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Contents Chapter 1: Preliminaries 1 1.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Concepts of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Almost sure (a.s.) convergence . . . . . . . . . . . . . . . . . . 3 1.2.2 Convergence in Probability . . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Convergence in Lq-norm. . . . . . . . . . . . . . . . . . . . . . . 6 1.2.4 Convergence in Distribution . . . . . . . . . . . . . . . . . . . . 7 1.3 Time Series Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Dependent and Identically Distributed Observations . . . . . 14 1.4.2 Dependent and Heterogeneously Distributed Observations. 15 1.4.3 Martingale Difference Process . . . . . . . . . . . . . . . . . . . 16 1.5 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.1 Dependent and Identically Distributed Observations . . . . . 17 1.5.2 Dependent Heterogeneously Distributed Observations . . . . 18 1.5.3 Martingale Difference Observations . . . . . . . . . . . . . . . . 18 1.6 Elements of Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . 19 Chapter 2: DSGE Models, Solutions and Approximations 27 2.1 Few useful models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1.1 A basic Real Business Cycle (RBC) Model . . . . . . . . . . . 28 2.1.2 Heterogeneous agent models . . . . . . . . . . . . . . . . . . . . 35 2.1.3 Monetary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Approximation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2.1 Quadratic approximations . . . . . . . . . . . . . . . . . . . . . . 45 2.2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2.3 Log linear Approximations . . . . . . . . . . . . . . . . . . . . . 51 2.2.4 Second order approximations . . . . . . . . . . . . . . . . . . . . . . 60 2.2.5 Parametrizing expectations . . . . . . . . . . . . . . . . . . . . . 62 2.2.6 A Comparison of methods . . . . . . . . . . . . . . . . . . . . . 65
Chapter 3: Extracting and Measuring Cyclical Information 67 3.1 Statistical Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1.1 Traditional methods . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1.2 Beveridge-Nelson (BN) decomposition . . . . . . . . . . . . . . 69 3.1.3 Unobservable Components (UC) decompositions . . . . . . . 72 3.1.4 Regime shifting decomposition . . . . . . . . . . . . . . . . . . . 75 3.2 Hybrid Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2.1 The Hodrick and Prescott (HP) Filter . . . . . . . . . . . . . . 79 3.2.2 Exponential smoothing (ES) filter . . . . . . . . . . . . . . . . . 86 3.2.3 Moving average (MA) filters . . . . . . . . . . . . . . . . . . . . 88 3.2.4 Band Pass (BP) filters . . . . . . . . . . . . . . . . . . . . . . . . 89 3.3 Economic Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3.1 Blanchard and Quah (BQ) Decomposition . . . . . . . . . . . 95 3.3.2 King, Plosser Stock and Watson (KPSW) Decomposition . 97 3.4 Time Aggregation and Cycles . . . . . . . . . . . . . . . . . . . . . . . 99 3.5 Collecting Cyclical Information . . . . . . . . . . . . . . . . . . . . . . 100 Chapter 4: VAR Models 105 4.1 The Wold theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.2 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2.1 Lag Length 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2.2 Lag Length 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2.3 Nonlinearities and nonnormalities . . . . . . . . . . . . . . . . . 116 4.2.4 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.2.5 Breaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.3 Moments and parameter estimation of a VAR(q) . . . . . . . . . . . 119 4.3.1 Companion form representation . . . . . . . . . . . . . . . . . . 119 4.3.2 Simultaneous equations format . . . . . . . . . . . . . . . . . . 121 4.4 Reporting VAR results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.4.1 Impulse responses . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.4.2 Variance decomposition . . . . . . . . . . . . . . . . . . . . . . . 124 4.4.3 Historical decomposition . . . . . . . . . . . . . . . . . . . . . . 125 4.4.4 Distribution of Impulse Responses . . . . . . . . . . . . . . . . 125 4.4.5 Generalized Impulse Responses . . . . . . . . . . . . . . . . . . 130 4.5 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.5.1 Stationary VARs . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.5.2 Nonstationary VARs . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.5.3 Alternative identification schemes . . . . . . . . . . . . . . . . . 139 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.7 Validating DSGE models with VARs . . . . . . . . . . . . . . . . . . . 151
