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Essays in empirical macroeconomcs Kano, Takashi 2003

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Essays in E m p i r i c a l  M a c r o e c o n o m i c s  by  Takashi K a n o B.A., M.A.,  M e i j i University, 1994 Hitotsubashi University, 1996  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE  REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY in THE  F A C U L T Y OF G R A D U A T E STUDIES (Department of Economics) We accept this thesis as conforming to the required standard  THE  UNIVERSITY OF BRITISH C O L U M B I A February 2003 © T a k a s h i K a n o , 2003  In  presenting this  degree at the  thesis in  University of  partial  fulfilment  of  the  requirements  British Columbia, I agree that the  freely available for reference and study. I further  this thesis for scholarly purposes may be granted  department  or  his  or  her  representatives.  an advanced  Library shall make  it  agree that permission for extensive  copying of  by  for  It  is  by the  understood  that  head of copying  my or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  ^CcSpK?  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  'c S  1  Abstract This thesis consists of three essays that contribute empirical macroeconomics. The first essay jointly tests several of the predictions of the intertemporal approach to the current account and one of its implications, the present value model of the current account (PVM). The intertemporal approach to the current account predicts that the current account of a small open economy is independent of.world common disturbances. The P V M predicts that the response of the current account to a country-specific shock depends on the persistence of the shock. This essay combines these predictions to identify a structural vector autoregression (SVAR). The identification exploits the orthogonality of the world real interest rate and country-specific shocks as well as the lack of a longrun response of net output to transitory shocks. Estimates of the SVAR show that the Canadian and U.K. data support the intertemporal approach with two puzzling exceptions . A recent study claims that habit formation in consumption improves the ability of the P V M of the current account to predict actual current account movements. The second essay shows that the habit-forming P V M of the current account is observationally equivalent to the canonical P V M augmented with a transitory consumption shock. To resolve the identification problem, this essay constructs a small open economy-real business cycle (SOE-RBC) models with habits and stochastic world real interest rates calibrated to Canadian postwar quarterly data. The results from Monte Carlo experiments reveal that to explain sample moments conditional on the habit-forming and standard PVMs, the SOE-RBC model with stochastic world real interest rates dominates the SOE-RBC  ii  model with habit formation. The third essay explores the ability of habit formation in consumption, in the context of the one-sector, closed economy-RBC model,'to account for the U.S. growth rates of consumption and output. Existing studies show that habit formation helps successfully explain the negative response of labour input to a positive, permanent technology shock as well as the empirical puzzles of asset pricing behavior. This essay shows that the R B C model with habit formation fails to mimic not only the persistence of output growth over business cycle frequencies but that of consumption growth at zero frequency as well. Further, the model yields counterfactually low volatility of equity returns.  iii  C o n t e n t s Abstract  ii  Contents  iv  List of Tables  viii  List of Figures  x  Acknowledgements  xii  Dedication  xiii  1  Overview and Summary  1  2  A Structural V A R Approach to the Intertemporal Model of the Current Account  5  2.1  Introduction  5  2.2  The Model and Its Predictions  10  2.2.1  A n Intertemporal, Small Open Economy Model  11  2.2.2  Derivation of the Predicted Responses  16  2.3  The S V M A and Identification Issues .  20  2.4  Empirical Results  25  iv  2.5 3  2.4.1 , Data and Reduced-Form VAR Estimation  25  2.4.2  Joint Test of the PVM's Restrictions  26  2.4.3  SVAR Estimation and Test Statistics  28  2.4.4  Impulse Response Analysis  31  2.4.5  Testing the Hypotheses  2.4.6  Forecast Error Variance Decomposition Analysis  ......  .  Conclusion  35 37  Habit Formation, the World Real Interest Rate, and the Present Value Model of the Current Account  39  3.1  Introduction  39  3.2  The PVMs with Habit Formation and Transitory Consumption: Observational Equi valence  3.3  3.4  4  32  Sample Moments Conditional on the Habit-Forming and Standard PVMs  44 47  3.3.1  Econometrics Issues  48  3.3.2  Empirical Results  52  Monte Carlo Investigation  56  3.4.1  The Small Open Economy Real Business Cycle Model  57  3.4.2  The Optimality Conditions and Interpretations  59  3.4.3  The Numerical Solution and Calibration  62  3.5  Results  64  3.6  Conclusion  68  Habit Formation and Aggregate Dynamics in Real Business Cycle  Models  70  4.1  Introduction  70  4.2  Empirical Facts of Consumption and Output Dynamics  74  4.3  The Model  75  4.3.1  A One-Sector R B C Model with Adjustment Costs of Investment and Habit Formation in Consumption  4.3.2 4.4  76  Numerical Solution, Calibration, and Evaluation  78  Results  80  4.4.1  A C F s and SDFs for the Output Growth Rate  80  4.4.2  IRFs of Output  82  4.4.3  A C F s and SDFs for Consumption Growth  83  4.5  Implications for Asset Prices  4.6  Conclusion  85 .  88  Bibliography  90  Appendices  97  A  97  Appendices for Chapter 2 A.l  Derivation of the Linear-Approximated Intertemporal Budget Constraint (2.7)  A.2  97  Derivation of the Approximated Solution of the Optimal Current Account Ratio (2.10)  99  vi  B  A.3  Derivation of Cross-Equation Restrictions fi  A.4  Data Description and Construction  101  A.5  Unit Root Tests  102  A. 6  Predicted Linear Restrictions on the Impact Matrix  103  cp  and fi  cs  Appendices for Chapter 3  100  107  B. l  Derivation of Eq. (3.6)  • 107  B.2  Derivation of Eq. (3.7)  108  B.3  The Two-Step G M M Estimation . . .  109  B.4  The State Space Representation of the Equilibrium Path  110  B.4.1  Deriving the Stationary System . .  Ill  B.4.2  The Deterministic Steady State  112  B.4.3  Derivation of the State Space Representation  113  vii  List o f Tables 2.1  Three Empirical Results of the Intertemporal Approach and the P V M of 118  the Current Account 2.2  Findings of the Paper  2.3  Identification Schemes  2.4  Calibrated Parameters and Joint Test of the Present Value Restrictions 121  2.5  Asymptotic Wald Test of the Cross-Equation Restrictions  122  2.6  The FEVDs of the CA under Identification Scheme I  123  2.7  The FEVDs of In NO under Identification Scheme I  124  3.1  The Sample Statistics of the PVMs  125  3.2  Empirical Facts of the Present Value Models  126  3.3  Calibrated Parameters of SOE-RBC Models  3.4  Sample Estimates and Empirical P-values under the Nulls of SOE-RBC  1 1 9  120  • • • • 127  128  Models 3.5  The Monte Carlo Experiments: Which SOE-RBC Model Mimics the 129  Empirical Facts? 4.1  Failures of the One-Sector RBC Model with Habit Formation and Ad130  justment Costs of Investment 4.2  Calibrated Parameters of the Model  131  4.3  Generalized Q Statistics  132  4.4  Asset Price Statistics  133  A.l  Unit Root Tests  134  viii  FEVDs of the CA under Identification Scheme II  ix  List of Figures 2.1  The PVM's Predictions on the Actual Current Account-Net Output Ratio 136  2.2  Impulse Responses of CA under Identification Scheme I  137  2.3  Impulse Responses of In NO under Identification Scheme I  138  2.4  Empirical Joint Distributions of the Statistics 7i  cp  and 7i under Identics  fication Scheme I . ;  139  3.1  The P V M Predictions on the Canadian Current Account  3.2  Theoretical Distributions of Test Statistics: The SOE-RBC with Habit Formation  3.3  141  Theoretical Distributions of Test Statistics: The SOE-RBC with Stochastic World Real Interest Rates  3.4  142  Sample Predictions and Theoretical Distributions: The SOE-RBC with Habit Formation  3.5  143  Sample Predictions and Theoretical Distributions: The SOE-RBC with Stochastic World Real Interest Rates .  4.1  144  The Sample Estimates of the ACFs and SDFs of the Growth Rates of Consumption and Output  145  4.2  The IRFs of Log of Output to Permanent and Transitory Shocks . . . .  4.3  The Sample Estimates and Theoretical Distributions of the ACFs and SDFs of the Output Growth Rate  4.4  140  146  . 147  The IRFs of Log of Output to Permanent and Transitory Shocks: The Sample Estimates and Theoretical Mean Responses  x  148  4.5  The Sample. Estimates and Theoretical Distributions of the A C F s and SDFs for Consumption Growth  149  A.l  Impulse Responses of C A under Identification Scheme II  150  A.2  Impulse Responses of In N O under Identification Scheme II  151  A.3  Empirical Joint Distributions of the Statistics ri  cp  fication Scheme II  and H  cs  under Identi152  xi  Acknowledgements I am greatly indebted to my thesis supervisor, Professor James Nason, for research direction and fruitful discussions. I would heartily appreciate his great kindness. I would like to thank Dr. Jeannine Bailliu, Professor Paul Beaudry, Professor Paul Bergin, Professor John Cragg, Professor Michael Devereux, Professor Francisco Gonzalez, Professor Chang Sik Kim, Professor Angela Redish, Dr. Lawrence Schembri, and Professor Akihisa Shibata for helpful suggestions and discussions at various stages of this dissertation. I am also grateful to my classmates, colleagues and friends, especially Matt Aharonian, Martin Berka, Matt Doyle, Gabi Dragan, Naoto Jinji, Lilia Karnizova, Shinya Kawahara, Hiroshi Ohashi, Rob Petrunia, Christoph Schleicher, Kazutaka Takechi, Genevieve Verdier, Steve Whelan and Jake Wong for helpful suggestions and discussions. I am solely responsible for any errors and misinterpretations.  xii  To Kazuko, and Our Parents  xiii  Chapter 1 Overview and Summary This thesis consists of three essays that contribute empirical macroeconomics. Essay 1. A Structural V A R Approach to the Intertemporal M o d e l of the Current Account  In the first essay, I jointly test several of the predictions of the intertemporal approach to the current account and one of its implications, the present value model of the current account (PVM). Given homogeneity across economies, the intertemporal approach predicts that the current account will not be affected by global disturbances; hence, country-specific shocks dominate current account fluctuations. The P V M predicts that the response of the current account to a country-specific output shock depends on the persistence of the shock. This essay develops schemes to identify three shocks: global, country-specific permanent, and country-specific transitory shocks. The assumption of a small open economy requires that a country-specific shock be orthogonal to the world real interest rate.  1  This orthogonality condition, as well as the lack of a. long-run response of output to a transitory shock, provides identification schemes for a structural vector autoregression (SVAR).. The identified SVAR in turn makes it possible to test the predicted responses of the current account to the three shocks. Using data of Canada and the U.K., I find support for the intertemporal approach to the current account in several dimensions. However, this essay reveals two puzzles that challenge the intertemporal approach. First, the impact response of the current account to a country-specific transitory shock is greater than that implied by the P V M . This is a puzzle because it implies that consumption responds negatively to a positive income shock. The second puzzle is that current account fluctuations are dominated by country-specific transitory shocks, which themselves explain very little of the fluctuations in income. This finding is inconsistent with the P V M . Essay 2. Habit Formation, the World Real Interest Rate, and the Present Value Model of the Current Account  Habit formation in consumption is often employed to resolve puzzles between macro models and aggregate data. One example is the present value model of the current account (PVM) that includes habit formation. A recent study argues that habit formation improves the ability of the P V M to predict actual current account movements. This essay shows that the habit-forming P V M is observationally equivalent to the canonical P V M augmented with a transitory consumption component that is serially correlated. This means that given the data, any test statistic constructed from the former P V M takes the same value as that from the latter P V M . Hence, by looking at  2  the sample test statistics, a researcher cannot identify whether or not habit formation plays an important role in actual current account movements. To resolve this identification problem, this essay constructs two small open economyreal business cycle (SOE-RBC) models: one with habit formation and the other with a stochastic world real interest rate, respectively. The two SOE-RBC models are calibrated to postwar Canadian quarterly data, and are used to generate artificial data to replicate the test statistics of the habit-forming P V M . The idea is: if the sample test statistics of the habit-forming P V M really reflect habit formation in consumption, the theoretical test statistics replicated by the SOE-RBC model with habit formation should be closer to the sample test statistics than those replicated by the SOE-RBC model with a stochastic world real interest rate. Results from the Monte Carlo experiments reveal that to explain the sample test statistics of both the habit-forming and standard PVMs, the SOE-RBC model with the stochastic world real interest rate dominates the SOE-RBC model with habit formation; in other words, the former model does a better job of replicating the data. This suggests that future research in this literature should concentrate on the determinants of the world real interest rate rather than on alternative specifications of utility. Essay 3. Habit Formation and Aggregate Dynamics in Real Business Cycle Models  The third essay explores the ability of habit formation in consumption, in the context of the one-sector, closed economy-real business cycle (RBC) model, to account for the U.S. growth rates of consumption and output. Existing studies have shown that habit formation helps successfully explain the negative response of labour input to a  3  positive, permanent technology shock as well as the empirical puzzles of asset pricing behavior. This paper shows that this, type of model (i) fails to mimic the persistence of output growth over business cycle frequencies, (ii) fails tp mimic the hump-shaped impulse response of output growth to a transitory shock, (iii) overstates the persistence of consumption growth around zero frequency, and (iv) yields counterfactually low volatility of equity return. These failures of the one-sector, closed economy- RBC model'cast doubt on habit formation as an important data generating mechanism to generate the dynamics of the U.S. aggregate data.  4  Chapter 2 A Structural V A R Approach to the Intertemporal Model of the Current Account  2.1  Introduction  The intertemporal current account approach provides an analytical framework to study current account movements of a small open economy because it emphasizes forwardlooking behavior of economic agents . The key message of the intertemporal approach is 1  that domestic residents use the current account as a tool to smooth consumption against country-specific shocks by borrowing and lending in international capital markets. To the contrary, no global shock gives a small open economy an opportunity of consumption lr  The small open optimal growth model of Hamada(1966) is an explicit precursor of the intertemporal  approach to the current account. Obstfeld and Rogoff(1995) is an excellent review of this approach.  5  smoothing since all economies react symmetrically to a global shock. A global shock has no effect on the current account in a small open economy. The present value model of the current account (PVM) expresses this consumptionsmoothing motive in current account fluctuations as a linear closed-form solution of the intertemporal approach. With the assumption of the exogenous, constant world real interest rate, the P V M characterizes the current account to be negative of the discounted sum of expected future changes in net output . This present value formula implies that 2  when domestic residents expect future net output to increase temporarily by countryspecific shocks, they lend out to the rest of the world to smooth consumption. Therefore, the current account moves into surplus. On the other hand, if an increase in future net output is expected to be permanent, the current account should not change because the permanent shocks to net output cannot be smoothed away . 3  2  Sheffrin and Woo(1990), Otto(1992), Ghosh(1995) and Bergin and Sheffrin(2000) jointly test the  cross-equation restrictions the P V M formula imposes on an unrestricted vector autoregressive, by applying the methodology originally developed by Campbell(1987) and Campbell and Shiller(1987) to test theories of consumption and stock price. Their tests statistically reject the basic PVM's cross-equation restrictions in the G-7 economies except for the U.S. This formal rejection of the P V M , however, does not necessarily imply that the P V M and the intertemporal approach are not useful to explain current account movements in a small open economy. For example, as Obstfeld and Rogoff(1995) discuss, the predictions of the P V M track historical current account movements fairly closely in some economies. 3  More precisely, if net output follows a random walk, a country-specific shock permanently raises net  output by the same amount. Sachs(1981,1982) shows that the current account does not respond to the shock since both consumption and current net output rise by the same amount in this case. Moreover when net output follows a more persistent process than a random walk, like an A R I M A process, the P V M predicts a negative response of the current account to a positive, country-specific shock. 6  Recent studies test the predictions of the intertemporal approach and the P V M in many different dimensions. Table 2.1 summarizes the main results of the past studies. First, by decomposing the Solow residuals into global and country-specific components, Glick and Rogoff(1995) and its successor I§can(2000) observe in the post-1975 data of the Group of Seven (G-7) economies that the current account in fact responds little to a global technology shock. To the contrary, by exploiting a structural vector autoregression (SVAR) approach, Nason and Rogers(2002) show in the post-1975 Canadian data that the hypothesis of no response of the current account to a global shock is sensitive to identification . Nason and Rogers(2002) also observe, as the second result in Table 2.1, 4  that country-specific transitory shocks dominate current account fluctuations not only in the short run but also the long run. Glick and Rogoff(1995) argue there is another puzzling observation in the joint dynamics of investment and the current account. The authors observe across the G-7 data that investment responds to the identified country-specific technology shock greater in the absolute value than the current account does. However, their intertemporal model predicts that when a country-specific technology shock is permanent, the current account should respond to the shock greater in the absolute value than investment because saving negatively responds to the permanent technology shock. They propose as a resolution a highly persistent but not permanent, country-specific technology shock. Similarly, the permanent-transitory decomposition of Hoffmann(2001) based on the vector error correction model (VECM), as well as the introduction of nontradable goods by I§can(2000), 4  I n the appendix, they apply the same analysis to the other G-7 economies and obtain the almost  same results as in the Canadian data.  7  provides a potential resolution for Glick and Rogoff's puzzle. The purpose of this essay is to evaluate the predictions of the intertemporal approach and the P V M on responses of the current account to different shocks. This essay jointly tests the predictions of the intertemporal approach and the P V M on responses of the current account to three shocks to net output: global, country-specific permanent, and country-specific transitory shocks. This essay accomplishes this purpose by providing its own identification schemes. The three shocks are identified by a SVAR with two restrictions. The first restriction stems from the small open assumption maintained by the intertemporal approach. This assumption restricts the world real interest rate to be orthogonal to any country-specific shock at all forecast horizons. Together with the assumption of the small open economy, allowing the world real interest rate to vary stochastically makes it possible to identify global and country-specific shocks. The second identifying assumption this paper employs restricts transitory shocks to have no longrun effect on net output. This long-run restriction, based on Blanchard and Quah(1989), decomposes country-specific shocks into permanent and transitory components. The assumption of the small open economy and the long-run restriction provide two identification schemes for the SVAR that contains the world real interest rate, the first difference of log of net output, and the current account-net output ratio as the endogenous variables. The identified SVAR in turn makes it possible to test jointly the predictions on the responses of the current account to the three shocks. The predictions are given as the cross-equation restrictions the intertemporal approach and the P V M impose on the SVAR . 5  5  These cross-equation restrictions are conditional on the identification of the SVAR. Hence, this  8  This essay studies quarterly data of two proto-type small open economies, Canada and the U.K. The main results of this essay are summarized in Table 2.2. First, in Canada and the U.K., impulse responses of the current account to the identified shocks are consistent with the corresponding theoretical predictions. Second, tests of the crossequation restrictions (CERs) show that the hypothesis that the current account does not respond to a global shock is sensitive to the identification, while the impact responses of the current account to country-specific shocks match the PVM's prediction. The test of the CERs also rejects the joint hypothesis related to the impact responses of the current account measure to all the three shocks. Third, given the identification, the data support the observation that the response of the current account-net output ratio to country-specific transitory shocks are greater than implied by the P V M . Fourth, the forecast error variance decompositions (FEVDs) of the current account reveal that country-specific transitory shocks dominate current account fluctuations not only in the short run but the long run as well, while the shocks explain almost none of the fluctuations in net output. The first result supports the intertemporal approach and the P V M . This result adds to the literature that finds the intertemporal approach can explain many aspects of current account dynamics. The second result echoes Nason and Rogers(2002): the response of the current account to a global shock is sensitive to identification. The third result reveals a new aspect of Glick and Rogoff's(1995) puzzle: even when country-specific shocks are decomposed into permanent and transitory shocks, the impact response of the current joint test is different from that of the cross-equation restrictions imposed on the reduced-form VAR as in Sheffrin and Woo(1990), Otto(1992), and Ghosh(1995).  