Generalized Method of Moments Estimation by Laszlo Matyas


7559cc848009424.jpg Author Laszlo Matyas
Isbn 9780521660136
File size 7.22MB
Year 1999
Pages 332
Language English
File format PDF
Category mathematics


 

GENERALIZED METHOD OF MOMENTS ESTIMATION Editor: LASZLO MATYAS Budapest University of Economics CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521660136 © Cambridge University Press 1999 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Generalized methods of moments estimation / editor, Laszlo Matyas. p. cm. Includes bibliographical references and index. ISBN 0-521-66013-0 (hardbound) 1. Econometric models. 3. Estimation theory. HB141.G463 2. Moments method (Statistics) I. Matyas, Laszlo. 1998 330'.01'5195-dc21 ISBN 978-0-521-66013-6 hardback ISBN 978-0-521-66967-2 paperback Transferred to digital printing 2007 98-51529 CIP Contents Preface 1. Introduction to the Generalized Method of Moments Estimation David Harris and Laszlo Matyas 1.1. The Method of Moments 1.1.1. Moment Conditions 1.1.2. Method of Moments Estimation . . . 1.2. Generalized Method of Moments (GMM) Estimation 1.3. Asymptotic Properties of the GMM Estimator 1.3.1. Consistency 1.3.2. Asymptotic Normality 1.3.3. Asymptotic Efficiency 1.3.4. Illustrations 1.3.4.1. GMM With i.i.d. Observations . 1.3.4.2. Regression With Instrumental Variables 1.3.5. Choice of Moment Conditions . . . . 1.4. Conclusion References 2. G M M Estimation Techniques Masao Ogaki 2.1. GMM Estimation 2.1.1. GMM Estimator 2.1.2. Important Assumptions 2.2. Nonlinear Instrumental Variable Estimation 2.3. GMM Applications with Stationary Variables 2.3.1. Euler Equation Approach 2.3.2. Habit Formation 2.3.3. Linear Rational Expectations Models 1 3 4 4 7 9 11 12 17 21 22 23 24 25 29 29 31 31 31 34 35 36 36 37 39 iv Contents 2.4. GMM in the Presence of Nonstationary Variables 2.4.1. The Cointegration-Euler Equation Approach 2.4.2. Structural Error Correction Models . 2.4.3. An Exchange Rate Model with Sticky Prices 2.4.4. The Instrumental Variables Methods 2.5. Some Aspects of GMM Estimation ... 2.5.1. Numerical Optimization 2.5.2. The Choice of Instrumental Variables 2.5.3. Normalization References 3. Covariance Matrix Estimation Matthew J. Cushing and Mary G. McGarvey 3.1. Preliminary Results 3.2. The Estimand 3.3. Kernel Estimators of VT 3.3.1. Weighted Autocovariance Estimators 3.3.2. Weighted Periodogram Estimators . . 3.3.3. Computational Considerations . . . . 3.3.4. Spectral Interpretation 3.3.5. Asymptotic Properties of Scale Parameter Estimators 3.4. Optimal Choice of Covariance Estimator 3.4.1. Optimal Choice of Scale Parameter Window 3.4.2. Optimal Choice of Scale Parameter . 3.4.3. Other Estimation Strategies 3.5. Finite Sample Properties of HAC Estimators 3.5.1. Monte Carlo Evidence 3.5.2. Regression Model 3.5.3. Heteroskedastic but Serially Uncorrelated Errors 3.5.4. Autocorrelated but Homoskedastic Errors 41 42 44 45 51 55 55 56 57 59 63 63 64 67 68 69 71 72 73 77 77 78 80 82 82 84 86 88 Contents 3.5.5. Autocorrelated and Heteroskedastic Errors 3.5.6. Summary References 4. Hypothesis Testing in Models Estimated by GMM Alastair R. Hall 4.1. Identifying and Overidentifying Restrictions 4.2. Testing Hypotheses About E[f(xu60)] . . 4.3. Testing Hypotheses About Subsets of E[f(xt,60)] 4.4. Testing Hypotheses About the Parameter Vector 4.5. Testing Hypotheses About Structural Stability 4.6. Testing Non-nested Hypotheses 4.7. Conditional Moment and Hausman Tests 4.7.1. Conditional Moment Tests 4.7.2. Hausman Tests References 5. Finite Sample Properties of GMM estimators and Tests Jan M. Podivinsky 5.1. Related Theoretical Literature 5.2. Simulation Evidence 5.2.1. Asset Pricing Models 5.2.2. Business Cycle Models 5.2.3. Stochastic Volatility Models 5.2.4. Inventory Models 5.2.5. Models of Covariance Structures . . . 5.2.6. Other Applications of GMM 5.3. Extensions of Standard GMM 5.3.1. Estimation 5.3.2. Testing 5.4. Concluding Comments References 90 93 94 96 99 101 104 108 Ill 118 122 123 124 125 128 129 131 132 134 135 138 139 140 141 141 144 145 145 vi Contents 6. GMM Estimation of Time Series Models David Harris 6.1. Estimation of Moving Average Models . . 