;c + E , l n ^ v. (87) The persistence of the cyclical component of consumption is increasing in habit persistence parameter ac, 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 ac. 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 1 + 0 -In/.", + • i+0-EMI/ + 1 r a + / 5 ) •In 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" 1t+i \-B{\-S")] a \x\H^-aH\nHt lnC, + , - a c InC, \-al l-aL (89) 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,+l=(\-SH)lnHl+SH InO?,'"/,"); (90) where -ff = d"- 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 + 0 E , l n / f , +-1 ZK<\ + P) -In f AK ^ ' r>IK V " J (91) 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 % K . This approximate linear capital investment demand function is subject to 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 l n ^ - = 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 + l =(l-SK)lnKl+ 8K Inff/*/*), (93) where ^ - = SK. This approximate linear capital accumulation function is subject to capital 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: PGG, „„. ( Bl+X A l n ^ = C In P.Y. + 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£, =^ rln B,+\ 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: n, = - Y M 1 + Y P ^ - 1 + P i+rMp E , {\-a)M){\-coMB) coM(i+rMP) CpYJ 1 -In 0, M (96) The persistence of the cyclical component of import.price inflation is increasing in indexation parameter yM, 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 parameter coM and indexation parameter yM. This import price Phillips curve is subject to 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 -v Yf Y v. 4.C d - 0 C / ) f In Y v: f/ (97) •MJ where — = 1 - <bc — - <f>'" ~-d> Y Y y * Y Y a G —. 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, ^ ( 1 - ^ ) | + / ' ( 1 ' ) ^ + ^ ( l - ^ ) ^ + ^ ( l - ^ ) | (98) where = (l-f)^+ ([-/' )!l + (i-0'K )Il + (i-0G)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 y/. This approximate linear import demand function is subject to import technology shocks. 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 ^ ^ htP_f+_P_E/-c pf 1 + pf \ + p pf 1 + '-' 1 + \ + p (99) 1 In L, - a L ln Z,_, 1 ln C, - a ° ln C,_, . T . „ , W. + {\-G)L)(\-6>Lp) + '• - r—'— + Inf, - l n ^ r f - " — Intf rj I - a a \ - a 1-r 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 yL, to 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 coL, and to changes in the cyclical component of adjusted employment is decreasing in elasticity 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 1 ln&, = ln-rr-^ lnf, R A, 1- r ' $ eY-\PY In 4 ^ . (100) AL. ••f"t 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, fcfx. 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 Et + ln Pf'f - ln Pf, while the cyclical component of the terms of trade satisfies ln Tt = ln Pf - In Pf, where ln Pf - ln Pf . The cyclical component of nominal output depends on the contemporaneous cyclical components of nominal consumption, housing investment, capital investment, government consumption, exports, and imports according to approximate linear aggregate resource constraint: ta(«) = £ l i i ( ^ < ^ Iff Y If_ Y Y +^ln(PfXl)-y\n(P,MMl). (103) 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 of nominal government consumption according to approximate linear government dynamic budget constraint V , ln(-73,G+l) = ^[ln(-73,G) + ] + ^ rWtPfYl)--\n(PfGt) (104) where — = ~~ffpy~Y]' ^ m s a P P r o x i m a t e 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 of export revenues, and the contemporaneous cyclical component of import expenditures according to approximate linear national dynamic budget constraint 147 P r ( B 1 -1 H-B,) + i,_Xj + ^ l n ( y f l j - ^ l n ( ^ M , ) (105) where = _ T ^ ( y ' a P P r o x i m a t e l i n e a r 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 = pvC ln vf + ef, ef ~ iid /V(0, of), In 4 = pA l n i , _ , + ef, sf ~ iid Af(0,o2A), lni?/" =pv,„ Invf +ef", ef - i i d jV(0,of), \nvf =pv,< \nvf +ef, ef ~ iid /V(0,of), ln vf = pvM In vf + ef, ef ~ iid A^(0, of), \n0f=per\ndf+ef, ef ~ iid A ^ O , ^ ) , l n f f = ^ lnt% + sf, ef ~ iid AA(0,CT^ ), l n ^ = ^ ln^_ , + ef, ef ~ iid A ^ O , ^ ), ^ = / V t > + < ' , < ' ~ i i d ^ ( 0 , C T 2 ) , y?=Pvcvf.l+ef, ef ~M A T ( 0 , a 2 c ) , v ; = p ^ ; _ 1 + < ' , < r ~ i i d AA(0,cr 2 r ) . (106) (107) (108) (109) (110) (111) (112) (113) (114) (115) (116) 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, coY = coM = coL = 0 and o2, = 0 . In deterministic steady state 1 4 8 equilibrium, vc = v' =v' = vM = 1 and of = a2A = <r2;„ = a2/K = af = cr2gY = of = a]L = o-2, = a2G = erf = 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 = Kl+\nP/+ef, Mid M(0,a2r), ( 1 1 7 ) \nPf = K, + In Pf + ef, -iid M(0,cr2pc), ( 1 1 8 ) l n / f = x, + ]nPf"+ef" - i i d / V ( 0 , o - J , „ ) , ( 1 1 9 ) In i f = nt + \n.Pt'_x + ef - i i d AT(0,(71,.), ( 1 2 0 ) / h\P,G = K,+]nPf+sf, -iid Af(0,crlc), ( 1 2 1 ) \nPf = +In pM ~iid M(0,CT2PM). ( 1 2 2 ) 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,_x + ef ef ~ iid / V ( 0 , ^ ) , ( 1 2 3 ) InC, =g,+n, + \nC,_{+ef, ef - iid Af(0,a2£), ( 1 2 4 ) In I" = g, + + In If + ef, ef ~ iid / V ( 0 , a2H), ( 1 2 5 ) In 7? =g, +n, +\nJf + ef, ef - iid Af(0,cr2K), ( 1 2 6 ) 149 InG, =g,+nl+ InG,_x +£?,e?~M JV(0,a\), (127) In x, = g, + n, + In Xt_x + ef, ef - iid Af(0, a\), (128) InM, =g, + «, + l n M ( , + , ~ iid .A/(0,e4). (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 of 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 nt: \nW, = nt +g, + \nW,_x + ef, ef - iid N{Q,a2w), (130) l n l , = n, +lnl,_, +ef, ef ~ iid Af(0,a2z). (131) 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 0t = In 0, while the trend component of the rate of capital utilization satisfies In w, = 0. The trend component of the shadow price of capital satisfies In Qf = In Pf , while the trend K K component of the capital stock satisfies In = 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,<T2J), (132) lnr; = lnv, +ef, ef - iid A/"(0,c^), (133) In£, = In St_x + ef, ef - iid /V(0,a\). (134) 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. The trend component of the output based real interest rate satisfies 7tY = {-E, nf+x, while the trend component of the consumption based real interest rate satisfies rtc = it - E, 7rf+x. The trend component of the real exchange rate satisfies InQ = l n £ ( +lny^ K y - kiPf, while the trend component of the terms of ) 150 trade satisfies ln7^ = InPf -]nP*, where lnP t x = 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,_x + < , e't ~ iid Af (0 ,a\ \ (135) g, e?