9  account remains puzzling. Moreover, the third result implies that consumption negatively responds to a positive income shock. This implication is hard to be reconciled with the standard macroeconomics literature. The final result confirms the observation of Nason and Rogers(2002) with different identification. This result violates the P V M since the basic present value formula requires current account fluctuations need to be explained by the shocks that dominate net output fluctuations in the short run as well as the long run. The following section introduces the model and develops the predictions of the intertemporal approach and the P V M as cross-equation restrictions on a structural V M A . Identification issues are discussed in section 3. Section4 reports the empirical results. Section 5 contains conclusions.  2.2  T h e M o d e l a n d Its Predictions  This essay considers a world that consists of many small open economies. Following Glick and Rogoff(1995), assume that all the economies are homogeneous with respect to preferences, endowments and technologies. Furthermore, the international financial market is assumed to be incomplete in the sense that no household in a small open economy can buy or sell state-contingent claims to diversify away country-specific shocks. Only riskless bonds, which are denominated in terms of the single consumption good, are traded internationally . 6  Incompleteness in the international financial market is one of the maintained assumptions in the  G  intertemporal approach [see, for example, Obstfeld and Rogoff(1995) and Glick and Rogoff(1995)]  10  2.2.1  A n Intertemporal, Small Open Economy Model  Consider an infinitely lived representative consumer in a representative small open economy. The assumption of the small open economy implies that this economy faces the world real interest rate r determined in the international financial market. The stant  dard P V M of the current account, for example, Sheffrin and Woo(1990), Otto(1992) and Ghosh(1995), assumes the world real interest rate to be exogenous and constant. Instead, this essay allows the world real interest rate to vary stochastically, as in Bergin and Shefffin(2000). The reason for this extension is that this essay exploits stochastic variations in the world real interest rate to identify global and country-specific shocks. In addition, this essay assumes the world real interest rate is covariance stationary. Let C be consumption at period t, u(C) be the period utility function of the cont  sumer, and (3 be the subjective discount factor taking a value between 0 and 1, respectively. The consumer's expected lifetime utility function at period t is then given as oo  (2.1) i=0 where E is the conditional expectation operator upon the information set at period t  t. Further defining B , Qt, h and G to be the international bond holding, output, t  t  investment and government expenditure at period t, respectively, gives the consumer's and the small open R B C models [see, for example, Mendoza(1991) and Cardia(1991)]. By contrast, the two-country R B C models [see, for example, Backus, Kehoe, and Kydland(1992) and Baxter and Crucini(1993)] assume the complete financial market. In this literature, agents in two countries can pool all idiosyncratic risks by trading any contingent claims.  11  budget constraint B  = (1 + r )B  t + 1  t  + Q  t  t  - It ~ G - C . t  t  (2.2)  The optimization problem of the representative consumer is then to maximize eq.(2.1) subject to eq.(2.2). The first order conditions of this problem comprise the budget constraint (2.2), the Euler equation u'(C ) = /3E (l + r )u'(C ), t  t  t+1  (2.3)  t+1  and the transversality condition lim E R B t  t>i  =0  t+i  (2.4)  i—>oo  where  R i t t  is the ex post market discount factor at period t for period t + i consumption,  which is defined as '  V  1 For simplicity, let N O at period t: N O  t  (2-5) if i  =  0.  denote output net of investment and government expenditure  t  = Q — I — G . Taking the infinite sum of the consumer's budget t  t  t  constraint (2.2) toward the future and using the transversality condition (2.4) yield the ex ante intertemporal budget constraint of the consumer oo  oo  ER C t  tti  t+l  = (1 +  r )B t  t  +  E t R t j N O t + i .  (2.6)  i=0  i=0  To derive the present value representation of the current account measure, this essay takes a log-linear approximation of the Euler equation (2.3) and a linear approximation  12  of the intertemporal budget constraint (2.6) . The approximation begins by dividing 7  the intertemporal budget constraint (2.6) by NO . After several steps of simple algebra, t  eq.(2.6) can be rewritten as  1 +  NO  t  Y ^ i=\  t+i  e  x  P  \Y {j=t+l  (  A  l  n  i  C  ~  R  +  J'))  = exp{ln(l + r ) - A ln NO } t  oo  (  Et  +  1  t  1 +  e  x  p  t-\-i  \ Yl  (  A  l  n  i  NO  ~  +  Let c, b, 7 , 7 and /i denote the means of the consumption-net output ratio C /NO , C  t  net foreign asset-net output ratio B /NO -i, t  t  the  the first difference of log of consumption  t  A l n C , the first difference of log of net output A m N O , and log of the gross world real t  t  interest rate ln(l + r ), respectively. Eq.(2.6) is then linearly approximated by taking a t  first-order Taylor expansion around these means. Appendix A . l shows the steps of the linear approximation of the intertemporal budget constraint in detail. For any variable Xt, let Xt denote deviation from its mean value. The linear-approximated intertemporal 7  B e r g i n and Sheffrin(2000) also conduct a linear approximation of the intertemporal current account  model in order to involve stochastic variations of world real interest rates and terms of trade into the standard P V M . While they follow Huang and L i n ' s (1993) log-linear approximation, this essay develops an alternative linear approximation to derive a closed-form solution of the optimal current account-net output ratio.  13  budget constraint is given as I-a NO,  «  B  t  N O t ^  +  1- a  1- a  -  Mn(l + r ) t  K  bA In NO,  OO  1-  A K  h7NO  t+i  - ln(lT^+i)} (2.7)  ^  i=l  where a = exp(7° — fi) < 1 and K = exp(7 — /i) < 1 . 8  Notice that eq.(2.7) makes the consumption-net output ratio depend on the expected future path of consumption growth. To characterize the process of consumption growth, the Euler equation (2.3) is approximated log-linearly. Suppose that the period utility function is given as a power function  where a is the elasticity of intertemporal substitution. This specification of the utility function yields the Euler equation  As shown in Campbell and Mankiw(1989) and Campbell(1993), when the world real interest rate and consumption are jointly conditionally homoscedastic and log-normally 8  T h e conditions a < 1 and n < 1 are required to satisfy boundedness of the expected present  discounted value terms of eq.(2.7). Through the following analysis, this essay assumes these conditions: the mean growth rates of consumption and net output are lower than the mean of world real interest, rates, respectively. These conditions imply that on the balanced growth path the economy is dynamically efficient.  14  distributed, the above Euler equation can be rewritten as E A In C +i = 5 + a In (3 + oE ln(l + t  t  r )  t  t+1  = 5 + a(ln/3 + u) + a£ [ln(l + r ) - y\ t  (2.8)  t+l  where 5 is a constant term including the variances of A l n C  t + 1  and ln(l + r +i) and the t  covariance between the two terms . 9  Finally, to derive an approximated solution of the current account-net output ratio, recall the current account identity CA  t  = r B + NO -Ci. t  t  (2.9)  t  By assuming that the economy possesses a balanced growth path, a = K, and using the approximation ln(l + r ) ~ r , Appendix A.2 shows that eqs.(2.7), (2.8) and (2.9) t  t  together give the present value representation of the current account-net output ratio: rvf  °°  = W + [(a - l)c +l]J2  —L  t  ' 9  &  Ki  °°  r  t+i  ^ Et A In NO .  -  t+i  (2.10)  i=l.  i=l  I t is important to note from the log-linearized Euler equation (2.8) that perfect consumption smooth-  ing as i n previous studies is not the case in this model. First, unless 5+a(\n /3+fJ.) = 0, log of consumption has a deterministic trend, as shown by the first two constant terms in the R H S of (2.8). Second, the last term shows that the substitution effect of variations of world real interest rates on the consumption profile. A rise in the world real interest rate makes current consumption more expensive in terms of future consumption. Hence the representative consumer is induced to shift consumption toward the future with elasticity a. These two effects together produce consumption profile that deviates from perfectly smoothed one. Furthermore, a caveat of the log-linearized Euler equation (2.8) is that it only cares about first moments of logs of consumption and the world real interest rate. Higher moments of two series are assumed to be fixed.  15  Eq.(2.10) is the desired schedule of the current account-net output ratio, which is represented as a linear present value relation among the current account-net output ratio, the first difference of log of net output and the world real interest rate. Eq.(2.10) says that the optimal current account-net output ratio is determined by three factors. The third term of the RHS of eq.(2.10) captures the consumption-smoothing motive. It implies that the representative consumer changes the current account-net output ratio to smooth consumption in response to expected changes in future path of net output growth. The second term represents a consumption-tilting factor due to expected variation of the world real interest rate. The coefficient (a — l)c+ 1 on the second term implies the intertemporal substitution effect, the income effect and the wealth effect, respectively. If the world real interest rate is expected to change in future, the small open economy wants to deviate consumption from its smoothed, random walk path through the three effects. The first term of the RHS of eq.(2.10) is an additional consumptiontilting factor. When there is a change in the world real interest rate, net interest payment from abroad is changed given the net international asset position. For example, a rise in the world real interest rate increases net interest payment from (to) abroad if the country is a net creditor (debtor). This change in net interest payment prompts the consumer to alter the current account-net output ratio beyond its consumption-smoothing level.  2.2.2  Derivation of the Predicted  Responses  This subsection derives the testable restrictions the present value formula (2.10) imposes on the responses of the current account measure to three orthogonal shocks to net output:  16  global, country-specific permanent and country-specific transitory shocks. Let ef, ef\ and £j denote global, country-specific permanent, and country-specific transitory shocks, 5  respectively, and be orthogonal each other. This essay assumes that the first difference of log of net output is linearly decomposed into three infinite-order M A components attributed to the three orthogonal shocks: AWNO  t  = r-(L)e? + r - ( L K  P  + rr  (LK*  (2.11)  where r™°(L) for % — {g,cp,cs} is an invertible, infinite-order polynomial with respect to the lag operator L, in which the impact coefficient r™°(0) is not restricted to one . 10  Similarly, the process of the world real interest rate is linearly decomposed into three infinite-order MA components attributed to the three orthogonal shocks:  r = r (Ly t  g  t  + Yl {L)e7 + r ( L ) e r . p  c s  (2.12)  Given the processes of the first difference of log of net output and the world real interest rate, eqs.(2.11) and (2.12), the present value formula (2.10) yields the predictions on the impulse responses of the current account-net output ratio to the three shocks. The following structural moving average (SMA) representation of the current accountnet output ratio represents the predictions (Appendix A.3 contains the details of this derivation.): CA  ^ 10  = r;(L)ef  + T%(L)e?  + T™{L)e?  (2.13)  N o t e that eq.(2.11) is a structural moving average ( S M A ) representation of the process A l n i V O t ,  rather than the Wold representation with the impact coefficient equal to one. Instead of being restricted to one, the impact coefficient is estimated.  17  where Tf (L) for an index i € {g,cp,cs} is an invertible, infinite-order polynomial with L  respect to the lag operator. The SMA (2.13) provides the testable hypotheses this paper studies. The first hypothesis predicts that a global shock does not matter for the current account at any forecast horizons. Under the homogeneity assumption across economies, every economy has the same excess demand for international riskless bonds. In this case, as argued by Razin(1993) and Glick and Rogoff(1995), no economy can alter its net foreign asset position to a global shock because all the other economies react to the shock symmetrically. Therefore, a global shock has no effect on the current account at any forecast horizons. All that occurs is that the world real interest rate adjusts. Let H denote the impulse response of C A to ef_ Then the first null hypothesis is given as g  t  H : 0  H = ^ir-  =  g  v  0  f o r  a n  y - °-  (Hypothesis 1)  1  To test this hypothesis, this essay recovers the impulse response functions (IRFs) of the level  of the current account to a global shock from the IRFs of the current account-net  output ratio and log of net output . 11  Next consider the impact responses of the current account-net output ratio to the two country-specific shocks af and ef: T™(0) and T™(0) in eq.(2.13). To derive the second and third hypotheses, recall the small open economy assumption of the intertemporal approach. This assumption requires that a small open economy have no influence on 1 1  T b the contrary, the response of the current account-net output ratio to a global shock is ambiguous.  For example, if a global shock has a positive impact on \nNO  t  and the mean value of CA /NO  positive, then the current account-net output ratio should respond negatively to the shock.  18  t  t  is  the world real interest rate: a country-specific shock does not matter for the world real interest rate at any forecast horizons. In other words, this assumption implies that zero restrictions are imposed on the coefficients of the infinite-order polynomials related to the two country-specific shocks in the world real interest rate process (2.12): for any i > 0, T  R  where  T  R CPI  and  T  CPI  R  = r^ = 0  (Small Open Economy Assumption)  s>i  are the i-th coefficients of the infinite-order polynomials T  (2.14) (L)  R  CSI  CP  and  in eq.(2.12), respectively.  T (L) R  CS  As shown in Appendix A.3 in detail, under the small open economy assumption (2.14), r^(0) and T^(O) should satisfy the following cross-equation restrictions, respectively: r^(o) = r-(o) - r £ ( « )  (n )  rS(o) = r-(o) - r - ( « )  (n )  cp  and  where for an index i G {cp, cs}, Z =  cs  is the infinite polynomial r™°(z) evaluated at  T1°(K)  K.  The cross-equation restrictions lZ  cp  and 1Z state that the impact response of the CS  current account-net output ratio to a country-specific shock should be given as the difference between the impact and the discounted long-run responses of A\n.NO  t  to the  shock. The current account identity (2.9) restricts the current account-net output ratio to be negatively related to the consumption-net output ratio. Therefore, if a countryspecific shock raises net output above (below) consumption, the current account-net output ratio rises (falls). r™°(0) in 1Z captures the impact effect of the shock ef on cp  19  net output, while r™°(«;) shows the impact effect of the shock on consumption . Hence 12  the impact effect of the shock on the current account-net output ratio, 1^(0), is given as the difference F"°(0) —  T™°(K).  Define the statistics H  cp  The same explanation is applicable for 7Z . CS  and H  cs  as H  = T™(0) - T™(0) + T^(K) and H  cp  cs  Fcs(0) ~ r"°(0) + r™°(/c), respectively. The cross-equation restrictions 1Z and 1Z cp  CS  =  then  provide the following null hypotheses: H :H 0  cp  = 0  (Hypothesis 2)  and H : Hcs = 0. 0  (Hypothesis 3)  By construction, if Hi ^ 0 for i £ {cp,cs}, the prediction of the P V M on the impact response of the current account-net output ratio to the shock e\ is rejected because the observed response is considered to be greater or lesser than the prediction.  2.3  T h e S V M A a n d Identification Issues  Hypotheses 1-3 are constructed conditionally on identification of the three shocks. Testing the null hypotheses discussed in the last section requires the three shocks to be identified. To do so, this essay exploits the SVAR methodology, as in Nason and Rogers(2002). The most important difference in identification between this essay and the existing literature is that this paper allows the world real interest rate to vary stochastically and 1 2  T h e underlying fact that consumption is determined by permanent net output makes the impact  response of consumption be given as the discounted long-run response of the first difference of log net output. See, for example, Quah(1990).  20  combine the small open economy assumption with the stochastically varying world real interest rate to identify global and country-specific shocks. In this essay, as implied by the small open economy assumption, country-specific shocks are identified as shocks that are orthogonal to the world real interest rate in either the short-run or the longrun. Furthermore, country-specific shocks are decomposed into permanent and transitory components by Blanchard and Quah's(1989) long-run restriction. To see this, consider a stationary column vector X — [r AlnNO t  t  t  CA /NO }'. t  t  Let  the probability distribution of the vector X be characterized by a p-th order unrestricted t  VAR. Since the vector X is stationary, it has a Wold-Vector Moving Average (VMA) t  representation, VMA(oo), X = C(L)v t  (2.15)  t  where C(L) is an invertible, infinite-order matrix polynomial with respect to the lag operator L, and in particular the coefficient matrix of L° is the identity matrix. The reduced-form disturbance vector v has a symmetric positive definite variance-covariance t  matrix E. Stacking eqs.(2.11), (2.12) and (2.13) vertically implies that the vector X has the t  following structural V M A (SVMA) representation:  r (L)  n Ah^NOt  C A / N O t  r (L) r: (L)  g  —  cp  H  s  r»°(L) r - ( L ) r - ( L )  r»(L) r - ( L ) r - ( L )  cp  t  fc  c  cs  or simply X = T(L)e t  t  21  (2.16)  where  e  t  is the structural shock vector given as e  e }'. In particular,  = [e ef a  t  c s  t  t  following  the standard exercise in the SVAR literature, this essay assumes that the variancecovariance matrix of the structural shock vector is given as the identity matrix:  Ee e' — t  t  I . 13  The small open economy assumption (2.14) implies  T  r  (L)  =  T (L) r  cs  = 0 in the  SVMA (2.16). This means that any country-specific shock has no influence on variations in the world real interest rate at any forecast horizons. Moreover, to decompose countryspecific shocks into permanent and transitory components, this paper imposes on the SVMA (2.16) a restriction that the country-specific transitory shock  e°  s t  has no long-run  effect on log of net output. This long-run restriction is given as 17(1) = 0.  (Long-Run Restriction)  (2.17)  Imposing the small open economy assumption (2.14) and the long-run restriction (2.17) makes the impact and long-run matrices, T(0) and T(l), of the SVMA (2.16) be  170) r(o) =  o  o  117(0) 17(0) 17(0) 17(0)  17(0)  r-(o)  r:;(i)  0  0  17(1)  17(1)  0  17(1)  17(1) r - ( i )  | ,  (2.18)  and  r(i)  1 3  (2.19)  T h a t is, the structural shocks are orthogonal at all leads and lags, and each shock has a unit variance.  Therefore, in this essay, the impulse response function of a variable is interpreted as the response to a unit standard error shock.  22  Notice that the SVMA with the impact and long-run matrices (2.18) and (2.19) is overidentified. To see this, comparing the reduced-form V M A (2.15) with the SVMA (2.16) immediately provides the following relationships: E = r(o)r(o)'  (2.20)  C(L)r(0) = T(L).  (2.21)  and  Moreover eq.(2.21) can rewrite eq.(2.20) as E = C7(l)" r(l)r(l)'C(l)'" . 1  1  (2.22)  Given estimates of E and C ( l ) , there are six linear independent equations and nine unknowns in eq.(2.22). Therefore, in general, three additional restrictions are needed for the SVMA (2.16) to be just-identified. On the other hand, the small open economy assumption (2.14) and the long-run restriction (2.17) impose an infinite number of restrictions on the coefficients in the SVMA (2.16): two impact restrictions, three long-run restrictions, and an infinite number of restrictions on IRFs. Since three restrictions are needed to just-identify the structural parameters, the SVMA (2.16) is an overidentified system. Following the identification strategy examined by King and Watson(1997) and Nason and Rogers(2002), this essay investigates two different identification schemes consisting of three restrictions from all the overidentifying restrictions in order to justidentify the system, and checks the robustness of the empirical results by comparing two identification schemes. The first identification comes from the lower triangularity of the long-run matrix (2.19). The maintained assumptions in this paper provide three long-run restrictions. 23  The zero restrictions on the (1, 2)th and (1, 3)th elements of r(l) reflect the small open assumption that requires country-specific permanent and transitory shocks to have no long-run effect on the world real interest rate, respectively. The zero restriction on the (2, 3)th element of T(l) implies that a country-specific transitory shock has no long-run effect on log of net output, which is explicitly shown as the long-run restriction (2.17). Therefore, the lower triangular long-run matrix (2.19) is just-identified and the impact matrix can be recovered through eq.(2.21). Hereafter, this Blanchard and Quah's (1989) style identification is called identification scheme I. Another identification scheme in this paper exploits together two impact restrictions in eq.(2.18) and the long-run restriction (2.17). The zero restrictions on the (1, 2)th and (1, 3)th elements of T(0) reflect the small open assumption that requires countryspecific permanent and transitory shocks to have no instantaneous effect on the world real interest rate. The zero restriction on the (2, 3)th element of T(l) implies that a country-specific transitory shock has no long-run effect on log of net output . 14  Notice that the long-run restriction (2.17) can be rewritten as an impact restriction. To show this, let A j denote the (i, j)th element in any matrix A. The zero restriction it  on the (2,3)th element in T(l) together with the zero restriction on the (1, 3)th element in T(0) implies the restriction c(i)2 r(o) ,3 + c(i) , r(o)3,3 - o. l2  2  2 3  (2.23)  Since C(l) ,2 and C(l) 3 are estimated, eq.(2.23) can be considered as an impact re2  1 4  2)  T h e reason for choosing this long-run restriction from the others is that the restriction is essential  for decomposing country-specific shocks into the permanent and transitory components.  24  striction. Together with the two impact restrictions shown in r(0), eq.(2.23) makes it possible to just-identify T(0) in eq.(2.18). Hence, the second identification scheme of this paper follows Galf's(1992) method that exploits the impact and long-run restrictions in concert. Hereafter, this identification is referred to as identification scheme II. Table 2.3 summarizes the two identification schemes of this essay.  2.4  E m p i r i c a l Results  This section discusses the data, estimation methods, tests, and empirical results of this essay.  