6.1.1. A Simple Estimator of an MA(1) Model 6.1.2. Estimation by Autoregressive Approximation 6.1.2.1. The Durbin Estimator 6.1.2.2. The Galbraith and Zinde-Walsh Estimator 6.1.3. A Finite Order Autoregressive Approximation 6.1.4. Higher Order MA Models 6.2. Estimation of ARMA Models 6.2.1. IV Estimation of AR Coefficients . . 6.2.2. The ARMA(1,1) Model 6.2.2.1. A Simple IV Estimator 6.2.2.2. Optimal IV Estimation 6.3. Applications to Unit Root Testing . . . . 6.3.1. Testing for an Autoregressive Unit Root 6.3.2. Testing for a Moving Average Unit Root Appendix: Proof of Theorem 6.1 References 149 7. Reduced Rank Regression Using GMM Frank Kleibergen 7.1. GMM-2SLS Estimators in Reduced Rank Models 7.1.1. Reduced Rank Regression Models . . 7.1.2. GMM-2SLS Estimators 7.1.3. Limiting Distributions for the GMM-2SLS Cointegration Estimators 7.2. Testing Cointegration Using GMM-2SLS Estimators 7.3. Cointegration in a Model with Heteroskedasticity 7.3.1. Generalized Least Squares Cointegration Estimators 171 150 150 153 154 155 155 156 157 157 158 159 160 162 162 165 167 169 173 173 174 178 184 187 187 Contents 7.4. Cointegration with Structural Breaks 7.5. Conclusion Appendix References vii . . 8. Estimation of Linear Panel Data Models Using G M M Seung C. Ahn and Peter Schmidt 8.1. Preliminaries 8.1.1. General Model 8.1.2. GMM and Instrumental Variables . . 8.1.2.1. Strictly Exogenous Instruments and Random effects 8.1.2.2. Strictly Exogenous Instruments and Fixed Effects 8.2. Models with Weakly Exogenous Instruments 8.2.1. The Forward Filter Estimator . . . . 8.2.2. Irrelevance of Forward Filtering . . . 8.2.3. Semiparametric Efficiency Bound . . 8.3. Models with Strictly Exogenous Regressors 8.3.1. The Hausman and Taylor, Amemiya and MaCurdy and Breusch, Mizon and Schmidt Estimators 8.3.2. Efficient GMM Estimation 8.3.3. GMM with Unrestricted £ 8.4. Simultaneous Equations 8.4.1. Estimation of a Single Equation . . . 8.4.2. System of Equations Estimation . . . 8.5. Dynamic Panel Data Models 8.5.1. Moment Conditions Under Standard Assumptions 8.5.2. Some Alternative Assumptions . . . . 8.5.3. Estimation 8.5.3.1. Notation and General Results . . 8.5.3.2. Linear Moment Conditions and Instrumental Variables 8.5.3.3. Linearized GMM 8.6. Conclusion 191 196 197 209 211 212 213 214 217 217 218 219 220 221 222 223 226 229 230 231 234 235 236 238 241 241 243 244 245 viii Contents References 9. Alternative GMM Methods for Nonlinear Panel Data Models Jorg Breitung and Michael Lechner 9.1. A Class of Nonlinear Panel Data Models 9.2. GMM Estimators for the Conditional Mean 9.3. Higher Order Moment Conditions . . . . 9.4. Selecting Moment Conditions: The Gallant-Tauchen Approach 9.5. A Minimum Distance Approach 9.6. Finite Sample Properties 9.6.1. Data Generating Process 9.6.2. Estimators 9.7. Results 9.8. An Application 9.9. Concluding Remarks References 10. Simulation Based Method of Moments R o m a n Liesenfeld and Jorg Breitung 10.1. General Setup and Applications 10.2. The Method of Simulated Moments (MSM) 10.3. Indirect Inference Estimator 10.4. The SNP Approach 10.5. Some Practical Issues 10.5.1. Drawing Random Numbers and Variance Reduction 10.5.2. The Selection of the Auxiliary Model 10.5.3. Small Sample Properties of the Indirect Inference 10.6. Conclusion References 11. Logically Inconsistent Limited Dependent Variables Models J. S. Butler and Gabriel Picone 11.1. Logical Inconsistency 246 248 250 253 255 256 258 259 259 260 263 265 267 273 275 277 281 286 290 292 292 294 295 296 297 301 302 Contents 11.2. Identification 11.3. Estimation 11.4. Conclusion and Extensions References Index ix 305 309 310 311 313 Contributors SEUNG C. AHN, Arizona State University JOERG BREITUNG, Humboldt University, Berlin J.S. BUTLER, Vanderbilt University MATTHEW J. CUSHING, University of Nebraska-Lincoln ALASTAIR HALL, North Carolina State University and University of Birmingham DAVID HARRIS, University of Melbourne FRANK KLEIBERGEN, Erasmus University MICHAEL LECHNER, Mannheim University ROMAN LIESENFELD, University of Tubingen LASZLO MATYAS, Budapest University