~M jV(0,o-2g), (136) « , = « , _ , + < , < ~ i i d Af(0,cr 2). (137) 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. 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 xt 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 second order stochastic linear difference equation 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 where vector stochastic process v, consists of the cyclical components of K exogenous variables. This vector stochastic process satisfies stationary first order stochastic linear difference equation (138) B0x, = Bxx,_x + B2E,xl+x + B,vn (139) (140) where sx, ~ iid Af(0,27,). The trend components of vector stochastic process x, satisfy first order stochastic linear difference equation 152 D0x, =D{+D2u,+ Z>3*,_, + e2j, (141) where e2l ~ iid A/"(0,272). Vector stochastic process «, consists of the levels of L common stochastic trends, and satisfies nonstationary first order stochastic linear difference equation = " , - i + £ 3 , , > (142) where £ 3 , ~ iid J\f(0,£3). Cyclical and trend components are additively separable, which implies that x,=xt+ x, and xt = xt + J t , , where i t , = x,. If there exists a unique stationary solution to multivariate linear rational expectations model (138) , then it may be expressed as: x, =Slxt_l+S2xt_,+Sivl. (143) If there exists a unique stationary solution to multivariate linear rational expectations model (139) , then it may be expressed as: i , = + r2v (. (144) These solutions are calculated simultaneously with the matrix decomposition based algorithm due to Klein (2000). Let yt denote a vector stochastic process consisting of the levels of M observed nonpredetermined endogenous variables. Also, let z, denote a vector stochastic process consisting of the levels of N-M 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 y, = F,zn (145) z,=Gi+ G2zt_t + G,e4>,, (146) where « 4 , ~ iid A/"(0,274) and z0 ~ N(z0\0,P0\0). Let wt 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 153 w, = Hxz,+s5<„ (147) where e 5, ~ iid Af(0,£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 proposed by Vitek (2006a, 2006b), which adapts the filter due to Kalman (1960) to incorporate prior information. Given initial conditions z0l0 and P 0 | 0 , estimates conditional on information available at time t -1 satisfy prediction equations: Z,\t-\ G \ + G2Zt-\\l-\' Pl\,-\ = G 2 P , - l \ , - \ G 2 + G J ^ 4 G i y,\,-= F P F1 1 \rty-\* i ' - H\P,\,-\H\ +Z5-(148) (149) (150) (151) (152) (153) 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 Pt\t - Pt\,-\ - K y , F \ P , \ l - \ - K w , H \ P t \ , - \ , (154) (155) where Ky = P,V_XFX Q~_x and Kw< = Pl[t__xHx R~f. Given terminal conditions zT}T and PT]T obtained from the final evaluation of these prediction and updating equations, estimates conditional on information available at time T satisfy smoothing equations ZI\T + J,(Z,+X,T Z,+u,). (156) 154 P,LT=P,LL+J,(PL+]LT-PL+N,)JJ, (157) where / , = P^G7 P~+\LT. Under our distributional assumptions, these estimators of the unobserved state vector are mean squared error optimal. Let 0 e0cz~RJ 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\lT)<xf(lT\0)f(0), (158) where Jt = {{ys}'s^,{ws}'s^}. Under the assumption of multivariate normally distributed signal and state innovation vectors, together with conditionally contemporaneously uncorrelated signal vectors, conditional density function / ( 1 T \ 0) satisfies: f(TT\0) = flf(y,\l^,0)-flf(w,\l,_],0). (159) Under our distributional assumptions, conditional density functions f(yt |2~_ p0) and f{w,\l,_x,0) satisfy: f(y, | J f _„0) = ( 2 / r ) ^ | p e x p | - i ( j , - y , ^ ) 7 Q ^ i y , - J,,,-,)}, (160) f(w, | Z,_„0) = (2n)^ | /?,„., r exp j-ito - »v,|(_,)T Rffw, - (161) Prior information concerning parameter vector 0 is summarized by a multivariate normal prior distribution having mean vector 0X and covariance matrix Q: 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. Under regularity conditions stated in Geweke (2005), posterior mode 0T satisfies 155 (163) where 0o<E0 denotes the pseudotrue parameter vector. Following Engle and Watson (1981), Hessian may be estimated by 1 ^ 1 1 (164) + - V , V > / ( < U where E,_, [vgVTg In f(y, | J M , 0 r ) ] = - V ^ . ^ V ^ , , , . , - { v . f i j . . ^ , ® g ^ , ) V , e , M , E,., [ V S V J lnf(w, | J,_.,0 r)] = - V F L W J _ , J ? - 1 _ 1 V F L W ( | ( . I - { V , * J _ , ( * - ' _ . ® , 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 SH is restricted to equal 0.01, implying an annualized deterministic steady state equilibrium depreciation rate of approximately 0.04, while depreciation rate parameter 8K 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 \-(/> c, housing investment import share \-(f>' , capital investment import share X — jr1 , and government consumption import share \-(f>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 * s restricted to equal 0.11. In deterministic steady state equilibrium, the foreign consumption import share 1 - (j>CJ, foreign housing investment import share 1 - <fi' J , foreign capital investment import share 1 - </>' J , and 156 foreign government consumption import share \-<fP'f are restricted to equal 0.02. 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 of great ratios Ratio Value Ratio Value CIY 0.6000 WLIPY 0.6500 I" IY 0.0600 HIY 1.5000 JK.IY 0.1138 K/Y 1.4224 GIY 0.2000 BG IPY -0.4950 XIY 0.3183 BIPY -0.6487 MIY 0.2921 Note: Deterministic steady state equi l ibr ium 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 of 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 yY is 0.75, implying considerable output price inflation inertia, while the prior mean of nominal rigidity parameter a>Y implies an average 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 ac 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 aH is 0.95, while the prior mean of housing investment adjustment cost parameter x" 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 xK is 5.75, implying moderate sensitivity of capital investment to changes in the relative shadow price of capital. The prior mean of indexation parameter yM is 0.75, implying moderate import price inflation inertia, while the prior mean of nominal rigidity parameter eoM 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 yL is 0.75, implying considerable sensitivity of the real wage to changes in consumption price inflation, while the prior mean of nominal rigidity parameter mL implies an average duration of wage contracts of two years. The prior mean of habit persistence parameter aL 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 %Y is 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. 