Über die Autorin bzw. den Autor

Fabio Canova

Von der hinteren Coverseite

"The last twenty years have witnessed a revolution in macroeconomic modeling. Yet an integrated and accessible treatment of the new methods has been notably lacking. Fabio Canova's book fills that gap magnificently. It is surely destined to be an indispensable reference for both students and researchers for years to come."--Charles Bean, Bank of England

"This book will become an invaluable reference for applied macroeconomists as well as a much-needed teaching tool for graduate macroeconomic courses. Anybody who has an interest in quantitative macroeconomics, either as an academic or as a practitioner, should buy it."--Lucrezia Reichlin, European Central Bank

"Dynamic general equilibrium models have become regular tools for policy analysis in central banks and other policy institutions. This book is a wonderful source for those who want to bring those models to the data. It is thorough and comprehensive, it has a great set of examples and exercises, and, above all, it provides many practical tips. A must-read for any applied macroeconomist."--Frank Smets, European Central Bank

"To be able to describe and interpret business-cycle fluctuations using modern methods developed by researchers is crucial to economists who want to make and evaluate forecasts and policy advice. Fabio Canova has a long experience from research at the frontier, but also from teaching and from applied work at policy institutions such as central banks. His book provides an indispensable toolbox for any researcher that wants to have an influence on practical policy work."--Anders Vredin, Sveriges Riksbank

"The material covered in this book is extensive, and the author always strives to provide an in-depth analysis and discussion for every topic, complete with the most up-to-date developments in the literature. The combination of DSGE macroeconomics and econometrics makes this book a unique product, likely to become an essential reference for empirical macroeconomists and policymakers."--Marco Del Negro, FRB Atlanta

"This book is unprecedented among econometrics books for the way it incorporates careful and sophisticated macroeconomic theory. It is unprecedented among books on dynamic macroeconomics for its level of practical statistical advice and econometric sophistication. There is simply nothing close to this book available. Many of the best young researchers will want to study and teach from it."--Thomas J. Sargent, New York University

"This book treats econometric, computational, and macroeconomic substantive issues jointly. Nearly all existing books in this area are either strictly econometric, strictly computational, or focus on substance without taking up econometric and computational issues. The need for a treatment like this on the part of applied researchers means there will be wide interest in it."--Christopher Sims, Princeton University

Aus dem Klappentext

"The last twenty years have witnessed a revolution in macroeconomic modeling. Yet an integrated and accessible treatment of the new methods has been notably lacking. Fabio Canova's book fills that gap magnificently. It is surely destined to be an indispensable reference for both students and researchers for years to come."--Charles Bean, Bank of England

"This book will become an invaluable reference for applied macroeconomists as well as a much-needed teaching tool for graduate macroeconomic courses. Anybody who has an interest in quantitative macroeconomics, either as an academic or as a practitioner, should buy it."--Lucrezia Reichlin, European Central Bank

"Dynamic general equilibrium models have become regular tools for policy analysis in central banks and other policy institutions. This book is a wonderful source for those who want to bring those models to the data. It is thorough and comprehensive, it has a great set of examples and exercises, and, above all, it provides many practical tips. A must-read for any applied macroeconomist."--Frank Smets, European Central Bank

"To be able to describe and interpret business-cycle fluctuations using modern methods developed by researchers is crucial to economists who want to make and evaluate forecasts and policy advice. Fabio Canova has a long experience from research at the frontier, but also from teaching and from applied work at policy institutions such as central banks. His book provides an indispensable toolbox for any researcher that wants to have an influence on practical policy work."--Anders Vredin, Sveriges Riksbank

"The material covered in this book is extensive, and the author always strives to provide an in-depth analysis and discussion for every topic, complete with the most up-to-date developments in the literature. The combination of DSGE macroeconomics and econometrics makes this book a unique product, likely to become an essential reference for empirical macroeconomists and policymakers."--Marco Del Negro, FRB Atlanta

"This book is unprecedented among econometrics books for the way it incorporates careful and sophisticated macroeconomic theory. It is unprecedented among books on dynamic macroeconomics for its level of practical statistical advice and econometric sophistication. There is simply nothing close to this book available. Many of the best young researchers will want to study and teach from it."--Thomas J. Sargent, New York University

"This book treats econometric, computational, and macroeconomic substantive issues jointly. Nearly all existing books in this area are either strictly econometric, strictly computational, or focus on substance without taking up econometric and computational issues. The need for a treatment like this on the part of applied researchers means there will be wide interest in it."--Christopher Sims, Princeton University

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Methods for Applied Macroeconomic Research

Princeton University Press

Chapter One

This chapter is introductory and intended for readers who are unfamiliar with time series concepts, with the properties of stochastic processes, with basic asymptotic theory results, and with the principles of spectral analysis. Those who feel comfortable with these topics can skip directly to chapter 2.

Since the material is vast and complex, an effort is made to present it at the simplest possible level, emphasizing a selected number of topics and only those aspects which are useful for the central topic of this book: comparing the properties of dynamic stochastic general equilibrium (DSGE) models to the data. This means that intuition rather than mathematical rigor is stressed. More specialized books, such as those by Brockwell and Davis (1991), Davidson (1994), Priestley (1981), or White (1984), provide a comprehensive and in-depth treatment of these topics.

