University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2015 Essays on Macroeconometrics Kotbee Shin University of Pennsylvania, kotbee.shin@gmail.com Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Economics Commons Recommended Citation Shin, Kotbee, "Essays on Macroeconometrics" (2015). Publicly Accessible Penn Dissertations. 1133. https://repository.upenn.edu/edissertations/1133 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/1133 For more information, please contact repository@pobox.upenn.edu. Essays on Macroeconometrics Abstract This dissertation presents two essays on macroeconometrics. In the second chapter, I empirically compare alternative specifications of time-varying volatility in the context of linearized dynamic stochastic general equilibrium models. I consider time variation in the volatility of structural innovations in two ways: one in which the logarithm of the volatility is assumed to follow a simple autoregressive process (stochastic volatility) and the other in which the volatility follows a Markov-switching process. A comprehensive simulation study is presented to assess the fit and performance of two specifications. I show that modeling heteroscedasticity in a highly synchronized fashion across shocks may lead to distorted estimation of the volatility. In the empirical application to the United States data, stochastic volatility model delivers the best-fit and accounts for the heteroscedasticity present in the data well. In the third chapter, I conduct a quantitative evaluation of the potential role of adaptive expectations in a two-country dynamic stochastic general equilibrium model. Under the learning mechanism economic agents are assumed to form their expectations of forward-looking variables using a simple vector autoregressive forecasting model. The agents estimate their vector autoregression based on past model variables and update the estimates every period via a constant gain learning algorithm. I show in a simulation study that the learning mechanism increases the volatility and persistence of the endogenous variables and that as the constant gain parameter grows larger, so do these increases. The two-country DSGE model is then estimated with data from the United States and Euro area. A comparison based on log marginal data densities favors the learning over the rational expectations specification. The learning mechanism generates more persistent responses of variables to the monetary shocks. The improvement in terms of fitting the observed Dollar-Euro exchange rate dynamics is limited. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Economics First Advisor Frank Schorfheide Keywords Adaptive Expectations, Exchange Rate Dynamics, Regime Switching Model, Stochastic Volatility, Timevarying Volatility Subject Categories Economics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/1133 ESSAYS ON MACROECONOMETRICS Kotbee Shin A DISSERTATION in Economics Presented to the Faculties of the University of Pennsylvania in Partial Ful…llment of the Requirements for the Degree of Doctor of Philosophy 2015 Supervisor of Dissertation Frank Schorfheide Professor of Economics Graduate Group Chairperson George J. Mailath Professor of Economics, Walter H. Annenberg Professor in the Social Sciences Dissertation Committee Cecilia Fieler, Professor of Economics Urban Jermann, Professor of Finance ESSAYS ON MACROECONOMETRICS COPYRIGHT 2015 Kotbee Shin To my beloved family iii Acknowledgements I am immensely indebted to my advisor, Frank Schorfheide. His insightful advice and support have inspired my research and helped me strive through every stage of my graduate studies. I am especially grateful that he has been patient with me even when I made painfully-slow progress. I would also like to thank my dissertation committee, Cecilia Fieler and Urban Jermann. Their valuable comments have helped shape my research greatly. I have also bene…ted from seminar participants at the University of Pennsylvania. I owe thanks to my undergraduate advisor, Chang-Jin Kim. His enthusiasm in economics motivated me to become a passionate researcher. I was so fortunate to be surrounded by smart colleagues and friends at McNeil. I am also thankful to Kyungmin Kim, Soojin Kim, Eunice Yang, Eun-young Shim, Cezar Santos, Dave Weiss, Chen Han, Sophie Shin, and Minchul Shin for their friendship. Special thanks go to Dongho Song for substantive conversations about research. Last and most importantly, I would like to thank my family. I cannot thank enough to my parents and parents-in-law for their unconditional love and sacri…ce. My deepest gratitude goes to my husband, Kihwan. Without his love and encouragement, I could not have concluded this endeavor. Also I thank my daughter, Elyse to …ll my life with great joy and love. iv ABSTRACT ESSAYS ON MACROECONOMETRICS Kotbee Shin Frank Schorfheide This dissertation presents two essays on macroeconometrics. In the second chapter, I empirically compare alternative speci…cations of time-varying volatility in the context of linearized dynamic stochastic general equilibrium models. I consider time variation in the volatility of structural innovations in two ways: one in which the logarithm of the volatility is assumed to follow a simple autoregressive process (stochastic volatility) and the other in which the volatility follows a Markov-switching process. A comprehensive simulation study is presented to assess the …t and performance of two speci…cations. I show that modeling heteroscedasticity in a highly synchronized fashion across shocks may lead to distorted estimation of the volatility. In the empirical application to the United States data, stochastic volatility model delivers the best-…t and accounts for the heteroscedasticity present in the data well. In the third chapter, I conduct a quantitative evaluation of the potential role of adaptive expectations in a two-country dynamic stochastic general equilibrium model. Under the learning mechanism economic agents are assumed to form their expectations of forward-looking variables using a simple vector autoregressive forecasting model. The agents estimate their vector autoregression based on past model variables and update the estimates every period via a constant gain learning algorithm. I show in a simulation study that the learning mechanism increases the volatility and persistence of the endogenous variables and that as the constant gain parameter grows larger, so do these increases. The two-country DSGE model is then estimated with data from the United States and Euro area. A comparison based on log marginal v data densities favors the learning over the rational expectations speci…cation. The learning mechanism generates more persistent responses of variables to the monetary shocks. The improvement in terms of …tting the observed Dollar-Euro exchange rate dynamics is limited. vi Contents Contents vii List of Tables viii List of Figures ix I 1 II Introduction Regime Switching and Stochastic Volatility in DSGE Models 4 1 Introduction 4 2 A Benchmark Model 9 2.1 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Market Clearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Exogenous Stochastic Process . . . . . . . . . . . . . . . . . . . . . . 14 2.6 Steady State and Model Solution . . . . . . . . . . . . . . . . . . . . 15 3 Simulation Study 16 3.1 Simulation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Application to U.S. data 22 vii 4.1 Estimation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Conclusion 27 6 Tables and Figures 28 III Bayesian Estimation of a New Open Economy Model with Adaptive Expectations 50 7 Introduction 50 8 The Model 53 8.1 Domestic Households . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8.2 Domestic Producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.2.1 Domestic Final Producers . . . . . . . . . . . . . . . . . . . . 55 8.2.2 Domestic Intermediate Producers . . . . . . . . . . . . . . . . 58 8.2.3 Domestic Importers . . . . . . . . . . . . . . . . . . . . . . . . 60 8.3 Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 8.4 Foreign Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 8.5 Market clearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8.6 Exogenous Stochastic Process . . . . . . . . . . . . . . . . . . . . . . 64 9 Learning Model 65 9.1 Learning Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 9.2 Exchange Rate Determination under Learning . . . . . . . . . . . . . 66 10 Simulation Study 68 viii 11 Empirical Application 70 11.1 Posterior Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 11.2 Marginal Data Density Comparison . . . . . . . . . . . . . . . . . . . 73 11.3 Impulse Response Function . . . . . . . . . . . . . . . . . . . . . . . 74 11.4 Variance Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 75 11.5 Posterior Predictive Checks . . . . . . . . . . . . . . . . . . . . . . . 75 12 Conclusion 76 13 Tables and Figures 77 A Appendices 90 A.1 Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 A.1.1 Stochastic Volatility in DSGE Models . . . . . . . . . . . . . . 90 A.1.2 A Four Regime-Switching in DSGE Models . . . . . . . . . . . 94 Bibliography 97 ix List of Tables 1 Summary of Simulation Study . . . . . . . . . . . . . . . . . . . . . . 28 2 Log Median Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Summary of Prior Densities and Posterior Estimates . . . . . . . . . 29 4 Summary of Parameters for Simulation . . . . . . . . . . . . . . . . 78 5 Standard Deviation of Simulated Models . . . . . . . . . . . . . . . . 79 6 Correlation of Depreciation to In‡ation and Interest Rate Gap . . . . 79 7 Prior and Posterior Distribution . . . . . . . . . . . . . . . . . . . . . 82 8 Marginal Data Density . . . . . . . . . . . . . . . . . . . . . . . . . . 83 9 Variance Decomposition from the Rational Expectations Model . . . 83 x List of Figures 1 True and Estimated Volatilities for the model SM1 : : : : : : : : : : : : : : 30 2 True and Estimated Volatilities for the model SM2 : : : : : : : : : : : : 31 3 True and Estimated Volatilities for the model SM3 : : : : : : : : : : : : 32 4 Variance Decomposition for the model SM1 : : : : : : : : : : : : : : 33 5 Variance Decomposition for the model SM2 : : : : : : : : : : : : : : 34 6 Variance Decomposition for the model SM3 : : : : : : : : : : : : : : 35 7 True and Estimated Volatilities for the model FM1 : : : : : : : : : : : 36 8 Posterior Distributions of non-Volatility Parameters: FM1 : : : : : : 37 9 True and Estimated Volatilities for the model FM2 : : : : : : : : : : : : : 38 10 Posterior Distributions of non-Volatility Parameters: FM2: : : : : 39 11 Posterior Probability of the High Volatility Regime: FM2 : : : : : 40 12 Rolling Standard Deviations for U.S. Data : : : : : : : : : : : : : : : 41 13 Estimated Standard Deviations: SV-DSGE : : : : : : : : : : : : : : 42 14 Estimated Standard Deviations: RS(2)-DSGE : : : : : : : : : : : : : 43 15 Estimated Standard Deviations: RS(4)-DSGE : : : : : : : : : : : : : 44 16 Variance Decomposition for U.S. Data : : : : : : : : : : : : : : : : : 45 17 Posterior Probability of the High Volatility Regime: RS(2)-DSGE : : : 46 18 Posterior Probability of the High Volatility Regime: RS(4)-DSGE : : : 47 19 Posterior Probability: RS(4)-DSGE : : : : : : : : : : : : : : : : : : 48 20 Posterior Density of High- and Low- Volatility Regime Duration : : : 49 21 Autocorrelation Function of the Real Exchange Rate from Simulation 80 22 Depreciation and In‡ation, Interest Rate Gap : : : : : : : : : : : : : : 81 23 Impulse Response to U.S. Monetary Policy Shock for the Learning Model 84 24 Impulse Response to Euro Monetary Policy Shock for the Learning xi Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 85 25 Impulse Response to U.S. Monetary Policy Shock : : : : : : : : : : : : 86 26 87 Impulse Response to Euro Monetary Policy Shock : : : : : : : : : 27 Variance Decomposition for Depreciation in the Learning Model : : : : 88 28 89 Posterior Predictive Checks : : : : : : : : : : : : : : : : : : : : : xii Chapter I Introduction The estimation of dynamic stochastic general equilibrium (DSGE) models has been an important subject in macroeconomics. DSGE models are useful to understand the propagation mechanism of structural shocks to business cycle ‡uctuations and provide a tool for the quantitative analysis of policy experiments. Over the past few decades, there has been remarkable advance in theoretical and empirical DSGE models. DSGE models with various frictions and di¤erent types of shocks have been developed and they seem to reproduce the key features of data well in many dimensions. Despite the signi…cant progress in DSGE model literature, there remains open issues on which speci…cations DSGE model should take to characterize macroeconomic observations. This paper aims to contribute to this literature. In this dissertation, I investigate and empirically compare the speci…cations of DSGE models to answer two macroeconometric questions: (1) what is the better speci…cation between regime switching model and stochastic volatility model to represent the overall volatility reduction, quoted as “Great Moderation”?; (2) can adaptive expectations instead of rational expectations explain the exchange rate dynamics in the general equilibrium framework better? In the second chapter, I compare the regime switching models and stochastic volatility models. Regime switching models have been a standard approach to identify the key source of a large decline in aggregate volatilities. The proponents of regime switching approach …nd it appealing since it is a parsimonious way of modeling the discrete jumps and can be potentially related to economic regimes with meaningful interpretations. An alternative time-varying volatility model is the stochastic volatility model that allows the volatility processes randomly ‡uctuate through time. Stochas1 tic volatility model is distinct from the regime switching model in two ways. First, while the volatility discontinuously shifts from one level to another in the regime switching model, it continuously changes with persistence in the stochastic volatility model. Second, the timing of volatility shifts across innovations is restricted in regime switching models, but stochastic volatility models can allow the independent movement of volatility processes. It is of importance to take a deeper look at the empirical performance of these two modeling approaches since a di¤erent speci…cation of variance could lead to a di¤erent conclusion on the source of macroeconomic ‡uctuations. To do so, I simulate the large-scale DSGE models with regime switching volatility and with stochastic volatility by assuming several possible episodes of underlying data generation processes for volatility dynamics. The simulation study shows that the stochastic volatility speci…cation provides results comparable to or better than regime switching models regardless of the underlying volatility patterns. In the third chapter, di¤erent approaches to modeling the expectation formation is explored in the open economy context. Standard open economy DSGE models with rational expectations have had challenges in the exchange rate determination because the exchange rates are too volatile and persistent to be justi…ed by economic fundamentals. The empirical shortcoming of rational expectations models arise from the tight link between the exchange rates and economic fundamentals. Researchers have attempted to modify the open economy models with di¤erent ingredients to relax the link of the exchange rates from the rest of the economy. In this chapter, I focus on the expectation formation mechanism. Rational expectations hypothesis assume that agents have complete knowledge of economic environment and have model consistent expectations. By relaxing this strong informational assumption, adaptive expectations has been of growing interest to study many macroeconomic observed behaviors that has been hard to reconcile in the closed economy lit2 erature. Under the adaptive expectations mechanism, economic agents are assumed to form subjective expectations with limited information and learn the structure of the economy over time. Since the agents’ learning process relax the tight restrictions on the relationships of model variables by forecast biases, the short-run dynamics of the model variables under adaptive expectations could be substantially di¤erent from those under rational expectations. To quantify the role of adaptive expectations in the open economy model, I conduct a simulation study and estimate the model using Bayesian techniques. A simulation shows that adaptive expectations mechanism increases the volatility and persistence of endogenous variables and allows the exchange rates to drift away from the uncovered interest parity equation. The estimation results provide evidence that the adaptive expectations is a potential channel to explain the persistence of variables. I also …nd that adaptive expectations also improve the …t of data. For example, see Bullard and Mitra (2001), Evans and Honkapojha (2001). 