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, 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. As 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 of interest RINTC_P (APR) RINTC_F (APR) RINTC_S (APR) Note: Estimated levels are represented by black lines, whi le blue and red lines depict estimated f lexible price and wage equi l ibr ium components and trend components, respectively. Symmetr ic 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 Economic Cyc le 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. As 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 L R E X C H _ P L R E X C H _ F L R E X C H _ S Note: Observed levels are represented by black lines, whi le blue and red lines depict estimated f lexible price and wage equi l ibr ium components and trend components, respectively. Symmetr ic 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 Economic Cyc le 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 st ~ i id JV(0, I). Vector stochastic process yt consists of domestic output price inflation nf, domestic output ln Yt, domestic consumption price inflation nf, domestic consumption ln C , , domestic housing investment price inflation n\ , domestic housing investment ln if, domestic capital investment price inflation nf, domestic capital investment ln if, domestic import price inflation nf, domestic exports ln Xt, domestic imports ln M,, domestic nominal interest rate z,, nominal 163 exchange rate In £ t , foreign output price inflation nf'f, foreign output In Y/, foreign consumption I n C / , foreign housing investment l n / , w / , foreign capital investment \nlff, and foreign nominal interest rate if. 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 in im ized values of 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 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 of tests of overidentifying restrictions Test Statistic P Values Asymptotic Parametric Bootstrap Nonparametric Bootstrap 359.2387 0.0000 1.0000 1.0000 Note: This l ikel ihood 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 DLPGDP (APR) DLPCON (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 30 35 40 5 10 15 20 25 30 35 40 DLPHINV (APR) DLPKINV(APR) 5 10 15 20 25 30 35 40 0 5 10 15 2 0 2 5 3 0 3 5 4 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 DLPIMP (APR) NINT(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 2 0 2 5 3 0 3 5 4 0 0 5 10 15 2 0 2 5 3 0 3 5 4 0 15 20 25 30 35 40 DLPGDPF (APR) 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 NINTF (APR) 0 5 10 15 20 25 30 35 40 Note: Theoretical impulse responses to a 50 basis point monetary pol icy shock are represented by black lines, whi le blue lines depict empir ical impulse responses to a 50 basis point monetary pol icy shock. Asymmetr ic 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) LRGDP DLPCON (APR) LRCON 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 X 35 40 O S 1 0 1S 20 25 30 35 40 DLPHINV(APR) LRHINV DLPKINV(APR) LRKINV 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 2 0 2 5 3 3 3 5 4 0 0 5 10 15 2 0 2 5 3 0 3 5 4 0 DLPIMP(APR) LREXP LRIM 3 NINT(APR) 0 5 10 15 20 25 30 35 40 0 5 10 15 2O 25 30 35 4O 0 5 10 15 2 0 2 5 3 0 3 5 4 0 0 5 10 15 20 25 3 0 3 5 4 0 LNEXCH DLPGDPF (APR) LRGDPF LRCONF 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 2 0 2 5 3 0 3 5 4 0 0 5 10 15 20 25 3 0 3 6 4 0 LRHINVF LRKINVF NINTF (APR) 0 5 10 15 2 0 2 5 3 0 3 5 4 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 3 3 3 5 4 0 Note: Theoretical impulse responses to a 50 basis point monetary pol icy shock are represented by black lines, whi le blue lines depict empir ical impulse responses to a 50 basis point monetary pol icy shock. Asymmetr ic 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 DSGE 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 DSGE 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 DSGE 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 Prior Distribution Posterior Distribution Mean Standard Error Mode Standard Error ac 0.950000 0.000950 0.942380 0.000880 a" 0.950000 0.000950 0.947530 0.000651 aL 0.950000 0.000950 0.940960 0.000905 x" 1.250000 0.001250 1.249300 0.001250 xK 5.750000 0.005750 5.746900 0.005750 1 0.750000 0.000750 0.750400 0.000750 K 0.100000 0.000100 0.099995 0.000100 V 1.500000 0.001500 1.500100 0.001500 a 2.750000 0.002750 2.751400 0.002749 9 0.750000 0.000750 0.750000 0.000750 f 0.750000 0.000750 0.750100 0.000750 rM 0.750000 0.000750 0.750140 0.000750 rL 0.750000 0.000750 0.750080 0.000750 0.875000 0.000875 0.876850 0.000864 wM 0.875000 0.000875 0.874030 0.000873 coL 0.875000 0.000875 0.879820 0.000868 <T 1.500000 0.001500 1.499300 0.001500 ? 0.125000 0.000125 0.124940 0.000125 ia -0.100000 0.000100 -0.099998 0.000100 <T 1.000000 0.001000 0.999100 0.001000 P,.c 0.850000 0.000850 0.850280 0.000850 PA 0.850000 0.000850 0.851190 0.000848 PV,~ 0.850000 0.000850 0.850130 0.000850 />„,- 0.850000 0.000850 0.851230 0.000850 P„« 0.850000 0.000850 0.849990 0.000850 Po< 0.850000 0.000850 0.850140 0.000850 p„* 0.850000 0.000850 0.850020 0.000850 0.850000 0.000850 0.850020 0.000850 p>> 0.850000 0.000850 0.853240 0.000840 0.850000 0.000850 0.849960 0.000849 0.850000 0.000850 0.849970 0.000849 - 00 0.204890 0.062569 - 00 0.319060 0.050146 a1,. - 00 0.270540 0.058117 <r\ - 00 0.245970 0.051460 o-\, - 00 0.237440 0.035607 4 - 00 0.232260 1.361800 - 00 0.258260 2.058700 - 00 0.251000 12.298000 - 00 0.208960 0.028450 - 1 CO 0.260570 0.038933 a1, *{, - o o 0.254010 0.031108 - 00 0.273060 0.066943 °\< - 00 0.097028 0.012105 -I,, - 00 0.242180 0.051732 °\, -V, - 00 0.241410 0.052788 - OO 0.259430 0.537140 - OO 0.092639 0.106830 -I - CO 0.227650 13.374000 - 00 0.248370 2.473000 °\„ - o o 0.003574 0.000491 - OO 0.116640 0.015587 - o o 0.268170 0.029474 - o o 0.617930 0.077548 - 00 0.059436 0.008994 4 - OO 0.582020 0.073046 172 Parameter Prior Distribution