2.4.1  Data and Reduced-Form V A REstimation  This essay studies two proto-type small open economies, Canada and the U.K. All data used in this essay are quarterly, span the period Ql:1960-Q4:1997, and are seasonally adjusted at annual rates. The estimation is based on the Q2:1963-Q4:1997 sample, with data prior to Q2:1963 used to construct lags. The world real interest rate is a weighted average of ex ante real interest rates across the G-7 economies. This follows the way in which Barro and Sala-i-Martin(1990) and Bergin and Sheffrin(2000) construct r . Net t  output and the current account are generated from the appropriate national accounting data. Appendix A.4 provides detailed information on the source and construction of the data. The standard augmented Dickey-Fuller (ADF) tests provide evidence that the vector  25  X follows a stationary process . Since the V M A (2.15) is invertible, it has an infinite15  t  order VAR representation. The infinite-order VAR is approximated by truncating at a finite lag length. To select an optimal lag length, both the AIC and BIC criteria are calculated with a maximum lag length of fifteen. Both criteria select a lag length of one for each country. The first-order reduced-form VAR (RFVAR), X = BX -\ +v , is t  t  t  estimated by OLS. Let B, £ and C(l) denote the estimates of the RFVAR coefficient matrix B, the variance-covariance matrix £ and the implied infinite sum of the V M A coefficient matrices C(l) = [73 — £?]  2.4.2  _1  through the following analysis.  Joint Test of the P V M ' s Restrictions  Before estimating the SVMA (2.16), this essay conducts the traditional joint test of the cross-equation restrictions the P V M (2.10) imposes on the RFVAR, by following Sheffrin and Woo(1990), Otto(1992), Ghosh(1995) and Bergin and Sheffrin(2000). Let a 1 x 3 vector  be the ith row of the 3 x 3 identity matrix I . The P V M (2.10) then implies 3  the following cross-equation restrictions on the RFVAR coefficient matrix B conditional on the parameters b, c, K and a: e = ei {6+ [(cr - l ) c + l]/d3[7 3  1 5  - e nB[h - riB}~ . 1  2  3  (2.24)  T h i s essay constructs the demeaned series- of the world real interest rate, the change i n log of net  output and the current account ratio, i.e. r , t  L\\nNO  and CAt/NOt,  t  and perform unit root tests  for them based on the A D F T-test. Appendix A.5 summarizes the method and the results of the unit roots tests. The A D F tests reject the unit root null i n all series at least at the 5 percent significance level. From this evidence, the series r , A l n N O t  t  and CA /NO t  following analysis.  26  t  are considered to be stationary i n the  To test the cross-equation restrictions (2.24), define a statistic k(B) such that k{B) = ei {b+[(a - 1)C+1}KB[I  - KB}' }  - e nB[h - KB}'  1  3  1  2  - e. 3  Under the null of k(B ) = 0, the Wald statistic 0  W = k(B)  dk{B)^dk{B)'^ dB  k(B)'  dB  (2.25)  asymptotically follows the % distribution with the third degree of freedom. 2  Recall that the Wald statistic W is constructed conditional on the parameters K, C, b, and a. This paper calibrates K, C, and b directly from the data; K = 0.993, c = 0.983, b = -0.712 for Canada; K = 0.990, c = 0.988, b = 0.377 for the U.K. The elasticity of intertemporal substitution a is calibrated by matching the predictions of the P V M (2.10) on the current account-net output ratio with the actual series. The predictions CAjHO[  are constructed as a function of a by = T{a)X  CA/NO{  (2.26)  t  where T{a) = ei {b+[(a - l)c + 1]KB[I - KB}' } 1  3  - e KB[I 2  - KB}~ . 1  3  The elasticity of intertemporal substitution a is then calibrated by minimizing the mean squared error of the prediction : T T~ Y (CA/N0 1=1  T  2  l  t  - CA/Afofy  =T  - 1  ^  [CA/NO  t  -  T{a)X }  2  t  t=i  The resulting a is 0.001 for Canada, and 0.08 for the U.K. The small values of the elasticity of intertemporal substitution are close to the estimates of Bergin and Sheffrin(2000)  27  in their two goods model. The first four rows of Table 2.4 summarize the calibrations in this paper. The last two rows of Table 2.4 report the Wald statistics (2.25) for the joint test of the cross-equation restrictions (2.24), and the corresponding p-values based on the %  2  distribution for Canada and the U.K. In the two economies, the Wald statistics are so large that the cross-equation restrictions are jointly rejected at any standard significance level. Figures 2.1(a) and (b) show the actual series of the current account-net output and the PVM's predictions CA/MO[  for Canada and the U.K., respectively. Even though a  is chosen to minimize the mean squared error, the PVM's predictions are much smoother than the actual series in Canada. The result is much better in the U.K., but the P V M still cannot capture the huge deficits happened in the end of the 1980s. In summary, the cross-equation restrictions the P V M imposes on the RFVAR is jointly rejected across the two economies. The predictions of the P V M closely tracks the U.K. series of the current account-net output ratio with the exceptional periods of the end of the 1980s, while those are still too smooth to match the Canadian series. This result suggests that especially in Canada, the source of the rejection of the P V M be attributed to something other than the fluctuations in net output as well as the world real interest rate.  2.4.3  SVAR Estimation and Test Statistics  The OLS estimates £ and C(l) make it possible to identify the impact matrix F(0) with each of the identification schemes. This paper recovers the impact matrix T(0) by the  28  full information maximum likelihood (FIML) procedure . 16  Tests of Hypotheses 1, 2 and 3 are constructed as the Wald statistics. To do that, this essay exploits the fact that all restrictions provided by the hypotheses can be rewritten as linear restrictions on the impact matrix T(0). Let [A]l and [A}1 denote the i th row and column vectors of a matrix A, respectively. Furthermore, let R and R^ for an index i > 0 be 1 x 3 row vectors such that CA  and R = [C(«).'2,1  where d, CA/NO,  CA  C(«) , -l 2  2  C(K) , 2  3  +  1]  and C{n)ij denote the coefficient matrix of IS in the V M A (2.15),  the mean of the current account-net output ratio, the mean of the current account, and the (i,j)th element of the matrix C(K), respectively. It can be then easily shown that the statistics H , ri g  ri  cs  =  cp  and fi  cs  are given as H = g  Ri[T(0)]\  for i > 0, ri  cp  = i?[r(0)] and 2  Appendix A.6 discusses derivation of the statistics in detail.  i?[r(0)J3.  Let W i , W and W3 denote the Wald statistics for the null hypotheses fi 2  fi  cp  = 0, and fi  cs  g  — 0,  = 0. In addition, let W 4 and W 5 be the Wald statistics for the joint  null hypotheses H° = H g  cp  = H  cs  - 0 and H° = H] = H] = VS] = 0. In particular, g  W 5 is based on the null hypothesis that a global shock does not matter for the current 1G  Because of the lower triangular long-run matrix a numerical maximization procedure is not needed  to recover the impact matrix i n identification scheme I. In identification scheme II, the impact matrix is numerically recovered through the F I M L procedure. See Amisano and Giannini(1997) and Hamilt o n 1994, chapter 11) for the F I M L estimation of the S V A R models.  29  account up to a year after impact. For example, the Wald statistic Wi for Hypothesis 1 is constructed as 0 fl-uO'l dH^dH^  dB  where  dB  1  1  9  is the point estimate of the statistic H . The asymptotic theory states that 0  Wi is distributed the x distribution with one degree of freedom . 2  17  To derive the Wald statistic W 4 for the joint null hypothesis Ti° = H = TL = 0, cp  construct a row vector A = [hi° H g  cp  CS  Hcs}- Then the Wald statistic for the joint null  is given as W =A 4  dB  A'.  dB  According to the asymptotic theory, W 4 asymptotically follows x (3). The same argu2  ment is applicable for the construction of the Wald statistic W 5 . As in the standard exercise of the SVAR literature, the IRFs and the FEVDs of the endogenous variables to the identified shocks are estimated. The empirical standard errors of the IRFs and the FEVDs are calculated by generating 10,000 nonparametric bootstrapping replications based on the reduced-form disturbances. The 10,000 replications of the statistics Hcp and H. generated by the bootstrapping exercise provide the cs  empirical joint distribution of H and H . cp  17  N o t i c e that the statistics H*, Ti , and 7i  cs  cp  cs  are constructed by the I R F s from the just-identified  S V A R . Since the I R F s are nonlinear functions of the R F V A R parameters, as shown i n Hamilton(1994, section 11.4), the asymptotic standard errors of the statistics H' , H , and 7i ;J  CI  cs  are obtained by using the  asymptotic standard errors of the R F V A R parameters and the Delta method. Similarly, the asymptotic X  2  statistics for the hypotheses can be constructed from knowledge of the asymptotic distribution of the  R F V A R parameters. Of course, the asymptotic x test depends on identification, as the I R F s do. 2  30  2.4.4  Impulse Response Analysis  Recall from the introduction that the basic response predictions of the intertemporal approach and the P V M are (i) a global shock does not matter for the current account at all forecast horizons, (ii) a country-specific permanent shock to net output has no or a negative impact on the current account, and (iii) a country-specific transitory shock to net output has a positive impact on the current account. This subsection examines the IRFs of the current account to check whether or not these predictions are supported by the Canadian and the U.K. data. Figure 2.2 shows the IRFs of the current account across the two economies under identification Scheme I. In each window, the dark line represents the point estimate and the dashed lines exhibit 95% confidence bands constructed by a nonparametric bootstrapping exercise. The results of the impulse response analysis are summarized as follows: In Canada and the U.K. • The IRFs of the current account to a global shock are not significant at any of the 40 periods after impact.  18  • The IRFs of the current account to a country-specific permanent shock are positive but insignificant. • The IRFs of the current account to a country-specific transitory shock are positive and significant. The positive responses remain significant for at least three years. 1 8  A  caveat is that the I R F s and the associated confidence bands are not a joint test statistic for  hypothesis 1. They provide pointwise information about the response of the current account to a global shock.  31  As reported in Table 2.2, the results support the basic predictions of the intertemporal approach and the P V M : no response of the current account to a global shock, no response to a country-specific permanent shock, and a positive response to a country-specific transitory shock. Figure 2.3 shows the IRFs of log of net output in Canada and the U.K. under identification scheme I. Notice that the responses of log of net output to a country-specific permanent shock are almost flat after jumps at impact. This observation is consistent with the PVM's prediction that if a country-specific shock is random walk, the current account has no response to the shock. The impulse response analysis, therefore, qualitatively supports the basic predictions of the intertemporal approach and the P V M : The predicted shapes of the impulse responses of the current account to the three shocks are consistent with the data. Although not reported, the same results are also observed even under identification scheme II . 19  Hence, this empirical result is robust for the two identification schemes.  2.4.5  Testing the Hypotheses  Notice that the qualitative validity of the predictions does not necessarily mean that the quantitative requirements of the intertemporal approach and the P V M - the crossequation restrictions imposed on the SVMA - are supported at the same time. Testing Hypotheses 1-3 provides information about the validity of the cross-equation restrictions. Tables 2.5(a) and (b) report the results of the asymptotic Wald tests under identification schemes I and II, respectively. Each table shows the Wald statistics and the 1 9  T h e results under  identification scheme II are available as Figures A . l ,  32  A.2, and A.3 and Table A.2.  corresponding p-values generated by asymptotic x distributions for the null hypotheses. 2  The following results are observed: • The single null ri = 0 is not rejected in Canada and the U.K. in identification g  scheme I, but rejected in the two economies in identification scheme II. • The single null H  cp  — 0 is not rejected in Canada and the U.K. across the two  identification schemes. • The single null ri  cs  = 0 is not rejected in Canada and the U.K. across the two  identification schemes. • The joint null ri° — H g  cp  —H  cs  — 0 is rejected in Canada and the U.K. across the  two identification schemes. • In Canada, the joint null  = H = ri — H = 0 is rejected across the two l  2  g  g  identification schemes. These results lead to the following inferences: (i) the validity of the hypothesis that the current account does not respond to a global shock is sensitive to the identification and the economy being studied, (ii) the P V M succeeds in making quantitative predictions on the impact responses of the current account to country-specific shocks, and (iii) the response predictions of the intertemporal approach and the P V M are jointly rejected. Recall that the IRFs support the hypothesis that the current account do not respond to a global shock. From the two different tests, it is safe to say there is no robust evidence for this hypothesis. This confirms the inference drawn by Nason and Rogers(2002) that the hypothesis is sensitive to identification. On the other hand, the IRFs and the 33  asymptotic W a l d tests consistently support the predictions of the P V M o n the responses of the current account to the country-specific shocks. F i n a l l y , the observation that the predictions of the P V M o n the impact responses of the current account to the three shocks are j o i n t l y rejected reinforces the rejection of the cross-equation restrictions the P V M imposes o n the R F V A R ; see section 2.4.2. A potential weakness of the W a l d test is that it depends o n the asymptotic x  2  dis-  t r i b u t i o n , a n d w i t h a small sample the W a l d statistic does not necessarily follow the x  2  distribution. F i g u r e 2.4 shows the scatter plots of 10,000 pairs of the statistics 7i a n d cp  H  cs  replicated b y nonparametric bootstrapping resamples under identification scheme I.  In each window, the darkest square represents the point estimate a n d the joint null is given b y the origin. Observe that i n the two economies the scatter plots have strikingly similar shapes a n d almost a l l replicated pairs are concentrated on the upper regions of the windows. Therefore, the empirical distributions of the statistics H  and H  provide  B y construction, the observation that the empirical joint d i s t r i b u t i o n of H  and H  cp  cs  information against the null hypothesis Ti. = 0. cs  cp  cs  is concentrated i n the upper region means that i n C a n a d a and the U . K . ,  r£(o)>r£(o)-o«). Under Hypothesis 3 the above equation must be satisfied w i t h equality.  Hence, this  paper reveals that in Canada and the U.K., the impact responses of the current account-  net output ratio to a country-specific transitory shock are too large to support the PVM. A g a i n the same observation is obtained even i n identification scheme II. Since the calibrated values of n i n the two economies are very close to one (see Table 34  2.3), the long-run restriction (2.17) requires the term r™°(«;) to be almost zero. Hence the above inequality says that the impact response of the current account to a countryspecific transitory shock is greater than that of net output. This observation is actually a puzzle. The current account identity requires that the impact response of the current account to a country-specific shock be the difference between the responses of net output and consumption. Thus, the greater response of the current account to a country-specific transitory shock than the response of net output implies that consumption responds negatively to a positive country-specific shock to net output. The basic intertemporal  approach to the current account is not built on the prediction that consumption responds negatively to an positive income shock. This puzzle is a challenge to the current account literature.  2.4.6  Forecast E r r o r V a r i a n c e D e c o m p o s i t i o n A n a l y s i s  Another way to examine the effects of the three shocks on the current account is to look at the forecast error variance decompositions (FEVDs) of the current account. The FEVD provides information about the share of current account fluctuations that can be explained by an identified shock. Table 2.6 provides the FEVDs of the current account attributed to the three shocks in Canada and the U.K. under identification scheme I. The table shows that at impact a country-specific transitory shock can explain almost 70 % of fluctuations in the current account across the two economies. Even at a year after impact, the shock can significantly explain 81 °/„ and 71 °/„ of fluctuations in the current account in Canada and the U.K.,  35  respectively. Therefore, the country-specific transitory shock can be considered as the dominant driving force of the current account in the short run. A striking fact revealed by the FEVDs is that even in the long-run the country-specific transitory shock dominates fluctuations in the current account in the two small open  economies. For example, at 40 quarters (10 years) after impact, about 80 % of fluctuations in the Canadian current account is attributed to the country-specific transitory shock. Similarly, at the same forecast horizon, the shock explains 72 °/„ of fluctuations in the U.K. current account. This observation is also obtained under identification scheme II. The result that country-specific transitory shocks dominate current account fluctuations not only in the short run but the long run as well echoes the finding of Nason and Rogers (2002). In their SVAR approach to study the joint dynamics of investment and the current account, they report the persistent dependence of the current account on country-specific transitory shocks across the G-7 economies. As they argue, at present there is no consensus intertemporal model that generates persistence in the current account to country-specific transitory shocks. Table 2.7 shows the FEVDs of log of net output. Observe that in the two economies a country-specific transitory shock cannot significantly explain fluctuations in log of net output at any forecast horizons. The observation that a country-specific transitory shock having no significant effect on net output dominates fluctuations in the current account in the short run as well as the long run is the second puzzle of this essay. This observation violates the standard P V M as well as the augmented P V M with the stochastic world real interest rate because in these models current account fluctuations should be explained by a country-specific shock that dominates the fluctuations in net 36  output. Combining with the joint rejection of the full cross-equation restrictions the P V M (2.10) imposes on the RFVAR, this puzzling observation suggests the importance of the consumption-tilting motive induced by country-specific shocks, rather than the consumption-smoothing behavior, to explain current account movements in the small open economies.  2.5  Conclusion  When the world real interest rate is allowed to vary stochastically, the intertemporal approach and its well-known closed-form solution, the P V M of the current account, jointly provide new identification for a SVAR. The small open assumption of the intertemporal approach gives the SVAR a restriction to identify global and country-specific shocks because the assumption requires any country-specific shocks to be orthogonal to the world real interest rate. By exploiting this orthogonality condition as well as the Blanchard and Quah's decomposition, this essay is able to develop two identifying schemes for the SVAR and recover its global, country-specific permanent and country-specific transitory shocks. The identified SVAR based on the Canadian and the U.K. data then yields tests of the predictions the intertemporal approach and the P V M make on the responses of the current account to the three shocks. A part of the results of these tests reaffirms the result of the past studies. Even though the test jointly rejects the PVM's cross-equation restrictions on the RFVAR, the intertemporal approach and the P V M are still useful to explain some aspects of current account movements. In fact, the IRFs of this essay  37  are consistent with the theoretical counterparts of the intertemporal approach and the P V M . Thus, this essay contributes to the current account literature by providing further evidence that small open economy models based on forward-looking economic agents are useful to understand current account dynamics. This paper reveals two puzzles that challenge the intertemporal approach. First, the response of the current account-net output ratio to a country-specific transitory shock is too large to support the P V M . This observation in turn draws a puzzling inference that consumption negatively responds to a positive income shock. The second puzzling aspect this paper observe is that current account fluctuations are dominated by country-specific transitory shocks that explain almost none of the fluctuations in net output in the short run as well as the long run. This puzzle implies that the consumption-tilting motive induced by country-specific shocks, rather than the consumption-smoothing behavior that the past studies emphasize, is important to account for current account movements. These failures of the intertemporal approach to the current account suggest that more research about its theoretical structure is needed. For example, more general utility functions, non-tradable goods and endogenous risk premia may yield resolution of these puzzles. Seeking valid modifications of the basic intertemporal approach is a future task of the current account literature.  38  Chapter 3 Habit Formation, the World Real Interest Rate, and the Present Value Model of the Current Account  3.1  Introduction  A small open economy model endowed with rational, forward-looking agents serves as a benchmark for studying current account dynamics in the recent literature. This model, as known as the intertemporal approach to the current account, stresses the consumptionsmoothing behavior of economic agents in the determination of the current account in a small open economy . When they expect changes in future income, forward-looking 1  agents smooth their consumption by borrowing or lending in international financial mar1  Obstfeld and Rogoff(1995) provide a recent and most detailed survey of the intertemporal approach  to the current account.  39  kets and hence by generating current account movements. This role of consumptionsmoothing behavior in current account determination is clearly expressed by the present value model (PVM) of the current account, which is a closed-form solution of the intertemporal approach. For example, the P V M predicts that the current account moves into deficit when a country's income is expected to decline temporarily, while no change in the current account occurs if the decline in income is expected to be permanent . 2  Many empirical studies including Sheffrin and Woo(1990), Otto(1992), Ghosh(1995) and Bergin and Sheffrin(2000), however, fail to find empirical support for the standard P V M of the current account in postwar data of the G-7 economies. The cross-equation restrictions the standard P V M imposes on the unrestricted vector autoregression (VAR) are statistically rejected for all of the G-7 economies except the U.S. Moreover, the forecasts of the standard P V M are too smooth to track actual current account movements. The empirical failures of the standard P V M have led some researchers to explore the role of consumption-tilting motives in current account movements: the current account might be adjusted to factors that deviate consumption away from the random-walk, permanent income level, for example, stochastic variations in the world real interest rate . 3  2  A crucial prediction of the P V M is that only country-specific shocks matter for the current account  of a small open economy. A global shock does not give a small open economy an opportunity to borrow or lend in international financial markets because all economies have identical preferences, technologies and endowments and hence react to a global shock symmetrically. A l l that occurs is that the world real interest rate adjusts to the global shock. 