3 Chapter II Regime Switching and Stochastic Volatility in DSGE Models y 1 Introduction In recent years, economists have produced a collection of methods to account for heteroscedasticity present in the U.S. aggregate data. The most notable example is the “Great Moderation” episode when the U.S. economy experienced a general reduction in macroeconomic volatilities. A branch of macro literature has presented empirical evidences in favor of heteroscedasticity in the shock variances. (Kim and Nelson, 1999; McConnel and Perez-Quiros, 2000; Stock and Watson, 2002; Sensier and van Dijk, 2004; Koopman, Lee, and Wong, 2006). One class of time-varying models are Markov switching models. These models allow the time series to be in any of a …nite number of distinct regimes, see Hamilton (1989) and Kim and Nelson (1999). The choice is attractive because of the parsimonious ‡exibility it provides in the speci…cation of the distributions of the underlying structural shocks and requires fewer parameters to estimate. Markov switching models are especially appealing for characterizing the Great Moderation if there has been a discrete and comprehensive volatility reduction in macroeconomic variables (Stock and Watson, 2002; Chauvet and Potter, 2001; Sensier and van Dijk, 2004). Initiated y This chapter is based on the joint work with Dongho Song. 4 by Hamilton (1989), Markov switching models are used extensively in business cycle analysis to characterize discrete changes in the volatilities (Sims and Zha, 2006; Davig and Doh, 2009; Liu, Waggoner, and Zha, 2010). A more ‡exible speci…cation is considered in stochastic volatility models, which allow the continuous change of volatility processes having the potential for moving one or two steps closer to complex reality. An advantage of stochastic volatility speci…cation is that it can characterize continuous shifts in variance and does not restrict the system to switch between the same con…guration. Methodologically, it is related to the statistics literature on stochastic volatility models (Jacquier, Polson, and Rossi, 1995; Kim, Shephard, Chib, 1998; Chib, Nardari, and Shephard 2006), but the recent contribution of Fernandez-Villaverde and Rubio-Ramirez (2007) and Justiniano and Primiceri (2008) indicates that stochastic volatility in general equilibrium models has been exploited in the business cycle literature. If the degree of time variability di¤ers across volatility processes and the structural disturbances hitting the economy display substantial stochastic volatility, stochastic volatility speci…cation is an e¤ective way to accommodate changes in the volatility of the U.S. economy (Cogley and Sargent, 2005; Primiceri, 2005; Fernandez-Villaverde and Rubio-Ramirez, 2007; Justiniano and Primiceri, 2008; Creal, Koopman, and Zivot, 2010). While numerous studies have found signi…cant time variation in shock variances, the magnitude, the number of structural breaks, as wells as the underlying causes of the Great Moderation still remain as one of the main open questions in macroeconomics. A lively debate unfolded between proponents of sudden change in volatility (Stock and Watson, 2002) versus gradual reduction in volatility (Blanchard and Simon, 2001). Some people are also concerned whether one or more structural breaks exist (see the discussion in Sensier and van Dijk, 2004). However, much of the disagreement also comes from the di¤erences in the model framework and in the empirical 5 approach. In the absence of actual knowledge of the underlying structure of volatility processes, it is often di¢cult to decide which estimation algorithm is the preferred route to pursue. What is left unsaid in the literature is how model-dependent the conclusions are when identifying the sources of macroeconomic ‡uctuations. In more general terms, good modeling practice requires investigation of the robustness of a conclusion when the study includes some form of economic modeling. To my best knowledge, not much research has been done to minimize the sensitivity to modeldependent analyses of the sources of macroeconomic ‡uctuations by estimating a variety of structural models that assumes time-varying shock variances. This provides a clear motivation to investigation. This paper incorporates time varying volatility structure in large-scale linearized DSGE economies and compares the Markov Regime-Switching (henceforth RS-DSGE) and Stochastic Volatility DSGE (henceforth SV-DSGE) models using both simulation study and empirical application with a strong emphasis on the speci…cation of volatility dynamics. The goal of this paper is to provide a systematic examination of the performance of two competing models in explaining the driving sources of macroeconomic ‡uctuations. Investigation is important since macroeconomic implications can be seriously distorted if the competing models produce di¤erent volatility estimates. First, I investigate whether the outcomes of two models are similar and second, which model is more reliable. I believe that only after taking account of the model sensitivity, it is possible to draw a …rm conclusion about the sources of macroeconomic ‡uctuations. This paper also tries to address the danger of relying exclusively on a model selection criterion that favors models that …t the data well. A common practice in the empirical DSGE literature is comparing models using marginal likelihood. From a Bayesian perspective, the marginal data density is the most comprehensive and accurate measure of …t and is needed for the comparison of non-nested Bayesian models. I illustrate with the 6 aid of simulation examples and empirical application that a situation where volatility dynamics are spuriously estimated but it survives the Bayesian model selection criterion for o¤ering a parsimonious approximation and delivering a better time-series …t is possible I propose a simulation study. The architecture of simulation study is designed to replicate salient features of U.S. business cycles and is implemented by using arti…cial dataset of 200 observations generated using a large scale DSGE model of Justiniano and Primiceri (2008) (henceforth, JP). The details are discussed in Section 3. Application to the simulated dataset will provide guidance on how well each competing model performs when the true model is in hand. In a simulation study, model performances are measured in three di¤erent ways. I compare the estimated volatility components to the true Data Generating Process (henceforth DGP), perform variance decomposition to examine the consequence of volatility misspeci…cation, and compute the log marginal data density to measure the data …t. Next, I repeat the same steps with the aggregate U.S. data and use the simulation results as a benchmark to understand and interpret the empirical performance of each model. The main empirical …ndings in the experiments are as follows. A two regimeswitching DSGE (henceforth, RS(2)-DSGE) model seems inappropriate for drawing inferences about the volatility processes of business cycles. A common disadvantage of RS(2)-DSGE models is that they assume complete synchronization of Markov states across volatilities. Estimated volatilities can be very crude if the true DGP exhibits contrasting patterns of ‡uctuations. The natural step is to amend RS(2)-DSGE to allow additional degree of ‡exibility in the movement of volatilities. A four regimeswitching DSGE (henceforth RS(4)-DSGE) model nests RS(2)-DSGE in this regard. Hence, I argue that RS(4)-DSGE may perform better than RS(2)-DSGE model for drawing inference about the volatility processes. Due to technical limitations, provid7 ing extra degree of ‡exibility in the RS-DSGE model is a challenging task. SV-DSGE model, on the contrary, promises great ‡exibility in modeling volatility dynamics and its performance is certainly not inferior to RS-DSGE’s. However, the price for this ‡exibility is an increase in dimension. Aside from the over-parametrization problem, SV-DSGE estimates may exaggerate or discount time variation and stochastic movement in volatility. Its performance may deteriorate as the oscillation between volatility regimes increases and the di¤erence between the regimes decreases. In the application to U.S. aggregate data, I estimate the large scale DSGE model in JP using RS(2)-DSGE, RS(4)-DSGE, and SV-DSGE models. I have grouped a subset of shock variances having the same Markov processes. I allow regime associated with the variances of the monetary policy shock to be independent of the regime switching processes of the other shock variances. The empirical motivation is from Sensier and van Dijk (2004) and Cecchetti, Hooper, Kasman, Schoenholtz, and Watson (2007) that the volatility of in‡ation has undergone multiple structural breaks. I …nd that SV-DSGE model delivers a best-…t and accounts for the heteroscedasticity present in the data well. RS(4)-DSGE performs better than RS(2)-DSGE model in volatility speci…cation, but does not improve upon the data …t. By synchronizing shifts in variances across two regimes, RS(2)-DSGE model may detect the timing of the volatility regime wrong and provide imprecise estimates for volatility processes while …tting to the data better than RS(4)-DSGE model. In order to provide robustness of empirical …ndings, I perform the variance decomposition and compute the marginal data density using the modi…ed harmonic mean method of Geweke (1999). Liu, Waggoner, and Zha (2010) is the closest paper to ours. They examine the sources of macroeconomic ‡uctuations by estimating a number of alternative regimeswitching models using Bayesian methods in a uni…ed DSGE framework and compare the …t to the time series data in the post war U.S. economy. Based on marginal data 8 density and Schwarz criterion, they …nd strong evidence in favor of the RS(2)-DSGE model where regime shifts in the variances are synchronized. They then use RS(2)DSGE model to identify shocks that are important in driving macroeconomic ‡uctuations. My approach di¤ers from Liu et al. (2010) since I extend the investigation by including stochastic volatility speci…cation and examine the e¤ectiveness of Bayesian model-selection criterion. This chapter is organized as follows. Section 2 brie‡y explains the DSGE model framework, Section 3 constitutes a simulation study. Section 4 illustrates an application to U.S. aggregate data and Section 5 concludes. Technical details are summarized in Appendix. 2 A Benchmark Model The model is based on JP and exhibits a number of real and nominal rigidities which has been shown to …t the data fairly well. For additional details, see JP. The basic elements of the model include a continuum of households, perfectly competitive …nal goods producers, and a continuum of monopolistic intermediate goods producers. Monetary policy follows a Taylor type rule and …scal policy is assumed to be fully Ricardian. Here is the illustration of the JP model. 2.1 Firms A monopolistic intermediate goods producing …rm i 2 [0; 1] produces output according to: Yt (i) = maxfAt1 Kt (i) Lt (i)1 9 At F; 0g where At is an exogenous measure of productivity that is the same across …rms and F represents a …xed cost of production. As usual, Kt (i) and Lt (i) denote, respectively, the capital and labor input for the production of good i. At follows a unit root process, log AAt t 1 ) that follows: with a growth rate (zt zt = (1 z) + z zt 1 + zt "zt (1) Firms follow a Calvo pricing mechanism when they set their prices. At the start of each period, a randomly selected fraction of …rms cannot reoptimize and set p their prices according to: Pt (i) = Pt 1 (i) where t is de…ned as Pt Pt 1 and p t 1 1 p is the steady-state value. Remaining fraction 1 p of …rms choose their prices by maximizing the present value of future pro…ts: Et 1 X s s t+s p s=0 where t+s n [Pet (i)( p s j=0 t 1+j 1 p )]Yt+s (i) [Wt+s Lt+s (i) + o k Rt+s Kt+s (i)] is the marginal utility of consumption, and Wt and Rtk denote, respectively, the wage and the rental cost of capital. There is a representative …nal goods producing …rm that produce the consumption goods using the intermediate goods and the following constant-returns-to-scale technology: 2 1 31+ Z 1 Yt = 4 Yt (i) 1+ pt di5 pt 0 where pt follows the exogenous stochastic process log pt = (1 p ) log p 10 + p log pt 1 + pt "pt (2) Pro…t maximization problem for the …nal goods producer yields a demand for each intermediate good given by Yt (i) = (1+ pt ) pt Pt (i) Pt Yt and the zero pro…t condition imply 2 1 Z 4 Pt (i) Pt = 0 2.2 1 pt 3 pt di5 : Households Firms are owned by a continuum of households, indexed by j 2 [0; 1]: As in JP, while each household is a monopolistic supplier of specialized labor, a number of "employment agencies" combine households’ specialized labor into labor services available to the intermediate …rms: 2 1 31+ Z 1 Lt = 4 Lt (j) 1+ w dj 5 w 0 Pro…t maximization problem for the employment agencies yields a demand for each labor given by (1+ w) w Wt (j) Wt Lt (j) = Lt and the zero pro…t condition imply 2 1 Z Wt = 4 Wt (j) 0 11 1 w 3 dj 5 w : Household j’s preferences are representable by a lifetime utility functions: Et 1 X s bt+s log(Ct+s (j) hCt+s 1 (j)) 't+s s=0 Lt+s (j)1+ 1+ which is separable in consumption, Ct (j); and labor Lt (j): h is the degree of habit formation, 't is a preference shock that a¤ects