Posterior 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 4 - 00 0.023448 0.002263 4 - 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 a2. - CO 0.000138 0.000019 °l - 00 0.000004 0.000006 o-] - 00 0.000025 0.000009 <> - 00 0.000068 0.000012 - 00 0.000023 0.000011 - 00 0.000029 0.000010 Note: A l l observed endogenous variables are rescaled by a factor o f 100. Figure 3.5. Predicted cyclical components of observed endogenous variables 173 1 z***^ 1975 198 0 1985 1990 1995 1000 20C 1990 198S 2000 2005 Note: Estimated cycl ical components are represented by blue lines, whi le red lines depict estimated deviations from f lexible price and wage equil ibr ium 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 Economic Cyc le Research Institute reference cycle. Figure 3.6. Filtered cyclical components of observed endogenous variables 174 L^i (Mit LPKINV f \ r\ f | A / wm mm u -• ft Note: Estimated cycl ica l components are represented by blue lines, whi le red lines depict estimated deviations from f lexible price and wage equi l ibr ium 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 Economic Cyc le Research Institute reference cycle. Figure 3.7. Smoothed cyclical components of observed endogenous variables 175 LPKINV L A . i\ A . (V A Afote: Estimated cycl ica l components are represented by blue lines, whi le red lines depict estimated deviations from f lexible price and wage equi l ibr ium 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 Economic Cyc le Research Institute reference cycle. 176 Figure 3.8. Predicted trend components of observed endogenous variables LPGDP -20. LRGDP z.y LPCON 11J0 LRCON ... LPHINv LRHINV LPKINV LRKINV LPGOV LRGOV -w a r / ... LREXP LPIMP LRIIvP LNWAGE LEW I1M NINT (APR) LTAXRATE LNEXGH LPGDPF LRGDPF / LRCONF LRHINVF LRKINVF LRGOVF LPIWF MO. Jt*S l l l . l l . , , LNWAGEF LE W F NINTF (APR) LTAXRATEF S 1*75 19*0 1965 1MO 1»5 2000 200 US Note: Observed levels are represented by black lines, whi le blue and red lines depict estimated f lexible price and wage equi l ibr ium components and trend components, respectively. Symmetr ic 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 Economic Cyc le Research Institute reference cycle. Figure 3.9. Filtered trend components of observed endogenous variables 177 ' 1240 E 1280. J Note: Observed levels are represented by black lines, whi le blue and red lines depict estimated f lexible price and wage equi l ibr ium components and trend components, respectively. Symmetr ic 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 Economic Cyc le Research Institute reference cycle. 178 Figure 3.10. Smoothed trend components of observed endogenous variables J .40. A J J E / / 13.0 i f ' Note: Observed levels are represented by black lines, whi le blue and red lines depict estimated f lexible price and wage equi l ibr ium components and trend components, respectively. Symmetr ic 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 Cyc le Research Institute reference cycle. Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equi l ibr ium are represented by blue lines, whi le green lines depict theoretical impulse responses to a unit standard deviation innovation under flexible price and wage equi l ibr ium. Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equi l ibr ium are represented by blue lines, whi le green lines depict theoretical impulse responses to a unit standard deviation innovation under f lexible price and wage equi l ibr ium. Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equi l ibr ium are represented by blue lines, whi le green lines depict theoretical impulse responses to a unit standard deviation innovation under f lexible price and wage equi l ibr ium. Figure 3.14. Theoretical impulse responses to a foreign output technology shock 182 Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equi l ibr ium are represented by blue lines, whi le green lines depict theoretical impulse responses to a unit standard deviation innovation under f lexible price and wage equi l ibr ium. 183 Note: Theoretical impulse responses to a unit standard deviation innovation under sticky price and wage equi l ibr ium are represented by blue lines, whi le green lines depict theoretical impulse responses to a unit standard deviation innovation under f lexible price and wage equi l ibr ium. 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 equi l ibr ium are represented by blue lines, whi le green lines depict theoretical impulse responses to a unit standard deviation innovation under f lexible price and wage equi l ibr ium. 185 Figure 3.17. Theoretical forecast error variance decompositions under sticky price and wage equilibrium •NO TAIJ BUM! er •OMRMLff •lTWETAt_MT •LTMCTAi.U7 mVUJT sNU a 'i' TAUT 187 Figure 3.19. Mean squared prediction error differentials for levels .'l«Ni-| • WOO. 5 6 7 800-5 6 7 2 3 4 5 6 7 LRGDPF 2 3 4 5 6 7 LRKINV 2 3 4 5 6 7 NINT 5 6 7 1(100. 5 6 7 5 6 7 8 Note: Mean 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. Symmetr ic 9 5 % confidence intervals account for contemporaneous and serial correlation o f forecast errors. Figure 3.20. Mean squared prediction error differentials for ordinary differences 188 5 6 7 5 6 7 5 6 7 8 1 2 3 5 6 7 2 3 4 5 6 7 DLRIMP 2 3 4 5 6 7 8 DNINT 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 DLRCONF -0 4 -o.a. 5 6 7 5 6 7 1 2 3 4 5 6 7 8 5 6 7 8 Note: Mean 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. Figure 3.21. Mean squared prediction error differentials for seasonal differences 189 SDLPCON 2 3 4 5 6 7 8 2 3 4 5 6 7 8 SDLPIMP 111!. SDLNEXCH 2 3 4 5 6 7 SDLPGDPF 2 3 4 5 6 7 2 3 4 5 6 T SDLRCONF 2 3 4 S 6 7 1 2 3 Note: Mean 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. Figure 3.22. Dynamic forecasts of levels of observed endogenous variables 190 Note: Real ized outcomes are represented by black lines, whi le blue lines depict point forecasts. Symmetr ic 9 5 % confidence intervals assume multivariate normally distributed and independent signal and state innovation vectors and known parameters. 