Chapter 5: GMM and Simulation Estimators 157 5.1 Generalized Method of Moment and other standard estimators . . 158 5.2 IV estimation in a linear model . . . . . . . . . . . . . . . . . . . . . . 161 5.3 GMM Estimation: An overview . . . . . . . . . . . . . . . . . . . . . . 167 5.3.1 Asymptotics of GMM estimators . . . . . . . . . . . . . . . . . 168 5.3.2 Estimating the Covariance Matrix . . . . . . . . . . . . . . . . 170 5.3.3 Optimizing the Asymptotic covariance matrix . . . . . . . . . 174 5.3.4 Sequential GMM Estimation . . . . . . . . . . . . . . . . . . . . 175 5.3.5 Two-Step Estimators . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.3.6 Hypotheses Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.4 GMM estimation of DSGE models . . . . . . . . . . . . . . . . . . . . 181 5.4.1 Some Applied tips . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.5 Simulation Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.5.1 The General Problem . . . . . . . . . . . . . . . . . . . . . . . . 188 5.5.2 Simulated Method of Moments Estimator . . . . . . . . . . . 191 5.5.3 Simulated Quasi-Maximum Likelihood/ Indirect Inference . 192 5.5.4 Matching impulse responses . . . . . . . . . . . . . . . . . . . . . . . 196 Chapter 6: Likelihood methods 201 6.1 The Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.2 The Prediction error decomposition of likelihood . . . . . . . . . . . 209 6.2.1 Some Asymptotics of ML estimators . . . . . . . . . . . . . . . 213 6.3 Numerical tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.4 ML estimation of DSGE models . . . . . . . . . . . . . . . . . . . . . . 218 6.5 Two examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 6.5.1 Does monetary policy react to technolocy shocks? . . . . . . 227 6.5.2 Does fiscal policy help to stabilize the cycle? . . . . . . . . . . 233 Chapter 7: Calibration 235 7.1 A Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.2 The Uncontroversial parts . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.3 Choosing parameters and stochastic processes . . . . . . . . . . . . . 239 7.4 Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.4.1 Watson’s R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7.4.2 Measure of fit based on simulation variability . . . . . . . . . 253 7.4.3 Measures of fit based on sampling variability . . . . . . . . . 256 7.4.4 Measures of fit based on sampling and simulation variability 259 7.5 The sensitivity of the measurement . . . . . . . . . . . . . . . . . . . . 265 7.6 Savings, Investments and Tax cuts: an example . . . . . . . . . . . . 268
Chapter 8: Dynamic Macro Panels 273 8.1 From economic theory to dynamic panels . . . . . . . . . . . . . . . . 274 8.2 Panels with Homogeneous dynamics . . . . . . . . . . . . . . . . . . . 276 8.2.1 Pitfalls of standard methods . . . . . . . . . . . . . . . . . . . . 278 8.2.2 The Correct approach . . . . . . . . . . . . . . . . . . . . . . . . 280 8.2.3 Restricted models . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 8.2.4 Recovering the individual effect . . . . . . . . . . . . . . . . . . 285 8.2.5 Some practical issues . . . . . . . . . . . . . . . . . . . . . . . . . 286 8.3 Dynamic heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 8.3.1 Average time series estimator . . . . . . . . . . . . . . . . . . . 290 8.3.2 Pooled estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 8.3.3 Aggregate time series estimator . . . . . . . . . . . . . . . . . . 294 8.3.4 Average Cross sectional Estimator . . . . . . . . . . . . . . . . 295 8.3.5 Testing for dynamic heterogeneity . . . . . . . . . . . . . . . . 297 8.4 To Pool or not to Pool? . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 8.4.1 What goes wrong with two-step regressions? . . . . . . . . . . 302 8.5 Is Money superneutral? . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Chapter 9: Introduction to Bayesian Methods 309 9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 9.1.1 Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 9.1.2 Prior Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 9.2 Decision Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 9.3 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.3.1 Inference with Multiple Models . . . . . . . . . . . . . . . . . . 322 9.3.2 Normal Approximations . . . . . . . . . . . . . . . . . . . . . . . 323 9.3.3 Testing hypotheses/relative fit of different models . . . . . . 325 9.3.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 9.4 Hierarchical and Empirical Bayes models . . . . . . . . . . . . . . . . 328 9.4.1 Empirical Bayes methods . . . . . . . . . . . . . . . . . . . . . . 332 9.4.2 Meta analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 9.5 Posterior simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 9.5.1 Normal posterior analysis . . . . . . . . . . . . . . . . . . . . . . 336 9.5.2 Basic Posterior Simulators . . . . . . . . . . . . . . . . . . . . . 337 9.5.3 Markov Chain Monte Carlo Methods . . . . . . . . . . . . . . 340 9.6 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 9.7 Estimating Returns to scale: Spain (1979-1999) . . . . . . . . . . . . 352 Chapter 10: Bayesian VARs 355 10.1 The Likelihood function of an m variable VAR(q) . . . . . . . . . . 356 10.2 Priors for VARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 10.2.1 Least square under uncertain restrictions . . . . . . . . . . . . 358 10.2.2 The Minnesota prior . . . . . . . . . . . . . . . . . . . . . . . . . 359