the marginal disutility of labor, and bt is a discount factor shock a¤ecting both marginal utility of consumption and the marginal disutility of labor. Both shocks follow exogenous stochastic processes log bt = log 't = (1 b log bt ' ) log ' + 1 + bt "bt ' log 't 1 (3) + 't "'t : (4) The jth household’s budget constraint is given by: Pt+s Ct+s (j) + Pt+s It+s (j) + Bt+s (j) Rt+s 1 Bt+s 1 (j) + Qt+s 1 (j) + k +Wt+s (j)Lt+s (j) + Rt+s (j)ut+s (j)K t+s 1 (j) t+s Pt+s a(ut+s (j))K t+s 1 (j) where It (j) is investment, Bt (j) denotes government bonds holding, Rt is gross nominal interest rate, Qt (j) is the net cash ‡ow from participating in state contingent securities, and t is the per capita pro…t that households get from owing the …rms. Households own capital and choose the capital utilization rate that transforms physical capital K t (j) into e¤ective capital Kt (j) = ut (j)K t 1 (j); which is rented to …rms at the rate Rtk (j): The cost of capital utilization is a(ut+s (j)) 12 per unit of physical capital. Following JP, I assume ut = 1 and a(ut ) = 0 in steady state. In this partially nonlinear approximation of the model solution, only the curvature of the function in steady state needs to be speci…ed, = a00 (1) : a0 (1) The usual physical capital accumulation equation is described by K t (j) = (1 where )K t 1 (j) + t (1 S( It (j) ))It (j); It 1 (j) denotes the depreciation rate and S captures the investment adjustment cost, with S0 > 0 and S 00 > 0. t can be interpreted as an investment-speci…c technology shock following Greenwood, Hercowitz, and Krusell (1997). Assume that this investment shock follows the exogenous stochastic process log t = log t 1 + t" t: (5) Households follow a Calvo pricing mechanism when they set wages. At the start of every period, a randomly selected w of households cannot reoptimize wages and set their wages according to the indexation rule: Wt (j) = Wt 1 (j)( The remaining 1 w zt t 1e 1 ) w ( e )1 w : of households set their wages by maximizing Et 1 X s w s bt+s 't+s s=0 Lt+s (j)1+ 1+ subject to Lt (j) = Wt (j) Wt 13 (1+ w) w Lt : 2.3 Policy The monetary authority sets the short-term nominal rate using the following rule Rt Rt 1 =( ) R R ( R t ) ( Yt =At ) Y =A 1 R e Y Rt "Rt ; (6) where R is the steady state for the gross nominal interest rate and "Rt is a monetary policy shock. Fiscal policy is, by assumption, fully Ricardian, and public spending is given by 1 )Yt gt Gt = (1 where gt is an exogenous disturbance following the exogenous stochastic process log gt = (1 2.4 g ) log g + g log gt 1 + gt "gt : (7) Market Clearing The resource constraint is given by Ct + It + Gt + a(ut )K t 2.5 1 = Yt Exogenous Stochastic Process With regard to exogenous stochastic process in (1)-(7), "jt ~iid N (0; 1) where j 2 fz; p; b; '; ; R; gg and t denotes time: As for the standard deviations, that is t in stochastic volatility model and 14 j , I assume 2 f#regimesg in regime switching models: For instance, 2 fHigh volatility regime; Low volatility regimeg in two Markov regime-switching model. Following Kim, Shepard, and Chip (1998), I assume that each element of jt evolves independently according to the following stochastic processes: log jt = (1 j jt ~ 2.6 iid ) log j + j log jt 1 + jt N (0; ! 2j ) Steady State and Model Solution Since technology process At is a unit root process, after detrending consumption, investment, capital, real wages, and output, I are able to compute the nonstochastic steady state. De…ne the vector of relevant model endogenous variables Solution of the linear rational expectations system, Et [f ( where t is a vector of exogenous disturbances and t+1 ; t; t 1; t; t. )] = 0; is a vector of structural para- meters, is obtained by running Chris Sims’s code gensys.m. Then, the observable t yt = [4 log Yt ; 4 log Ct ; 4 log It ; log Lt ; 4 log W ; Pt t ; Rt ] can be expressed as a linear function of the endogenous model variables xt yt = D + Z t = T( ) t 1 (ME) t + R( ) t Further details regarding the model solution is discussed in the appendix. 15 (TE) 3 Simulation Study 3.1 Simulation Design In this section I propose a simulation study for the performance evaluation of RS(2)DSGE, RS(4)-DSGE, and SV-DSGE model. I consider three simulation scenarios in which the key qualitative features of the Great Moderation are replicated. In the …rst set of simulation, labeled as SM1, whereas monetary shock displays a double hump-shaped pattern. A double humpshaped pattern of monetary shock has its ground in the growing literature on Markov switching DSGE models (see, for more information, Davig and Doh, 2009; Bianchi, 2010; Liu, Waggoner, and Zha, 2010). All other structural shocks exhibit a one time permanent reduction in the volatility. Many empirical studies document a discrete drop in conditional variance of macroeconomic time series. For example, Stock and Watson (2002) argue that changes in the volatility around 1984 are comprehensive and best characterized as discrete break. The second simulation scenario (henceforth, SM2) complexi…es volatility process one step further by allowing a subset of volatilities to have independent regimes. By doing so, I are able to examine the ability of each estimation method to recover the true volatility speci…cation. The last simulation scenario (henceforth, SM3) is based on the argument that the high volatility regime is often associated with recession (French and Sichel, 1993; Hamilton and Susmel, 1994). Since there are relatively fewer number of recessions after 1990s, this may appear as volatility reduction. I let high volatility regimes manifest themselves at the NBER recession dates. In all scenarios, I assume the true DGP to follow Markov processes since the volatility dynamics can be easily con…gured to replicate salient features of Great Moderation. Due to substantial computational demand and time constraint, during 16 a simulation study I shut down Metropolis-Hastings algorithm and give the true parameter values for non-volatility parameters. A full-blown estimation requires roughly three (two) days to complete 200,000 iteration for RS-DSGE (SV-DSGE) models which is the minimum of iteration number to achieve convergence. Since I are interested in the estimation of volatility dynamics, without loss of generality, I assume true values for non-volatility parameters are recovered through Metropolis-Hastings algorithm. All simulations are based on arti…cial dataset generated from a large scale DSGE model of JP and the posterior median estimates for non-volatility parameters in JP are used. Next, I conduct some full-blown estimation to check the validity of the assumption that true values for non-volatility parameters are recovered through MetropolisHastings algorithm. I extend SM1 to a full-blown estimation scheme and denote as FM1. I also consider a case (henceforth, FM2) in which the true DGP follows stochastic volatility process. Here, I use the posterior volatility estimates of SV-DSGE in JP as DGP and try to estimate with RS(2)-DSGE model. This exercise centers on the following idea: if the true DGP of macroeconomic variables follow stochastic volatility process, how well RS(2)-DSGE models estimate the volatility dynamics? Table 1 summarizes the simulation study. 3.2 Model Comparison Model performance is measured in three di¤erent ways; I compare the estimated volatility components to the true DGP, perform variance decomposition to examine the consequence of volatility misspeci…cation, and compute the log marginal data density to measure the data …t. All …gures are obtained from the remaining 10,000 posterior draws after discarding the initial 40,000 draws. Figure 1 through …gure 3 17 plot the time-varying standard deviations for the structural shocks of each model. Top panels of …gure 1, …gure 2 and …gure 3 juxtapose the volatility estimates from RS(4)DSGE, SV-DSGE model, and true DGP. Bottom panels are constructed similarly but carry the estimates from RS(2)-DSGE model. Figure 4 to …gure 6 present the evolution of the variance shares of GDP growth attributed to each structural shock. Figure 1 is based on SM1. Note that the true DGP and the estimates from RS(4)-DSGE model are almost identical in the top panel. However, in the bottom panel, the estimated monetary shock demonstrates a one-time reduction in volatility as the other shocks do. This is inevitable because RS(2)-DSGE assumes all shock variance to switch regimes simultaneously. Due to limited space, I do not report in this paper but depending on the parameterization, I have cases where all estimated shock variances follow a double hump-shaped. This double hump-shaped pattern has been consistently documented in previous literatures (see Davig and Doh, 2009; Bianchi, 2010; Liu, Waggoner, and Zha, 2010). From this example, I cast doubt on the estimated volatility components from the RS(2)-DSGE model. I believe that when the model assumes synchronized regime shifts in the variances, it is more likely to produce spurious estimates. SV-DSGE model captures the time variation in volatility well. Some interesting …ndings are shown in …gure 2. I slightly modify SM1 by allowing independent regimes for government spending and price mark-up shocks. It is now called SM2. The estimates from RS(2)-DSGE, presented in the bottom panel, are misleading in that they spuriously detect the number of structural breaks in volatility. They can be very crude as the true DGP exhibits contrasting patterns of ‡uctuations. Considering that the number of structural breaks in macroeconomic time series is still controversial, this result deserves attention. RS(4)-DSGE allows additional degree of ‡exibility in the movement of volatilities, and hence performs better in capturing 18 volatility movements than RS(2)-DSGE. One remark is that when grouping a subset of shock variances to have the same Markov processes, I have to rely on empirical evidences to …nd the right combination. I will discuss this in more detail in the application to real data. A common disadvantage of RS-DSGE models is that they assume perfect synchronization of Markov states across volatilities. SM1 and SM2 show how easily RS-DSGE model can be misleading when the evolutions of shock volatilities are separated. On the other hand, SV-DSGE performs quite well in both scenarios. Since I employ random-walk speci…cation for the stochastic volatility processes, SV-DSGE can account for both a gradual decline and a sudden change in volatility. In SM3, I try to identify cases when SV-DSGE model performs poorly as shown in …gure 3. SV-DSGE estimates may exaggerate or discount time variation in volatility. Because SV-DSGE has a tendency to smooth out the patterns, when there are frequent oscillations the estimates can be misleading. I do not report in this paper, but when the di¤erences between the two volatility regimes are small, the posterior credible interval of SV-DSGE widens and the time-invariant volatility hypothesis cannot be rejected. What can go wrong if the estimated volatility processes are misspeci…ed? I would like to address this issue by performing variance decomposition. This exercise is important since conclusions drawn from the estimated volatility components are dependent on the estimation methods and consequently variance decomposition results will di¤er. Variance decomposition is obtained by solving the following discrete Lyapunov equations: V ar( t j ; Qt ) = T ( )V ar( t j ; Qt )T ( )0 + R( )Qt R( ) V ar(yt j ; Qt ) = Z( )V ar( t j ; Qt )Z( )0 19 where Qt is a regime-dependent variance-covariance matrix in RS-DSGE and is per se a stochastic volatility variance-covariance matrix in SV-DSGE. The contribution of shock i is obtained by setting to zero the volatility of all disturbances but one, 2 it : Fig- ure 4 through Figure 6 report the evolution of the variance shares of GDP growth attributed to each structural shocks. Variance decomposition results are mostly similar across models. Especially, SV-DSGE and RS(4)-DSGE models produce qualitatively similar results in all scenarios. However, SV-DSGE model slightly exaggerates the role of government spending shock in explaining the variability of GDP growth. The variance decomposition from RS(2)-DSGE relies on the poorly estimated volatilities and tend to exaggerate the role of monetary policy shock. Table 2 reports the log likelihood for all combination of experiments. Since my simulation study shut down Metropolis-Hastings algorithm for non-volatility parameters, I use log median likelihood in model comparison. Note that I am integrating out the unobserved volatilities nor penalizing the likelihood with the number of parameters, instead I compute L(Yjvolatilities; true non-volatility parameters). Although this approach will favor models with many parameters, it may be a primitive way to understand the e¤ectiveness of each model. I use marginal data density approach in the empirical application. SV-DSGE model outperforms others with the exception of SM3. RS(2)-DSGE model performs poorly in general, but if there is a synchronized regime switching in shock volatility, RS(2)-DSGE model is the best-…t model. Note that the performance of RS(4)-DSGE model is certainly better than RS(2)-DSGE model in all scenarios. SV-DSGE model delivers best-…t in two out of three scenarios that I considered. In sum, I argue that when there is not enough knowledge about the volatility process, RS(2)-DSGE model may not be a good choice since it cannot minimize the impact of misspeci…cation for the volatility dynamics. Simulation examples show that 20 allowing for additional degree of ‡exibility can cure this problem. SV-DSGE model can be a good candidate since it promises great ‡exibility in modeling volatility dynamics and delivers data-…t. A Full-blown Estimation I report the …ndings from a full-blown estimation. Two sets of full-blown estimations are conducted by generating 200,000 draws. All …ndings are reported after discarding the initial 150,000 posterior draws. First, I modify the estimation algorithm to make inferences about the non-volatility parameters in SM1 setting. Notice that volatility estimates from SV-DSGE in …gure 7 are very similar to those in …gure 1. Estimation of non-volatility parameters does not change the inference of volatility processes. Figure 8 shows the kernel density estimation of Bayesian posterior distributions of non-volatility parameters. Except few parameters, the Bayesian credible set includes the true values. Second, I use the posterior median estimates of volatility processes from JP as the true DGP and estimate with RS(2)-DSGE model. Figure 9 plots the estimated time-varying volatility components. As suggested in the simulation study, RS(2)DSGE model detects spurious structural break in some volatility components. This is because regime shifts in the variances are synchronized. Figure 10 shows the kernel density estimation of Bayesian posterior distributions of non-volatility parameters. Compared to …gure 8, more true parameter values is not contained in the Bayesian credible set. It is not clear at this point what roles volatility speci…cations play in consistent estimation of non-volatility parameter values. It might be that volatility misspeci…cation a¤ects Kalman gain and in turn compromises the validity of the Kalman …lter. Figure 11 displays the posterior expected values of the high-volatility regime probability. This will be explained in more detail in Section 4. 