191 Figure 3.23. Dynamic forecasts o f ordinary differences of observed endogenous variables N— ' Note: Real ized outcomes are represented by black lines, whi le blue lines depict point forecasts. Symmetr ic 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: Real ized outcomes are represented by black lines, whi le blue lines depict point forecasts. Symmetr ic 9 5 % confidence intervals multivariate normally distributed and independent signal and state innovation vectors and known 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. Al t ig , D. , L . Christiano, M . Eichenbaum and J. Linde (2005), Firm-specific capital, nominal rigidities and the business cycle, Federal Reserve Bank of Chicago Working Paper, 1. A o k i , K . , J. Proudman and G. 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Woodford (1997), A n optimization-based econometric framework for the evaluation of 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 of 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 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. 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 of 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. Yun , T. (1996), Nominal price rigidity, money supply endogeneity, and business cycles, Journal of Monetary Economics, 37, 345-370. 196 C H A P T E R 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. As 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 of 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. As 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. As 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 of 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 DSGE 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 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(. v and labour supply f s represented by intertemporal utility function (1) where subjective discount factor B satisfies 0 < B < 1. The representative household consists of Ns identical members, and has intratemporal utility function: u(CKs,L.s) = Nsv (2) The intratemporal utility function of the representative household member is multiplicatively separable: i l - l / o -1-l/cT exp -x l - l / o -I + I/77 (3) 200 In order to ensure the existence of a well defined balanced growth path, the marginal utility of 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 s which completely spans all relevant uncertainty. It also holds a diversified portfolio of shares {xtJ s}^=0 in domestic intermediate good firms which pay dividends {17'. s }\=0. The representative household supplies final labour service L{ s , earning labour income at nominal wage Ws. These sources of wealth are summed in household dynamic budget constraint: E, il.^B,,, + | y,^,,,,di = Bu + J (77 ; v + Vjs)xL.sdj + WtLu - PscC.s. (4) According to this dynamic budget constraint, at the end of period 5 , the representative household purchases a diversified portfolio of state contingent bonds Bt J + 1 , where Qs 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. It also purchases a diversified portfolio of shares {JC,. . v + 1 } .^=0 at prices {Vjv}].=0. Finally, the representative household purchases final consumption good C ; s at price Pf. In period t, the representative household chooses state contingent sequences for consumption {QJ™, , labour supply {Lis}™=l, bond holdings {Bis+X}™=l, and share holdings {{xi,j,s+\)Xj=tM=t t 0 maximize intertemporal utility function (1) subject to dynamic budget constraint (4) and terminal nonnegativity constraints B: T+j > 0 and xt J J + l > 0 for T - » oo . In equilibrium, selected necessary first order conditions associated with this utility maximization problem may be stated as uc(C„L,)^PfA„ (5) -uL(C„Lt) = WtXn (6) fi,+i4=M+1, (7) F y V / l , = ^ E , ( 7 7 . , + , + F . , + 1 H + 1 , (8) where Ajs 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 l i m £ A ^ Q B 0 , (9) l i m ff^Ly• X =0. (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 satisfy ^ - = E, QLL+]. Combination of this equilibrium asset pricing relationship with necessary first order conditions (5) and (7) yields intertemporal optimality condition uc(Cl,Ll) = p^,(\ + il)^uc(Cl+l,Ll+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 '~uc(c„Ly Pr { 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 }!=0 to domestic households at price Vjj. Recursive forward substitution for VjJ+s 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. s , defined as revenues derived from sales of differentiated intermediate output good Y. s at price Pjs less expenditures on final labour service L,, nj,s = Pj,sYj,s-WsLUs. (15) The representative intermediate good firm rents final labour service Ljs given labour augmenting productivity coefficient As to produce differentiated intermediate output good y. 5 according to production function Y , , - A ^ (16) where As > 0. 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 {Lis}™=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 PsY@JiS denotes the Lagrange multiplier associated with the period s production technology constraint. This necessary first order condition equates real marginal cost 0t to the 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 Yjt supplied by intermediate good firms in a monopolistically competitive output market to produce final output good Yt according to constant elasticity of 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- \plYj<tdj, (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: (PY\ P> V ' J Y. (20) 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 of intermediate good firms adjust their price to account for past steady state output price inflation according to indexation rule: p' p.\ i f - v (22) 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: e 0-1 CO f pY p Y \ pY pY pYy s s E , 5 > s=t ( p Y p Y \ rt-\  rs (23) PYY 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 = (i~a>)(pry-d+co ( pY \ rt-\ pY pY rt-\ \rt-2 J \-0 (24) 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 \ Pr=~hp'^ (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, £ PYJ Qs=^y-, (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 Cfl to produce final consumption good C, according to constant elasticity of substitution production function 206 C. = _!