10.2.3 Adding other prior restrictions . . . . . . . . . . . . . . . . . . 363 10.2.4 Some Applied tips . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 10.2.5 Priors derived from DSGE models . . . . . . . . . . . . . . . . 366 10.2.6 Probability distributions for forecasts: Fan Charts . . . . . . 370 10.3 Structural BVARs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 10.4 Time Varying Coefficients BVARs . . . . . . . . . . . . . . . . . . . . 379 10.4.1 Minnesota style prior . . . . . . . . . . . . . . . . . . . . . . . . . 380 10.4.2 Hierarchical prior . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 10.5 Panel VAR models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 10.5.1 Univariate dynamic panels . . . . . . . . . . . . . . . . . . . . . 385 10.5.2 Endogenous grouping . . . . . . . . . . . . . . . . . . . . . . . . . 388 10.5.3 Panel VARs with interdependencies . . . . . . . . . . . . . . . 392 10.5.4 Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 10.5.5 Impulse responses . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Chapter 11: Bayesian time series and DSGE models 399 11.1 Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 11.1.1 Arbitrage Pricing (APT) Models . . . . . . . . . . . . . . . . . 403 11.1.2 Conditional Capital Asset Pricing models (CAPM) . . . . . 406 11.2 Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . 408 11.3 Markov switching models . . . . . . . . . . . . . . . . . . . . . . . . . . 414 11.3.1 A more complicated structure . . . . . . . . . . . . . . . . . . . 415 11.3.2 A General Markov switching specification . . . . . . . . . . . 418 11.4 Bayesian DSGE Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 11.4.1 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 11.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 11.4.3 A few applied tips . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 11.4.4 Comparing the quality of models to the data . . . . . . . . . 433 11.4.5 DSGEs and VARs, once again . . . . . . . . . . . . . . . . . . . 438 11.4.6 Non linear specifications . . . . . . . . . . . . . . . . . . . . . . . 439 11.4.7 Which approach to use? . . . . . . . . . . . . . . . . . . . . . . . 440 Appendix 443

Preface There has been a tremendous improvement over the last twenty years in the mathematical, statistical, probabilistic and computational tools available to applied macroeconomists. This extended set of tools has changed the way researchers have approached the problem of testing models, validate theories or simply collect regularities from the data. The rational expectation and the calibration revolutions have also forced researchers to try to build a more solid bridge between theoretical and applied work, a bridge which was often missing in much of the applied exercises conducted in the 1970s and the 1980s. This books attempts to bring together dynamic general equilibrium theory, data analysis, advanced econometric and computational methods to provide a comprehensive set of techniques which can be used to address questions of interest to academics, business and central bank economists in the fields of macroeconomics, business cycle analysis, growth theory, monetary, financial, and international economics. The point of view taken is the one of an applied economist facing time series data (at times a panel of them, coming from different countries), who is interested in verifying the prediction of dynamic economic theories and in advising model builders and theorists on how to respecify existing constructions to obtain better match between the model and the data. The book illustrates a number of techniques which can be used to address the questions of interest, agnostically evaluates their usefulness in bringing out information relevant to the users, provides examples where the methods work (and where they don’t) and points out problems when approaches developed for microeconomic data are used in time series frameworks. Unavoidably, a modern treatment of such a complex topic requires a quantitative perspective, a solid dynamic theory background and the development of both empirical and numerical methods. A quantitative perspective is needed to give empirical content to theories; empirical methods must provide an effective link between economic theory and the data; numerical techniques help us to solve complicated dynamic stochastic general equilibrium (DSGE) models and to implement advanced econometric estimators, both in the classical and Bayesian tradition. In some cases empirical methods are intimately linked with the numerical procedure chosen to solve the model. In others, they are only constrained by the restrictions economic theory imposes on the data. Given this background, the structure of this book is quite different from the typical graduate textbook both in macroeconomics and in econometrics. Rather than listing a series of estimators and their properties for different data generating processes, this book starts from a class of DSGE models, finds an approximate (linear) representation for the decision rules and describes methods needed to estimate/choose their parameters, to examine their fit to the data and to conduct interesting policy exercises. The first three chapters of the book are introductory and review material extensively used in later chapters. In particular, chapter 1 presents basic time series and probability concepts, a list of useful law of large numbers and central limit theorems, which are employed in the discussions of chapters 4 to 8, and gives a brief overview of the basic elements of spectral analysis, heavily used in chapters 3, 5 and 7. Chapter 2 presents a number of macroeconomic models currently