of Economics and Erudite, Universite de Paris XII MARY McGARVEY, University of Nebraska-Lincoln MASAO OGAKI, Ohio State University GABRIEL PICONE, University of South Florida JAN M. PODIVINSKY, University of Southampton PETER SCHMIDT, Michigan State University PREFACE The standard econometric modelling practice for quite a long time was founded on strong assumptions concerning the underlying data generating process. Based on these assumptions, estimation and hypothesis testing techniques were derived with known desirable, and in many cases optimal, properties. Frequently, these assumptions were highly unrealistic and unlikely to be true. These shortcomings were attributed to the simplification involved in any modelling process and therefore inevitable and acceptable. The crisis of econometric modelling in the seventies led to many well known new, sometimes revolutionary, developments in the way econometrics was undertaken. Unrealistically strong assumptions were no longer acceptable. Techniques and procedures able to deal with data and models within a more realistic framework were badly required. Just at the right time, i.e., the early eighties when all this became obvious, Lars Peter Hansen's seminal paper on the asymtotic properties of the generalized method of moments (GMM) estimator was published in Econometrica. Although the basic idea of the GMM can be traced back to the work of Denis Sargan in the late fifties, Hansen's paper provided a ready to use, very flexible tool applicable to a large number of models, which relied on mild and plausible assumptions. The die was cast. Applications of the GMM approach have mushroomed since in the literature, which has been, as so many things, further boosted recently by the increased availability of computing power. Nowadays there are so many different theoretical and practical applications of the GMM principle that it is almost impossible to keep track of them. What started as a simple estimation method has grown to be a complete methodology for estimation and hypothesis testing. As most of the best known "traditional" estimation methods can be regarded as special cases of the GMM estimator, it can also serve as a nice unified framework for teaching estimation theory in econometrics. The main objective of this volume is to provide a complete and up-to-date presentation of the theory of GMM estimation, as well as insights into the use of these methods in empirical studies. 2 Preface The editor has tried to standardize the notation, language, exposition, and depth of the chapters in order to present a coherent book. We hope, however, that each chapter is able to stand on its own as a reference work. *** We would like to thank all those who helped with the creation of this volume: the contributors, who did not mind being harassed by the editor and produced several quality versions of their chapters ensuring the high standard of this volume; the Erudite, Universite de Paris XII in Prance and the Hungarian Research Fund (OTKA) in Hungary for providing financial support; and the editors' colleagues and students for their valuable comments and feedback. The camera-ready copy of this volume was prepared by the editor with the help of Erika Mihalik using T^X and the Initbook (Gabor Korosi and Laszlo Matyas) macro package. Laszlo Matyas Budapest, Melbourne and Paris, August 1998 Chapter 1 INTRODUCTION TO THE GENERALIZED METHOD OF MOMENTS ESTIMATION David Harris and Laszlo Matyas One of the most important tasks in econometrics and statistics is to find techniques enabling us to estimate, for a given data set, the unknown parameters of a specific model. Estimation procedures based on the minimization (or maximization) of some kind of criterion function (M-estimators) have successfully been used for many different types of models. The main difference between these estimators lies in what must be specified of the model. The most widely applied such estimation technique, the maximum likelihood, requires the complete specification of the model and its probability distribution. The Generalized Method of Moments (GMM)- does not require this sort of full knowledge. It only demands the specification of a set of moment conditions which the model should satisfy. In this chapter, first, we introduce the Method of Moments (MM) estimation, then generalize it and derive the GMM estimation procedure, and finally, analyze its properties. 