3  F o r example, by using a structural V A R approach to identify global and country-specific shocks,  the second chapter of this thesis shows that almost all of Canadian current account movements are dominated by country-specific shocks unrelated to variations in the smoothed, permanent income. This  40  One way to introduce the consumption-tilting motive into the standard P V M is habit formation in consumption. Habit formation makes optimal consumption decisions depend not only on permanent income but also on past consumption. The household tends to maintain its past consumption level against unexpected shocks to permanent income; therefore, habit formation makes consumption smoother and more sluggish than in the basic permanent income hypothesis (PIH). The sluggishness of consumption in turn implies more volatile current account movements than the standard P V M predicts. Gruber(2000) uses habit formation in consumption to improve the ability of the P V M to track actual current account movements in the postwar quarterly data of the G-7 economies, of the Netherlands, and of Spain. He concludes that habit formation plays an important role in determining current account dynamics. This essay shows that the habit-forming P V M is observaMonally equivalent to the canonical P V M augmented with a serially-correlated transitory consumption shock. In other words, given the information set studied by Gruber(2000), the two PVMs yield the same values of the sample test statistics. Because of this identification problem, Gruber's tests of the habit-forming P V M are not informative to detect the role of habit formation in current account movements. In this essay, the source of the serially-correlated transitory consumption shock is specified with stochastic movements in the world real interest rate because of two reasons . 4  First, the stochastic real interest rate is a well-known way to introduce a  result empirically suggests the importance of consumption-tilting motives in Canadian current account movements. 4  A n o t h e r sources of the transitory consumption shocks are a transitory government expenditure  41  consumption-tilting motive into the P V M of the current account as well as the permanent income hypothesis of consumption . Expected future changes in the world real 5  interest rate tilt the consumption path away from the random-walk, permanent income level and, as a result, introduce the consumption-tilting component into the P V M of the current account. Second, recent studies on small open economy-real business cycle (SOE-RBC) model, Blankenau, Kose and Yi(2001) and Nason and Rogers(2003), provide evidence that the world real interest rate shocks play a crucial role in explaining net trade balance/current account movements in a small open economy. To solve the identification problem, this essay conducts Monte Carlo experiments based on a small open-real business cycle model (SOE-RBC) that incorporates with either habit formation or the stochastic world real interest rate. To this end, the SOERBC model of Nason and Rogers(2003) is extended by introducing habit formation. The extended model is then used to generate artificial data that yield theoretical distributions of "moments" to be explained in this essay. As in a standard calibration exercise, moments of the artificial data generated by SOE-RBC models are compared with their sample counterparts. However, as examshock affecting the utility function and the stochastic terms of trade. 5  See Campbell and Mankiw(1989) for tests of the permanent income hypothesis (PIH), and Bergin  and Sheffrin(2000) and Kano(2003) for tests of the current account P V M . In particular, Bergin and Sheffrin(2000) extend the standard P V M by introducing stochastic variations in the world real interest rates as well as real exchange rates, which yield a serially-correlated transitory consumption component independent of permanent income. They observe that the extension improves the P V M prediction in Canada. The second chapter also shows the P V M of the current account in the presence of the stochastic world real interest rate using a different approach.  42  ined by Nason and Rogers(2003), the "moments" this essay studies are not standard unconditional variances and covariances of the sample. Instead, they are the sample statistics conditional on the habit-forming and standard PVMs of the current account: the sample estimate of the habit-formation parameter, the cross-equation restrictions implied by the habit-forming and standard PVMs, and the current account forecasts of the habit-forming and standard PVMs. It is worth noting that by construction, the theoretical distributions have the null hypothesis of the underlying SOE-RBC model as the data-generating process (DGP) of the moments. This essay generates the theoretical distributions under two different null hypotheses. First, setting the structural parameters of the SOE-RBC model to rule out stochastic variations of the world real interest rate derives the theoretical distributions under the null of the SOE-RBC model with habit formation. Second, setting the habit parameter equal to zero provides the theoretical distributions under the null of the SOERBC model with the stochastic world real interest rate. The two different SOE-RBC models are evaluated from the viewpoint of classical statistics; that is to say, the sample statistics are used as critical values to derive empirical p-values. For example, if a sample statistic drops into the five percent tail of the theoretical distribution, the null is rejected at the five percent significance level. The results from the Monte Carlo experiments support the SOE-RBC model with stochastic world real interest rates. Although the SOE-RBC model with habit formation can replicate a part of the empirical facts of the habit-forming P V M , the SOE-RBC model with the stochastic world real interest rate mimics all the relevant sample moments. The superiority of the SOE-RBC model with stochastic world real interest rates casts doubt 43.  on habit formation as the significant source of the consumption-tilting behavior needed to explain Canadian current account movements. The structure of this essay is as follows. The next section introduces the habitforming PVM and discusses the observational equivalence problem. The sample moments conditional on the habit-forming and standard PVMs are reported in section 3.3. Section 3.4 introduces the SOE-RBC models of this essay to mimic the sample moments. Section 3.5 reports the results of the Monte Carlo experiments. Concluding, remarks are made in section 3.6.  3.2  T h e P V M s w i t h H a b i t F o r m a t i o n a n d Transitory C o n s u m p t i o n : Observational Equivalence  Gruber (2000) extends the standard P V M by introducing habit formation in consumption. Let C , B and NO denote consumption, international bond holding, and net t  t  t  output at period t, respectively. As in the standard literature, net output, which is defined as output minus domestic investment minus government expenditure, follows a nonstationary process having a country-specific, random-walk technology shock as the driving force . The period utility function is specified as a quadratic form 6  u(C  t+l  6  - hCt+i-i) = C +i - hCt+i-! - \{C t  - hCt+i-i) , 2  t+i  0<h<l  T h e basic S O E - R B C model, which is well-known as the intertemporal approach to the current  account, is a single-shock model containing a country-specific, unit-root technology shock. See Obstfeld and Rogoff(1995), Glick and Rogoff(1995), and Nason and Rogers(2003). Under this assumption, the intertemporal approach has the standard P V M as a closed-form solution. 44  where h represents the habit parameter. C represents aggregate consumption unaffected t  by any representative household decision. This specification of habit formation is related to external habit formation or the catching up with the Joneses, as in Abel(1990) and  Campbell and Cochrane(1999) . Note that C = C in equilibrium. 7  t  t  The problem the representative household faces is to maximize its expected discounted lifetime utility oo  i=0  subject to the budget constraint B  t+i  = (1 + r)B + NO ~ C t  t  t  where r is the world real interest rate assumed to be constant and equal to the subjective discount rate. In this case, the first-order necessary conditions together with the transversality condition yield an optimal consumption decision rule. Letting e det  note a disturbance orthogonal to information at period t-1 and adding e to the optimal t  consumption decision rule provide  (3.1) where the equilibrium condition C = C is imposed . With habit formation, consump8  t  t  tion is determined by a weighted average of permanent income and past consumption 7  I f habits are internal, as in Constantinides(1990), they depend on the household's own consumption  and the household takes habits into account when choosing the amount of consumption. 8  Campbell(1987) argues that a transitory consumption error uncorrelated with lagged information  improves the ability of the P I H to fit the U.S. data.  45  with the weight. h/(l + r). This fact makes adjustments of consumption to permanent income shocks more sluggish than in the standard PIH. Substituting the resulting consumption equation into the current account identity CA  = rB + NO - C produces the P V M with habit formation  t  t  t  t  E ANO t  t+i  — e. t  (3.2)  Notice that the current account depends on its own past value. This makes the process of the current account more persistent than in the standard PVMs of Sheffrin and Woo(1990) and Otto(1992). Furthermore, the current account becomes sensitive to the current change in net output: the current account depends on not only the expected present value of future declines of net output but the current change of net output as well. This makes the current account more volatile than in the standard P V M . An important point is that the present value formula (3.2) is observationally equivalent to the P V M derived from a multiple-shock model. Let Cf denote arbitrary transitory consumption that follows an exogenous AR(1) process C? = C?_ +u Pe  1  \ \<l  t  Pc  (3.3)  where Cf may be observable or may not, and uj is a white noise shock. Assume that cont  sumption C is linearly decomposed into the transitory consumption Cf and permanent t  income Cf:  9  C =C +C T  t  9  p t  (3.4)  Because the underlying S O E - R B C model has the unique stochastic trend, i.e. the country-specific,  permanent, technology shock, it is possible to decompose consumption into a random-walk component Cf  and a transitory component Cf: see King, Plosser and Rebelo(1988).  46  where permanent income Cf is determined by the standard PIH formula  cr  E  ( r h )  h  r  )  S  ,  t  (3.5)  N O , t+i  fi(^)  +  Appendix B . l shows that the non-habit-forming, multiple-shock model specified by eqs.(3.3), (3.4), and (3.5) has the following present value representation of the current account  C A  t  =  PcCAt-i  +  {l + r )  A  N  O  Y  t  l + ) f j ( l + r)  1  Ei  •t+i - v  A N O ,  t  (3.6)  r  where v is a disturbance orthogonal to information at period t-1, which satisfies t  E -iV t  t  =  0 for i > 1. Notice that the non-habit-forming P V M (3.6) is equivalent to the habit-forming P V M (3.2). Therefore, given the data of C  A  t  and  A N O  t  ,  any statistics based on eq.(3.2), for  instance, an estimate of h, take the same values as those statistics from eq.(3.6). The habit-forming P V M is observationally equivalent to the non-habit P V M augmented with the AR(1) transitory consumption component. This implies that the statistics based on the habit-forming P V M (3.2) are not informative to identify whether or not habit formation plays an important role in explaining current account movements.  3.3  Sample M o m e n t s C o n d i t i o n a l on the H a b i t - F o r m i n and Standard P V M s  This section reports the sample moments conditional on the habit-forming and the standard PVMs. As mentioned in the introduction, this essay considers the sample test 47  statistics of the two PVMs as the sample "moments" explained by SOE-RBC models. The next subsection discusses econometric issues related to estimation and test of the habit-forming P V M . The following subsection reports the sample moments.  3.3.1  Econometric Issues  Gruber(2000) exploits the generalized method of moments (GMM) procedure to estimate the habit parameter h in the habit-forming P V M (3.2). Define a variable D = CA — t  ANO  t  - (1 + r)CA -i t  t  and rewrite the P V M (3.2) as  D = hDt-x - e + (1 + r)e _! + e t  t  t  (3.7)  t  where e , e _i and e are disturbances orthogonal to the information set at period t — 2, t  t  t  Q -2 [See Appendix B.2 for the detailed derivation of eq.(3.7).]. Let W t  t  2  denote a f c x l  vector that contains k different variables in Q _ - Eq.(3.7) then implies unconditional t  2  moment conditions - /iA-i) - 0  EW - (D t  2  t  (3.8)  where E is the unconditional expectation operator. Eq.(3.8) makes it possible to estimate h by the GMM/two step-two stage least square (2SLS) procedure by West(1988). Let hasLS  be the 2SLS estimate of  h.  When  k >  1, I\SLS is overidentified. The J-statistic  of Hansen(1982) tests the orthogonality conditions (3.8). Given k(> 1) instruments, the J-statistic is asymptotically distributed x with k — 1 degrees of freedom. 2  This essay proposes a more efficient estimate of the habit parameter than the 2SLS estimate h^sLs- In addition to the unconditional moment conditions (3.8), other theoretical restrictions the habit-forming P V M imposes on a p-th order bivariate vector autoregres48  sive (VAR) of CA and ANO t  are used to estimate the habit parameter. Recall that a  t  VAR(p) process has a corresponding first-order representation with a companion matrix  A: y = Ay -i+u t  t  (3.9)  t  where U is a 2p x 1, zero mean, homoskedastic, serially uncorrelated error vector such t  that U = [uf  0  NO  t  •••  0 uf  0 •••  A  0]', and y is a 2p x 1 vector constructed t  as y = [ANO t  t  AiVO _!  ••• ANO _  t  t  CA  p+1  t  CA ^ t  ••• CA,t-p+ij  By assumption of the VAR, y -i is orthogonal to the VAR disturbances U — [u: t t  t  uf }. That is, the following unconditional moment conditions are satisfied: A  Ey -i t  ®U = 0  (3.10)  t  where (g> is the operator of the Kronecker product. Define a 1 x 2p vector  that includes zeros except for the ith element equal to 1, i.e. e  ?; =  [0---^0^ i—1st  ^  s  ith  _0^---0]. i+lst  The habit-forming P V M (3.2) then implies that under the null hypothesis, the following cross-equation restrictions should be the case: e Ay p+1  = hC Ay  (3.11)  h  t  t  where hZ is a 1 x 2p vector such that h  I - [ J - ) A 1+ r  49  -l  Note that the cross-equation restriction (3.11) can be considered as an unconditional moment condition E(e  - IC )Ay  = 0.  h  p+1  t  (3.12)  Eq.(3.12) holds under the null hypothesis of the habit-forming P V M (3.2). As a result, if the joint probability distribution of CA and ANO t  t  is specified by the  unrestricted VAR (3.9), the habit-forming P V M (3.2) yields the unconditional moment conditions (3.10) and (3.12) in addition to (3.8) . Construct a (4p + k + 1) x 1 vector 10  gt(0) such that  r 9t(9)  y -i  =  ®  t  (e  -  p+1  u  t  hC )Ay h  t  where 9 is a vector constructed by stacking the habit parameter h and the elements of the companion matrix A, i.e. 9 = [h vec(A)'}'. The sample analogs of the theoretical moment conditions (3.8), (3.10), and (3.12) are given as T  G ^ T "  1  $>(<?) = 0  where T is the sample number. To obtain an efficient estimate of 9, this essay conducts the two-step GMM procedure of West(1988) . Let n  GMM estimate of 10  6  6 M GM  be the resulting two-step  with the asymptotic covariance matrix Vg . GMM  In this case, the  Gruber(2000) does not use the moment conditions (3.10) and (3.12) to estimate h. This fact makes  Gruber's estimation and specification test based only on the over-identifying restrictions (3.8) inefficient since his procedure does not use all of information the model provides potentially. n  A p p e n d i x B.3 reviews the two-step G M M estimation in detail.  50  J-statistic JT for the overidentifying restriction test, which satisfies  JT = TG(9 )'  M*G(6CMM)  GMM  under the optimal weighting matrix M*, asymptotically follows the x distribution with 2  degrees of freedom k. Notice that the J-statistic jointly tests the overidentifying restrictions implied by the unconditional moment conditions (3.8), (3.10), and (3.12), but does not test the exact cross equation restrictions (3.11). To do so, define a 1 x 2p vector T{9) as T{9) = (IC — e )-4 + p+i- Let #o denote the true parameter vector under the null of the habitH  e  p+1  forming P V M . Eq.(3.11) implies that T{9Q) = e i.e.  p + 1  under the true parameter vector 9 , 0  the p + 1st element of the vector T{9 ) should be one, while the others should 0  be zero. The G M M estimate of the vector J-{9), T(6GMM),  makes possible piecewise  tests of the 2p cross-equation restrictions by the standard t-statistics, as well as joint test of those restrictions by the Wald statistic. The asymptotic standard error of the estimate T{9GMM) is calculated from its covariance matrix numerically derived by the Delta method CV(#GMM) > - Va 89' '  0 G M M  Let k(0) = e i — F{9). p +  9J (§GMM) ' 89' S  T  Then the estimates 9cMM and Ve  GMM  yield the Wald statistic  WT satisfying W  T  =  k(9  ) 8k(9cMM) v> 89' °  GMM  e  MM  , n -1 9k(9cMM, k(0GMJV/)'89'  Under the null hypothesis of k(# ) = 0, the Wald statistic WT asymptotically follows 0  the x with degrees of freedom 2p. 2  51  Finally, the predictions of the habit-forming P V M on actual current account movements, denoted by CA{, are constructed as CA{  = ^(^GMM)^-  Under the null, it is the  case that CA{ = CA . Therefore, comparing the predictions with actual current account t  series provides another information to test the null hypothesis of the habit-forming P V M (3.2).  3.3.2  Empirical Results  This essay studies the quarterly, real, seasonally-adjusted Canadian data that spans the sample periods Ql:1963 and Q4:1997. The data construction follows Otto(1992) and Nason and Rogers(2003) . The current account series and the first difference series of net 12  output are demeaned to construct the sample vector y . The fourth lag p = 4 is chosen t  as the optimal lag by the general-to-specific likelihood ratio (LR) tests. To construct the series D , this essay uses the calibrated value of the constant world real interest rate t  r = 0.0091 [or equivalently 3.70 percent point on an annual basis: r = (1.037) ' — 1]. 0 25  A crucial point for conducting the GMM/2SLS estimation is how to choose the instrument variables W ~i- Theoretically, any variables in the information set fi _2 can be t  t  included in W _ . This essay lags the instruments more than one period and includes in t  2  M _ the fourth and fifth lagged values of CA and ANO /  t  2  t  t  to avoid potential correlation  between D — hD -\ and any variable at period t — 2 or t — 3. In this case, W -2 is a t  t  t  4 x 1 vector satisfying  W -2 — [ A J V 0 _ t  12  All  4  4  ATO-5  the data are distributed by Statistics Canada.  52  CA -  t 4  CA-s]'.  Therefore, p = k = 4 are chosen in the following analysis. Table 3.1(a) summarizes the empirical results. First, the two estimates of the habit parameter, basis and lie,MM, are reported in the first two columns. The 2SLS estimator based only on the unconditional moment conditions (3.8) yields h^sLS = 0.931 with the asymptotic standard error 0.192. This number is close to the estimate Gruber(2000) obtains (h^sLS = 0.902 and s.e. = 0.257, respectively). On the other hand, the G M M estimator based on the full moment conditions (3.8), (3.10), and (3.12) provides h MM = G  1.002 with the asymptotic standard error 0.152. Therefore, the G M M estimate based on the full moment conditions draws an inference of a larger habit parameter than the 2SLS estimate . Although it is safe to claim that h is non-zero, either h^shs or h 13  G M M  has a 95 % confidence interval including h = 1 . This inference violates the constraint 1 4  h<l.  The statistic JT is 0.455 with a p-value of 0.978, which means that the overidentifying restrictions out of the unconditional moment conditions (3.8), (3.10), and (3.12) cannot be jointly rejected even at 97.8 % significance level. However, the Wald statistic W T for the cross-equation restrictions is 37.128 with a small p-value. This means that the crossequation restrictions k(6* ) — 0 are jointly rejected at any standard significance levels. 0  Furthermore, the piecewise tests of the eight elements in the vector J-(0) reflect this joint 13  It is worth while mentioning that the standard error of the GMM estimate is smaller than that of  the 2SLS. This means that the sampling uncertainty of the GMM estimate is smaller that that of the 2SLS estimate. 14  If h = 1, the utility function implies that the household wants to smooth change in consumption,  rather than level of consumption, across periods.  53  rejection of the cross equation restrictions. Recall that under the null, the fifth element T should be one, while all the other elements should be zero. The table reports that the 5  G M M estimate Ts is 1.276 with the asymptotic standard error 0.226. Hence, the estimate is not significantly different from one. The observation that two estimates T\ = —0.302 and T = —0.400 are statistically significant, however, violates the respective single null 6  hypotheses. All the other estimates T for i ^ 1, 5, 6 are statistically insignificant based on the two standard error rule. Figure 3.1(a) plots the actual current account series, the predictions of the habitforming P V M CA{, and the corresponding asymptotic two standard error band. Observe that the predictions of the habit-forming P V M track the actual current account fairly closely. The narrow standard error band reflects small sampling uncertainty attached to the predictions. The standard error band includes the actual current account in all the sample periods. These observations support the inference that the habit-forming P V M explains actual movements of the Canadian current account fairly well, as Gruber(2000) reports. Comparing the empirical results of the habit-forming P V M (3.2) with those of the standard P V M demonstrates how introducing habit formation improves the ability of the P V M to track actual current account movements. Setting h = 0 and e = 0 in the t  habit-forming P V M (3.2) provides the following cross-equation restrictions imposed on the unrestricted VAR (3.9) under the null of the standard P V M  k*{e ) = 0  e + -F{0 ) P  54  1  0  =0  where T*{9) = -  ( l + r)- A[h  - (1 +  l  e  i  Note that 9 includes only the V A R parameters.  r)- A]-\ l  Hence, the unbiased estimate of 9 is  obtained by OLS. Let 9Q-LS denote the O L S estimate. Table 3.1(b) reports the Wald statistic W  T  to test the cross-equation restrictions  k*(6> ) = 0 jointly, and the estimates of the eight elements of the vector T*(9OLS) to 0  test the cross-equation restrictions piecewisely. First, the Wald statistic W  T  is 20.589  with the asymptotic p-value 0.009. Therefore, the cross-equation restrictions are jointly rejected at any standard significance levels. The failure of the standard P V M is clearer in the piecewise tests of the null hypotheses. If the standard P V M holds, the fifth element of the vector T*(9OLS) should be one, while the other elements be zero. The estimate of the fifth element T£ is -0.115 with the asymptotic standard error 0.408. Hence, the single null Tt, = 1 is strictly rejected by the standard t-statistic. A l l of the other estimates are statistically insignificant. Figure 3.1(b) plots the actual Canadian current account series, the predictions of the standard P V M CA\* = T(9oLs)yt,  and the asymptotic two standard error band.  The predictions are too smooth to track the actual series. The standard error band excludes the actual series at almost all periods. Hence, the standard P V M cannot predict the position of the Canadian current account.  