used in the profession and discusses numerical methods needed to solve them. Most of the examples and exercises of this book are based on versions of these models. Chapter 3 discusses procedures used to obtain interesting information about secular and cyclical fluctuations in the data. In the remaining chapters we present various methodologies to confront models to the data and discuss how they can be used to address other interesting economic questions. Given our empirical perspective, formal results are often stated without proofs and emphasis is given to their use in particular macroeconomic applications. Chapter 4 describes minimalist vector autoregressive (VAR) approaches, where a limited amount of economic theory is used to structure the data. Chapter 5 presents limited information methodologies such as Generalized Methods of Moments (GMM), Simulated Method of Moments (SMM) and general simulation approaches. Chapter 6 examines full information Maximum Likelihood and in chapter 7 Calibration techniques are discussed. In chapter 8, we then branch into dynamic macro panel methods, which can be used to effectively study cross-country issues, and conclude the book with an extensive description of Bayesian methods and their use for VAR and panel VAR models, for advanced time series specifications, and for DSGE models (Chapters 9 to 11). The approach of this book differs, for example, from the one of Hamilton (1994) or Hayashi (2002), both of which are primarily directed to econometricians and are not directly concerned with the question of validating dynamic economic models. The emphasis also differs from more macroeconomic oriented books like Sargent and Liungqvist (2001) or computationally oriented books like Judd (1998) or Miranda and Fackler (2002) in that empirical methods play a larger role and the connection between theory, numerical and empirical tools is explicitely spelled out. The book is largely self-contained but presumes a basic knowledge of modern macroeconomic theory (say, one or two quarters of a Ph.D. course in macroeconomics), of standard econometrics (say, a quarter of a Ph. D. course in econometrics) and assumes that the reader has or will acquire in the process some programming skills (e.g., RATS, Matlab, Gauss). The book is thought for a year long sequence starting from second semester of a first year econometric/ applied macroeconomics course and continuing with the first semester of a second year macroeconometric course. Roughly, the first 5 chapters and the seventh could be thought in first part, chapter 6 and the last four in the second part. This is the setup I have used in teaching this material over a number years and it seems the natural division between what I consider basic and advanced material. Ph. D. students at Brown University, University of Rochester, UniversitatPompeu Fabra, Universita’ di Napoli, University of Porto, University of Southampton, London Business School, Bocconi University, Universita’ Milano-Bicocca; participants in various editions of the Barcelona Summer School in Macroeconomics (BSSM), of the European Economic Association (EEA) Summer school in Macroeconomics, Paris, of the Center for Financial Studies (CFS) Summer school in Macroeconomics, Eltville (Germany), of the ZEI Summer School in Bonn, of the course for Central Bankers in Genzersee (Switzerland); and attendants of various intense and short courses at the ECB, Bank of England, Bank of Italy,
Bank of Canada, Bank of Hungary, Riksbank, Bundesbank and European Business Cycle Network (EABCN) have passed through several versions of this book and played around with some of the codes which implement the procedures discussed in the book with some practical examples. Some suffered; some enthusiastically embraced the philosophy of this book; some were critical; some made useful comments and helped in debugging the codes, all of them were encouraging. To all goes my thanks. I have learned a lot through the process of writing this book and teaching its material, probably as much as students have learned from the lectures and practical sessions. Three people taught me to approach empirical problems in a sensible but rigorous way, combining economic theory with advanced statistical tools and numerical methods, and to be suspicious and critical of analyses which leave out one of the main ingredients of the cake. Christopher Sims and Tom Sargent where of crucial importance in making me understand that the grey area at the crossroad between theory and econometrics is a difficult but exciting place to be and their uncompromising intellectual curiosity, their stern intuition and their deep understanding of economic and policy issues has been an extraordinary lever behind this book. Adrian Pagan shaped my (somewhat cynical) view of what should and can be done with the data and the models. I always like to argue with him because his unconventional views helped to bring out often forgotten methodological and practical aspects. And on most issues of interest to applied macroeconomists he was more often right than wrong. This book would not have been possible without their fundamental inputs. As mentors, there was no one comparable to them. I also have an intellectual debit with Ed Prescott. It was his brusc refusal to follow the traditional econometric track that made me understand the need to create a different and more solid link between theory, econometric and statistical techniques and the data. Several of my collegues, in particular Albert Marcet and Morten Ravn, Jordi Gali, Lucrezia Reichlin, Harald Uhlig, Carlo Favero, Marco Maffezzoli and Luca sala contributed to form and develop some of the ideas presented in the book. A special thanks goes to Tom Doan, Marco del Negro, Chris Sims, Kirdan Lees and Adrian Pagan, who spotted mistakes and imprecisions in earlier versions of the manuscript.