21 4 Application to U.S. data 4.1 Estimation Approach I estimate SV-DSGE, RS(2)-DSGE, and RS(4)-DSGE using the same prior distributions and dataset in JP 2008. The data comprises of seven series of U.S. quarterly aggregate variables; the growth rate of output, consumption, investment, real wage, the log of hours worked, annualized in‡ation, and nominal interest rates (for more details on data description, see JP 2008). I use the same priors for non-volatility parameters across three speci…cations in order to treat them equal a priori. I refer to Liu, Waggoner, and Zha (2010) for the choice of the prior distributions for the volatility parameters. As discussed previously, a careful investigation is required to determine how to group a subset of shocks. I allow regime associated with the variance of monetary policy shock to be independent of the regime switching processes of the other shock variances based on previous literatures and on the rolling estimation of standard deviations presented in …gure 12. Sensier and van Dijk (2004) …nd that 83% of the U.S. macroeconomic time series variables have experienced a break in the (un)conditional volatility, and in particular nominal variables such as in‡ation and interest rates experienced multiple volatility breaks. Cecchetti et al. (2007) report that the level and volatility of in‡ation display coincident hump-shaped patterns that allow us to date the start of the Great In‡ation in the late-1960s and a synchronized In‡ation Stabilization in the mid-1980s. The rolling estimation of standard deviations in …gure 12 depicts the overall volatility movements. Consistent with Cecchetti et al. (2007), I also witness multiple ‡uctuations in two nominal variables, in‡ation and interest rates. This motivates us to assume that the regime associated with nominal shock variances is independent of the regime switching processes of the other shock variances. From 22 here on, I denote RS(4)-DSGE as the model with the regime associated with monetary shock being independent of other regimes. Table 3 reports posterior medians and …fth and ninety-…fth percentiles of a model estimated with SV-DSGE, RS(2)-DSGE, and RS(4)-DSGE model. All posterior estimates are obtained by running a single block random-walk MH algorithm (RW-MH) for 400,000 iterations following a burn-in of 350,000 iterations. Calibrated parameters are capital share ( ) at 0.3, depreciation rate ( ) at 0.025, SS government spending share (g) at 0.22, and persistent of mark-up shock ( ) at zero. Chib and Ramamurthy (2010) argue that the results from the RW-MH algorithm are not satisfactory due to slow convergence and often the algorithm does not work in many circumstances. According to Sims, Waggoner, and Zha (2008), due to the complexity inherent in high-dimensional Markov-switching models, the RW-MH algorithm can be very costly and sometimes take a couple of weeks to obtain an estimate that is close to the peak of the likelihood. Indeed, RW-MH algorithm needed around …ve days to complete 400,000 iterations for RS-DSGE models. I assess the convergence of RS-DSGE model using some di¤erent starting values and found that they delivered roughly similar results when looking at medians. However, due to substantial computational burden, the (informal) convergence test was limited. For SV-DSGE model, I veri…ed the robustness of the algorithm by obtaining almost identical posterior median values in JP 2008. 4.2 Estimation Results Parameter estimates are not entirely identical. While most of the parameter estimates from SV-DSGE are similar to ones reported in JP 2008, labor-related parameters of RS-DSGE models are inconsistently estimated. For instance, labor-related 23 parameters of RS-DSGE models are somewhat di¤erent. For example, labor disutility coe¢cient is higher in both RS(2)-DSGE and RS(4)-DSGE models. This may somehow generate lower labor disutility shock estimates than one from SV-DSGE model. Although this variation in estimates may be important, I do not explore it any more. Instead, I would like to focus on volatility estimates. Figure 13 through …gure 15 plot the time-varying standard-deviations for the seven structural shocks of SV-DSGE, RS(2)-DSGE, and RS(4)-DSGE models, respectively. Note that …gure 13 is roughly identical to the results in JP 2008. In …gure 14, observe that the estimates of monetary policy, investment speci…c, and government spending shocks are roughly similar to those in SV-DSGE model, but the estimates of technology and intertemporal preference shocks behave very di¤erently. While these two estimates from SV-DSGE model tend to show gradual decline over the time periods, corresponding estimates from RS(2)-DSGE model are characterized by multiple Markov-shifts. Since the true volatility process is unknown, I do not know which is closer to the truth. However, it is very unlikely that all volatility processes change magnitude and shape simultaneously. The estimates from RS(4)-DSGE model look quite similar to those from RS(2)-DSGE model. But two things stand out in …gure 15. Since I allow the regime associated with the exogenous disturbance showing the largest degree of time variation (monetary policy shock) to be independent of the regime switching processes of other shock variances, the double-hump shaped pattern is most notable in monetary policy shock estimates. Also, relatively fewer high volatility regimes are realized since mid 1980s. This enables us to replicate the great reduction in volatility of the remaining disturbances around mid 1980s. The fact that estimates from RS(2)-DSGE and RS(4)-DSGE models are somewhat di¤erent indicates that independent movement across volatility processes are evident. This shows why one should be careful about using RS(2)-DSGE model to identify the sources of the changes in the volatility of 24 U.S. macroeconomic variables. Figure 16 presents the variance decomposition for output growth. With some exception, each model delivers roughly similar results. Note that SV-DSGE model assumes variance shares attributed to each shock are more time-varying. Figure 17 and …gure 18 show posterior expected values of the high volatility regime in RS(2)DSGE and RS(4)-DSGE model. Notice that …gure 17 looks as if the two posterior expected values in Figure 18 are combined in one …gure. According to RS(2)-DSGE model, the high volatility regimes for monetary shock are observed in the mid-1960s and in the beginning of 2000s. (See also the …gure 2 in Liu, Waggoner, and Zha (2010), page 40. They have the same …gure like I do.) Figure 18 tells us that there was no high volatility regime for monetary shock at that time. Observe that the starting period of high volatility regime for monetary shock and that for the others do not coincide in early 2000s. By allowing additional degree of ‡exibility in RS-DSGE model, I am able to detect the timing of each volatility regime shift better. Figure 19 separately plots the regime probabilities in RS(4)-DSGE model. The second and third rows in Figure 19 imply that taking account of these two possible regimes can be important since both of them are signi…cantly greater than zeros in probabilities. I try to address the drawback of RS(2)-DSGE model in di¤erent direction. Using the posterior median values of SV-DSGE model, reported in Table 3 and …gure 13, I generate an arti…cial dataset. The thought experiment centers on the following idea: provided that SV-DSGE model ideally captures the volatility dynamics, how is the performance of RS(2)-DSGE model. Figure 13 is now used as the true DGP and the estimates of monetary shock disturbance in the …gure tell us that low volatility regime was present in the mid 1960s. However, the estimates from RS(2)-DSGE model detects the presence of high volatility regime for monetary shock at that period. This is shown in …gure 9. Posterior expected values of the high volatility regime are 25 displayed in …gure 11. This …gure conveys wrong impression that monetary shock was in high volatility state around 1960s as well as other shocks. These evidences show that RS(2)-DSGE model can detect the timing of the high volatility regime wrong and provide imprecise estimates for volatility processes. As it is standard in the literature, I assess the …t by computing the marginal data density as suggested in Geweke (1999). From a Bayesian point of view, the marginal data density comparison gives a comprehensive measure of …t on non-nested competing models. Details of the computation of the marginal data density is relegated to the technical appendix in JP 2008. However, I would like to point out that JP do not integrate out all latent variables numerically. In fact, JP choose f ( ; H T ) = f ( )f (H T ) = f ( ) (H T ) and assume ( ; H T ) = ( ) (H T ) for computational convenience: m(Y ) = Z f ( ; HT ) p( ; H T jY )d( ; H T ) L(Y j ; H T ) ( ; H T ) 1 Then, the marginal data density can be approximated by: " N f ( j) 1 X ) mN (Y ) = N j=1 p(Y j j ; HjT ) ( j ) where j # 1 and HjT are from posterior distribution. I acknowledge the possible approx- imation error from following JP’s method. But, at the same time I understand that their assumption allows us to compute mN (Y ) very easily. The last two rows of Table 3 reports the log-marginal data density and the median likelihood values for SV-DSGE, RS(2)-DSGE, and RS(4)-DSGE model, respectively. Consistent with JP, I …nd that the values of the log-marginal likelihood are in favor of SV-DSGE model. I also get results in line with Liu, Waggoner, and Zha (2010) 26 that between the two speci…cations of RS-DSGE models, the data favor the parsimoniously parameterized model with shock variances switching regimes simultaneously. I suspect that volatility dynamics are spuriously estimated in RS(2)-DSGE model but …nd out that it performs pretty well in terms of the Bayesian model selection criterion for o¤ering a parsimonious approximation and delivering a better data …t. Consistent with the simulation study, I …nd that the median likelihood value for SV-DSGE model is the highest while RS(4)-DSGE model is the lowest. 5 Conclusion I have estimated a variety of large-scale DSGE models in which the variance of the structural shocks is time-varying. In particular, I evaluated the e¤ectiveness of Markov-switching and stochastic volatility speci…cation of volatility dynamics. Results from simulation indicate that allowing for too few regimes in the RS-DSGE model leads to poor volatility estimates. SV-DSGE model promises great ‡exibility in modeling volatility dynamics in a sense that it does not restrict the number of changes nor synchronization across shocks. In empirical application, SV-DSGE model delivers best-…t and accounts for the heteroscedasticity present in the data well. Among RS-DSGE models, the one with synchronized shifts in shock variances …ts best, but may provide imprecise estimates for volatility processes. I have shown that model comparison based on the marginal likelihood approach can be misleading since parsimoniously parameterized model will always be favored regardless of its ability to capture volatility dynamics. The …ndings imply that DSGE models extended with stochastic volatility are good alternatives for understanding the evolving volatility dynamics of U.S. aggregate data since it can minimize the impact of misspeci…cation for the volatility dynamics. 27 6 Tables and Figures SM1 SM2 SM3 FM1 FM2 Table 1: Summary of Simulation Study Description True DGP Monetary vs the Rest RS(4) Monetary vs Gov’t and Price Mark-up vs the Rest RS(8) High volatility in recession RS(2) Monetary vs the Rest RS(4) Stochastic Volatility Process SV Note. Restricted+ : I only estimate volatility parameters. Table 2: Log Median Likelihood DGPnEstimation RS(2)-DSGE RS(4)-DSGE SM1 -1892.25 -1793.39 -1823.29 -1729.91 SM2 -1429.50 -1429.50 SM3 28 SV-DSGE -1783.69 -1710.28 -1469.68 Estimation Restricted+ Restricted+ Restricted+ Full-blown Full-blown Table 3: Summary of Prior Densities and Posterior Estimates coe¢cient p w h p w Lss 29 r gss p w S 00 p y R z g p ' b median likelihood MDD Prior Density Mean 0.30 0.025 B 0.50 B 0.50 N 0.50 B 0.50 N 0.15 N 0.15 N 396.83 N 0.50 N 0.50 0.22 G 2.00 B 0.75 B 0.75 G 5.00 G 3.00 N 1.70 G 0.13 B 0.60 B 0.40 B 0.60 B 0.60 0.00 B 0.60 B 0.60 Std 0.00 0.00 0.15 0.15 0.03 0.10 0.05 0.05 0.50 0.10 0.10 0.00 0.75 0.10 0.10 1.00 0.75 0.30 0.10 0.20 0.20 0.20 0.20 0.00 0.20 0.20 SV-DSGE Median 90% CI 0.83 [0.75, 0.91] 0.07 [0.04, 0.13] 0.43 [0.39, 0.47] 0.84 [0.80, 0.88] 0.22 [0.16, 0.28] 0.16 [0.09, 0.23] 397.03 [396.31, 397.79] 0.55 [0.37, 0.72] 1.02 [0.92, 1.13] 1.49 [0.92, 2.37] 0.91 [0.89, 0.93] 0.66 [0.57, 0.73] 7.04 [5.42, 8.78] 3.19 [2.38, 4.22] 1.87 [1.65, 2.14] 0.08 [0.06, 0.11] 0.84 [0.80, 0.87] 0.33 [0.23, 0.43] 0.99 [0.99, 0.99] 0.93 [0.88, 0.96] 0.90 [0.82, 0.95] 0.81 [0.70, 0.88] -1718.8 -1846.9 RS(2)-DSGE Median 90% CI 0.91 [0.83, 0.97] 0.01 [0.00, 0.03] 0.40 [0.36, 0.44] 0.81 [0.78, 0.84] 0.25 [0.19, 0.31] 0.17 [0.11, 0.26] 397.09 [396.81, 397.45] 0.60 [0.44, 0.72] 1.23 [1.11, 1.36] 1.28 [0.72, 1.80] 0.90 [0.88, 0.93] 0.54 [0.40, 0.65] 7.68 [7.27, 8.16] 3.01 [2.44, 3.68] 2.01 [1.87, 2.21] 0.07 [0.05, 0.09] 0.84 [0.81, 0.87] 0.33 [0.25, 0.41] 0.98 [0.98, 0.98] 0.95 [0.92, 0.97] 0.96 [0.93, 0.98] 0.90 [0.87, 0.93] -1737.7 -1867.0 RS(4)-DSGE Median 90% CI 0.82 [0.75, 0.89] 0.01 [0.00, 0.06] 0.40 [0.36, 0.44] 0.80 [0.76, 0.82] 0.26 [0.20, 0.31] 0.18 [0.10, 0.24] 396.47 [396.31, 396.89] 0.63 [0.44, 0.94] 1.12 [1.00, 1.21] 0.66 [0.50, 0.91] 0.91 [0.89, 0.94] 0.54 [0.46, 0.59] 5.08 [4.95, 5.39] 1.98 [1.87, 2.15] 1.80 [1.63, 1.95] 0.07 [0.05, 0.09] 0.81 [0.77, 0.84] 0.27 [0.17, 0.34] 0.98 [0.98, 0.98] 0.93 [0.90, 0.95] 0.98 [0.97, 0.98] 0.89 [0.83, 0.92] -1730.6 -1904.7 Figure 1: True and Estimated Volatilities for the model SM1 Note: In this …gure I plot the true volatility processes (bold), median standard deviation and 90% credible intervals generated in the estimation of SV-DSGE (dashed), RS(2)-DSGE (black-dotted), and RS(4)-DSGE (pink-dashed). 30 Figure 2: True and Estimated Volatilities for the model SM2 Note: In this …gure I plot the true volatility processes (bold), median standard deviation and 90% credible intervals generated in the estimation of SV-DSGE (dashed), RS(2)-DSGE (black-dotted), and RS(4)-DSGE (pink-dashed). 31 Figure 3: True and Estimated Volatilities for the model SM3 Note: In this …gure I plot the true volatility processes (bold), median standard deviation and 90% credible intervals generated in the estimation of SV-DSGE (dashed), RS(2)-DSGE (black-dotted), and RS(4)-DSGE (pink-dashed). 32 Figure 4: Variance Decomposition for the model SM1 Note: This …gure presents the contribution of each shock to the variability of GDP growth. Variance decomposition under RS(4)-DSGE (pink), SV-DSGE (blue), and RS(2)-DSGE model (blackdotted) are included. 33 Figure 5: Variance Decomposition for the model SM2 Note: This …gure presents the contribution of each shock to the variability of GDP growth. Variance decomposition under RS(4)-DSGE (pink), SV-DSGE (blue), and RS(2)-DSGE model (blackdotted) are included. 