These observations clearly reveal the  superiority of the habit-forming P V M to the standard P V M at least in the predicting ability. The empirical results of this essay track those of Sheffrin and Woo(1990), Otto(1992),  55  and Gruber(2000). Tables 3.2(a) and (b) summarize the empirical facts - the sample moments - of both the habit-forming and standard PVMs. In particular, this essay shares with Gruber(2000) the observation that taking habit formation into account greatly improves the PVM's prediction on the Canadian current account. The empirical results of both Gruber and this essay appear to support the claim that habit formation helps to explain Canadian current account movements. However, the observational equivalence between the PVMs with habit formation and serially-correlated transitory consumption makes a researcher unable to identify whether the successful aspects of the habit-forming P V M are actually attributed to habit formation or other factor that generate consumption-tilting motives. A leading example for a small open economy is the stochastic world real interest rate. The next section discusses this essay's strategy to solve the identification problem.  3.4  M o n t e C a r l o Investigation  Facing the identification problem, this essay conducts calibration-Monte Carlo exercises based on the SOE-RBC models with habit formation and the stochastic world real interest rate. The first task is to extend the SOE-RBC model of Nason and Rogers(2003) by introducing habit formation in consumption, as discussed in the next subsection.  56  3.4.1  The Small Open Economy Real Business Cycle Model  The lifetime utility function of the representative household is oo  U = E YPMC; ,L ) i=0 t  t  +i  (3.13)  t+l  where C * = C — hC -i and L is leisure at period t. Eq.(3.13) implies that the lifetime t  t  t  t  utility is non-separable not only across periods but also between consumption and leisure in each period. In particular, the period utility function u(C*,L) is parameterized as a constant relative risk aversion type (C^L -*) -? - 1 1  u(C ,L) = for 7  7^  1  1-7  1. For 7 = 1, u(C*, L) =  In C* + (1 - <f>) ln L  and in either case 0 < (j) < 1. Therefore, in the case of 7 = 1 the preferences are separable between consumption and leisure. Define Y , I , G and r to be output, investment, government consumption expent  t  t  t  diture, and the real interest rate the representative household faces at period t. The household's budget constraint is B  t+1  = (1 + r )B t  + Y - I - G - C.  t  t  t  t  t  (3.14)  Output Y is produced by a Cobb-Douglas production function t  Y = Kt[AN ] -* 1  t  where K , A and t  t  t  0 < V < 1  (3.15)  are capital stock, county-specific, labor-augmenting technology,  and labor input at period t. Since the household is endowed with a unit hour to allocate 57  between labour and leisure, the restriction L + N = 1 must be satisfied. The law of t  t  motion for capital is represented as (3.16) where 0 < 5 < 1 is the depreciation rate. Eq.(3.16) includes adjustment costs of investment with the parameter (p. This specification of the adjustment costs follows Baxter and Crucini(1993). As studied by Nason and Rogers(2003) and Schmitt-Grohe and Uribe(2003), the real interest rate r is decomposed into two components. The first component q is the t  t  exogenous and stochastic return that is common across the world. In this essay, q follows t  a covariance stationary process. The other component is the risk premium specific to this small open economy. The risk premium is given as a linear function of the economy's bond-output ratio. Following Nason and Rogers(2003), this essay specifies the stochastic real interest rate r to be t  (3.17) t  Eq.(3.17) implies that if the small open economy is a debtor (i.e. B < 0), the economy t  must pay a premium above q .  15  t  The processes of the three exogenous variables G , A and q are specified as follows. t  t  t  Government consumption expenditure G is proportional to output Y with a constant t  1 5  t  T h e endogenous risk premium i n eq.(3.17) excludes an explosive/unit root path of international  bonds in the linearized solution of the equilibrium.  Moreover i t solves the famous problem i n the  S O E - R B C model that the deterministic steady state depends on the initial condition.  58  ratio g:  16  G = gYt-  (3.18)  t  The country-specific, labor-augmenting technology A is a random walk with drift t  A = A -! exp(a + e"), t  t  a > 0,  e ~ i.i.d.N(0,0-  ( -19)  a  3  t  Finally, the world real interest rate c/ follows an AR(1) process t  1 + q = (1 + g*)( -"«)(l + g^)" exp(e?), 1  t  \ \ < 1,  e? ~ i.i.d.N(0, a*)  Pq  (3.20)  where c/* is the deterministic steady state value of q . In the following analysis, e" and t  e[ are assumed to be uncorrelated at all leads and lags.  3.4.2  The Optimality Conditions and Interpretations  The problem of the representative household is to maximize eq.(3.13) subject to eqs.(3.14)(3.17), given the processes of the exogenous variables, eqs.(3.18)-(3.20), and the initial conditions C -i > 0, K > 0, and B = 0. The optimality conditions are t  t  t  (C i-hCt\  W^-  1 = ET  1+ r  1  t+  t  t+1  (1-N  - T)  t + 1  \  (3.22)  Yt+i t  1 6  F o r example, consider the government budget that G is financed by lump-sum tax T satisfying t  t  T = gY . This assumption means that G and Y share not only a common trend but also a common t  cycle.  t  t  t  Although this restriction is strict, it is reasonable for the Monte Carlo exercise in this essay  because any shock to G can be considered as a shock to induce the consumption-smoothing motive, t  rather than the consumption-tilting motive.  59  1 -  C  t  -  1  hC -X  21 1 + r,  (i -  t  - TV,  (3.23)  and 1 \K ,  1-V  t  Y  (  EtT, t 1 t+i  1 + 7/ t+1  5,  t+1 t+i  +  1-5 1 - ip  +  <p  i-v>  /t+i  1-ip  \K  't+i  (3.24)  t + l i  Recall that in equilibrium, the level of aggregate consumption must equal that of the representative household's consumption: C — C . Any equilibrium path must satisfy t  t  the optimality conditions (3.21)-(3.24), the constraints (3.14)-(3.17), and the exogenous processes (3.18)-(3.20) with the transversality conditions lim p Et\ ,t+iBt i+i i  B  +  = 0  and  lim p E XK,t+iK +i i i  t  i—>oo.  t  +  = 0  i—>oo  where As and \x,t are the shadow prices for the constraints (3.14) and (3.16), respec)t  tively. Eq.(3.21) shows the stochastic discount factor, which turns out to be a familiar form (3(C i/C )~  when h = 0 and 7 = 1. When h ^ 0 and 7 ^ 1 , the stochastic discount  l  t+  t  factor depends further on past consumption C -\ and leisure at periods t and t+1, L and t  L \.  The higher C _\ is, the lower r  t+  t  t + 1  t  is because the marginal utility of consumption  at period t rises due to habit formation and the marginal rate of the intertemporal substitution falls . Similarly, the higher L is, the lower T 17  t  t + 1  is because the marginal  utility of consumption at period t positively depends on leisure. Eq.(3.22) is the optimality condition for holding the international bonds, i.e. the Euler equation. Notice that if rj = 0, h = 0, 7 = 1, and the world real interest is 7  A rise in C increases the stochastic discount factor r i as in the standard case. t  t +  60  constant, under the assumption of (3(1 + r) = 1, the Euler equation requires perfect smoothness of consumption across periods. Habit formation h > 0, the non-separable period utility over consumption and leisure  and stochastic variations in the world  real interest rate tilt consumption from the perfectly smoothed level through their effects on the stochastic discount factor . The optimal consumption deviates away from the 18  perfect smoothed level, i.e. permanent income. Hence, the deviation can be considered as the consumption-tilting motive or the transitory consumption component. Eq.(3.23) is the optimality condition for the intratemporal substitution between consumption expenditure and leisure. It implies that the marginal rate of substitution between C and L should be equal to the marginal product of labour gross of the ret  t  sponse of the endogenous risk premium to a change in labour. The Euler equation for capital, (3.24), has the interpretation that the expected loss of holding one more capital (represented by the LHS) should be equal to the expected benefit of the additional capital (represented by the RHS). The benefit consists of increased production gross of the risk premium, depreciation and smaller future adjustment costs of investment. On the other hand, the household needs to pay the cost that consists of the current utility loss due to investment in capital. 18  Habit formation makes the household want to smooth not only consumption level but also con-  sumption growth. The non-separable utility over consumption and leisure makes the household desire to smooth not only consumption but also leisure. Finally, if the real interest rate is expected to rise the future, the household wants to tilt consumption toward the future by lending out in international capital markets.  61  3.4.3  The Numerical Solution and Calibration  To derive the numerical solution of the equilibrium path, this essay takes linear approximation of the equilibrium conditions. First, all of the endogenous variables except for N and T are stochastically detrended by dividing them by the random walk technology t  t  shock A. Define the stochastically detrended variables c = C /A , t  w = C -i/A -\, t  t  k = K /A -\  t  t  t  and b = B /A -i.  t  t  t  t  t  i = It/A , y = t  t  t  Y /A , t  t  Next, a first-order Taylor expansion  t  of each of the equilibrium conditions (3.14)-(3.17) and (3.21)-(3.24) is taken around the deterministic steady state. Let x = x — x and x = x / x — 1 for any variable x with t  t  t  t  t  the steady state x. Define vectors Vt and S by t  V = [c t  t  it  yt  N }'  and  t  S = [w t  t  k  b  t  AhiA  t  ln(lTg )]'. t  Then the solution method of Sims(2000) shows that there exists the unique equilibrium path and the vectors V and S follow the processes t  t  Pt = H S 1  t  and  St = H St2  l  + H et 3  (3.25)  where et = [e" e'l]. Eq.(3.25) is the state space representation of the SOE-RBC model of this essay(see Appendix B.4 in detail). Recall that there are fourteen structural parameters in the model. Table 3.3 gives the calibrated values of the structural parameters used in Monte Carlo experiments. This essay conducts two types of Monte Carlo experiments as discussed below. The baseline parameters [3, 7 , </>, tjj, 93, 5, 77, g, a, o and q* are fixed across the experiments a  and set as the mean values of the prior distributions of Nason and Rogers(2003). In particular, across the experiments, the risk premium parameter 77 is chosen to be a very 62  small number 0.000071 in order to cut the effect of the endogenous risk premium on the consumption-tilting motive/the transitory consumption component. In this case, the real interest rate r is almost equivalent to the world common real interest rate q 19  t  t  The first Monte Carlo experiment is related to the SOE-RBC model with habit formation. This case sets the habit parameter depending on the estimated value. Although there are two candidates from two different estimations, the G M M estimate from the full moment conditions, he MM, is suitable because it is more efficient than OQSLS- The problem is that he MM is greater than one, under which there exists no steady state in the SOE-RBC model. Therefore, in this experiment, the habit parameter is chosen to be 0.990, which is close to the estimate and included in the corresponding 90 % confidence interval. This experiment does not allow the world real interest rate to vary stochastically in order to maintain the assumptions of the habit-forming P V M : there is only a countryspecific, unit-root technology shock. To this end, the persistence of the world real interest rate, p  and its standard deviation a are set to be negligible: p = o~ — 1.00 x 10~ . r  (p  q  q  q  Therefore, the resulting theoretical distributions of the text statistics of the PVMs have the SOE-RBC model with habit formation as the null hypothesis. The second experiment is related to the SOE-RBC model with the stochastic world real interest rate. In this case, the world real interest rate is allowed to vary stochastically. Nason and Rogers(2003) also estimate the persistent parameter p and the standard q  deviation a of the common component of the world real interest rate . They give 0.903 20  q  19  A s Nason and Rogers(2003) study, the specific number 0.000071 implies that the risk premium in  Canada is one basis point at an annual rate at the steady state. 20  They calculate the world real interest rate by using Fisher's equation, the three-month Euro-dollar  63  and 0.004 as the means of the prior distributions of p and a , respectively. This essay q  q  uses these values, and also set the habit parameter to zero to rule out the effect of the habit formation. The resulting theoretical distributions of the statistics of the PVMs have the null hypothesis of the multi-shock SOE-RBC model - the SOE-RBC model with the stochastic world real interest rate. Each of the experiments generates 1000 sets of artificial data by which' the theoretical distributions of the test statistics, hqsLS,  ^GMM,  VVr, F{0GMM),  VV and T)  T*(0OLS),  are constructed. The G M M procedure is repeatedly applied to the sets of the artificial data, and the resulting 1000 replications of OGMM are used to construct the theoretical distributions of the statistics. The matching of the theoretical moments with the sample moments is evaluated as in Christiano(1989) and Gregory and Smith(1991). That is, taking the sample statistics as critical values, this essay counts the proportion of times that the simulated number exceeds the corresponding sample point estimate. This proportion is considered as the empirical p-value of the corresponding sample point estimate under the null hypothesis that the data generating process - the underlying SOE-RBC model - is true. Extreme values below 5 % or above 95 %imply a poor fit in the dimension examined.  3.5 Results This section reports the results of the Monte Carlo experiments. The first experiment is related to the SOE-RBC model with habit formation. Three successful aspects of deposit rate, the Canadian dollar-U.S.dollar exchange rate, and the implicit GDP deflator of Canada.  64  the habit-forming SOE-RBC model should be mentioned. The third column of Table 3.4 summarizes the empirical p-values of the sample estimates. First, observe that the p-values of h^sLS  a n  d ^ G M M are 0.7245 and 0.3824, respectively. Figures 3.2(a) and (b)  show the nonparametrically smoothed theoretical distributions of h^sLS d he MM a n  21  Notice that the modes of the theoretical distributions are close to the sample estimates, especially in 3~(6GMM)  hcMM-  Second, Table 3.4 reveals that there are no elements of the vector  that take extreme p-values above 0.95 or below 0.05. The third successful  aspect is observed in the predictions of the habit-forming P V M , CA{. Figure 3.4(a) plots the estimated predictions of the habit-forming P V M and the 90 °/ theoretical confidence 0  band. Note that all the point estimates fall inside the confidence band. The probability that the sample predictions are inside the band through the whole periods is actually equal to 1. Hence at least from these observations, it is hard to reject an inference that the true distributions of h^sis,  hGMMi  J~(@GMM)  a R  d  CA{  are the theoretical distributions  under the null of the SOE-RBC model with habit formation. The habit-forming SOE-RBC model, however, fails to replicate the sample estimates W T , W , J-*(§OIS) T  and  CA . F  T  The third column of Table 4 reports that the empirical p-  values of the Wald statistics for both the habit-forming and standard PVMs, W T and Wf, are 0.0696 and 0.0141, respectively. The p-value of W implies that at the significance r  level of 5 %, the sample estimate rejects the habit-forming SOE-RBC model as the underlying DGP, while the p-value of W T means rejection of the habit-forming SOERBC model on boundary and at least at 10 °L significance level. The nonparametrically 21  T h e smoothed distribution is obtained by the nonparametric kernel density estimation with the  normal kernel.  65  smoothed theoretical distributions of WV and  in Figures 3.2(c) and (d) visually show  the failure of the habit-forming SOE-RBC model to replicate the test statistics of the habit-forming and standard PVMs, WT and Wf: the sample estimates are at the far right tails of the theoretical distributions. Moreover, all the p-values of the elements of the vector  T*(6OLS)  take extreme values above 0.95 or below 0.05, except for  T%  equal  to 0.0605. Finally, Figure 3.4(b) plots the sample predictions of the standard P V M and the corresponding 90 "L theoretical confidence band. Observe how frequently the sample predictions fall outside the confidence band. The probability that the sample predictions are inside the confidence band through the whole period equals to 0.3972. The next Monte Carlo experiment is based on the SOE-RBC model with the stochastic world real interest rate. The surprising result of this experiment is that there is no clear evidence to reject the null hypothesis that the true DGP is the SOE-RBC model with the stochastic world real interest rate. The fourth column of Table 3.4 reports the empirical p-values of the sample estimates in this experiment. First, note that the empirical p-values of  h^sis  and  IIGMM  are 0.115 and 0.1070, which in turn imply that the  underlying SOE-RBC model cannot be rejected even at 10 °/„ significance level. Figures 3.3(a) and (b) draw the smoothed theoretical distributions of liasis and he MM • A l though the dispersion of the theoretical distribution of h^sLS is large, and the theoretical distribution of he MM is heavily skewed toward the left, their modal values are close to the sample estimates. Regarding the vector  J-(0 MM), G  the empirical p-values of all the  elements except for the first one support the SOE-RBC model with the stochastic world real interest rate as the true DGP. As shown in Figure 3.5 (a), even with a couple of exceptions, almost all of the sample predictions on the current account, CA{, fall inside 66  the theoretical 90 % confidence band. The probability that the sample predictions are inside the band is equal to 0.9858. The result of the Wald statistic WT is the first clear difference between the two Monte Carlo experiments. In the SOE-RBC model with the stochastic world real interest rate, the empirical p-value of the Wald statistic WV is 0.5499. This implies that the sample estimate is fairly close to the median of the theoretical distribution, and the underlying null cannot be rejected at any standard significance levels. Its smoothed theoretical distribution in Figure 3.3(c) visually repeats this inference. Furthermore, striking differences are observed regarding the sample statistics related to the standard P V M . The empirical p-value of the Wald statistics for the standard P V M , Wf, is 0.3259, which in turn implies together with the smoothed theoretical distribution in Figure 3.3(d) that the null of the SOE-RBC model with the stochastic world real interest rate cannot be rejected in this dimension. Except for JF^, all the estimates of the elements of the vector T*(6OLS) have the p-values between 0.05 and 0.95. Moreover, Figure 3.5(b) shows that the sample predictions are inside the 90 % theoretical confidence band in greater number of periods than in the case of the habit-forming SOE-RBC model. Indeed, the probability that the sample predictions are inside the band through the whole periods is 0.8156. This observation echoes the main finding of Nason and Rogers(2003): stochastic variations in the world real interest rate can explain the rejections of the standard P V M observed in the literature. The results of the two Monte Carlo experiments are summarized in Table 3.5. This essay therefore reveals the superiority of the SOE-RBC model with the stochastic world real interest rate to the habit-foming SOE-RBC model to explain the broad empirical 67  facts of the habit-forming and standard PVMs. Better than habit formation in consumption, stochastic variations in the world real interest rate explain the transitory consumption component/the consumption-tilting behavior, which is a crucial factor of the DGP of the Canadian current account.  3.6  Conclusion  This essay issues a caution about interpreting the empirical results from the habitforming P V M as evidence that habit formation in consumption plays a significant role in explaining current account movements. One reason is that the habit-forming P V M is observationally equivalent to the non-habit P V M associated with serially correlated transitory consumption. This makes identification of the habit-forming P V M of the current account problematic. Monte Carlo simulations based on SOE-RBC models are one to avoid this identification problem. The simulation exercises study the ability of different SOE-RBC models to mimic the sample moments or the empirical facts conditional on the habit-forming and standard PVMs. Two SOE-RBC models are hypothesized as the true DGPs of the sample moments: the one with with habit formation and the other with the stochastic world real interest rate. The Monte Carlo simulations make it possible to construct the theoretical distributions of the sample moments from the two hypothesized DGPs. The results of the matching exercise based on the post-war Canadian data support the SOE-RBC model with the stochastic world real interest rate. The model matches all the key sample moments of the habit-forming and standard PVMs. The SOE-RBC  68  model with habit formation mimics only a part of the empirical facts of the habitforming P V M . This model fails to mimic the cross-equation restrictions predicted by the habit-forming P V M and all the empirical facts related to the standard P V M . Thus, the SOE-RBC model with a world real interest rate shock dominates the habit forming SOE-RBC model. Recent studies of Lettau and Uhlig(2000) and Otrok, Ravikumar and Whiteman(2002) claim counterfactual predictions of habit formation on several aspects of macroeconomics, e.g. consumption volatility and the equity premium puzzle. This essay also casts doubts on habit formation as an important source for the Canadian current account movements.  69  Chapter 4 Habit Formation and Aggregate Dynamics in Real Business Cycle Models  4.1  Introduction  Habit formation in consumption is proposed as a way of resolving the empirical puzzles in behavior of asset prices. The habit-forming consumer takes care of past consumption in determining current consumption: having consumed a good deal in the past, she also tends to consume a good deal in the current period. Therefore, habit formation makes a consumption process smoother. The equity premium puzzle and the risk-free rate puzzle are solved by introducing habit formation simply because smoother consumption implies the higher marginal rate of intertemporal substitution, which in turn yields a lower risk-free rate even under moderate curvature of the utility function. 