_ \_ if/-\ (27) where 0 < </> < 1 and ^ > 1. The representative final consumption good firm maximizes profits derived from production of the final consumption good nf =PfC,~PjCKl-EtPjfCf!, (28) 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: pC V ~ J (29) (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: Ptc =[<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: chj = (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 /3uc(Ct+„LM) Pf Q ' M ~ uc(C„Ll) Pff ( 3 4 ) _/3uc(Cf+l,L{+i) Pff £, ucicf,L{) pffel+f ( 3 5 ) respectively. Combination of these intertemporal optimality conditions with real exchange rate definition (26) yields international risk sharing condition: uc(C„L,) 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,) + vl, (37) where £, > 1 and ^ > 0. As 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,=ChJ + CffJ. (38) The assumption that the domestic economy is of negligible size relative to the foreign economy is represented by parameter restriction <f>f = 1, under which PjJ = PICJ 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, xl+s denotes the rational expectation of variable x ( + ! , conditional on information available at time t. Also, x, denotes the cyclical component of variable xt, xt 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 co - + -A,N, </> f yf ^ l n ^ + O/zQ + ^ - ^ l n Q AN. ,(39) where /?' = /?(l + «)(l + g ) M / C T . 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^ — + — 0 n ln- 1-0 A,N, (j> Yf l n ^ ^ + O/zO + ^ - ^ l n g ; +(1 - (/>) In - P - - p\\ - 0)E, In % (40) 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 A,N, A,+XN, \ + (/>(o-\) 0 - 1 9 or, -E,ln-P-i -< \ + (/>(o~\) 0-1 0 E, A ln A ,+\ + y/(\ + 0)E, ln ^ + (41) 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 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: ^ 2CT + K 1 + M 1 - < * ) ) I + ^ C T - I ) 0-1 ln A,N, A! Nf + l n ^ r + ( l - ^ ) l n - ^ -Af Nf .(43) The cyclical component of the real interest rate satisfies r, = it - E ( 7rf+x, while the cyclical component of the real exchange rate satisfies InQ = ln£ ( + \nPfJ - XnPf. 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: In A, = p.A ln 4_, + ef, ef ~ iid /V(0, o\), (44) lnNt=p.]nN,_l+ef, ef ~M M(0,o]\ (45) v,=pvv,-i +<> < - iid M(Q,o2v). (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 <j2 = 0. 4.3.1.2. Trend Components The trend components of the prices of output In Pf and consumption In Ptc are driven by common deterministic or stochastic trend In Pt, while the trend component of output In Yt is driven by common deterministic or stochastic trends In A, and In Nt: In Pf = 7i + \nPtf+ef, (47) l n T f =/r + ln/>51+<?f, (48) \nY,=g + n + \nY,_x+ef+ef. (49) 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 it and nominal exchange rate ln£, are time and state independent: W - P (50) l n ^ = ln^_ , . (51) The trend component of the real interest rate satisfies Tt = it - E, nf+x , while the trend component of the real exchange rate satisfies ln(j> = ln£, + \nPlY'f - InPf. Long run balanced growth is driven by three common deterministic or stochastic trends. The trend components of the price level In Pt, productivity In A T , and population In TV, follow random walks with constant drifts: XnPt = n + ln/j_, + ef, ef ~ iid N(0,<T\), (52) In A, - g + In 4_, + ef, ef ~ iid M(Q,CJ\), (53) In N, = n + In /Y ,_ , + ef, ef ~ iid AA(0, a\). (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 A,N, </> l n ^ ^ + a/(l + 0 - 0 1 n Q 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 — + — \4> vj ln-A,N, <f> l n ^ r - + GKl + 0-0)ln£> + ( l - ^ ) l n - S — / ? ( 1 - ^ ) E , I n % -(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, e err - E , In-4 i-< l + ^(o--l) 0 - 1 E ( A In . Yh + ¥(\ + 0)E, In % • 9 (57) 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 of consumption price inflation and output according to monetary policy rule: 1,-1= £ 0 r c - kf) + CQn Yt — \nYl) + vr (58) 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: 1 + ^(cr - 1 ) 0-1 Y Yf A,N, A!N{ < J + l n A + ( 1 _ ^ ) l n A A! NF (59) The cyclical component of the real interest rate satisfies rr = it - E, nf, while the cyclical component of the real exchange rate satisfies InQ = ln£, + \aPjJ - \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 = PA In + ^ , sf ~ iid /V(0, CT? ), (60) lnA> = / > . t o A r _ 1 + < , e * ~ i i d / v - ( 0 , t 7 ? ) , (61) K = PvK-i + < , < ~ i i d /V(0,a 2 v). (62) 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 Yt follows a random walk with time varying drift g, +nt: ln Pf = nt + ln Pf + sf, sf ~ iid /V(0, a1-, \ (63) ln 7 f = x, + ln i f + sf, sf ~ iid Af(0, a2pc), (64) ln Y, = g l + n, + ln + sf, ~ iid /V(0, cr^). (65) 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), (66) IncT = ln^_ , + ef, sf ~ iid /V(0,<r|). (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 7t = it - E r nf, while the trend component of the real exchange rate satisfies InQ = l n ^ + l n ^ r / - l n ^ y . 215 Long run balanced growth is driven by three common stochastic trends. Trend inflation nt, productivity growth g, , and population growth nt follow random walks without drifts: nt = + < , < ~ iid Af(0, a]), (68) g, =&_,+*,*, ef~x\AM^a\\ (69) n, = «,_,+ s:, < ~ iid /V(0, o-„2). (70) 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.1 Furthermore, under the first best approximation to this DSGE model, the exogenous stochastic processes associated with the stochastic trend component specification nest commonly employed types of exogenous stochastic processes under parameter restrictions.2 It follows that 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 xt denote a vector stochastic process consisting of the levels of /V nonpredetermined endogenous variables, of which M are observed. The cyclical components of this vector stochastic process satisfy second order stochastic linear difference equation AQx, = AlX^+A2E,xl+l +A3x, + A4v„ (71) where vector stochastic process 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 B0xl=Blxl_:+B2Elxl+l+B3v„ (72) where vector stochastic process v, consists of the cyclical components of K exogenous variables. This vector stochastic process satisfies stationary first order stochastic linear difference equation v,=Cxv,_x+eu, (73) ' Under the first best approximation wi th common deterministic trends al, = alc = a~ = al = al = a\ = a2 = a2„ = 0 and Op. , = al, = al, = a2, = a2, = a2, = 0 . Under the first best approximation with common stochastic trends al, = alf = al , a1- =O 1- JKJ 1- , t£=o\=al= ol =o-2=0, o i ,=<£,, cr1-, = a\, + a2-, and al, = a1, = a2. =a2, = 0 . 