34 Figure 6: Variance Decomposition for the model SM3 Note: This …gure presents the contribution of each shock to the variability of GDP growth. Variance decomposition under RS(4)-DSGE (pink), SV-DSGE (blue), and RS(2)-DSGE model (blackdotted) are displayed. 35 Figure 7: True and Estimated Volatilities for the model FM1 Notes: Bold lines represent true volatility processes. Median standard deviations(solid) and 90% credible intervals (dotted) are generated in the estimation of SV-DSGE model. 36 Figure 8: Posterior Distributions of non-Volatility Parameters: FM1 Note: This …gure plots kernel density estimation of posterior distributions. True parameter values are indicated by vertical lines. DGP is RS(4)-DSGE and I estimate with SV-DSGE model. 37 Figure 9: True and Estimated Volatilities: FM2 Notes: Bold lines represent true volatility processes, SV-DSGE. Median standard deviations (solid) and 90% credible intervals (dotted) are generated in the estimation of RS(2)-DSGE model. 38 Figure 10: Posterior Distributions of non-Volatility Parameters: FM2 Note: This …gure plots kernel density estimation of posterior distributions. True parameter values are indicated by vertical lines. DGP is SV-DSGE and I estimate with RS(2)-DSGE model. 39 Figure 11: Posterior Probability of the High Volatility Regime: FM2 Notes: Shaded bars indicate NBER recessions and solid line represents posterior expected value of the high volatility regime in RS(2)-DSGE model. DGP is SV-DSGE and I estimate with RS(2)DSGE model. The results are based on 200,000 posterior draws. 40 Figure 12: Rolling Standard Deviations for U.S. Data Notes: Rolling estimation window is twenty quarters for this …gure. 41 Figure 13: Estimated Standard Deviations: SV-DSGE Notes: Median (bold) and 90% credible intervals (dotted) for the time-varying volatility of each disturbance computed with the draws generated in the estimation of SV-DSGE model. 42 Figure 14: Estimated Standard Deviations: RS(2)-DSGE Notes: Median (bold) and 90% credible intervals (dotted) for the time-varying volatility of each disturbance computed with the draws generated in the estimation of RS(2)-DSGE model. 43 Figure 15: Estimated Standard Deviations: RS(4)-DSGE Notes: Median (bold) and 90% credible intervals (dotted) for the time-varying volatility of each disturbance computed with the draws generated in the estimation of RS(4)-DSGE model. 44 Figure 16: Variance Decomposition for U.S. Data Notes: This …gure presents the contribution of each shock to the variability of GDP growth. Variance decomposition under RS(4)-DSGE (pink), SV-DSGE (blue), and RS(2)-DSGE model (blackdotted) are displayed. 45 Figure 17: Posterior Probability of the High Volatility Regime: RS(2)-DSGE Notes: Shaded bars indicate NBER recessions and solid line represents posterior expected value of the high volatility regime in RS(2)-DSGE model. The results are based on 300,000 posterior draws. 46 Figure 18: Posterior Probability of the High Volatility Regime: RS(4)-DSGE Notes. Solid line represents posterior probability of the high volatility regime for monetary shock and the other shocks respectively in RS(4)-DSGE model. 47 Figure 19: Posterior Probability: RS(4)-DSGE Notes: Shaded bars indicate NBER recessions and bold lines are posterior probability of each regime in RS(4)-DSGE model and dashed lines are 90% credible intervals. 48 Figure 20: Posterior Density of High- and Low- Volatility Regime Duration Notes: Top …gure is posterior density of each regime for RS(2)-DSGE and two bottom …gures are for RS(4)-DSGE. 49 Chapter III Bayesian Estimation of a New Open Economy Model with Adaptive Expectations 7 Introduction Explaining real exchange rate dynamics has been an long-lasting challenge in international economics. Exchange rates are more volatile and persistent than standard open-economy models can account for, and they appear to be disconnected from fundamentals in the short run. In one strand of the literature, there are a large number of theoretical studies which incorporate the endogenous sources that can make consumption unresponsive to the exchange rate. Some examples include nominal rigidities, pricing-to-market, introduction of durable goods and investment, and alternative asset market structure, e.g., the work of Betts and Devereux (2000), Chari, Kehoe and MaGrattan (2002), Monacelli (2005), Benigno (2009), and Engel and Wang (2011). In the other strand of the literature, relatively few empirical assessments of structural models have been done, as seen in Smets and Wouters (2002), Adolfson et al. (2001, 2007), Bergin (2003, 2006), Lubik and Schorfheide (2005). These papers estimate structural models with di¤erent frictions and speci…cations and document the importance of incomplete pass-through. A variety of open economy models are built on is the rational expectations hypothesis that expectations have to be model-consistent. Motivated by the criticism 50 on rational expectations that assumes too much knowledge by economic agents, recent research has formulated ways of deviating from rational expectations. The most common approach is adaptive learning, which assumes that economic agents make their forecasts based on past observations and update the forecast every period, like an econometrician following Sargent (1993) and Evans and Honkapohja (2001). In the closed-economy general equilibrium framework, a variety of scholars shows that the learning mechanism ampli…es the e¤ects of stochastic shocks and gives a plausible explanation for in‡ation persistence. (Milani, 2005, 2007; Huang, Liu and Zha, 2009; Eusepi and Preston, 2011; Slobodyan and Wouters, 2012) Learning breaks the tight link between fundamental variables imposed by general equilibrium models and therefore can potentially reconcile business cycle patterns that were di¢cult to explain under rational expectations. In the open economy literature, the learning mechanism succeeds in replicating some aspects of exchange rate, e.g., Mark (2009), Lewis and Markiewicz (2009), Dieppe et al. (2013). Mark (2009) …nds that the learning paths match the volatility and actual movement of the real deutschemark-dollar exchange rate from 1973 to 2005 better in a partial equilibrium model. Lewis and Markiewicz (2009) adapt a simple monetary model with dual learning and reproduce the excess volatility of the exchange rate return. Dieppe et al. (2013) use a multi-country euro area model with a limited information learning approach, and document di¤erent responses to an expansionary …scal policy under learning and rational expectations. The contribution of this paper is that I introduce adaptive expectations rather than rational expectations to relax the tight link between the exchange rate and fundamentals imposed by a two-country open economy model with nominal price rigidities, i.e., New Open Economy Macroeconomics. The model is the open-economy version of the canonical New Keynesian dynamic stochastic general equilibrium (DSGE) model extended with the existence of monopolistically competitive importers. The 51 importers’ price-setting behavior introduces endogenous deviation from purchasing power parity, allowing for incomplete exchange rate pass-through. Under the learning mechanism, economic agents are assumed to form their expectations of forwardlooking variables using a simple vector autoregressive forecasting model. The agents estimate their vector autoregression based on past model variables and update the estimates every period via a constant-gain learning algorithm. Constant-gain learning is widely used in learning literature due to its appealing features, namely, that a single parameter regulates the departure from rational expectation, and that the learning model nests the rational expectation model. I conduct simulation exercises to compare the equilibrium paths implied by rational expectations and the learning mechanism. The simulation results show that the learning mechanism increases the volatility and persistence of the endogenous variables. By increasing the gain parameter in a constant gain algorithm, these increases become more pronounced. Since agents’ subjective views of the economy change over time, the belief-updating process increases the overall volatilities. When agents observe a higher realization of variables than expected, the perceived persistence will be revised upward, leading to the additional persistence in the data generating process. The gradual and ongoing adjustment of beliefs is an endogenous source of persistence in learning models. The learning mechanism also improves the model in terms of uncovered interest rate parity by allowing the interest rate gap to deviate from the exchange rate depreciation by the expectational di¤erence between rational expectations and the expectations formed from the learning process. For some values of the gain parameter, the learning mechanism reduces the correlation between the exchange rate depreciation and the interest rate gap as far as is found in the data. The two-country open economy model is estimated with U.S. and Euro area data from 1983:Q3 to 2012:Q4 under the two di¤erent speci…cations of expectations: ra52 tional expectations and learning. The Bayesian estimation provides three key …ndings. First, the posterior distributions under the learning mechanism show that price stickiness parameters for the U.S. are much smaller than they are under rational expectations and that the learning mechanism reduces the importance of habit formation. This implies that the learning mechanism replaces the endogenous source of persistence in line with …ndings of Milani (2005, 2007) for closed economy DSGE models. Second, I compute posterior probabilities for the learning and the rational expectations speci…cations, and …nd that the data favors the model with constant gain learning. More speci…cally, the learning model better …ts the output growth comovement between the U.S. and the Euro area and the correlation between output growth and in‡ation of the U.S., although the improvement in terms of …tting observed Dollar-Euro exchange rate dynamics is limited. Third, the model with the learning mechanism produces lagged and persistent responses of in‡ations and exchange rate depreciation to the monetary policy shocks, whereas the shocks dissipate quickly in the rational expectations model. This chapter is organized as follows. Section 8 describes a small-scale two-country general equilibrium model and Section 9 illustrates the learning model and how the learning mechanism changes the exchange rate determination. Section 10 presents the simulation study, and the empirical application to U.S. and Euro-area data will be followed in Section 11. I conclude in Section 12. 8 The Model In this section, I specify a small-scale two-country general equilibrium model, the extension of Monacelli (2005). Home country and foreign country are assumed to be symmetric in terms of preference and technology and will be denoted by H for 53 the home country and by F for the foreign country. Each economy is populated by a continuum of households, …nal good producers, intermediate good producers, importers and a monetary authority. As a matter of notation, subscripts H and F refer to the country where the good is produced and a asterisk ’*’ indicates the foreign variables. When it comes to log-linearization, non-stationary worldwide technology shock induces the non-stationary trend on some variables, so I express the model with respect to detrended variable beforehand. I let the tilded variables denote the logxt x deviation from the steady state, i.e. x~t = ln 8.1 : Domestic Households The domestic representative household consumes Dixit-Stiglitz aggregate Ct of domestic goods CH;t and imported goods CF;t . Aggregate consumption is de…ned as Ct = (1 1 1 ) CH;t + 1 1 1 CF;t (8) The household is a monopolistic labor supplier to the …rms and derives the disutility from hours worked and maximizes max E0 1 X t=0 t ( (Ct h Ct 1 ) =AW;t 1 1 Lt ) (9) subject to PH;t CH;t + PF;t CF;t + Et [Qt;t+1 Dt+1 ] where is a discount factor, W t L t + Dt Tt (10) is a relative risk-aversion parameter, h is the habit formation parameter and Wt is the nominal wage. The household is assumed to derive utility from e¤ective consumption relative to the level of non-stationary world-wide 54 technology, AW;t , so that the economy evolves along a balance growth path even if the utility is additively separable in consumption and leisure. I also assume the existence of complete asset markets so that households have access to a complete set of the state-contingent claims denominated in the home country currency, which will be evaluated by the stochastic discount factor Qt;t+1 at time t. Tt is the lump-sum tax imposed by the government to …nance its purchase. The household maximization problem implies the following optimality conditions: Pt t 1 = AW;t 1 = t Pt Qt;t+1 = t Ct h Ct AW;t 1 Et " h AW;t+1 Ct+1 h Ct AW;t+1 Wt Pt # (11) (12) t+1 (13) Pt+1 Under the complete asset market assumption, the price of risk-free government bond is constructed by averaging the price of state-contingent claims as: 1 = E [Qt;t+1 ] = E Rt t+1 t Pt Pt+1 (14) where Rt is the nominal interest rate, which is the instrument of monetary authority. 