When trying to provide background material, there is always the risk of going too far back to the basics, of trying to reinvent the wheel. To avoid this, we assume that the reader is familiar with simple concepts of calculus such as limits, continuity, and uniform continuity of functions of real numbers, and that she is familiar with distributions functions, measures, and probability spaces.

The chapter is divided into six sections. The first defines what a stochastic process is. The second examines the limiting behavior of stochastic processes introducing four concepts of convergence and characterizing their relationships. Section 1.3 deals with time series concepts. Section 1.4 deals with laws of large numbers. These laws are useful to ensure that functions of stochastic processes converge to appropriate limits. We examine three situations: a case where the elements of a stochastic process are dependent and identically distributed; one where they are dependent and heterogeneously distributed; and one where they are martingale differences. Section 1.5 describes three central limit theorems corresponding to the three situations analyzed in section 1.4. Central limit theorems are useful for deriving the limiting distribution of functions of stochastic processes and are the basis for (classical) tests of hypotheses and for some model evaluation criteria.

Section 1.6 presents elements of spectral analysis. Spectral analysis is useful for breaking down economic time series into components (trends, cycles, etc.), for building measures of persistence in response to shocks, for computing the asymptotic covariance matrix of certain estimators, and for defining measures of distance between a model and the data. It may be challenging at first. However, once it is realized that most of the functions typically performed in everyday life employ spectral methods (frequency modulation in a stereo, frequency band reception in a cellular phone, etc.), the reader should feel more comfortable with it. Spectral analysis offers an alternative way to look at time series, translating serially dependent time observations into contemporaneously independent frequency observations. This change of coordinates allows us to analyze the primitive cycles which compose time series and to discuss their length, amplitude, and persistence.

Whenever not explicitly stated, the machinery presented in this chapter applies to both scalar and vector stochastic processes. The objects of interest in this book are defined on a probability space ([??], F, P), where [??] is the space of possible state of nature x, F is the collection of Borel sets of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [[??].sup.m]psi]] x [[??].sup.m]psi]] x ..., and P[??]s a probability function for x that determines the joint distribution of the vector of stochastic processes of interest. The notation [{[y.sub.t]](x)}.sup.[infinity].sub.t] = -[infinity] indicates the sequence {..., [y.sub.0](x), [y.sub.t] (x), ..., [y.sub.t] (x), ...}, where, for each t, the random variable [y.sub.t] (x)[psi] is a measurable function of the state of nature x, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [??] is the real line. We assume that at each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] belongs to [F.sub.t], so that any function h([y.sub.[tau]]) will be "adapted" to [F.sub.t]. To simplify the notation, at times we write {[y.sub.t] (x)} or [y.sub.t]. A normal random variable with zero mean and variance [[summation].sub.y]psi]] is denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and a random variable uniformly distributed over the interval [a.sub.1], [a.sub.2]] is denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII][??][a.sub.1], [a.sub.1]. Finally, "i.i.d." indicates identically and independently distributed random variables and a white noise is an i.i.d. process with zero mean and constant variance.

1.1 Stochastic Processes

Definition 1.1 (stochastic process). A stochastic process [{[y.sub.t](x)}.sup.[infinity].sub.t=1[psi]] is a probability measure defined on sets of sequences of real vectors (the "paths" of the process).

The definition implies, among other things, that the set [??] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for arbitrary [??] [member of] [??] and t]psi] fixed, has well-defined probabilities. In other words, choosing different [??] [member of] [??] for a given t, and performing countable unions, finite intersections, and complementing the above set of paths, we generate a set of events with proper probabilities. Note that [y.sub.t] [psi]] is unrestricted for all [tau][psi][less than or equal to] t: the realization need not exceed [??] only at t. Observable time series are realizations of a stochastic process {[y.sub.t](x)}, given [x.sup.2]. Two simple stochastic processes are the following.

Example 1.1. (i) {[y.sub.t](x)} = [e.sub.1] cos (t x[e.sub.2]), where [e.sub.1[psi]] and [e.sub.2[psi]] are random variables, [e.sub.1[psi]] > 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], t > 0. Here [y.sub.t]]psi]] is periodic: [e.sub.1[psi]] controls the amplitude and [e.sub.2[psi]] the periodicity of [y.sub.t].

(ii) {[y.sub.t](x)}is such that P][y.sub.t]]psi]] = [+ or -] 1][psi]=0.5 [psi]for all t. Such a process has no memory and flips between -1 and 1 as t]psi] changes.

Example 1.2. It is easy to generate complex stochastic processes from primitive ones. For example, if, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [??](0, 1), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [??](0, 1), and [e.sub.1t]psi]] and [e.sub.2t]psi]] are independent of each other, [y.sub.t] [psi]] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a stochastic process. Similarly [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] i.i.d. (0, 1) is a stochastic process.

1.2 Convergence Concepts

In a classical framework the properties of estimators are obtained by using sequences of estimators indexed by the sample size, and by...