70  In spite of the success of habit formation in solving the two asset pricing puzzles, it is still controversial what implications habit formation has for aggregate economic dynamics in the context of the real business cycle (RBC) models. Francis and Ramey(2002) argue that the one-sector RBC model with habit formation and adjustment costs of investment can replicate the negative response of hours worked to a positive permanent technology shock, which Gali(1999) finds by applying his structural VAR (SVAR) identification to the U.S. data. To the contrary, in their one-sector RBC model with the habit-forming utility function of Campbell and Cochrane(1999), Lettau and Uhlig(2000) show that their model generates an extremely smoothed consumption path, which cannot match the sample volatility of the H-P filtered U.S. consumption. Furthermore, they find that habit formation tends to dampen volatilities of output and investment counterfactually. Finally, Boldrin, Christiano and Fisher(2001) develop the two-sector RBC model with habit formation and inflexible labour mobility across sectors, and show that their model is successful in explaining broad business cycle dimensions in the U.S. data . One exception 1  is that their model cannot replicate the negative response of labour input to a positive, permanent technology shock. This essay evaluates Francis and Ramey's(2002) one-sector RBC model with habit formation and adjustment costs of investment by examining the model's ability to account for sample moments representing aggregate dynamics of the U.S. data. The main question this essay asks is whether or not the habit-forming RBC model resolving Galf's(1999) ]  F o r example, their model is successful in explaining the sample first and second moments of asset  prices, output, consumption and investment, the comovement of employment across sectors, the excess sensitivity of consumption to income, and the inverted leading indicator phenomenon.  71  observation can explain the dynamics of consumption and output in the U.S. data. The dynamics of consumption and output are characterized by three moments of the sample: (i) autocorrelation functions (ACFs) of the growth rates of consumption and output, (ii) spectral density functions (SDFs) of the growth rates of consumption and output, and (iii) impulse response functions (IRFs) of log of output to permanent and transitory shocks. As studied by Cogley and Nason(1995), the IRFs of log of output are identified by applying Blanchard and Quah's(1989) long-run restriction to a bivariate, secondorder SVAR including the growth rate of output and hours worked. The equilibrium path of the RBC model is log-linearly approximated around the deterministic steady state. The resulting linear rational expectation model is solved to obtain the state space representation, which is used to conduct Monte Carlo experiments. The results from the matching exercise are summarized in Table 4. 1. First, the habit model fails to mimic the significantly positive, first and second order ACFs of output growth in the sample. Second, the habit model cannot replicate the maximum power spectrum observed over business cycle frequencies in the sample. Third, the habit model fails to generate the hump-shaped IRFs of output to a transitory shock. Fourth, the habit model overstates the higher order ACFs of consumption growth. Fifth, the habit model overstates the power spectrum around zero frequency. The first three results confirm Cogley and Nason's(1995) conclusion: the propagation mechanisms embodied in standard RBC models do not generate the right kind of output dynamics. This essay reveals that this conclusion is also applicable to the habit-forming R B C model. The next two results echo the observation of Lettau and Uhlig(2000): habit formation in consumption makes the consumption path counterfactually smooth. 72  Moreover, this essay examines the implications of the habit model for the asset pricing puzzles. As many past studies show, habit formation in consumption can generate a high equity premium and a low risk-free rate on average; hence, the equity premium puzzle and the risk-free rate puzzle are solved by introducing strong habit formation. However, as the sixth result in Table 4.1, the habit model fails to yield the high volatility of the rate of returns on equity observed in the sample. This is because collaborating with adjustment costs of investment and elastic labour supply, habit formation in consumption dampens volatilities of output and investment. That is to say, as emphasized by Francis and Ramey(2002) in accounting for Gali's(1999) observation, the mechanism that generates the negative correlation between labour input and a permanent technology shock leads to a wrong implication for an aspect of asset pricing behavior. Therefore, it is hard for the one-sector RBC model with habit formation and adjustment costs of investment to survive as a restricted data generating process of the aggregate dynamics of the postwar U.S. economy. The next section reviews the empirical facts of consumption and output dynamics. Section 4.3 introduces the habit RBC model of this essay. Sections 4.4 and 4.5 discuss the results summarized in Table 4.1 in detail. Finally, Section 4.6 makes conclusion.  73  E m p i r i c a l Facts of C o n s u m p t i o n a n d O u t p u t D y -  4.2  namics This section introduces the sample moments related to consumption and output dynamics. As in the standard RBC literature, this essay uses real, seasonally-adjusted GNP as output, and divides it by total population to obtain per capita output. The data of consumption are constructed by taking the sum of real, seasonally-adjusted personal expenditures on nondurable goods and services and dividing the result by total population. The data of hours worked are constructed from the average weekly hours of production workers . The sample period spans between Ql:1954 and Q2:2002. 2  Figures 4.1(a) and (b) show the sample estimates of the ACFs and SDFs for the growth rate of output . The figures repeat the well-known empirical fact regarding GNP 3  growth: the GNP growth rate is positively and significantly autocorrelated over short horizons. At lags of 1 and 2 quarters, the sample ACFs are significantly positive. Furthermore, the SDF for output growth has its maximum power at roughly 14 quarters or 3.5 years per cycle. As discussed by Cogley and Nason(1995), this means that a relatively large portion of the variance of output growth occurs at business cycle frequencies. 2  D R I Basic Economics distributes all the data. In particular, this essay uses the civilian noninstitu-  tional population as total population. A l l the data series are seasonally adjusted at annual rates. 3  T h e A C F s are estimated by the G M M procedure with the optimal weighting matrix calculated by  the heteroskedasticity-autocorrelation consistent estimator of Newey and West(1987). Following Cogley and Nason(1995), this essay estimates the S D F s by smoothing the sample periodogram using a Bartlett window.  74  On the other hand, Figures 4.1(c) and (d), which respectively plot the ADFs and SDFs of the growth rate of consumption, show no clear evidence that the growth rate of consumption is persistent at business cycle frequencies. Although the ACFs of the first 6 quarter lags are positive and about 0.15 on average, all sample ACFs are insignificant except for the lag of 6. Furthermore, the maximum power of the estimated SDFs is at zero frequency. Figures 4.2(a) and (b) plot the sample estimates and the corresponding 90 percent confidence band of the IRFs of log of output to both permanent and transitory shocks identified by Blanchard and Quah's(1989) long-run restriction . Figure 4.2(b) repeats the 4  most important observation of Blanchard and Quah(1989) and Cogley and Nason(1995): output has a significant, hump-shaped response to a transitory shock over the shorthorizon. This observation implies that output appears to have an important trendreverting component.  4.3  The Model  This section introduces a closed-economy, one-sector RBC model with adjustment costs of investment and habit formation in consumption, and asks whether or not the RBC model can mimic the empirical facts of the consumption and output dynamics found in the last section. 4  T h e 90 percent confidence bands are calculated by 1000 non-parametric bootstrapping resamples.  75  4.3.1  A One-Sector R B C M o d e l w i t h A d j u s t m e n t C o s t s of Investment a n d H a b i t F o r m a t i o n i n C o n s u m p t i o n  Let C and N denote consumption and labour supply at period t, respectively. As in t  t  Boldrin, Christiano and Fisher(2001) and Francis and Ramey(2002), the lifetime utility function of the representative household is oo  0 < /5 < 1,  (4.1)  0<h<l  where E and (3 are the conditional expectation operator on the information set at period t  t and the subjective discount factor. The parameter h characterizes habit formation in consumption: if 0 < h < 1, the representative household forms consumption habits, while if h — 0 the utility function turns out to be time-separable with unit relative risk aversion, as in the standard RBC model. This essay adopts the "internal habit" specification. In this case, current utility depends on household's own past consumption, rather than aggregate past consumption as in the "external habit" or "catching-up-with-the-Joneses" specification studied by Abel(1990). As discussed in Constantinides(1990) and Boldrin, Christiano and Fisher(2001), the internal habit specification makes it possible to derive a high equity premium even under moderate levels of risk aversion. Eq.(4.1) also shows that the utility function is defined as the logarithm of the difference C — hC -\. This difference specit  t  fication, as in Campbell and Cochrane(1999), yields time-varying risk aversion . Habits 5  depend on only 1 lag of consumption. 5  O n the other hand, the ratio specification studied by Abel(1990) and Fuhrer(2001) yields constant  risk aversion. 76  The representative household owns capital and technology to produce consumption goods. Let Y , K , I , and A denote output, capital, investment, and the aggregate state t  t  t  t  of technology at period t. The production function is Cobb-Douglas: Y = Kf(A N y-^ t  t  0 < V < 1  t  (4.2)  where tp implies the capital share. The aggregate state of technology A follows an t  exogenous random walk with drift in log term: A = A _ exp(a + e") t  t  x  e ~ i.i.d.N(0, a ). a  2  t  a  (4.3)  The law of motion of capital is K  t+1  = (l-5)K +  (jfflt  t  0<5<1  (4.4)  where 5 is the depreciation rate of capital. The second term of the RHS of eq.(4.4) implies that the representative household faces adjustment costs of investment. Jermann(1998) and Boldrin, Chiristiano and Fisher(2001) find that when the utility function is habitforming, adjustment costs of investment improve the ability of a one-sector RBC model to account for behavior of asset prices. Moreover, Francis and Ramey(2002) argue that the combination of habit formation and the adjustment costs of investment yields the negative response of N to a permanent technology shock, which Gali(1999) observes in t  his SVAR identification. This essay follows Baxter and Crucini(1993) in specifying the adjustment costs of investment. The aggregate resource constraint is Y = C +I +G t  t  t  77  t  (4.5)  where G is government consumption spending that is assumed to be exogenous and t  stochastic .  This essay follows Nason and Rogers(2002) in specifying the stochastic  6  process of G : G shares a common trend with Y , and the ratio of government spending t  t  t  to output g = G /Y t  t  t  follows an exogenous stationary process in log term  .gt = G /Y = { *) - "9 tU exp(e?) l  t  p  4 ~/./. /..\(<). a ).  P  2  9  (  g  (4.6)  Hence, as in Cogley and Nason(1995), the model is driven by two exogenous shockstechnology shocks and government spending shocks. The equilibrium allocation is found by solving the household's optimization problem at period t: maximizing the lifetime utility (4.1) subject to the production function (4.2), the law of motion of capital (4.4), the budget constraint (4.5), two exogenous driving forces (4.3) and (4.6), and the initial conditions K > 0 and C _x > 0 given. Together t  t  with the transversality conditions for the state variables K and C -\, the first-order t  t  necessary conditions characterize the equilibrium path of the economy.  4.3.2  Numerical Solution, Calibration, and Evaluation  This essay log-linearly approximates the equilibrium path around the deterministic steady state. Solved by Sims's(2000) method, the resulting linear rational expectation model 6  T h e reason stochastic variations i n government consumption expenditure are allowed is that this  essay repeats the S V A R exercise of Cogley and Nason(1995). The authors identify the I R F s of A In Y to t  both permanent and transitory shocks by applying the Blanchard and Quah's(1989) long-run restriction to a bivariate S V A R including A l n Y i and N as the endogenous variables. W i t h o u t the government t  spending shock, the standard, single-shock R B C model with the permanent technology shock makes the bivariate S V A R stochastically singular.  78  derives the state space representation of the equilibrium path, which in turn is used to conduct Monte Carlo simulations to generate artificial data of aggregate variables. The model is calibrated by the parameter values of Chiristiano and Eichenbaum(1992), Cogley and Nason(1995), Boldrin, Christiano and Fisher(2001), and Nason and Rogers (2002). Table 4.2 summarizes the parameter values this essay uses. In particular, g* is calibrated to the U.S. data by taking the sample average of the government spendingGNP ratio. Given the other calibrated parameters, the habit parameter h = 0.985 is obtained by maximizing the ability of the model to account for the risk-free rate . This 7  essay conducts two Monte Carlo experiments: one with habit formation (h = 0.985) and the other without habit formation (h = 0). In other words, this essay considers the one-sector RBC model with adjustment costs of investment as the benchmark model, and compares the moment-matching performance of the benchmark model with that of the habit-forming RBC model. The ability of the models to replicate the sample moments is evaluated from the viewpoint of classical statistics. The model is considered to be restricted data generating processes (DGP) for the sample moments. The synthetic data generated by Monte Carlo simulations yield the theoretical distributions of the sample moments under the null hypothesis that the RBC model is the restricted DGP. The sample moments are used as critical values to evaluate the null hypothesis: if a sample moment drops outside 5 percent of the corresponding theoretical distribution, the null hypothesis-the RBC 7  T h e risk-free rate is calculated as the inverse of the expected stochastic discount factor(i.e. the  marginal rate of intertemporal substitution of consumption) minus one. It is defined i n detail in section  3.5.  79  model-is rejected by two side test at 10 percent significance level. In addition, this essay constructs the generalized Q statistics for the ACFs for the growth rates of consumption and output, and for the IRFs of output to permanent and transitory shocks, as proposed by Cogley and Nason(1995). For example, the generalized Q statistic for the ACFs of the growth rate of output has the null hypothesis that all replicated ACFs of the first 8 quarter lags match their sample counterparts. The same is true for the generalized Q statistic for the ACFs of the growth rate of consumption. The generalized Q statistic for the IRFs of output has the null hypothesis that the replicated IRFs at the first 8 periods after impact match the sample counterparts. Under the null hypothesis, each Q statistic asymptotically follows the % distribution with 8 degrees of 2  freedom. Hence, the matching performance of the model is also evaluated by the x test 2  statistics . 8  4.4  Results  This section reports the results of the matching exercise.  4.4.1  ACFs and SDFs for the Output Growth Rate  First, the upper two windows of Figure 4.3 show the sample estimates of the ACFs and SDFs of the growth rates of output, and the corresponding 90 percent confidence bands constructed by 1000 artificial data generated under the null of the benchmark, non-habit model. Notice that the sample ACFs at lags of 1 and 2 are clearly outside the 90 percent 8  F o r detailed derivation of the generalized Q statistics, see Cogley and Nason(1995).  80  confidence band: the benchmark model fails to mimic the sample first and second ACFs. Moreover, the maximum power spectrum observed around 14 quarters in the sample is also outside the confidence band: the benchmark model fails to replicate the empirical fact that a relatively large portion of the variance of output growth occurs at business cycle frequencies. Table 4.3 shows that for the ACFs of the growth rate of output, the benchmark model yields the generalized Q statistic of 40.994 with zero asymptotic pvalue. Hence, the null hypothesis that the model-generated ACFs up to 8 lags match the sample ACFs is strictly rejected. The failure of the benchmark, non-habit model in mimicking output dynamics echoes the observations of Cogley and Nason(1995). Can habit formation in consumption improve the matching performance of the RBC model with respect to output dynamics? The answer is no. Again, the lower two windows of Figure 4.3 show the sample estimates of the ACFs and SDFs of the growth rates of output, and the corresponding 90 percent confidence bands constructed by 1000 artificial data generated under the null of the habit model in this experiment. Notice that all of the above results of the benchmark model can be applied to the habit model: the habit model cannot replicate the sample ACFs with lags of 1 and 2 and the maximum power spectrum at the business cycle frequencies. The generalized Q statistics for the ACFs of the growth rate of output, which is given in Table 4.3, is slightly smaller than that of the benchmark model; however, the asymptotic p-value is still zero.  81  4.4.2  IRFs of Output  The next matching exercise involves the IRFs of output to permanent and transitory shocks. Figure 4.4 summarizes the results for the IRFs of log of output to permanent and transitory shocks. The upper window is related to the IRFs to a permanent shock, while the lower window is related to those to a transitory shock. In each window, the solid line shows the sample IRFs, the dashed line shows the mean of the IRFs generated by the benchmark, non-habit model, and the dotted line shows the mean of the IRFs generated by the habit model. It is important to mention the following three observations. First, on average, the habit model overstates the IRFs of output to a permanent shock, while the benchmark model understates them. Second, the two models fail to mimic the humpshaped response of output to a transitory shock observed in the sample. Hence, habit formation is not a reliable propagation mechanism in explaining the humped-shaped response of output. This inference is further strengthened by the third observation: the habit model generates a transitory shock that falsely has a very persistent effect on output. In addition to Figure 4.4, the generalized Q statistics related to the IRFs of output in Table 4.3 statistically indicate the poor matching performance of the habit and benchmark models in this dimension. For the IRFs of output to a permanent shock, the habit and benchmark models respectively yield the generalized Q statistics that are 112.525 and 40.152 with zero p-values, while for the IRFs of output to a transitory shock, the two models yield the generalized Q statistics of 236.037 and 24.189 with zero p-values. Although these chi-squared test statistics strictly reject both the habit and benchmark  82  models as the restricted DGP of the IRFs of output, the habit model yields the generalized Q statistics about three times larger than those of the benchmark model. This is indirect evidence that habit formation deteriorates the ability of the RBC model to explain an important aspect of the output dynamics.  4.4.3  ACFs and SDFs for Consumption Growth  The clearest implication of habit formation in consumption is that it produces a smoother consumption path than that of the time-separable utility function. Consequently, checking the matching performance of the habit model for consumption dynamics is the most direct and helpful exercise in evaluating the habit model. The upper two windows of Figure 4.5 illustrate the sample estimates of the ACFs and SDFs of the growth rates of consumption, and the corresponding 90 percent confidence bands constructed under the null of the benchmark model. Observe that in the two windows, the 90 percent confidence bands include almost all of the sample ACFs and SDFs with an exception of the sample SDFs around 28 quarters per cycle. Therefore, it is safe to say that the benchmark model can mimic the ACFs and SDFs of the growth rate of consumption fairly well. This successful aspect of the benchmark model is also confirmed by the generalized Q statistic in Table 4.3. This chi-squared test statistic is 7.917 with p-value 0.442. This means that at 10 percent significance level, the benchmark model cannot be rejected as the restricted DGP of the sample ACFs of the growth rate of consumption. Notice in Table 4.3 that the generalized Q statistic is not able to reject the habit  83  model as the DGP of the sample ACFs of consumption growth up to 8 lags: the corresponding generalized Q statistic is 6.94 with p-value 0.543. On the other hand, the lower two windows of Figure 4.5 show the sample estimates of the ACFs and SDFs of the growth rates of consumption as well as the corresponding 90 percent confidence bands constructed under the null of the habit model. Observe that the theoretical confidence band for the ACFs is shifted up relative to that of the benchmark model. As a result, the sample ACFs are almost on the lower(left) boundary of the confidence band. In particular, the sample ACFs of lags of 7, 8, 9, 13, 15, 16, and 17 are outside the the confidence band: these ACFs reject the habit model at least at 10 percent significance level. These facts lead to an inference that the habit model overstates the ACFs of consumption growth. Next, the sample SDFs around zero frequency are below the theoretical 90 percent confidence band. This means that the habit model overemphasizes volatilities of the growth rate of consumption around zero frequency. The above inference that the habit model overstates the ACFs of consumption growth is shared with Lettau and Uhlig(2000) who argue that the habit-forming utility of Campbell and Cochrane(1999) yields a counterfactually smooth consumption path. Moreover, by using the concept of the spectral utility function, Otrok, Ravikumar and Whiteman(2002) show that the habit-forming utility (4.1) makes the household more averse to high-frequency fluctuations of consumption than to low frequency fluctuations. The model-generated SDFs of consumption growth are excessively concentrated around zero frequency, as shown in Figure 4.5, reflect that the habit-forming household prefers low frequency fluctuations of consumption growth to high-frequency fluctuations. However, because the sample SDFs around zero frequency are below the theoretical 90 percent 84  confidence band of the habit model, this aspect of the habit model is simply rejected.  4.5  Implications for Asset Prices  The last section shows the difficulties involved when using the RBC model with habit formation to explain the consumption and output dynamics in the U.S. data. Despite these difficulties, Jermann(1998) and Boldrin, Chiristiano and Fisher(2001) claim that the one-sector RBC model with habit formation and adjustment costs of investment can solve two empirical puzzles of asset pricing behavior: the equity premium puzzle and the risk-free rate puzzle . The habit model in this essay also solves the two asset pricing 9  puzzles. However, this essay reveals another difficulty of the habit model: it fails to explain the high volatility of the rate of return on equity in the sample. The risk-free rate and the rate of return on equity implied by the RBC model are calculated on the equilibrium path as in Boldrin, Christiano and Fisher(2001). First, the risk-free rate r{ is given as the inverse of the expected stochastic discount factor minus one: - 1  r{ = &v-  where F 9  t + 1  (4.7)  t+i  is the stochastic discount factor or the intertemporal marginal rate of sub-  T h e equity premium puzzle is the empirical fact that returns on the stock market exceed returns  on Treasury bills by an average of 6 percentage point. The standard consumption C A P M explains this phenomenon only by an extremely high risk aversion. Weil(1989) then points out the risk-free rate puzzle if, indeed, consumers are highly risk-averse. T h e return on Treasury bills is low on average, and consumption grows steadily. To reconcile these empirical facts with high risk-aversion requires that consumers be extremely patient with a low or even negative rate of time preference.  