8.2 8.2.1 Domestic Producers Domestic Final Producers The …nal goods producers combine home produced goods and imported goods according to a constant elasticity of substitution (CES) aggregation technology and sell 55 in a perfectly competitive market. Final good producers maximize the pro…t max Pt Yt PH;t YH;t PF;t YF;t (15) subject to Yt = (1 where 0 < 1 1 1 ) YH;t + 1 1 YF;t < 1 is a import share parameter and (16) determines the degree of substi- tutability of home goods and foreign goods. YH;t is the aggregate of home goods sold in home country and YF;t is the aggregate of imported goods sold in home country. The demand for home good and the demand for foreign good are given by ) PH;t Pt PF;t Pt Yt YH;t = (1 YF;t = Yt (17) (18) The zero pro…t condition implies the aggregate price index as Pt = (1 1 1 + PF;t ) PH;t 1 1 (19) The log-linearized version of (19) yields ~ t = (1 ) ~ H;t + ~ F;t = ~ H;t (~ H;t = ~ H;t q~t (20) ~ F;t ) where the log terms of trade qt is de…ned by the price of exports relative to the price 56 of imports in terms of the domestic currency (PH;t =PF;t ). When the exchange rate pass-through is complete, the exchange rate ‡uctuation is transmitted to the domestic in‡ation, scaled by the import share. Each home good aggregate and foreign good aggregate are composed of a continuum of di¤erentiated intermediate goods indexed by j2 [0; 1] YH;t = Z 1+ 1 1 YH;t (j) 1+ dj (21) 0 YF;t = Z 1+ 1 YF;t (j) 1 1+ dj (22) 0 The pro…t maximization problems of home and foreign …nal good producers yield the demands for intermediate goods YH;t (j) = PH;t (j) PH;t YF;t (j) = PF;t (j) PF;t 1+ YH;t (23) YF;t (24) 1+ and the zero pro…t conditions give the relationships between the aggregate good price and the individual intermediate home goods prices PH;t = Z 1 PH;t (j) 1 dj (25) dj (26) 0 PF;t = Z 1 PF;t (j) 0 57 1 8.2.2 Domestic Intermediate Producers The intermediate good producers are monopolistically competitive. They produce di¤erentiated goods according to a linear production function that uses only labor YH;t (j) = AW;t AH;t Lt (j) (27) where AW;t is a non-stationary world-wide technology shock and AH;t is a stationary and country-speci…c technology shock. Firms face the nominal rigidities following the Calvo-type pricing mechanism. At the beginning of the period, a fraction H of …rms cannot re-optimize their price and adjust the prices to steady state in‡ation, : PH;t (j) = PH;t 1 (j) The remaining fraction (1 H) (28) of …rms choose their prices by maximizing the present value of future pro…ts. max Et 1 X s H Qt;t+s fPH;t (j) s YH;t+s (j) Wt+s Lt+s (j)g (29) s=0 subject to s YH;t+s (j) = PH;t (j) PH;t+s 1+ YH;t+s (30) YH;t (j) = AW;t AH;t Lt (j) where Qt;t+s is the households’ stochastic discount factor and Wt+s stands for the nominal wage. The …rst part of bracket is the revenue when they cannot adjust the price optimally. Since intermediate good …rms are monopolistically competitive, they 58 face the downward-sloping demand function given by the pro…t maximization of home …nal good producers. The optimal price for the …rms who can reset the price is given by the …rst order condition: new PH;t (j) P s Et 1 s=0 H Qt;t+s Wt+s YH;t+s (j) P = (1 + ) s s Q Et 1 H t;t+s YH;t+s (j) s=0 (31) The aggregate price index for home good is de…ned as: 1 1 new H )PH;t (j) PH;t = (1 + H( P^H;t 1 (j)) 1 (32) By substituting the log-linear equation of (31) into log-linearized (32), one can derive the Phillips-curve relationship between domestic in‡ation and marginal cost ~ H;t = Et ~ H;t+1 + where H (1 H )(1 H) H ~t and mc ~ t = ~ t H mc (~ pH;t (33) p~F;t ) A~H;t : Using the house- hold optimality condition (12) and the de…nition of overall price equation (19), the marginal cost can be expressed as follows: mc ~ t = w~t p~H;t = ~ t + p~t = ~ t + (1 = ~t A~H;t A~H;t p~H;t (~ pH;t )~ pH;t + p~F;t p~F;t ) 59 A~H;t p~H;t A~H;t (34) 8.2.3 Domestic Importers The importers are assumed to purchase a homogenous good produced abroad at price sold in the foreign market and convert it into a di¤erentiated good for the home market. Analogously to the intermediate good producers, importers are monopolistically competitive and face the nominal rigidities subject to Calvo-pricing mechanism, inducing the incomplete pass-through. A fraction F of importers adjust the prices according to the steady state in‡ation and the remaining fraction (1 F) of importers set their prices optimally to maximize the present value of expected pro…ts. max Et 1 X s F Qt;t+s PF;t (j) s CF;t+s (j) et+s PF;t+s (j)CF;t+s (j) (35) CF;t (36) s=0 subject to PF;t (j) PF;t CF;t (j) = 1+ Importers pay et+s PF;t+s (j) in their home currency at the world market, so that the law of one price holds at the border. Under monopolistic competition, importers charge a mark–up, which generates the deviation from the law of one price in the short-run. The …rst-order condition yields: new PF;t (j) = (1 + ) Et P1 s Qt;t+s et+s PF;t+s (j)CF;t+s (j) PF1 s s Et s=0 F Qt;t+s CF;t+s (j) s=0 (37) The aggregate domestic import price is de…ned as: 1 PF;t = (1 new F )PF;t (j) 1 + F( P^F;t 1 (j)) 1 (38) Following Monacelli (2005), I de…ne the measure of the law-of-one-price gap (l.o.p 60 gap) as F;t = et PF;t PF;t (39) With incomplete pass-through, l.o.p gap captures the endogenous deviation from purchasing power parity. It will play a role in determining the dynamics of real exchange rate. Combining the log-linear equations of (37) and (38) yields the Phillips-curve relationship between import price in‡ation and the l.o.p gap: ~ F;t = Et ~ F;t+1 + where F (1 F )(1 F F) F ~ (40) F;t : The l.o.p gap acts as a marginal cost for the importers. When the nominal exchange rate rises, it creates the persistent increases of the l.o.p gap due to the nominal rigidities, which results in the persistent increase in import good in‡ation. 8.3 Policy Monetary authority sets the short-term nominal interest rate by an interest-rate feedback rule with smoothing. 1 ^t Rt = Rt R1 R R exp("Rt ) (41) ^ t is the target rate and "Rt is the monetary policy shock. The central bank where R is assumed to respond to in‡ation, output growth, exchange rate depreciation : ^ t = Rss R t ss Yt Yt 1 61 2 1 et et 1 3 (42) The government spending is exogenously given as log Gt = 8.4 g log Gt 1 + "gt (43) Foreign Economy Since domestic and foreign economy are symmetric, the budget constraint of a consumer in the foreign country is given by PH;t CH:t + PF;t CF;t + Et Qt;t+1 Dt+1 et W t Lt + Dt et Tt (44) where Dt denotes the foreign household’s holdings of the portfolio denominated in home country currency. The …rst-order condition with respect to portfolio choice is Qt;t+1 t+1 1 = Pt et Pt+1 et+1 t (45) The existence of complete international state-contingent claims implies the perfect risk sharing Pt t Pt+1 et t+1 Pt t Pt+1 et+1 t+1 Qt;t+1 = = (46) The real exchange rate is de…ned as: St = et Pt Pt 62 (47) and the log-linearization gives the purchasing power parity (PPP) relationship : s~t = e~t + t (48) t Using the de…nition of the real exchange rate, equation (46) implies that the ratio of marginal utilities of home and foreign countries adjusts the real exchange rate: s~t = ~ t ~t (49) Also, by substituting de…nitions of terms of trade, the overall in‡ations and the l.o.p gap into (47) , one can derive the evolution of the real exchange rate : s~t = e~t + p~t p~t = e~t + (1 )~ pF;t + p~H;t (1 = e~t + p~F;t p~F;t p~H;t ) = ~ F;t q~t (1 (~ pF;t )~ qt )~ pH;t (1 p~F;t )(~ pH;t p~F;t ) (50) Under complete international asset market, households can purchase the domestic and foreign government bond denominated in each country’s currency and by no arbitrage condition, the expected returns on domestic government bond and foreign government bond have to be equalized. Since the return on the foreign government bond will be the nominal interest rate and the expected exchange rate depreciation, this implies a log-linear version of an uncovered interest rate parity condition (UIP) : ~t R ~ = Et e~t+1 R t 63 (51) 8.5 Market clearing The resource constraints for domestic country and foreign country are given by YH;t = CH;t + CH;t + Gt (52) YF;t = CF;t + CF;t + Gt (53) That is, the goods produced in home country are consumed by domestic households, CH;t ; or exported to foreign country, CH;t , or consumed by government. 8.6 Exogenous Stochastic Process Non-stationary worldwide technology follows a random-walk with drift. A~W;t = ln + A~W;t z~t = ~t 1 zz 1 + z~t + "zt The monetary policy shocks for each country, "Rt and "R t ; are assumed to be i.i.d. Country-speci…c technology shocks and government spending shocks evolve according to the following stochastic processes : A~H;t = AH AH;t 1 A~F;t = AF AF;t 1 ~ H;t = G GH GH;t 1 ~ F;t = G GF GF;t 1 ~ + "AH t ~ + "AF t ~ + " GH t ~ + "GF t 64 9 Learning Model 9.1 Learning Algorithm The log-linearized model described in the previous section can be stacked in a form of 0( )Xt 1 + 1( )Xt + t + t = 0; where Xt is a vector of all endogenous variables and exogenous processes, of stochastic innovations, and t (54) t is a vector consists of the rational expectations forecast errors. By imposing the rational expectation condition, the state equation (54) has a solution written as: Xt = T ( )Xt 1 + H( ) t ; (55) where T and H are non-linear functions of structural parameters . Xt can be decomposed into state variables, Xts ; and forward variables, Xtf , which appear with a lead in the model. More speci…cally, in the model, agents have to forecast domestic in‡ation and import in‡ation for each country, domestic consumption for their home country, the real exchange rate, and the world-wide technology shock. In this paper, I relax the rational expectation assumption that agents have the equilibrium-consistent forecasts. Instead, I assume that agents forecast the value of forward variables, only using the limited information set fXjf 1 gtj=11 : Agents form their forecast following a small forecasting rule in a vector autoregressive regression form at each period with data available. The forecasting rule is assumed as: Xtf = at 1 + bt 1 Xtf 1 + et ; (56) where the coe¢cients of the forecasting rule are called the beliefs. The equation 65 (56) represents the “Perceived Law of Motion” (PLM) of the agents. As the new data get available, agents update their beliefs according to the constant gain learning algorithm: where Zt 1 [1 Xtf 1 ]; t = t 1 + g(Zt 1 Zt0 t = t 1 +g t 1 t (Xtf t 1 ); 1 Zt and 1 t 1 ); [at; bt ]0 stacks the beliefs, t (57) (58) denotes the second moments, and g is the gain parameter which determines the rate at which past observations are discounted. This algorithm places more weight on recent observations and geometrically discounts the weight to g(1 g)t after t periods. Orphanides and Williams (2005) refer to this as “perpetual learning” since agents are alert to potential structural change and forget the past. As g ! 0; the learning model converges to the rational expectation model. By substituting PLM into the state equation (54), I derive the “Actual Law of Motion” (ALM): Xt = Ct ( ; t 1) + Tt ( ; t 1 )Xt 1 Now, Tt and Ht are functions of agents’ beliefs + Ht ( ; t 1; t 1) t: as well as structural parame- ters . 9.2 Exchange Rate Determination under Learning Under the learning mechanism, model optimality conditions have the subjective expectations denoted by E^t in place of the rational expectations denoted by Et : I will discuss the equations that determine the exchange rate dynamics imposed by the 66 model and how the learning mechanism a¤ects the equilibrium path. The real exchange rate is determined by the relative marginal utility: ~t; st = ~ t where ~ t = 1 h (~ ct h~ ct 1 )+ 1 h h E^t [ (~ ct+1 h~ ct )+~ zt+1 ]. Due to the habit formation, the marginal utility will be a¤ected by the deviation from the rational expectations. Since the adaptive expectations are not …rmly tied to the equilibrium, the link between relative consumption and the real exchange rate will be loosened. The model also imposes the PPP equation that the price di¤erential determines the exchange rate, shown as: e~t = st p~t + p~t ; Since the expectation term does not directly appear in PPP equation, but only through the real exchange rate st ; the learning mechanism will not play a separate role. The linearized UIP condition under the learning mechanism adds an additional term to the UIP equation under rational expectations. This is shown as: ~t R ~ t = E^t e~t+1 R = Et e~t+1 + (E^t e~t+1 Et e~t+1 ) It is well known that the linearized UIP equation has been rejected by the data. To relax this equation, McCallum and Nelson (1999, 2000) derive the extra term in the UIP equation as a time-varying risk premium omitted by linearization, and Jeanne and Rose (2002) consider noise traders. Under the learning mechanism, the forecast 67 errors will disconnect the exchange rate dynamics from the interest rate di¤erential. 10 Simulation Study Throughout the simulation study and the empirical application, I consider the observables consisting of seven variables: output growth, in‡ation, interest rate for home country, output growth, in‡ation, interest rate for foreign country and the exchange rate depreciation. The state-space representations under rational expectations and under the learning mechanism are summarized in the following table. Rational Expectations Model Learning Model Yt = D( ) + Z( )Xt Xt = T ( )Xt 1 + H( ) Yt = D( ) + Z( )Xt t Xt = Tt ( ; t = t = t 1 t 1 t 1 )Xt 1 +g t 1 + Ht ( ; (Zt + g(Zt 1 Zt0 Zt 1 t 1) t 1 t 1) t 1) All simulations are based on the same structural parameters as the medians of Lubik and Schorfheide (2005) in Table 4. I generate 1000-period-long data sets using the rational expectations model solution and the learning model solution. In order to provide discipline on constant gain parameters, I consider di¤erent values of constant gain parameters, (0.001, 0.002, and 0.01) that impose the weight on the data by half after 172, 86, and 17 years. I focus on small values to ensure the stationary path around the rational equilibrium, and these gain parameters are in line with Eusepi and Preston (2011), who used the interval of 0.0013 to 0.003. To initialize the belief process, I generate presample data from the rational expectations model solution, following Huang, Liu, and Zha (2008). Table 5 shows that the learning mechanism a¤ects the volatilities in economics variables. For instance of “g =0.001”, the learning model moderately reduces the 68 volatilities of output growth, interest rate for home country and the volatility of in‡ation for foreign country, but increases the variability of other variables including the exchange rate depreciation. As I adapt higher values of constant gain parameters, there is a tendency for the standard deviations of variables to grow because the belief updating process ‡uctuates more. This result is in line with the …ndings of Slobodyan and Wouters (2012) that the standard deviations of variables do not increase much for small value of gain parameters, and tend to increase variability of the level variable by the so-called “exits” or “large deviations” for higher value of gain parameters. In terms of persistence, Figure 21 provides the autocorrelation of the real exchange rate from the rational expectation and learning models. The learning mechanism generates the additional persistence in the real exchange rate which is related to the increased persistence of foreign in‡ation. The reason for the additional persistence is that agents’ perceived in‡ation and the exchange rates are more persistent than under rational expectations. When I run regressions with forecasts of the exchange rates from the learning model and rational expectations model, the autoregressive coe¢cients on the past forecasts is 0.93 in the learning model whereas it is 0.79 in the rational expectations model. Table 6 gives an idea of how the learning mechanism disconnects the exchange rate from fundamentals. The UIP and PPP equation columns report the correlation of the interest rate di¤erentials and the exchange rate depreciation and the correlation of the in‡ation di¤erentials and the exchange rate depreciation, respectively. In the rational expectation model, the interest rate di¤erential shows a signi…cantly positive correlation, 0.26 to the exchange rate depreciation while the data implies the correlation is close to zero. The learning mechanism decreases the correlation as the gain parameter get increased. When g =0.002, the learning model can generate the closest correlation to the data. The correlations of in‡ation di¤erentials to depreciation are 69 similarly larger than found in the data. As I discussed in previous section, learning does not play a role in the PPP relationship. The Backus-Smith column shows the correlation between relative consumption and the real exchange rate. As documented in the literature, I …nd that this correlation is 0.03 in the data, and that the standard rational expectations model predicts a correlation of 0.97. Learning models also perform poorly in this dimension, but notice that the learning mechanism shows a tendency of decreases toward the data moment. 