2 Consider the exogenous stochastic process governing the evolut ion o f the level o f productivity. Under the first best approximation, this exogenous stochastic process has structural form representation ln A, = ln A, + ln A, , where ln A, = ln A,_, + with ~ i id JV (0 , a?), and ln A, = g + ln AIA + ef with sf M(0,al). Under the assumption that sf and ef are independent, it has reduced form representation A l n A, = (1 - p-A)g + pt A l n A,_t + ef + 9Aef_, with sf — i id A/"(0, a2A), where 0A and a\ are functions o f p-A , < 7 ? and al. 217 where e,, ~ i id 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 D0x, = Z>, + Z) 2 u, + Z) 3x + e2,, (74) where e2l — iid A/"(0,272). Vector stochastic process ut consists of the levels of L common stochastic trends, and satisfies nonstationary first order stochastic linear difference equation ",=«,_, +EXI, (75) where e3, ~ iid A/"(0,273). The parameter matrices differ depending on whether the first best or second best approximation is employed, in addition to whether common trends are deterministic or stochastic.3 Cyclical and trend components are additively separable, which implies that 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,=Six,_l+S2xt_}+S3vl. (76) 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 yt denote a vector stochastic process consisting of the levels of M observed nonpredetermined endogenous variables. Also, let z, denote a vector stochastic process consisting of the levels of N-M 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 common deterministic trends D2 = 0 , Z2 = 0 and I, = 0 . Under the first best approximation with common stochastic trends D2 = 0 and X , = 0 . 218 y,=Fxzt, (78) z, = G 1 + G 2 z / _ I + C 3 e 4 i „ (79) where «s4, ~ iid /V(0,274) and z0 ~ N(z0l0,P0]0). 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>, = / / , z , + e 5 „ (80) where e5j ~ i id A/"(0,275). 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 Pt 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 0 | 0 and PQ{0, estimates conditional on information available at time t-l satisfy prediction equations: ~ G \ + G 2 V l | , - l > (81) p 1 t\t-\ = G2P,-\\t-\Gl + G 3 ^ 4 G l i (82) •V,|.-i = FT (83) = F P FT (84) (85) ^1,-1 = HxPt^Hj +275. (86) 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 = V i + K y , U ~ ->Vi) + K», (w> ~ w « k - i ) ' (8 7> 219 P,\t = P , \ , - \ ~ K y l F l P , \ , - \ - K w , H \ P t \ , - \ ' (88) where K y< = P^Ff Qf_x and K = PlXl_xHxrR^_x. Given terminal conditions zw and PT[r obtained from the final evaluation of these prediction and updating equations, estimates conditional on information available at time T satisfy smoothing equations ZI\T = z<\t + (Zi+\\r ~ zi+\\t)' (89) P,T=P„+J,(P,+ ,T-PI+„)J,\ (90) where / , = P^Gj Pt~+\tl. Under our distributional assumptions, these estimators of the unobserved state vector are mean squared error optimal. Let ^ € 0 c R J 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(O\lT)Kf(TT\0)f(6), (91) where 2~ = {{ys}'s={,{ws}'s=\} • Under the assumption of multivariate normally distributed signal and state innovation vectors, together with conditionally contemporaneously uncorrected signal vectors, conditional density function / ( 1 T 19) satisfies: f(iT\o)-Ylf{y,\i^e).f[f(Wl | j , _ „ 0 ) , (92) Under our distributional assumptions, conditional density functions f(y,\l,_x,0) and f(w, \1,_X,G) satisfy: M 1 r j f(y,\Xt_x,e) = {2K) ' | g , M e x p | - ^ U - y^f QfM, - J>„_,)j, (93) — - 1 f 1 1 / ( W , | Z , _ „ 0 ) = (2;r) 2 | ^ H | ^ e x p ^ - ( w , - W l l , J R f x ( W , ~ W l l , _ x ) [ . (94) Prior information concerning parameter vector 0 is summarized by a multivariate normal prior distribution having mean vector 0X and covariance matrix Q: f{0) = (2ar)~* | Q f exp{- | (0 -0 X ) J Q~\0 - 0 , ) j . (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. Under regularity conditions stated in Geweke (2005), posterior mode 0T satisfies (96) where 0oe0 denotes the pseudotrue parameter vector. Following Engle and Watson (1981), Hessian ^ may be estimated by * r = 7 Z E M [ v f V > / ( ^ J ^ I , t f r ) ] + l j ; E M [ v f V ; i n / ( W | | I,_JT) 1 t=i 1 i=i + ^ v > M ) , (97) where ^eVTg\nf(0T) = -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. This sample size corresponds to 50 years of quarterly observations. 221 Table 4.1. True values of parameters Parameter Value Parameter Value Parameter Value P 0.9900 P\ 0.5000 0.5000 n 1.0000 Pi, 0.5000 g 0.2500 (O 0.8000 P , 0.5000 n 0.5000 <t> 0.7000 a2 A 0.2500 °? 0.2500 V 1.5000 °* 0.2500 0.1250 a 1.0000 a2v 0.2500 4 0.1250 0 7.6667 A1 0.2500 0.2500 $ 1.5000 0.2500 0.1250 i 0.1250 0.2500 4 0.1250 Note: The 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 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. 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 A S E i 1.000000 1.000000 0.100000 1.001700 0.002682 0.098380 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 a2 A 0.250000 - oo 0.247010 0.003462 0.042837 < 0.250000 - oo 0.248150 0.002833 0.032829 a] 0.250000 - oo 0.249110 0.001892 0.046709 0.250000 - oo 0.245890 0.004309 0.050068 0.250000 - oo 0.247250 0.003572 0.032892 0.250000 - oo 0.249250 0.002178 0.037266 4 0.000000 - 00 0.009145 0.009146 0.001130 0.000000 - OO 0.009100 0.009101 0.001106 0.000000 - OO 0.009897 0.009897 0.012641 a2- 0.000000 - OO 0.009766 0.009766 0.005081 4 0.000000 - 00 0.009765 0.009766 0.003443 0.000000 - oo 0.009404 0.009404 0.001445 0.000000 - 00 0.009900 0.009900 0.012376 of/ 0.000000 - OO 0.009650 0.009650 0.002895 al 0.000000 - OO l .OOxlO" 6 1.00x10"* 3 . 3 6 x l 0 " 6 a\ 0.000000 - OO 5 .00x l0" 7 5.00x10"' 7 .27x |0" 6 al 0.000000 - OO 5.00* 1(T7 5.00x10"' 7.27x10" 6 a], 0.000000 - CO l .OOxlO" 6 l .OOxlO" 6 2 . 8 7 x l 0 " 6 a2, s 0.000000 . - OO 5 .00X10 - 7 5.00x10"' 6.66x10" 6 al. 0.000000 - oo 5.00x10"' 5.00x10"' 6.66x10" 6 Note: The 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 a2g = al and cr*, = a2, . 224 Table 4.3. Experimental results under stochastic trend specification, parameters Parameter True Value Prior Distribution Posterior Distribution Mean SE Mode SE A S E 1 1.000000 1.000000 0.100000 1.000700 0.005004 0.099044 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 i 0.125000 0.125000 0.012500 0.124640 0.000718 0.012489 Pi 0.500000 0.500000 0.050000 0.527120 0.029400 0.034489 P/, 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 4 0.250000 - 00 0.257040 0.009317 0.050935 al 0.250000 — 00 0.252440 0.004238 0.071539 4 0.250000 — CO 0.251070 0.003634 0.071831 4 0.250000 — 00 0.253530 0.005909 0.051491 0.250000 — CO 0.251380 0.004521 0.054079 4 0.250000 — CO 0.251700 0.006655 0.026205 4 0.250000 - CO 0.252430 0.008012 0.026170 4 0.250000 — CO 0.243260 0.008393 0.055033 a2 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 4 0.250000 — 00 0.244940 0.007746 0.047782 4 0.000000 - 00 0.009455 0.009457 0.003662 a2, 0.000000 — CO l .OOxlO" 6 1.00x10-* 2.04x10" 5 0.000000 — CO 5 .00x l0" 7 5.00x10"' 2.05*10" 5 al 0.000000 — CO 5.00x10"' 5.00x10"' 2 . 