85  stitution of consumption. Second, the rate of return on equity r  is  e  t+i  t+i t+i )  —  +(  l-<p)  \K  t +  J\  +  t+l  (1-5)  Qt+\  -  (4.8)  1  Qt  where q is the relative price of capital to consumption, which is well-known as Tobin's t  q and calculated on the equilibrium path by  Intuitively, eq.(4.8) says that the rate of returns on equity equals the amount of goods the representative household can consume at period t+1 if investing a unit of consumption goods to capital at period t. To invest a unit of consumption goods to capital, the household has to pay q at period t. The invested capital, on the other hand, increases t  consumption at period t+1 by raising output and reducing adjustment costs [i.e. the first and second terms of the RHS of eq.(4.8)]. Moreover, if the relative price q  t+1  rises,  the household can consume capital gain net of depreciation [i.e. the third term]. The "Data" column of Table 4.4 shows estimates of the mean of the risk free rate Erf, the mean of the equity premium E(r*  +1  — r{), the standard deviation of the rate  of return on equity a <-., and the Sharpe ratio E(r^ r  +1  — r{)/a e, r  over the U.S. sample .  First, observe the "Benchmark" column in Table 4.4. This column implies that the benchmark, non-habit RBC model fails to solve the two asset pricing puzzles. That is to say, the replicated mean of the risk-free rate is too high to account for the sample mean, while the replicated mean of the equity premium is too low to match the sample mean. This model also fails to explain the high volatility of the equity return: the replicated 1  "These sample moments are provided by Ceccheti, L a m and Mark(1993). Boldrin, Chiristiano and  Fisher(2001) also use these sample moments.  86  10  standard deviation of the rate of return on equity is 0.82, while its sample counterpart is 19.4. Finally, this model yields a lower Sharpe ratio than that in the sample. The "Habit" column of Table 4.4, on the other hand, reports evidence that habit formation in consumption helps solve the two puzzles of asset pricing. First, the replicated mean of the risk-free rate takes a value close to the sample mean: the former is 1.28, and the latter is 1.19. Second, the habit model increases the equity premium by 3.6 percent more than the benchmark model. Although the habit model still understates the sample mean of the equity premium by about 3 percent, as discussed by Boldrin, Christiano and Fisher(2001), this discrepancy is not important because the gap can be closed if a slightly higher curvature were introduced into the utility function. Caveats, however, should be added to the above successful results of the habit model in the two asset pricing puzzles. First, this model fails to mimic the sample standard deviation of the rate of return on equity; indeed, habit formation makes the replicated standard deviation of the rate of return on equity lower than that of the benchmark model. Second, the high equity premium and the low standard deviation of the rate of return.on equity automatically imply an implausibly high Sharpe ratio, which is observed in the last column of Table 4.4. The failure of the habit model in explaining the high volatility of the rate of returns on equity stems from the following result of habit formation. Together with the adjustment costs of investment and elastic labour supply, habit formation in consumption conterfactually dampens the volatilities of output, investment, and capital, which jointly determine the volatility of the equity return through eq.(4.8). The habit-forming consumer desires an extremely smooth consumption path. To smooth consumption, he/she 87  can adjust investment and labour supply in the closed-economy RBC setting. However, since the consumer faces adjustment costs of investment, he/she does not want to change investment a lot. The consumption-smoothing enforced by habit formation is implemented mainly by adjusting labour supply in a countercyclical way. In particular, as shown in Boldrin, Christiano and Fisher(2001) and Francis and Ramey(2002), this model implies that a positive, permanent technology shock reduces labour input over the short-horizons because the income effect overcomes the substitution effect. This negative correlation between a permanent technology shock and labour input makes output less volatile . Hence, the volatilities of output and investment are counterfactually 11  dampened by strong habit formation.  4.6  Conclusion  Francis and Ramey(2002) consider the one-sector RBC model with habit formation in consumption and adjustment costs of investment as a candidate for the restricted DGP for the negative response of labour input to a positive, permanent technology shock, which Galf(1999) find in his SVAR identification. This essay reexamines other dimensions of their model, and shows that this type of the RBC model fails to replicate the dynamics of consumption and output in the postwar U.S. data.  As many past studies show,  habit formation can help solve the two asset pricing puzzles. However, habit formation 1 1  Jermann(1998) observes high volatility of the equity return i n a one-sector R B C model with habit  formation and adjustment costs of investment.  This is because the model assumes constant labour  supply. The representative household can adjust only investment to smooth consumption.  88  dampens volatilities of both output and investment and yield extremely low volatility of equity returns. Based on these results, this essay concludes that it is hard to find support for habit formation in consumption in the one-sector RBC model, even though the model is consistent with Gali's(1999) observation. One way of future research is to abandon habit formation in consumption. There are several non-habit models that can generate a negative response of labour input to a positive, permanent technology shock: e.g. the sticky price model in Gali(1999), the Leontief model with labour-saving technology shocks in Francis and Ramey(2002), and the home production RBC model in Campbell and Ludvigson(2001) to give examples. However, it is still unclear what implications these models have for the asset pricing behavior. 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Solving linear rational expectations models, manuscript, http://eco072399b.princeton.edu/yftp/gensys/.  Weil, P. (1989). The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary Economics, 24:401-421.  West, K. D. (1988). Dividend innovations and stock price volatility. 56:37-61.  96  Econometrica,  Appendices A : Appendices of Chapter 2  A p p e n d i x A . l : D e r i v a t i o n of the L i n e a r - A p p r o x i m a t e d Intertemporal B u d g e t Constraint  (2.7)  Dividing the intertemporal budget constraint (2.6) by NO gives t  ( *%£- + Z > *  (Al.l)  1+  i=0  i=0  Notice that for any variable X the relation %ti =  • • • -j^-  t  At  J  JVO,  A t At+i  holds. Therefore  Aj+j-i  eq.(Al.l) can be rewritten as 00  C  t-\-i  /  t  NO  i=l  t  j=t+i  v  J  1  7  (A1.2)  97  Notice that for any variable X , the relation t  rij=t+i -^j  =  ex  P{X^=t+i (^\?)} holds. m  From this relation and the definition of i? , eq.(A1.2) can be further rearranged as t>i  t+i  C NO  t  1+  Y Et exp I J2 ( i=i  t  J ~  A l n C  l n  ( l + r )) d  [j=t+i  = exp{ln(l + r ) - A ln NO } t  t  co  1+  Y Et  ( e x  P {  B NO; t-1 t  t+i  Y  [j=t+i  (  A l n N 0  J " ( hl  1  (A1.3)  + i)) r  Taking afirst-orderTaylor expansion of the LHS of eq.(A1.3) around the mean values gives ThTlHS «  —*- + 1— a  i V Ot  T a Et { AhrTcW, - l n ( l + ^ ) ) .  (.4.1.4)  1  + i  1—  OL  I  t—'  J  z=l  where a = exp(7 — fi) < 1. The RHS of eq.(A1.3) is also approximated as c  T h T R H S « --J^—  + - l n ( l + r,) - - A h T J V O ,  K NO -l  K  t  K,  oo  1  + 7 ^ I > ^ { A i n ^  t  +  i  - i n ( i T ^  +  l  ) } -  W  i=l  where K = exp(7 — \i) < \.  From the results of eqs.(A1.4) and (.45), the linear-  approximated intertemporal budget constraint (2.7) is finally given as C NOt t  1—a  B  t  +  1 - a , , -rp— . 61n(l + r ) t  1 — a , . -r~^rru 6AlniV0 4  oo  Y«'  Et {A^Ct+i  i=l  - ln(f+^ )} + l  co  1  +  Y  98  ' & \^AlnNOt+i - ln(l + n ) }  K  +i  A p p e n d i x A . 2 : D e r i v a t i o n of the A p p r o x i m a t e d Solution of the O p t i m a l C u r r e n t A c c o u n t Ratio(2.10) Substitute the log-linearized Euler equation (2.8) into the linear-approximated intertemporal budget constraint (2.7). For simplicity, assuming that the economy is around the balanced growth path; a = K and using the approximation ln(l + r ) ~ r ,1 can obtain t  t  the consumption-net output ratio equation as C  --77— = NO t  t  1—K  «  B  ' NO -!  +  t  1 — K ,„  1 — «;  br  A  t  t  K  t  -r^rT^  bA\nNO  K CO  CO  - (CT - l)c Y^ & ~ Y ^ & ^ ?t+i  i=l  t+i  i=l  oo +  EA\^NO .  (A.2.1)  t+i  To derive the optimal current account-net output ratio equation, consider the current account identity (2.9). Dividing eq.(2.9) by NO rewrites eq.(2.9) as t  CA  t  =  exp[ln(l + r )\ - 1 B exp(AlnATO ) NO -i  1  t  NO  t  C NO  t  t  t  t  t  Taking a first-order Taylor expansion of the above equation gives CA  t  NO,  1 K  B NOt-i t  exp(7)  +  K  ~r  t  1 K  exp( )J  bA ln NO  t  -  C NO ' t  (A.2.2)  t  7  Substituting the consumption equation (A.2.1) into (A.2.2), I can obtain the equation of the optimal current account-net output ratio: CA Wh t  1 exp(7)  B NO -! t  t  + br  t  1 bAlnNOt exp( )J 7  + [(a - l)c + 1] ]T K* Et i=l  99  r  t  +  i  -  & A ln~iVO . i+J  i=l  Since exp(7) takes a close value to one, it might be a reasonable approximation to set the coefficient [1 — 1/ exp(7)] to zero. Then the optimal current account-net output ratio equation (2.10) is constructed as CA y-j  °° = br + {(a - l)c + 1]  0 0 K <  t  Et r  t+i  -  i=l  4  K> Et A In  NO . t+i  i=l  A p p e n d i x A . 3 : D e r i v a t i o n of C r o s s - E q u a t i o n Restrictions H  cp  and H  cs  To derive the cross-equation restrictions ri  cp  and ri , I exploit the Wiener-Kolmogorov cs  formula, which is well-known as Hansen and Sargent's(1980) distributed predicted leads formula. For exposition, I give this formula as the following lemma without proof; Lemma (Hansen and Sargent (1980)). For a covariance-stationary process X with t  a Wold MA representation X — A(L)v t  t  and (3 £ (0,1) it is the case that ;  'A{L)-A{PY  L-{3  v. t  By using the present value relation (2.10), the maintained DGPs of the first difference of log of net output and the world real interest rate, (2.11) and (2.12), and the above lemma, I can derive a structural MA representation of the current account-net output ratio CA  j^-  = r»(L)ef + T%(L)e? + T™{L)eT  (A.3.1)  where T (L), r™(L) and F^(L) are infinite-order polynomials, respectively, which sata  g  100  isfy  r-(L)  r - ( L ) = ferj(L) + [ c ( a - l ) + l]«  L—K r%(L)  L  (A.3.2) — K  - r£(«)  L —  K  L—  K,  (A.3.3)  and (A.3.4)  under the assumption of the small open economy (2.14). Since the impact responses of the current account ratio to ef and e are given as c s t  and ri  cs  17(0)  and  17(0),  respectively, ri  cp  are obvious from (A.3.3) and (A.3.4).  Appendix A.4: D a t a Description and Construction This essay uses quarterly data of four G-7 economies, Canada, Japan, the U.K. and the U.S., which span the sample period Ql:1960-Q4:1997. All data are seasonally adjusted at annual rates and provided by Datastream and IFS. To construct a measure of the world real interest rate, r , I follow the method of t  Barro and Sala-i-Martin(1990) and Bergin and Sheffrin(2000). I collect short-term nominal interest rates, three-month Treasury bill rates or money market rates, on the G-7 economies from IFS. The inflation rate in each country is calculated by using that country's CPI and the expected inflation rate is constructed by regressing the inflation rate on its own eight lags. The nominal interest rate is then subtracted by the expected inflation rate to compute an ex-ante real interest rate. The world real interest rate is computed by taking a weighted average of ex ante real interest rates across the G-7 economies, 101  with time-varying weights for each country based on its share of real GDP in the G-7 total. To construct the net output and current account series, I use each country's national accounting data distributed by Datastream. All nominal series are converted to real series by using the GDP price deflators. The resulting real series are divided by population. Following definition, I construct the net output series, NO , by subtracting t  gross fixed capital formation, change in stocks and government consumption expenditure from GDP. Taking a log of the net output series and a first difference of the resulting logarithmic series provides the first difference of log net output A In NO . The current t  account series, CA , is constructed by subtracting gross fixed capital formation, change t  in stocks, government consumption expenditure and private consumption expenditure from GNP. Dividing CA by NO provides the series of the current account-net output t  ratio,  t  CAt/NOt.  Finally the three series, r , A In NO and CA /NO , t  series, r Aln NO and t)  t  t  t  t  are demeaned to construct the  CA /NO . t  t  A p p e n d i x A . 5 : U n i t R o o t Tests To check whether r , A l n N O t  t  and CA /NO t  are stationary, I apply the augmented  t  Dickey-Fuller test (the ADF test) for the three series. The ADF r-statistic for time series y is given as a t-statistic of the coefficient A in the following OLS regression t  n  (A.5.1)  102  where the lag length n is chosen to render rj white noise. Since the demeaned series r , t  A l n NO and CA /NO t  t  t  t  fluctuate around zero and have no clear time trend, I do not  include either constant or a time trend in the ADF regression (A.5.1). Davidson and MacKinnon(1993) provide asymptotic 10 %, 5 % and 1 % critical values for the DickeyFuller T-statistics equal to -1.62, -1.94 and -2.56, respectively. I perform this test for three choices of the lag length, one, three and five. Table A . l summarizes the results of the unit root tests. Except for CA /NO t  t  of the  U.S., the ADF tests reject the unit root null in all series at least at the 5 °L significance level for all cases of the lag length. In the case of CA /NO t  t  of the U.S., the ADF tests  reject the unit root null at the 10 °/ significance level for three and five lags, while the 0  unit root null cannot be rejected even at 10 % significance level for the case of one lag.  A p p e n d i x A . 6 : P r e d i c t e d L i n e a r Restrictions on the Impact M a t r i x In this appendix, I show that all the hypotheses can be rewritten as linear restrictions on the impact matrix T(0). For exposition, let [A]\ and [A]\ denote the i th row and column vectors of a matrix A, respectively. First of all, recall that two of three restrictions in identification scheme II are zero restrictions on the impact matrix, which means that they are linear restrictions on the impact matrix in nature. More precisely, let  denote a 1 x 3  row vector which has zeros as the j ^ i th elements and one as the i th element. Then two exclusion restrictions are rewritten as r(0)i = ei^O)]^ = 0 and F(0) = e [r(0)]g = 0, >2  103  ii3  1  respectively. Second, it is the case from eq.(2.22) that C(l)r(0) = T(l). This relation implies that r(l) - = [C(l)]£[r(0)]^ for any i,j = 1, 2, 3. Therefore a long-run restriction ij  r(l) j = 0 should be equal to an orthogonality condition between the ith row vector of i]  C(l) and the j th column vector of T(0) and can be rewritten as a linear restriction on the impact matrix. To find the impulse response functions (the IRFs) of CA to a global shock ef for t  —i  any i > 0, I take a derivative of the identity CA = (CA /NO )NO t  t  t  and obtain the  t  following relation dCA _ CA ~o^eJ~~CA/NO t  where CA/NO  DCA /NO del, t  and CA are means of CA /NO t  dh\NO ~WT  t  t  +  and the CA , respectively. In particular,  t  t  the last term in the RHS of the above relation can be given by the accumulated impulse response of AlnNO  t  to ef . Hence the IRF of CA to ef_ is given by —i  t  t  where r™° and r^" are the impulse responses of A l n N O and CA /NO t  t  t  to ef_ respecit  tively. Let Ci denote the coefficient matrix of L in the V M A (2.16). Since Fj = C;F(0) for l  any i > 0, the IRFs, r™° and T ^ , can be written as follows: = (c,r(o)) = (ar(o))  2il  =  murmt  3il  =  [c&Fio)} ! 0  These equations and eq.(A.6.1) rewrite hypothesis 1 as (^ra  +^E^lTOJ^^rai^O NO  s=0  104  Vz>0  (A.6.2)  where Ri is a 1 x 3 row vector such that Ri = { [ C ^ + [CA/NO)  E U o M J - Therefore  hypothesis 1 is also rewritten as a linear restriction on the impact matrix. Next notice from eq.(2.21) that T(K) = C(K)T(0) and thus 1\,(K) = [C(K)]-[r(0)]5 for any i,j = 1,2,3. By using this fact, I can rewrite hypothesis i for i=2,3 as r(o)  3li  = r(o)  2ii  -  [C{K)unm  or more compactly, with a 1 x 3 row vector R = [C («) 2jl  R[Y(<d)] = 0. c  i  C (KO — 1 £2,3(1) + 1], 2j2  (A.6.3)  Eq.(A.6.3) shows that hypothesis 2 and 3 are also given as the linear restrictions on the impact matrix. A striking fact is that under tfie joint null hypotheses the impact matrix F(0) should be singular. To show this, first consider identification Scheme I. Notice that there are three linear restrictions on [T(0)]§ under the null: [C(l)ft[r(0)]§ = 0, '[C(1)]5[T(0)]| = 0 and i?[r(0)]g = 0. Since these restrictions are linearly independent and [T(0)]§ is a 3 x 1 vector, a unique solution for [T(0)]§ exists and should be equal to zero. This implies then that the impact matrix [T(0)] should be singular under the null. The same result is obtained even with identification Scheme II. In this case, three linearly independent restrictions on [T(0)] under the null are given as ei[r(0)]^ = 0, [C(l)]^[r(0)]g = 0 and c  s  i?[r(0)]g = 0. Therefore a unique solution for [r(0)]g exists and equals to zero. The impact matrix should be singular under the null. The singularity of the impact matrix makes it impossible to examine the LR and L M tests for the null since these asymptotic tests depend on the restricted ML estimates of 105  the test statistics. On the other hand, the asymptotic Wald test, which exploits only the unrestricted ML estimates, is applicable for this situation.  106  B: Appendices for Chapter 3  A p p e n d i x B . l : Derivation of Eq.(3.6) Let CAf denote the standard P V M under h = 0 and C f = 0: /  0 0  t  1  =  CA[ = rB + NO -Cf t  V J S AiVO t  i=l ^  t + i  .  (B.l.l)  '  Substituting the decomposition C — C f + C f into the current account identity and t  using eq.(B.l.l) yield CA  t  + NO -  = rB  t  t  C  t  = rB + NO - C f - C t  t  T t  = CAf-Cf.  (B.1.2)  Applying the AR(1) process of C f to eq.(B.1.2) gives CA = CA  - Cf  p  t  t  =  CA[- Cl -u Pc  = CA _ Pc  t  x  1  t  + CAf - P c C A f i - ^  107  (B.l.3)  Several steps of algebra easily show that the term CAf — p CA^_ c  has the following  A  representation  C A f  -  04f_,  ANO, - (l - £ - )  =  ft  £  (  R  L . ) ' W 0  W  -(^KI+T)'^- -''^6  Note that the last term of the RHS represents revision of expectation for future changes in net output between periods t and t — 1. Let this term be, say, £ , and notice that t  expectation of £  t + s  conditional on the information set at period t is zero for any s > 1  by law of the iterated expectation. Substituting back the term CA — p CA -\ t  c  t  into  eq.(B.1.3) and setting v = t;t + u provide eq.(3.6). t  t  A p p e n d i x B.2: D e r i v a t i o n of Eq.(3.7) Substituting the P V M (3.2) into the definition of D yields t  A = CA - ANO t  t  - (1 +  r)CA -! t  = -(1 + r - h)CAt-, - ( l - J^j = hD ^ t  (^J  £ AiVO t  - hDt-, - (1 + r - / O C A - i - ( l - j - ^ - ; ) E  (+J  - e  t  (1T7)  ^NOt+i  - *e  (B.2.1) Substituting the definition D - i = C A - i — ANO -i t  t  — (1 + r)CA -2 t  into the second  term in the RHS of eq.(B.2.1) and using the P V M (3.2) to eliminate the resulting term  108  CA -i t  further rewrite eq.(B.2.1) as  A = KDt-i - e + (1 + r)e -i t  t  + (l + r - A ) g  A J V O ^ . , - (l - —  J  j £  j W  A  +  ,  .(B.2.2) Note that the fourth term in the RHS of eq.(B.2.2) equals  Therefore eq.(3.7) is the case: A  = hD -i - e + (1 + r ) e _ ! - ( l t  = /iA-i-e  t  J2 (jj^)  t  t  & -  Et-x)ANO  t+l  + (l+r)e _ + e .  Note that expectation of e  t  t+s  1  t  conditional on the information set at period t is zero for  any s > 1 because £ e t  t + s  = - (l -  (T+~T)  E t { E t + s  ~  t+s-i) °t+*+s  E  AN  =0 by the law of the iterated expectation.  A p p e n d i x B.3: T h e T w o - S t e p G M M E s t i m a t i o n In the first step, the criterion function J{6) = G(9)'MG(8)  is minimized with respect  to 9 under the restriction that the weighting matrix M is the identity matrix / . The 109  resulting estimate of 9, say 9*, is used to construct the optimal weighting matrix M* such that M* = t-i  when gt(9*) follows an i.i.cl. process. Because there is a possibility of serial correlation of gt(9*) in the first step, this essay exploits the heteroskedasticity-autocorrelation consistent estimator of Newey and West (1987) to calculate the optimal weighting matrix M*. In the second step, minimizing the criterion function J{9) under the optimal weighting matrix M* yields the second step estimate 9GMM with the asymptotic variance-covariance matrix 8GMM  v  9G(9GMM) 1  9G(9GMM)  89'  89'  A p p e n d i x B.4: T h e State Space Representation of the Equilibrium Path The purpose of this appendix is to explain in detail the derivation of the state space representation from the system of stochastic difference equations, which contains eqs.(3.14)(3.24). The first step is to convert the system to the stationary one. To do that, it is convenient to introduce a new variable w satisfying t  &t =  Ct-i/At-i.  That is, w is stochastically detrended consumption at period t — 1. t  110  (B.4.1)  B.4-1: Deriving the Stationary  System  Using the stochastically detrended variables and eqs.(3.17), (3.18) and (B.4.1) rewrites the system of equations (3.14)-(3.16) and (3.21)-(3.24) as the following stationary system: The Stationary System  H+i  exp(—Aln A  l + qt  t  exp(-Aln  y = kfNl^expi-^ k  = (1 — 8) exp(—Aln A )k +  t+1  t  /?exp {[</>(!- )- l ] A l n A } 7  + 1  t  Aln  t  c  A )b + {1 - g)y t  t  t  (3.15')  A) t  (3.16')  — ) i exp(—ipAln A,  t  it  t  1 - Nit+i (l-^)(l-7) l-Nt  ct i feexp(-AlnA+i)^t+i -i^(l-7)-l Ct — hexp(—Aln A )w +  t  t  r i t +  1+ q  i = ar  - 27?exp(-Aln  t+1  t  t  (1-V)  t  EtT,t+i  A ) t+1  "t+i  (3.21') (3.22')  Vt+i  Ct — hexp(—A ln A )w l-N  l-(p  (3.14')  (3.23')  l + '/?exp(-2AlnA) — Vt  N  exp((/?A In A ) —  \k  t  t  {1-5  -t- zr^-  1 - ip  exp[(l - yOAln A i ] f  + E r -(/»exp(AlnA+i t  t+1  * 1 exp^Aln A  +  1-ip  +  1  \k +ij  *  \kt+i.  t  Vt+i 1 + 7?exp(-2A InAt+i) kt+i  ) f ^  yt+i  (3.24')  and eq.(B.4.1). The stationary system contains the eight equations, the eight endogenous variables and the two exogenous variables following the processes (3.19) and (3.20).  Ill  B.4-2: The Deterministic Let  Steady State  c, y, i, N, k, b, T and va denote the deterministic steady state values of the cor-  responding variables. From the stationary system, the deterministic steady state is characterized as follows. First, from eq.(3.21'), the steady state value of the stochastic discount factor, T, is given as  r = /?exp{[</>(7-!)-!]«} where a is the unconditional mean of A l n A . Eq.(B.4.1) shows that the steady state t  value vo is equal to c vo = c. From eqs.(3.16') and (3.22'), the steady state ratios i/k and b/y are determined by — = [1 — (1 — 5) exp(—a)] ^ e x p ^ a ) ^ k 1  1  and q* - $ ' 2nexp(—a) i +  Given i/k and b/y, the steady state ratio y/k is determined as a solution of the equation  (£)  e x p ( v o )  r  =  { r ^ + r ^ i - " i {ff ' J ^ ' e x p  ( 1  ) a  (£f  + Tip exp (a) y I + 77 exp(—2a) k Because i/k and y/k have been already derived, the steady state ratio i/y can be constructed by dividing i/k by y/k. Eqs.(3.15') and (3.23') then yield the steady state ratios 112  k/N and c/y as  i  k k  N  0-1  and c  1 + <f - V [ - ) exp(-a)  y  exp(-a)  -  + (1 - g)  \y.  Finally, eq.