11 Empirical Application The model is …tted to U.S. data for the “home country” and Euro area data for the “foreign country”. The data are quarter-to-quarter output growths, annualized in‡ations, annualized interest rates for both countries, and bilateral nominal exchange rate depreciation in percentage terms from 1981:Q3 to 2012:Q4. U.S. data series are obtained from the database of the Federal Reserve Bank of St. Louis (FRED) and Euro area data are from the Area-Wide Model database. This latter database became a standard reference for Euro area data after Fagan, Henry, and Metre (2005). Euro area data are the weighted aggregates of individual countries based on constant gross domestic product (GDP) at market prices for 1995z . For the exchange rates data, I use the U.S./Euro exchange rate from FRED after the introduction of the euro in 1999:Q1 and construct the synthetic exchange rate from the bilateral exchange rates of countries in the Euro area for the historical data. The bilateral exchange rates of individual countries are obtained from FRED, and the historical exchange rate depreciation is given as the weighted sum of exchange rate depreciations of countries z These weighs are 0.036 on Belgium, 0.283 on Germany, 0.111 on Spain, 0.201 on France, 0.015 on Ireland, 0.195 on Italy, 0.003 on Luxembourg, 0.060 on Netherlands, 0.030 on Austria, 0.024 on Portugal, 0.017 on Finland and 0.025 on Greece. 70 with weights underlying in the AWM database: 4 ln Et = n X fi 4 ln Ei;t : i=1 Figure 22 plots the nominal depreciation and the in‡ation and interest rate di¤erentials. The shaded section in the top panel indicates the recent …nancial crisis, while the bottom panel magni…es the in‡ation di¤erential and the interest rate di¤erential. Notice the di¤erent patterns before and after the crisis. The nominal depreciation movement seems to be more correlated with fundamentals after the crisis. The correlation between the depreciation and in‡ation di¤erential goes from 0.18 up to 0.34, and the correlation between the depreciation and interest rate di¤erential changes even more, going from -0.11 to 0.17. Initialization of Estimation I use the prior density from Lubik and Schorfheide (2005) for both the rational expectations model and the learning model. The import share parameter, ; is …xed at 0.13 since the data do not have information that can …t the trade. The gain parameter is set to 0.002. To get regression-based initial beliefs as in the simulation study, I use the smoothed states from the Kalman …lter using pre-sample data from 1973:Q1 to 1981:Q2 f = a0 + b0 X f 1j0 ; X0j0 f = where X0j0 o0 n f Xtj0 t= k and X f 1j0 = n Xtf 1j0 o0 t= k . In order to get the smoothed states, I run the rational expectation model with pre-sample data and save the mean and variance of states from the Kalman …lter. In terms of Hamilton (1994), Xtjt = E(Xt jIt ), Ptjt = V ar(Xt jIt ); Xtjt 1 = E(Xt jIt 1 ); and Ptjt 1 = V ar(Xt jIt 1 ). Smoothed states are obtained by iterating the following algorithm from the end of 71 sample: 11.1 1 (Xt+1jt XtjT = Xtjt + Ptjt T 0 Pt+1jt T Xtjt ) 1 (Pt+1jT PtjT = Ptjt + Ptjt T 0 Pt+1jt 1 T Ptjt : Pt+1jt )Pt+1jt Posterior Distribution Posterior densities from a Bayesian estimation for the rational expectations model and the learning model are summarized in Table 7 along with the prior densities. Based on the random-walk Metropolis-Hastings algorithm, posterior distributions are explored by generating 100,000 draws and burning the initial 90% of draws. There are some remarkable changes in estimated parameters. First, I …nd that the Calvo parameters for U.S. drop signi…cantly in the learning model. The estimated nominal rigidity parameter for domestic good in the U.S. under rational expectations is 0.41, suggesting 1.7 quarters of price stickiness. Under constant-gain learning mechanism, that parameter is 0.16, suggesting 1.2 quarters of price stickiness. The nominal rigidity parameter for the import sector in the U.S. is signi…cantly reduced to 0.09 (1quarter price stickiness) with the learning mechanism, compared to 0.57 (2.3-quarter price stickiness) under rational expectations. The price stickiness parameters in the Euro area do not change much either in domestic production or in the import sector. These …ndings are in line with Vilagi (2007) that learning does not replace the structural sources of persistence in the Euro area. Second, the degree of habit formation signi…cantly drops from 0.23 to 0.02. Third, the autoregressive coe¢cients of structural shocks decrease with the exception of Euro-area speci…c technology shocks. As a ‡ip-side of the simulation study, learning replaces the structural sources of persistence and generates the endogenous persistence. Last, I obtain a reduction in the standard 72 deviations of four out of seven structural shocks. This implies that learning can add volatilities to variables as found in simulation exercises. The non-structural PPP shock which is designed to capture the model misspeci…cation decreases slightly, but not signi…cantly, suggesting that the improvement of the learning model is limited in explaining the excess volatility of the exchange rate. Also, two model speci…cations similarly match these moments. This can give support to the learning mechanism since the learning model predictions are generated with smaller structural persistence parameters and smaller exogenous shocks. 11.2 Marginal Data Density Comparison As is standard in the Bayesian framework, I evaluate which model …ts the data better by comparing the marginal data density. Due to the computational di¢culty of the marginal data density, I adapt the modi…ed Harmonic mean estimator suggested by Geweke (1999). The Harmonic mean estimator is based on the following identity p(Y ) = Z where p( ) is the prior density of 1 f( ) p( jY )d L(Y j )p( ) ; and f ( ) is a function satisfying 1: Then, the sample correspondence is given by: " J f ( j) 1X ) p^J (Y ) = J j=1 L(Y j j )p( j ) where j # Z f ( )d = 1 ; are draws from the posterior distribution, p( jY ): Table 8 reports the log marginal data densities (MDD) from the rational expectations and learning models with di¤erent gain parameters (0.001, 0.002, and 0.01). 73 The learning model with a constant gain of 0.002 delivers the log MDD of -1372.3, while the log MDD of the rational expectations model equals -1416. For the other values of constant gain parameters, I …nd the dominance of learning models. This implies that the data favor the constant gain learning speci…cation over the rational expectations speci…cation. Since the benchmark model with constant gain of 0.002 provides the best …t, it relieves the limitations of …xing the gain parameter instead of estimating it. 11.3 Impulse Response Function The impulse response functions to the U.S. monetary policy shock from rational expectation model and the learning model are shown in from Figure 23 to Figure 26. Note that all the variables react with signi…cant lags in learning model. It takes more than 20 quarters for the exchange rate depreciation to be adjusted in the learning model, visible as the hump-shaped response, while in the rational expectations model it instantly drops and dissipates. In the learning model, when the unanticipated monetary shock increases the interest rate, agents do not immediately recognize the source of the ‡uctuations. It takes time for agents to realize the nature of the shock based on the equilibrium relations of variables in their forecasting rules. Therefore, the responses are small at the beginning and propagate through the expectation formation. In the learning model, monetary policy shocks are absorbed in the expectation, and do not produce real e¤ects on consumption and output. Instead, expectation a¤ects the price movement, and thus the exchange rate. The responses of output to the monetary policy are contrary to that which closed-economy models predict. I also …nd that the e¤ects of the Euro area monetary policy shock on the U.S. economy and the exchange rate are not signi…cant. 74 11.4 Variance Decomposition I report the variance decomposition results from the rational expectations model in Table 8 and from the learning model in Figure 27. The unconditional variance decomposition results are calculated with the medians of posterior density. Variance decomposition is obtained by solving the following discrete Lyapunov equations: V ar(Xt j ) = T ( )V ar(Xt j )T ( )0 + H( )Q( )H( ); and V ar(Yt j ) = Z( )V ar(Xt j )Z( )0 ; where Q( ) is a variance-covariance matrix of structural shocks. The contribution of shock i is obtained by setting the volatility of all disturbances except the corresponding shock i to zero. Under both the rational expectations and learning speci…cations, the non-structural PPP shock explains most of the exchange rate ‡uctuation. The second biggest contribution in the rational expectation model is made by the Euro and U.S. monetary policy shocks, followed by world-wide technology shock. In contrast, learning model attributes the movement of depreciation to the real shocks, U.S. technology shock, and world-wide technology shock by approximately 6%. 11.5 Posterior Predictive Checks To assess the absolute …t of models, I conduct posterior predictive checks. Given the posterior densities from the rational expectations and the learning models, I simulate 10,000 samples of observations, each of which is 126 periods in length for each set of draws from posterior densities. These are known as predictive densities. I plot the kernel distributions to determine how well models with di¤erent expectations 75 speci…cations …t data statistics in Figure 28. In the …rst row, I show the correlation of output growths across countries, and the correlations of output growth and in‡ation in both the U.S. and Euro areas. Solid lines indicate predictive densities from the learning model, the dashed lines represent results from the rational expectations model, and the red dotted line denotes the actual data moments. The solid lines are clearly closer to the dotted "actual" moments than the dashed lines, indicating that the learning model explains better the business cycle ‡uctuation. The second row shows the exchange rate determination equations, the UIP relation, PPP relation, and Backus-Smith relation. Since I do not use consumption data, I calculate the correlation of real exchange rate depreciation and output growth di¤erentials for the Backus-Smith relation. The results from the rational expectations and learning models are similar, though rational expectations model sometimes performs slightly better. This is not inconsistent with the simulation study because simulations are conducted under di¤erent parameterization for predictive checks. Thus, the performance of the learning model results from di¤erent structural parameters as well as the di¤erent expectation mechanism. The last row displays the standard deviation and the persistence of the real exchange rate depreciation. 12 Conclusion I specify and estimate two-country New Open Economy Models under rational expectations and learning mechanism. A major feature of the learning model is that agents are assumed to use econometric models with historical data to make their forecasts as well as update their forecasts over time via a constant-gain learning algorithm. I …nd from a simulation study that the learning mechanism increases the volatilities of variables as the gain parameter rises. The self-ful…lling property of the belief-updating 76 process engenders a more persistent exchange rate and more persistent in‡ation series, as shown in Orphanides and Williams (2005). A Bayesian estimation with U.S. and Euro-area data implies that the learning mechanism substitutes the structural persistence sources of the model, e.g., the nominal price stickiness for the U.S. and habit formation, since learning generates additional persistence. Model comparison based on the marginal data density favors the learning speci…cation over rational expectations, although the improvement in terms of …tting the observed Dollar-Euro exchange rate dynamics is limited. The performance of the learning mechanism surpasses the rational expectations model in explaining some business cycle statistics, such as output growth comovement between the U.S. and the Euro area and the correlation between output growth and in‡ation of the U.S. Impulse response functions from the learning model are lagged and persistent, since agents realize the source of ‡uctuation with a lag and adjust their forecasts in a recursive manner. Yet the results from the learning model rely on the speci…cation of agents’ forecasting rules and the choice of initial belief. It is worth examining with di¤erent types of forecasting rules and di¤erent initial belief schemes. It would also be a plausible extension of the learning model to consider the endogenous switching of forecasting models based on the forecasting model performance, as in Lewis and Markiewicz (2009). 13 Tables and Figures 77 Table 4: Summary of Parameters for Simulation Coe¢cient H F H F h 1 2 3 1 2 3 AH R g AF R g z rA A A R g A R g z Description Calvo parameter of domestic goods for home country Calvo parameter of imported goods for home country Calvo parameter of imported goods for foreign country Calvo parameter of domestic goods for foreign country import share consumption habit relative risk aversion coe¢cient intratemporal substitution elasticity between home and foreign good response to in‡ation gap for home country response to output growth for home country response to exchange rate depreciation for home country response to in‡ation gap for foreign country response to output growth for foreign country response to exchange rate depreciation for foreign country AR coe¢cient for country-speci…c technology shock of H AR coe¢cient for monetary policy of H AR coe¢cient for government spending shock of H AR coe¢cient for country-speci…c technology shock of F AR coe¢cient for monetary policy of F AR coe¢cient for government spending shock of F AR coe¢cient for world-wide technology shock steady state interest rate world-wide technology growth rate steady state in‡ation standard deviation of country-speci…c shock for H standard deviation of monetary policy shock for H standard deviation of government spending shock for H standard deviation of country-speci…c shock for F standard deviation of monetary policy shock for F standard deviation of government spending shock for F standard deviation of world-wide technology shock 78 Value 0.66 0.56 0.86 0.76 0.13 0.41 3.76 0.43 1.41 0.66 0.03 1.37 1.27 0.03 0.83 0.76 0.90 0.85 0.84 0.94 0.60 0.86 0.39 3.16 1.66 0.18 0.50 2.61 0.18 0.62 0.35 Table 5: Standard Deviation of Simulated Models R.E. model Output growth for H In‡ation for H Interest rate for H Output growth for F In‡ation for F Interest rate for F Depreciation Real exchange rate 0.57 1.22 1.53 0.62 0.32 0.68 1.64 1.17 Learning model g = 0.001 g = 0.002 g 0.57 0.57 1.64 1.79 2.40 2.71 0.63 0.63 0.33 0.41 0.69 0.64 1.71 1.80 1.65 1.69 = 0.01 0.62 2.28 2.57 0.64 0.61 0.92 1.89 1.70 Table 6: Correlation of Depreciation to In‡ation and Interest Rate Gap UIP equation PPP equation Backus-Smith Data 0.03 0.19 0.03 Rational Expectations 0.26 0.54 0.97 Learning Model g = 0.001 0.23 0.52 0.95 0.05 0.58 0.93 g = 0.002 g = 0.01 -0.03 0.72 0.92 Note. UIP column reports the correlation between depreciation and the interest rate gap (i.e., Corr(4et+1 ; Rt Rt )). The PPP column reports the correlation between depreciation and the in‡ation gap (i.e., Corr(4et ; t t )). The Backus-Smith column shows the correlation between relative consumption and the real exchange rate (i.e., Corr(st ; ct ct )): 79 Figure 21: Autocorrelation Function of the Real Exchange Rate from Simulation 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 Note. The solid line shows results from the rational expectations model The dashed line denotes results from the learning model where g = 0.001; the dash-dot line for g = 0.002; and the dotted line for g = 0.01. 