0 5 x l 0 " 5 °\ 0.000000 - 00 l . O O x l O ' 6 1.00x10"* 4 . 0 3 x l 0 " 5 4 0.000000 - 00 5.00x10"' 5.00x10"' • 1.72xl0" 5 4 0.000000 - 00 5.00x10-' 5.00x10"' 1.72xl0" 5 Note: The 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 of posterior modes from true values, and asymptotic standard errors across 100 replications, respectively. The parameters are estimated subject to identifying restrictions a\ = al and a2, = a2, . 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 Cyclical Component Trend Component Bias R M S E A R M S E Bias R M S E A R M S E Bias R M S E A R M S E Predicted 0.002016 1.978920 1.977000 -0.000881 2.005960 1.921080 0.002897 0.517920 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: 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 uncondit ional 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 of interest Estimate Level Cyclical Component Trend Component Bias R M S E A R M S E Bias R M S E A R M S E Bias R M S E A R M S E Predicted Filtered Smoothed 0.006736 0.006227 0.016802 2.092360 1.362800 1.396280 2.033000 1.333400 1.197800 -0 .010660 -0 .012104 -0 .001545 2.094880 1.455360 1.439960 1.965280 1.441680 1.294040 0.017396 0.018331 0.018347 0.643840 0.647920 0.600920 0.856360 0.762360 0.587240 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 f Bias R M S E A R M S E Bias R M S E A R M S E Bias R M S E A R M S E Predicted Filtered Smoothed -0 .001269 -0.000612 0.000476 0.633730 0.236010 0.237530 0.656290 0.319740 0.293560 0.005225 0.006317 0.009998 0.646410 0.313410 0.280010 0.645090 0.402800 0.349460 -0 .006495 -0 .006930 -0.009522 0.146420 0.148620 0.151880 0.252810 0.188510 0.145340 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. Table 4.7. Experimental results under stochastic trend specification, natural exchange rate Estimate Level Cyclical Component Trend Component Bias R M S E A R M S E Bias R M S E A R M S E Bias R M S E A R M S E Predicted -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 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 of intermittent structural breaks in real data sets spanning a reasonably long period, irrespective of whether common trends are deterministic or stochastic. In order to analyze the robustness of the finite sample properties of the estimation procedure under consideration to forms of 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 of evaluating the implications of individual forms of model misspecification is high, while the exogenous stochastic processes associated with the estimated model nest commonly employed types of exogenous stochastic processes. References Calvo, G . (1983), Staggered prices in a utility-maximizing framework, Journal of Monetary Economics, 12, 983-- 998. Canova, F., M . Finn and A . Pagan (1994), Evaluating a real business cycle model, Non-stationary Time Series Analysis and Cointegration, Oxford University Press. Clarida, R., J. Gal i and M . Gertler (2001), Optimal monetary policy in open versus closed economies: A n integrated approach, American Economic Review, 91, 253-257. Clarida, R., J. Gal i and M . Gertler (2002), A simple framework for international monetary policy analysis, Journal of Monetary Economics, 49, 877-904. Clements, M . and D. Hendry (1999), Forecasting Non-stationary Economic Time Series, M I T Press. Engle, R. and M . Watson (1981), A one-factor multivariate time series model of metropolitan wage rates, Journal of the American Statistical Association, 76, 774-781. Gertler, M . , S. Gilchrist and F. Natalucci (2001), External constraints on monetary policy and the financial accelerator, Unpublished Manuscript. Geweke, J. (2005), Contemporary Bayesian Econometrics and Statistics, Wiley. Goodfriend, M . and R. King (1997), The new neoclassical synthesis and the role of monetary policy, NBER Macroeconomics Annual, 231-283. Kalman, R. (1960), A new approach to linear filtering and prediction problems, Transactions ASME Journal of Basic Engineering, 82, 35-45. King , R., C. Plosser and S. Rebelo (1988), Production, growth and business cycles: The basic neoclassical model, Journal of Monetary Economics, 21, 195-232. Kle in , P. (2000), Using the generalized Schur form to solve a multivariate linear rational expectations model, Journal of Economic Dynamics and Control, 24, 1405-1423. Kydland, F. and E. Prescott (1982), Time to build and aggregate fluctuations, Econometrica, 50, 1345-1370. Long, J. and C. Plosser (1983), Real business cycles, Journal of Political Economy, 91, 39-69. Lucas, R. (1976), Econometric policy evaluation: A critique, Carnegie-Rochester Conference Series on Public Policy, 1, 19-46. 229 Maddala, G . and I. K i m (1998), Unit Roots, Cointegration, and Structural Change, Cambridge University Press. McCal lum, B . and E. Nelson (1999), A n optimizing I S - L M specification for monetary policy and business cycle analysis, Journal of Money, Credit, and Banking, 31, 296-316. McCal lum, B . and E. 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. Ramsey, F. (1928), A mathematical theory of saving, Economic Journal, 38, 543-559. 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 of monetary policy, NBER Macroeconomics Annual, M I T Press. Solow, R. (1956), A contribution to the theory of economic growth, Quarterly Journal of Economics, 70, 65-94. 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. 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 of monetary policy in a closed economy: A dynamic stochastic general equilibrium approach, Unpublished Manuscript. Vitek, F. (2006f), Measuring the stance of monetary policy in a small open economy: A dynamic stochastic general equilibrium approach, Unpublished Manuscript. Woodford, M . (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. Yun , T. (1996), Nominal price rigidity, money supply endogeneity, and business cycles, Journal of Monetary Economics, 37, 345-370. 

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