(3.23') determines the steady state level of N as a solution of the equation  -^[l-fcexpC-aJl^a-^)-^  1 + 77exp(—2a)  Given iV, the steady state level k is obtained by multiplying the ratio k/N by AT. The steady state level y is obtained by multiplying y/k by k. Similarly, the other steady state levels c and i are constructed by multiplying c/y and i/y by y, respectively . 12  B.4-3:  Derivation  of the State Space  Representation  The next step is to take a first-order Taylor expansion of the system (B.4.1), (3.14')(3.16') and (3.21')-(3.24') around the deterministic steady state. Let x = x — x and t  t  x = x /x — 1 for any variable x with the deterministic steady state x. Note that the t  t  t  linear approximations of eqs.(3.15') and (3.23') are static equations. By using this fact, y and N can be solved as linear functions of c\, k , b and A In A , respectively, which t  t  t  t  t  in turn are used to solve out y and N in the other linear approximations of eqs.(3.14'), t  t  (3.21'), (3.22'), (3.24'). Furthermore, eq.(3.21') characterizes the process of the stochastic discount factor. Using the linear approximation of this equation can solve out the 1 2  I t is important to note that the above derivation of the steady state does not require solving a  nonlinear simultaneous equation system. This fact makes the following numerical exercise simple.  113  stochastic discount factor in the other equations. As a result, the linear approximations of (3.14'), (3.16'), (3.22'), (3.24') and (B.4.1) are given as linear stochastic difference equations with respect to the five endogenous variables Q, i , w , k and b , and the two t  t  t  t  exogenous variables A ln A and ln(l + q ) that follows the stochastic processes eqs.(3.19) t  t  and (3.20), respectively. Let X = \c\ i t  t  £>t fa bt A l n A  ln(l + q )]. Then it is shown that the system  t  t  of the linear stochastic difference equation has the matrix representation: Q X = e X - + *e + ILv 0  t  1  t  1  t  (B.4.2)  t  where Q and 0 i are 7 x 7, f and LT are 7 x 2, e = [e" e ]' and v is the vector of 0  t  t  t  expectational errors satisfying c\ - Et-idt Vt  =  it - Et-yit  The leading matrix 6 is non-singular and invertible. 0  This essay solves the linear rational expectation model (B.4.2) by following Sims(2000). Sims argues that the disturbance vector \I/e + Uu is not exogenous as e itself is, because t  t  t  v depends on the endogenous variables c and i and their expectations. Hence solving t  t  t  the linear rational expectation model (B.4.2) needs to determine v from e . t  t  Since the leading matrix is invertible, premultiplying eq.(B.4.2) by 0 Q yields 1  X = QiXt-i + #e + nV t  t  t  (B.4.3)  where 0 j = 0 Q © ! , 4> = O ^ ^ and n = © o ^ . The matrix 0\ has the eigenvalue 1  1  decomposition such that 0! =  VAV'  1  114  where A is the diagonal matrix that contains the eigenvalues of Qi in the descending order in absolute value, and V is the matrix constructed by the corresponding eigenvectors. Premultiplying eq.(B.4.3) by V~ and defining a new vector Z — V~ X yield l  l  t  Z = AZ - +V- $!e  + ilv }.  l  t  t  l  t  t  (B.4.4)  t  Let A i be the diagonal matrix that contains the eigenvalues greater than or equal to one in absolute value. Also let Z\ is the vector containing the elements of Z cort  responding to the explosive eigenvalues. Then eq.(B.4.4) implies that Z\ follows the process Z\ = A Z\_ X  X  + Bifiet + livt].  where B comes from the partition V~ = [B'  B']'. Since all the diagonal elements of  l  x  (B.4.5)  x  Ai are explosive eigenvalues, eq.(B.4.5) has a forward solution such that oo  Z\ = - E A r 5 i [ * e t i i + nVt+i+i].  (B.4.6)  i_1  +  +  i=0  Notice that E Z\ = Z\ because all the elements of X are included in the information t  t  set at period t. Since E e t  = E v +i — 0 for any i > 1,  t+i  t  t  oo  Z\ = E Z\ t  = -J2  AT^B^Etet+i+x  + Uu ] t+i+1  = 0.  (B.4.7)  Comparing eq.(B.4.6) and (B.4.7) shows that the following equality must be satisfied oo  £ A p i=0  1  ^ [*e  t + z + 1  + flut+i+i] = 0.  (B.4.8)  For eq.(B.4.8) to be satisfied it is the case that Billut = -B^e  t  115  (B.4.9)  for all t. Therefore, as Sims argues, the necessary and sufficient condition for the existence of a solution satisfying eq.(B.4.8) is that the column space of Bi^f be contained in that of BjII. That is, for any realization of e , there must exist some v satisfying eq.(B.4.9) t  t  for the existence of a solution. Next consider the stable part of eq.(B.4.4): Z? = A Zl 2  1  + B $e 2  + IIv ].  t  (B.4.10)  t  where A is the diagonal matrix that contains the eigenvalues of 0 , which are less than 2  2  one in absolute value. The problem, here is the uniqueness of the solution. Since the existence requires eq.(B.4.9), B\Ylv can be determined from a known stochastic process t  for e . However, eq.(B.4.10) requires that B YLv should be known at the same time. It t  2  t  is possible that knowing BiU.v is not enough to show B Tlv when the solution is not t  2  t  unique. Sims gives as the necessary and sufficient condition for the uniqueness that the row space of B Ii be contained in that of Bill. 2  In other words, it should be the case  that there exists some matrix <> f satisfying B tl = $ B i f i . 2  (B.4.11)  Suppose then that the necessary and sufficient conditions for the existence and uniqueness of the solution is satisfied, i.e. eq.(B.4.9) is satisfied and a matrix <> f satisfying eq.(B.4.11) exists. Then eq.(B.4.10) can be rewritten as Zl = A Zli 2  + (B - QBJVet. 2  (B.4.12)  To derive the state space representation, it is convenient to partition the vector X  t  116  and the matrices V and V  X = [X '  X?'}',  1  t  t  as  1  X} = [<k i ]', t  X? = {vo  k  t  Via  v  21  B  Bu  B\  B\  B  2  2  22  AteAt  t  =  v  b  ln(lT%)]',  2  22  Moreover assume that from eq.(B.4.11) the matrix 4> is obtained as  $ =  B Il(B Il)- . 1  2  1  For the inverse to exist, the matrix Bill must be square. This then implies that the row number of B\ must be 2 because the column number of LT is 2. Since by construction Bn is square, B  n  Z\ = B\X  t  and B  i2  should be 2 x 2 and 2 x 5 . Recall that eq.(B.4.7) requires  — 0. Hence it is the case that  Xl = -B^B X?  (B.4.13)  l2  Eq.(B.4.13) shows the cross-equation restrictions characterizing the saddle path. Using these cross-equation restrictions (B.4.13) can rewrite eq.(B.4.12) as the process of X^\  X = g-'A^Xl, 2  t  + g~\B  - SB^tf  2  where Q is a 5 x 5 matrix satisfying Q = B  - B iB Bi . l  i2  2  n  2  (B.4.14)  Let X? = S . Eq.(B.4.14) t  is then the transition equation of the state variables S in eq.(3.25). Recall that y and t  t  N are given as linear functions of X . This fact and eq.(B.4.13) yields the observation t  t  equation with respect to Vt of the state space representation eq.(3.25).  117  T a b l e 2.1: T h r e e E m p i r i c a l R e s u l t s o f t h e I n t e r t e m p o r a lA p p r o a c h  a n d  t h e P V M  o f t h e  C u r r e n t  A c c o u n t : How Does the Current Account Respond to the Shocks? 1.  Does a Global Shock Have No Impact on the Current A c count? • Yes: Glick and Rogoff(1995), i§can(2000) • Sensitive to Identification: Nason and Rogers(2002)  2.  Country-Specific Transitory Shocks Dominate the Current Account Fluctuations in the Short-Run as Well as the Long-Run: Nason and Rogers (2002)  3.  Glick and Rogoff's (1995) Puzzle • Persistent Country-Specific Shock: Glick and Rogoff(1995) • Permanent and Transitory Decomposition by the V E C M : Hoffmann(2001) • Nontradable Goods: I§can(2000)  118  T a b l e  2 . 2 : F i n d i n g s  o f T h i s  E s s a y  1. Impulse Responses of the Current Account to the Identified Shocks are Consistent with the Corresponding Theoretical Predictions 2. Tests for the Cross-Equation Restrictions on the SVAR Show • T h e Hypothesis the Current Account Does Not Respond to a Global Shock is Sensitive to the Identification. • The Impact Responses of the Current Account to Country-Specific Shocks M a t c h the P V M ' s Predictions. • The Joint Hypothesis Related to the Impact Responses of the Current Account to A l l the Three Shocks is Rejected.  3. The Data Support the Observation that the Current Account Responds to a Country-Specific Transitory Shock Greater than Net Output. 4. The FEVDs Show that Country-Specific Transitory Shocks Dominate Current Account Fluctuations Not Only in the Short Run But the Long Run As Well, While the Shocks Explain Almost None of the Fluctuations in Net Output.  119  T a b l e  2 . 3 : I d e n t i f i c a t i o n  S c h e m e s  (a) Identification Scheme I Economic  Meaning  Restriction  A Country-Specific Permanent Shock Has N o Long-Run  Effect  r(l)i, = o  Effect  r(i)i, = o  Effect  r(i)  2  on the W o r l d Real Interest Rate A Country-Specific Transitory Shock Has N o Long-Run  3  on the W o r l d Real Interest Rate A Country-Specific Transitory Shock Has No Long-Run  2l3  =o  on L o g of Net Output  (b) Identification Scheme II Economic  Meaning  A Country-Specific Permanent Shock Has N o  Restriction r(o)  Instantaneous  1)2  =o  Effect on the W o r l d Real Interest Rate A Country-Specific Transitory Shock Has N o  Instantaneous  r(o)i,  3  = o  Effect on the W o r l d Real Interest Rate A Country-Specific Transitory Shock Has N o Long-Run  Effect  r(i) ,3 = o 2  on L o g of Net Output  N o t e 1:  I n a d d i t i o n to three restrictions,  each i d e n t i f i c a t i o n scheme requires the  structural  shocks to be o r t h o g o n a l a n d have u n i t variances. N o t e 2: F ( 0 ) a n d T ( l ) are the i m p a c t a n d the l o n g - r u n m a t r i c e s of the S V M A , F o r a m a t r i x A, Aij  shows the ( i , j ) t h element of the m a t r i x  120  A.  respectively.  T a b l e  2 . 4 : C a l i b r a t e d P a r a m e t e r s a n d J o i n t T e s t t h e P r e s e n t V a l u e R e s t r i c t i o n s Canada  the U . K .  K  0.993  0.990  C  0.983  0.988  b  -0.712  0.377  a  0.001  0.080  W  18.193  23.224  p-value  0.000  0.000  o f  N o t e 1: T o c a l i b r a t e b requires the d a t a o f i n t e r n a t i o n a l b o n d h o l d i n g s Bt- T h i s essay uses as Bt the i n t e r n a t i o n a l net investment p o s i t i o n (IIP) i n the balance of p a y m e n t statistics. S t a t i s t i c s Canada (http://www.statcan.ca)  d i s t r i b u t e s the a n n u a l IIP for C a n a d a from 1926 to 2001.  T h i s essay converts the a n n u a l series to q u a r t e r l y series, d i v i d e s the r e s u l t i n g series b y n o m i n a l net o u t p u t a n d takes the s a m p l e average from Q l : 1 9 6 3 - Q 4 : 1 9 9 7 to c o n s t r u c t b. O n the other h a n d , N a t i o n a l S t a t i s t i c s ( h t t p : / / w w w . s t a t i s t i c s . g o v . u k ) provides the a n n u a l IIP series of the U . K . o n l y from 1966. Nevertheless, the value of b for the U . K . is c a l i b r a t e d b y a p p l y i n g the same m e t h o d as i n the C a n a d i a n case for the whole sample p e r i o d 1966-1997. N o t e 2: T h e e l a s t i c i t y of i n t e r t e m p o r a l s u b s t i t u t i o n a is c a l i b r a t e d b y m i n i m i z i n g the m e a n squared error of the P V M p r e d i c t i o n o n the current account-net o u t p u t r a t i o . N o t e 3: T h e W a l d s t a t i s t i c W is c a l c u l a t e d b y eq.(2.25) c o n d i t i o n a l o n the c a l i b r a t e d parameters ft, c, 6, a n d a. t h i r d degree of  T h e c o r r e s p o n d i n g p-value is based o n the chi-squared d i s t r i b u t i o n w i t h the freedom.  121  2 o OS  <  2 o  2 o  On  2 o O  2 ofQ  2 o  W  W  H  H H H H cr cr cr cr ft  CD  O  O  ET 3C  fi>  H cr  fD  un  05  3  ST  U>  5-  n> o o  CS  Uh  52.  «>  O  fa  TO  Bi o  fO  P M-j  O  ^  ES  w3  O  4^  U>  Vi  ff.  o  cr cr  <D  p.  cs  i— vi  £r cr Cr  O  O  ro  ft>  UJ  to  Ci  p o  Ci CI Ci  >°  Ci to  Ln  Os  ^  tO  Ci  O  N vo  as  Uj  Ci  °  so  ON o\  O  —  vo  3  S  CD  <-r  P &-  p  O  3  r-r  CD  o  P  P  l=h  a"a s-  O-  a-  r ' -  H  i—*•  P  p  Ci vo  bo  U> <-h  IO UJ  Ci Ci Ci Ci  U> to O  CD c*>  vo VO  Ci i — t o Ln C i oo Oo VO  Ci vo Oo U>  o o o  Ci  p  Oo -t\  oo  r-r  O  un  r-r  CD  o  f°  '  -  p c£  ty>" K>  >  GO  o Otr  ff ff  </>  to  ts  GO  ffi  vi  ff.  o  ro  CO •-»• t/i  cr  O  IO  Vi  <y>"  50  cr  o'  — I  tC X X X  O  fu  O  ^ 3 3 3  >5  e  O  M-j  3 cr br fcr cr &r cr jo jo a-  P  a*  to  ro  r+>  H  P  <  o  Eh ro  a.  OO  W  •S P  o ^3  § 3 g -  un  TO  <  §  ^—i  s  '—'  TO  TO  ^3  •  ^ UJ  K  a>  CD co  5i o  o  ts <-+  TO  P  r-f-  Ci Ci Ci o,  cs  Ci Ci Ci  oo  o  VO  tO 4^ 00  o  VO  Ci C i OS «vj o UJ  os  1—  Ci  bo  Uj  to —]  \1 Os  o oo to  Ci Ci Ci  o  o B  UJ  VO J>.  4*.  122  Ci Ci  UJ  Ui  to  KJ  Ci ivj  1— VO  Ci Ci Ci Ci  to 4^  o o  C/3  o  cr ct) p a- B P  Ci Ci Ci Ci  o ts  •  a>  r-r l-t H-* •  r-f  o !/3  2 2 o n  H  o Ft  2  H  OQ B* ~  rt  o  rt  o  ro cr> hrj tt <  ft  13 •O 03  ro 3  2 rt  Ou 3 O ro  o  rt B  o  o  CT 03  " o o c r  CD  3  n >  S-«3 a i *Q  2  3  o-  2. ai  P - 13  °1 n>  Cfi  _  IT)  ro  o  i-*  2  PCD  B  P  o>  <—f  JO  » O  5:  B  "  O  g  o  ° >3  B O B T3  ~  GO O  B* O  3 ST 3 o cr  ro  B  ^3 n,  2  §n r+ 03  rt  13  *o  5"  0Q  03  3  123  z; 2  o o rt- rt  ro  ro  H CTQ cr " Si 3 ^ c P m  Si  3  H  cr ro  13 P  ro •a ro  3  ~  Er <JQ £ o ro c r p ~  vi  O.  r  2 o 3 3  ° B  ro  3  ro cn  T3  Er3 '  2. 3 2 prt. P  w  °1 n. ro  -a 2  PT 3  S e o - 5 3 ,  S? p  ©  SS.  3 O 3  » ^  ©S  -  S o <  •a o P o  3 ^ 3 2  «  S-  ro  o o cr 3 § < rt  P 13  •  •a  5'  CTQ P  3  13  124  Table 3.1: The Sample Statistics of the P V M s (a) T h e H a b i t - F o r m i n g P V M h-GMM  h-2SLS  W  JT  T  0.931  1.002  0.455  37.128  (0.192)  (0.152)  [0.978]  [0.000]  PA  T  -0.302  -0.068  0.017  0.006  1.276  -0.400  0.062  0.138  (0.130)  (0.059)  (0.073)  (0.049)  (0.226)  (0.157)  (0.073)  (0.079)  (b)The S t a n d a r d P V M f{  h  *77* --5  T*  ••3  -77* --8  20.589  0.229  0.066  0.010  0.106  -0.115  0.046  -0.019  -0.095  [0.009]  (0.171)  (0.179)  (0.126)  (0.088)  (0.408)  (0.106)  (0.113)  (0.106)  Note: Table 3.1(a) reports the sample statistics of the P V M with habits. h^SLS is the 2SLS estimate of the habit parameter based on the single unconditional moment conditions (3.8) while he;MM is the G M M estimate of the habit parameter based on the full unconditional moment conditions (3.8), (3.10) and (3.12). JT is the x  2  statistic with the fourth degree of  freedom for the overidentifying restriction test. WT is the x  2  statistic with the eighth degree  of freedom for the cross-equation restrictions (3.11). The brackets below JT and W T show the corresponding asymptotic p-values. Ti represents the estimate of the i t h element in the vector T(6).  The numbers in parentheses give the asymptotic standard errors for the corresponding  estimates. Table 3.1(b) shows the sample statistics for the standard P V M . W  T  is the x  2  statistic  with the eighth degree of freedom for the cross-equation restrictions of the standard P V M . T* represents the estimate of the ith element in the cross-equation restrictions of the standard PVM.  125  Table 3.2: Empirical Facts of the Present Value Models  (a) T h e P V M with H a b i t F o r m a t i o n  1. The Habit Parameter is Close to One. 2. The Cross-Equation Restrictions are Jointly Rejected. 3. The Fifth Element of  J-(OGMM)  is Close to One.  4. The Predictions Tracks the Actual Series Closely.  (b) T h e S t a n d a r d P V M  1. The Cross-Equation Restrictions are Jointly Rejected. 2. The Fifth Element of T*(6 LS) 0  is Close to Zero.  3. The Predictions are Too Smooth.  126  Table 3.3: Calibrated Parameters of SOE-RBC Models  Baseline Parameters /?  qb  0.994  7  0.371  ip  2.000  V  0.350 g  0.071 x 10~  4  tp 0.050  0.020 a  a  0.230  5  a  0.0024 0.012  Monte Carlo Experiments with Habit Formation h  p  a  0.990  1.000 x I O "  q  q  7  1.000 x I O "  7  Monte Carlo Experiments with the World Real Interest Rate h  p  a  0.000  0.903  0.004  q  127  q  Table 3.4: Sample Estimates and Empirical P-values under the Nulls of SOE-RBC Models E m p i r i c a l P-values Sam,pleEstimates  Habit Formation  W o r l d Real Interest Rates  h2SLS  0.931  0.7245  0.1150  hGMM  1.002  0.3824  0.1070  W  37.128.  0.0696  0.5499  ti  -0.302  0.7326  0.9536  t  -0.068  0.6297  0.5217  fz  0.017  0.4733  0.4571  t  0.006  0.4904  0.6670  h  1.276  0.2593  0.1493  f  -0.400  0.7841  0.7215  f  0.062  0.4198  0.3885  Ts  0.138  0.3481  0.1766  W  20.589  0.0141  0.3259  ft  0.229  0.0000  0.7164  H  0.066  0.0000  0.8073  0.010  0.0071  0.7952  •r 4  0.106  0.0131  0.0363  <£•*  -0.115  0.9980  0.4773  tl  0.046  0.0151  0.7548  •> 7  -0.019  0.0605  0.8295  -0.095  0.0111  0.9072  T  2  4  6  7  T  J 5  Note: E m p i r i c a l p-values are c o n s t r u c t e d as the frequency of times t h a t the s i m u l a t e d n u m b e r exceeds the c o r r e s p o n d i n g sample p o i n t estimate.  128  Table 3.5: The Monte Carlo Experiments: Which SOE-RBC Model Mimics the Empirical Facts? 1. The S O E - R B C model with habit formation mimics the first, third and fourth facts of the habit-forming P V M . 2. The S O E - R B C model with habit formation fails to mimic the second fact of the habit-forming P V M : the Wald statistics for the cross-equation restrictions. 3. The S O E - R B C model with habit formation fails to mimic all the facts of the standard P V M . 4. The S O E - R B C model with stochastic world real interest rates mimics all the facts of the habit-forming P V M . 5. The S O E - R B C model with stochastic world real interest rates mimics all the facts of the standard P V M . In particular, the model does a better job in replicating the third fact of the standard P V M than the S O E - R B C model with habit formation does.  129  Table 4.1: Failures of the One-Sector R B C Model with Habit Formation and Adjustment Costs of Investment The Habit Model 1. Fails to Mimic the Significantly Positive, First and Second Order A C F s of Output Growth. 2. Fails to Mimic the Maximum Power Spectrum of Output Growth Over Business Cycle Frequencies. 3. Fails to Mimic the Hump-Shaped IRFs of Output to a Transitory Shock. 4. Overstates the Higher-Order A C F s of Consumption Growth. 5. Overstates the Power Spectrum of Consumption Growth around Zero Frequency. 6. Fails to Yield the High Volatility of Equity Return.  130  Table 4.2: Calibrated Parameters of the Model Parameter  C a l i b r a t e d Value  Source  P  0.992  CE, C N  a  0.004  CE, CN, B C F  0.360  BCF  5  0.021  CE, CN,B C F  9*  0.228  U . S . data  0.050  NR  0.960  CE, C N  0.018  BCF  0.021  CE  Pg  h  0.985  Notes: C E , C N , B C F , and N R denote Chiristiano and Eichenbaum(1992), Cogley and Nason(1995), Boldrin, Christiano, and Fisher(2001), and Nason and Rogers(2002b), respectively. In particular, g* is calibrated to the U.S. data. Given the other parameters, h is calibrated to maximize the ability of the model to account for the risk free rate.  131  Table 4.3: Generalized Q Statistics  ACFs  IRFs  Model  AlnY*  AlnC;  Permanent  Transitory  Benchmark  40.994  7.917  40.152  24.189  (0.000)  (0.442)  (0.000)  (0.002)  39.411  6.94  112.525  236.037  (0.000)  (0.543)  (0.000)  (0.000)  Habit  N o t e : I n the table, each n u m b e r denotes the generalized Q s t a t i s t i c , a n d the n u m b e r i n the parenthesis shows the c o r r e s p o n d i n g p-value. F o r d e r i v a t i o n of the generalized Q statistic, see C o g l e y a n d N a s o n ( 1 9 9 5 ) . A l l the generalized Q statistics i n the table a p p r o x i m a t e l y follow the chi-squared d i s t r i b u t i o n w i t h 8 degrees of freedom.  132  Table 4.4: Asset Price Statistics Statistic  Data  B e n c h m a r k ( / i = 0)  Habit (h = 0.985)  Er{  1.19  4.94  1.28  0.05  3.67  0.82  0.40  0.06  9.16  (0.81) E(rt  - r{)  +1  6.63 (1.78)  cv  19.4 (1.56)  E(rl -r{)/a e +1  r  0.34 (0.09)  N o t e s : (i) T h e " D a t a " c o l u m n reports estimates of the m e a n of the risk free rate, the m e a n of the e q u i t y p r e m i u m , the s t a n d a r d d e v i a t i o n s of the rate of r e t u r n of equity, a n d the S h a r p e r a t i o , w i t h s t a n d a r d errors i n parentheses, over the p e r i o d 1892-1987 for U . S . d a t a . T h e s e n u m b e r s are t a k e n from C e c c h e t t i , L a m a n d M a r k ( 1 9 9 3 ) a n d B o l d r i n , C h i r i s t i a n o a n d F i s h e r ( 2 0 0 1 ) . (ii) A l l statistics are a n n u a l i z e d a n d i n percent terms, (iii) T h e s t a t i s t i c from the m o d e l s are based o n 1000 M o n t e C a r l o e x p e r i m e n t s .  133  Table A . 1 Unit Root Tests no of lags r  1  3  5  -2.217 **  -2.572  -2.012 **  Canada A l n NO CA/NO  -9.520  -6.010 ***  -5.140 ***  -2.864  -2.167 **  -2.393 **  Japan A h NO CA/NO  -6.083 *** -2.502 **  -10.863 -3.018 ***  -4.427 *** -2.825 ***  U.K. A h NO CA/NO  -6.220 *** -2.448 **  -9.938 -2.325 **  -5.430 -2.672 ***  U.S. A h NO CA/NO  -7.090 ***  -5.060 ***  -4.756 ***  -1.605  -1.816  -1.911 *  Note 1: The unit root tests are based on the A D F t-test. Since each variable is demeaned, the A D F regression does not include both constant and trend. Note 2: ***, ** and * denote that the unit root null is rejected at 1%, 5% and 10% significance levels, respectively. Note 3: Asymptotic 1%, 5% and 10% critical values are provided by Davidson and MacKinnon( 1993) and equal to -2.56, -1.94 and -1.62, respectively.  134  z; z;  H  0 o F» rt  I ciT >  to —  1 *  tt W <  Ht  1g rt  O  t r era  rt fir* cn O n> cn  a. rt  O •-+)  cr p  o  3 ° S e rt 3  ro n  >  B-«3 P 2 cn 13 3  2.  a. HI 1-1 " o. -a  CJ  rt 3 2  P-  cr  O-  o  HH  P P 3  CD  cn  a-  <H-  1—" •  o 3  8 3 S -3 3  o h-» •  13 rt o  o  ^  11  2  GO O  3  o  3 13  P  3i o  tr CD  8*3  B  s «>  CD  * s p  -o  rt  5' J" 3 cro rt cn 13 U M  a *°  2 rt  £ 2  3 < 13  rt_  135  65  136  a  o a  41  to CL  o  i-t  CD  to  j-  \ 5 15 1 7 IO  \  /  /  >3  TO  >§ O  a  TO  Go*  O  fD  a si  Xi  29 31 33i 35 37  S~ Cr  £ a_ o  TO  CD to Xi  o  oo CD oo O  cc  >3  O > § fD >-! QfD  a  K  a  o &  TO TO  O  GO  TO o Ia fD a  TO  J a o  TO >3  O  a o a s a  <?  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O c T3  !*ure 4.2: The IRFs of Log of Output to Permanent and Transitory Shocks  (a) To a Permanent Shock 0.018 0.016 0.014 0.012 0.01 0.008 0.006 • Sample Estimates 90 % Confidence Band 90 % Confidence Band  0.004 0.002 0  J  1  L J 1—1 1—1 I I L_l_  3  5  7  9  _1 l_J I  I  LJ I 1 L  J  I I LJ  L  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39  (b) T o a Transitory Shock 0.01 0.008 0.006 0.004 0.002 0 -0.002  i„„i„„)„„,i,.,.i„..i  1 / 3 5 7 9 1 1  i i i  13 15 17 19 21 23 25 27 29 31 33 35 37 39  -0.004  146  gure 4.4: The IRFs of Log of Output to Permanent and Transitory Shocks: The Sample Estimates and Theoretical Mean Responses (a) To a Permanent Shock 0.018 0.016 0.014 0.012 0.01 0.008 0.006 • Sample Estimates Benchmark Model Habit Model  0.004 0.002 i i i i i i  0 1  3  5  i  7  9  i i  i  i i  i  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39  (b) To a Transitory Shock  1  3  5  7  9  11 13 15 17 19 21 23 25 27 29 31 33 35 37 39  148  o  o  o i—  o 1  1  i  p NJ i  o u>  o  i  i  ©  Ln  o  ^  Os  Ul  i  O O  P  •— 1  g Ul  1  1  NJ  NJ  u>  UJ  P  o NJ  i—  i  3  era'  c •t re *>.  1  1ft H sr  JS> Ui  Ul  re  os  os  CZ)  )  5 12.  -J  oo S  10 11  /  > o  ,  > o  O  NJ  re  •^^^UJ  /  It i  3  1  :  j  re o »i  ^ \  J  oo.iV .  a  /  \  sd  3  Tl  to  Ul/  re" M <-•  so  SO  /  re  -  I  re  SO^  re tt re  dd  SL  rt  35°  re s  5  sr  3 O SL  re  o  s  O P I - P N J P U J P ^ P  O U l ^ U l N J U l U > U l J > U l U l  o  a  re  sr re  > n S  a  CZ)  O  O  w 00  u  Tl  o  00  O Tl  "I  n o 3  C  3  T3  tt O S  0 o  149  c  CD  \  =  >3  >  ! 15 1 7 1 9 2 1 2:i 25 27  O  5"  CD CD OO  i  O  ts  OJ  CD oo  O  CD  o >  9r  89  s  th o o ts GO o tr CD CD  a  9 3 -I  Hcs i—* •  org  d  z; z:  o  o  KJ  h-•  >  H H cr =r  rt rt rv  DO  M g  CJ  rt  0)  rt  3  "d ^•  H  C/3  -Q C  o rt SS  O* P  a  o  s3s  CO  —  H  o  I—> •  3  OB  c± rt ^  n>  t- — 1  a. a  09 <—»• HJ H» •  o o  CT  ~  rt o3  d  <— p—> •  g P B » rt  o 3  3  o  o3.  cr  o o  CD  00 I—t-  CJ  Hcs  r;c  GO  o' GO  X o -cs  —  a* o  < >  GO  d d o  o  i-t HH  CP  r+  p—i  Eh o  •  «—>•  o' d 00 o d152  

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