80 Figure 22: Depreciation and In‡ation, Interest Rate Gap 10 5 0 -5 -10 -15 1980 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 3 2 1 0 -1 -2 -3 -4 1980 Note. The top panel shows three series: the solid line shows nominal depreciation; the dotted line shows the in‡ation gap; and the dashed line shows the interest rate gap. The bottom panel shows a magni…cation of the in‡ation gap and the interest rate gap. 81 Table 7: Prior and Posterior Distribution Coe¢cient H F H F h 1 2 3 1 2 3 AH R g AF R g z rA A AH R g AF R g z e Prior Density Mean Beta 0.50 Beta 0.50 Beta 0.75 Beta 0.75 Beta 0.30 Gamma 2.00 Gamma 1.00 Gamma 1.50 Gamma 0.50 Gamma 0.10 Gamma 1.50 Gamma 0.50 Gamma 0.10 Beta 0.80 Beta 0.50 Beta 0.80 Beta 0.60 Beta 0.50 Beta 0.80 Beta 0.66 Gamma 0.50 Normal 0.40 Gamma 7.00 InvGamma 1.00 InvGamma 1.00 InvGamma 0.40 InvGamma 0.40 InvGamma 1.00 InvGamma 0.20 InvGamma 0.50 InvGamma 3.50 Std 0.15 0.15 0.15 0.15 0.10 0.50 0.50 0.25 0.25 0.05 0.25 0.25 0.05 0.10 0.20 0.10 0.20 0.20 0.10 0.15 0.50 0.20 2.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 R.E. Model Median 90% Interval 0.41 [0.34 0.47] 0.57 [0.45 0.69] 0.84 [0.75 0.93] 0.62 [0.57 0.65] 0.23 [0.09 0.38] 3.39 [3.31 3.49] 0.52 [0.39 0.60] 1.01 [1.00 1.04] 0.53 [0.45 0.66] 0.02 [0.00 0.03] 1.02 [1.01 1.04] 1.16 [1.08 1.32] 0.06 [0.03 0.08] 0.98 [0.96 0.99] 0.77 [0.74 0.80] 0.97 [0.95 0.99] 0.87 [0.75 0.97] 0.82 [0.79 0.84] 0.97 [0.96 0.99] 0.83 [0.78 0.88] 0.49 [0.35 0.57] 0.37 [0.31 0.43] 3.24 [3.09 3.35] 1.13 [0.96 1.27] 0.23 [0.20 0.26] 0.75 [0.67 0.84] 1.99 [1.78 2.32] 0.19 [0.16 0.22] 0.57 [0.51 0.64] 0.25 [0.21 0.30] 4.49 [4.34 4.70] 82 Learning Model Median 90% Interval 0.16 [0.07 0.26] 0.09 [0.03 0.15] 0.84 [0.71 0.95] 0.60 [0.52 0.69] 0.02 [0.01 0.03] 4.64 [3.97 5.34] 0.11 [0.04 0.21] 1.53 [1.22 1.82] 0.77 [0.33 1.28] 0.04 [0.02 0.06] 1.43 [1.14 1.70] 1.01 [0.55 1.56] 0.03 [0.02 0.06] 0.88 [0.79 0.95] 0.93 [0.91 0.95] 0.95 [0.91 0.98] 0.92 [0.85 0.97] 0.94 [0.92 0.96] 0.92 [0.85 0.97] 0.82 [0.72 0.92] 0.52 [0.01 1.20] 0.44 [0.32 0.58] 5.09 [3.58 6.44] 1.31 [1.01 1.66] 0.18 [0.16 0.20] 0.60 [0.52 0.68] 2.48 [1.50 4.56] 0.12 [0.11 0.13] 0.64 [0.55 0.73] 0.21 [0.18 0.24] 4.45 [4.00 4.98] Table 8: Marginal Data Density R.E. Model MDD -1416.0 Learning Model g = 0.001 g = 0.002 g = 0.01 -1377.3 -1372.3 -1397.4 Table 9: Variance Decomposition from the Rational Expectations Model monetary policy technology govn’t spending monetary policy technology govn’t spending world-wide tech PPP shock output in‡ation int rate output in‡ation int rate Depreciation 0.007 0.067 0.918 0.002 0.001 0.001 0.002 0.000 0.320 0.614 0.083 0.006 0.036 0.009 0.053 0.002 0.122 0.752 0.191 0.013 0.092 0.016 0.076 0.000 0.019 0.069 0.002 0.151 0.046 0.386 0.307 0.011 0.0444 0.2029 0.2242 0.155 0.015 0.130 0.322 0.014 0.115 0.031 0.557 0.077 0.027 0.407 0.038 0.004 0.042 0.005 0.004 0.044 0.000 0.008 0.036 0.861 Note: The asterisks denote the foreign(Euro-area) variables. 83 Figure 23: Impulse Response to U.S. Monetary Policy Shock for the Learning Model U.S. Output U.S. In‡ation 0.016 0.15 0.014 0.1 U.S. Interest Rate 0.8 0.6 0.012 0.05 0.01 0.4 0 0.008 -0.05 0.2 0.006 -0.1 0.004 0 -0.15 0.002 0 0 -0.2 10 20 30 40 -0.2 0 Euro Output 5 x 10 10 20 30 40 0 Euro In‡ation 10 20 30 40 Euro Interest Rate -3 0.15 0 0.1 -5 0.05 -10 0 -15 -0.05 0.1 0.05 0 -0.05 -20 -0.1 0 10 20 30 40 Exchange Rate Depreciation -0.1 0 10 20 30 40 0 10 20 30 40 Exchange Rate Level 0.01 0.2 0 0.1 0 -0.01 -0.02 -0.1 -0.03 -0.2 -0.04 -0.3 -0.05 -0.4 -0.06 -0.5 0 10 20 30 40 0 10 20 30 40 Note. Solid lines represent the median and dotted lines represent the 90% con…dence interval of impulse response functions. All responses are in percentage and those for in‡ation and interest rate are annualized. 84 Figure 24: Impulse Response to Euro Monetary Policy Shock for the Learning Model U.S. Output U.S. In‡ation 0 -0.002 U.S. Interest Rate 0.1 0.1 0.05 0.05 0 0 -0.05 -0.05 -0.1 -0.1 -0.004 -0.006 -0.008 -0.01 -0.15 -0.012 0 10 20 30 -0.15 0 40 Euro Output 10 20 30 40 0 Euro In‡ation 0.15 0.6 0.012 0.1 0.5 0.01 0.05 0.4 0.008 0 0.3 0.006 -0.05 0.2 0.004 -0.1 0.1 0.002 -0.15 0 -0.2 -0.1 10 20 30 0 40 Exchange Rate Depreciation 10 20 20 30 40 Euro Interest Rate 0.014 0 0 10 30 40 0 10 20 30 40 Exchange Rate Level 0.03 0.3 0.025 0.2 0.02 0.015 0.1 0.01 0 0.005 0 -0.1 -0.005 -0.01 -0.2 0 10 20 30 40 0 10 20 30 40 Note: Solid lines represent the median and dotted lines represent the 90% con…dence interval of impulse response functions. All responses are in percentage and those for in‡ation and interest rate are annualized. 85 Figure 25: Impulse Response to U.S. Monetary Policy Shock U.S. Output U.S. In‡ation U.S. Interest Rate 0 0.8 -0.5 0.6 -1 0.4 -1.5 0.2 -2 0 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 -2.5 -0.06 0 2 4 6 8 10 -0.2 0 12 Euro Output 2 4 6 8 10 12 0 Euro In‡ation 2 4 6 8 10 12 Euro Interest Rate 0.1 0.12 0.08 0.12 0.1 0.1 0.08 0.06 0.08 0.06 0.04 0.04 0.02 0.06 0.02 0.04 0 0 0.02 -0.02 -0.02 0 2 4 6 8 10 12 -0.04 0 Exchange Rate Depreciation 2 4 6 8 10 12 0 0 2 4 6 8 10 12 Exchange Rate Level 0.4 0 0.2 -0.2 0 -0.4 -0.2 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Note. Solid lines correspond to results from the learning model, and dashed lines correspond to results from the rational expectations model. All responses are in percentage and those for in‡ation and interest rate are annualized. 86 Figure 26: Impulse Response to Euro Monetary Policy Shock U.S. Output U.S. In‡ation 0 U.S. Interest Rate 0.02 0.08 -0.005 0.06 0.01 -0.01 0.04 0 -0.015 0.02 -0.02 -0.01 0 -0.025 -0.02 -0.02 -0.03 -0.035 -0.03 -0.04 0 2 4 6 8 10 12 0 Euro Output 2 4 6 8 10 12 0 Euro In‡ation 0.05 0 0 -0.2 -0.05 -0.4 -0.1 -0.6 -0.15 -0.8 -0.2 -1 -0.25 -1.2 2 4 6 8 10 12 10 12 Euro Interest Rate 0.5 0.4 0.3 0.2 0.1 -0.3 -1.4 0 2 4 6 8 10 12 0 Exchange Rate Depreciation 2 4 6 8 10 12 0 0 2 4 6 8 Exchange Rate Level 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 -0.2 0 2 4 6 8 10 12 0 0 2 4 6 8 10 12 Note: Solid lines correspond to results from the learning model, and dashed lines correspond to results from the rational expectations model. All responses are in percentage and those for in‡ation and interest rate are annualized. 87 Figure 27: Variance Decomposition for Depreciation in the Learning Model U.S. Monetary Policy x 10 U.S. Technology U.S. Govn’t Spending -4 x 10 6.5 0.0236 6.4 0.0235 -4 1.3 1.2 6.3 0.0235 6.2 0.0234 1.1 6.1 0.0234 6 1 0.0233 5.9 0.0233 0.9 5.8 0.0232 5.7 0.8 0.0232 5.6 5.5 1980 1985 1990 1995 2000 2005 2010 2015 0.0231 1980 Euro Monetary Policy x 10 1985 1990 1995 2000 2005 2010 2015 0.7 1980 1985 Euro Technology -4 x 10 1.64 13 1.63 12 1.62 11 1.61 10 1.6 9 1.59 8 1.58 7 1990 1995 2000 2005 2010 2015 Euro Govn’t Spending -4 x 10 -5 2.1 2 1.9 1.8 1.7 1.57 1980 1985 1990 1995 2000 2005 2010 2015 1.6 6 1980 Worldwide Technology 1985 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 PPP 0.9435 0.0336 0.0334 0.943 0.0332 0.9425 0.033 0.0328 0.942 0.0326 0.9415 0.0324 0.0322 1980 1985 1990 1995 2000 2005 2010 2015 0.941 1980 1985 88 1.5 1980 1985 1990 1995 2000 2005 2010 2015 Figure 28: Posterior Predictive Checks corr( y; y ) corr( ; y) 5 5 4 4 3 3 corr( ; y ) 3.5 3 2.5 2 2 1 1 2 1.5 1 0.5 0 - 0 .6 - 0 .4 - 0 .2 0 0 .2 corr(R 0 .4 0 .6 0 .8 1 1 .2 0 -1 -0 .8 -0 .6 R ; e) -0 .4 -0 .2 0 0 .2 0 .4 corr( 0 .6 0 .8 ; e) 4 5 0 -1 -0.5 0 corr( y 0.5 1 y ; e) 5 3.5 4 4 3 2.5 3 3 2 2 2 1.5 1 1 1 0.5 0 -3 -2 -1 0 1 2 3 0 -0.5 std( s) 0 0.5 1 1.5 0 -0 .4 -0 .2 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 corr( st ; st 1 ) 5 1 .4 1 .2 4 1 3 0 .8 0 .6 2 0 .4 1 0 .2 0 2 4 6 8 10 12 14 16 18 0 -1 -0.5 0 0.5 Note: Solid lines represent kernel densities from the learning model; the dashed lines represent kernel densities from the rational expectations model; and the red dotted vertical lines are actual data moments. 89 A A.1 A.1.1 Appendices Estimation Algorithm Stochastic Volatility in DSGE Models A state-space representation of log-linearized DSGE model is given by: yt = D + Z t = T( ) t 1 (ME) t + R( ) (TE) t The stochastic volatilities for each structural shock (j) are, where j 2 fz; p; b; '; ; R; gg; log jt = jt "jt jt = (1 j ) log j + j log jt 1 + jt ; N (0; ! 2j ) jt In order to reduce the number of free parameters, I assume that the stochastic volatilities are random walk processes by setting the autoregressive coe¢cients the element of vector ht be hjt = log( jt ): = 1: Let j Then, H T = [h1 ; :::hT ]0 is a T-by-7 vector matrix. Similarly, denote the sample of structural shocks as T = [ 1 ; ::: T] 0 : Collect the remaining coe¢cients of the stochastic volatility processes by V; where j th element is ! 2j : Initialization of the Algorithm. Generate fH T 0 ; V T 0 ; 0 ; T0 g: H T 0 is constructed recursively using the posterior median value of the time-invariant standard deviations for the structural shocks as initial values. V T 0 is squared values of time-invariant standard deviations. 0 ~N (^; c 1 ) where ^ and 1 are posterior mode and variance respectively, obtained by using Chris Sims’s maximization algorithm. c is a scaling 90 parameter (set to 2). Finally, T0 is obtained by using simulation smoother developed by Durbin and Koopman (2002), which will be explained later. Simulation Smoother by Durbin and Koopman (2002). Consider the linear Gaussian form where "t ~N (0; Ht ) and t ~N (0; Qt ): Note that I assume yt = yt D and Ht = 0: yt = Z t t + "t = T( ) The distribution of w is w~N (0; ); t 1 ( ) + R( ) t = diag(H1 ; Q1 ; :::HT ; QT ): Step 1 Draw a random vector w+ from density p(w) and use it to generate y + by means of recursion ( ) with w replaced by w+ ; where the recursion is initialized by the draw + 1 ~N (a1 ; P1 ): Step 2 Compute w b = E(wjy) and w b+ = E(w+ jy + ) by means of standard Kalman Filtering and disturbance smoothing: Kalman F ilter 1 vt = yt Zat Ft = ZPt Z 0 + Ht Kt = T Pt Z 0 Ft Lt = T Kt Z at+1 = T at + Kt vt Pt+1 = T Pt L0t + RQt R0 w bt rt Step 3 Take w e=w b 1 Disturbance Smoothing 2 1 30 1 0 H t Kt 7 B v t C 6 Ht Ft = 4 A 5@ 0 0 Qt R0 = Z 0 Ft 1 vt + L0t rt w b+ + w + : 91 Random Walk Metropolis Hastings Algorithm. At the beginning of each iteration g, given fH T g 1 ; g 1 g Step 1 Draw a candidate, r = min ; c1 1 ) , conditional on H T g Step 2 Evaluate the likelihood at the candidate 1 : L(Y j ; H T g 1 ) ;1 L(Y j g 1 ; H T g 1 ) g 1 accept the proposal with probability r or keep Step 3 Sample the structural innovations tional on g 1 from proposal distribution; N ( Tg with 1 r probability using simulation smoother condi- g Tg Step 4 Draw the stochastic volatilities using : Denote ejt = log[b2jt + 0:001] and ejt = log("2jt ): This leads to the following ap- proximating state-space form: ejt = 2hjt + ejt hjt = hjt 1 + jt Since the innovations in the measurement equation follow log 2 (1); I transform the system in a Gaussian one using a mixture of normal approximations as described in Kim, Shephard, and Chib (1998): f (ejt ) = 7 X qk fN (ejt jsjt = k) k=1 where sjt is the indicator variable selecting which member of the mixture of normals has to be used at time t for the innovation j, qk = Pr(sjt = k); and fN is the pdf of 92 a normal distribution. w qj = Pr(w = j) mj rj2 1 0:00730 10:12999 5:79596 2 0:10556 3:97281 2:61369 3 0:00002 8:56686 5:17950 4 0:04395 2:77786 0:16735 5 0:34001 0:61942 0:64009 6 0:24566 1:79518 0:34023 7 0:25750 1:08819 1:26261 Source : Kim; Shepard; and Chib (1998) Conditional on sT g ; the system has an approximate linear Gaussian state-space form. Hence, standard algorithm can be applied. A new draw for the complete history of H T g can be obtained recursively with the standard Gibbs sampler for state-space forms using the forward-backward recursion of Carter and Kohn (1994). Having generated H T g , the elements of the vector V T g can be generated from Normal inverseGamma distributions. A new sample of indicators sgjt is obtained conditional on Tg and H T g : Pr(sgjt = ij Step 5 Set g g g jt ; hjt ) / qj fN (egjt j2hgjt + mi 1:2704; ri2 ); i = 1; ::::; 7 1 = g and go to Step 1 and repeat until convergence is achieved 93 A.1.2 A Four Regime-Switching in DSGE Models A state-space representation of log-linearized DSGE model is given by: yt = D + Z t = T( ) t 1 (ME) t + R( ) t (TE) The volatility for each structural shock, j; follow Markov process: jt jst = jst "jt 2 f "jt jH ; N (0; 1) jL g While the majority of the existing Markov regime-switching DSGE models assumes regime-switching behavior to be synchronized across shocks, I create ‡exibility by allowing some shock processes to be independent of the rest. For instance, in RS(4)DSGE model, I separate monetary policy shock from the remaining shocks. Suppose the regime governing the volatility of the shock process, sit ; follows a two-state Markov chain: 2 6 Pi = 4 piHH 1 1 piLL where piHH = Pr[sit = Highjsit 1 piHH piLL 3 7 5 i 2 fM oney; Restg = High]: Let P M oney and P Rest denote two transition matrices respectively and assume that each regime switching system consists of two regimes. The number of all the possible regimes from two independent structures is four and the resulting transition matrix is P = PM 94 PR where denotes the Kronecker product and R if sM t = High; st = High st = HH R st = HL if sM t = High; st = Low R if sM t = Low; st = High st = LH R st = LL if sM t = Low; st = Low Inference on Volatility Regimes. The following procedure by Kim and Nelson (1999) provides the probability of st conditional on the information up to t and appropriate approximation of likelihood as a by-product Step 1 Run the Kalman …lter to get i;j Ptjt 1( V (at jst = j; st 1 i;j tjt 1 ( E(at jst = j; st 1 = i; It 1 ) and = i; It 1 )) Step 2 Update the probability of the current regime after observing data at time t: f (yt ; st ; st 1 jIt 1 ) f (yt jIt 1 ) f (yt jst ; st 1 ; It 1 ) Pr[st ; st 1 jIt 1 ] = XX f (yt jst ; st 1 ; It 1 ) Pr[st ; st 1 jIt 1 ] Pr[st ; st 1 jIt ] = s t st Pr[st jIt ] = X st 1 Pr[st ; st 1 jIt ] 1 95 Step 3 Using these probability terms, collapse 4 = 4 X p;jtjt = 4 X ;j tjt I sample sT Pr[sit = j; st 1 = ijIt ] 4 posteriors into 4 1: i;j tjt 1 i=1 Pr[st = jjIt ] Pr[sit = j; st 1 i;j +( = ijIt ] Ptjt j tjt j i;j tjt )( tjt i;j 0 tjt ) i=1 Pr[st = jjIt ] [s1 ; :::; sT ]0 by backward recursion, using Pr[st jIt ]: Gibbs-Sampler for Transition Probability. 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