4 Introduction to the Generalized Method of Moments Estimation 1.1 The Method of Moments The Method of Moments is an estimation technique which suggests that the unknown parameters should be estimated by matching population (or theoretical) moments (which are functions of the unknown parameters) with the appropriate sample moments. The first step is to define properly the moment conditions. 1.1.1 Moment Conditions DEFINITION 1.1 Moment Conditions Suppose that we have an observed sample {xt : t = 1,..., T} from which we want to estimate an unknown p x 1 parameter vector 9 with true value 90. Let f(xt,9) be a continuous g x l vector function of 9, and let E(f(xt,9)) exist and be finite for all t and 9. Then the moment conditions are that E(f(xt,90)) = 0. Before discussing the estimation of 90, we give some examples of moment conditions. EXAMPLE 1.1 Consider a sample {xt : t = 1,...,T} from a gamma 7(p*, q*) distribution with true values p* = p^ and g* = q^. The relationships between the first two moments of this distribution and its parameters are: In the notation of Definition 1.1, we have 9 = (p*, q*)' and a vector of q* = 2 functions: Then the moment conditions are E(f(xt,9o)) = 0. The Method of Moments EXAMPLE 1.2 Another simple example can be given for a sample {xt : t = 1,...,T} from a beta j3(p*, q*) distribution, again with true values p* — p^ and q* = q£. The relationships between the first two moments and the parameters of this distribution are: E{xt) = ^Szz ,2, E(x2t) = PoiPo + 1) (Po + Qo)(Po With e = (p*,q*)' we have f(x ff\-(x- P x2 - * ^ C ^ + l) V EXAMPLE 1.3 Consider the linear regression model yt = x't(3Q + ut, where xt is a p x 1 vector of stochastic regressors, /?0 is the true value of a p x 1 vector of unknown parameters /?, and i£t is an error term. In the presence of stochastic regressors, we often specify E{ut\xt) = 0, so that E(yt\xt) = x't(30. Using the Law of Iterated Expectations we find E(xtut) = = E(xtE(ut\xt))- =0 The equations E(xtut) = E(xt(yt - x%)) = 0, 6 Introduction to the Generalized Method of Moments Estimation are moment conditions for this model. That is, in terms of Definition 1.1, 9 = (3 and f({xt,yt),0) = xt(yt - x't(3). m Notice that in this example E(xtut) = 0 consists of p equations since xt is a p x 1 vector. Since (3 is a p x 1 parameter, these moment conditions exactly identify (3. If we had fewer than p moment conditions, then we could not identify /?, and if we had more than p moment conditions, then (3 would be over-identified. Estimation can proceed if the parameter vector is exactly or over-identified. Notice that, compared to the maximum likelihood approach (ML), we have specified relatively little information about ut. Using ML, we would be required to give the distribution of uu as well as parameterising any autocorrelation and heteroskedasticity, while this information is not required in formulating the moment conditions. However, some restrictions on such aspects of the model are still required for the derivation of the asymptotic properties of the GMM estimator. It is common to obtain moment conditions by requiring that the error term of the model has zero expectation conditional on certain observed variables. Alternatively, we can specify the moment conditions directly by requiring the error term to be uncorrelated with certain observed instrumental variables. We illustrate with another example. EXAMPLE 1.4 As in the previous example, we consider the linear regression model yt = x't(30 + ut, but we do not assume E(ut\xt) = 0. It follows that ut and xt may be correlated. Suppose we have a set of instruments in the g x 1 vector zt. These may be defined to be valid instruments if E(ztut) = 0. Thus the requirement that zt be a set of valid instruments immediately provides the appropriate moment conditions E(ztut) = E(zt(yt-x't0o))=O. That is, ,y u zt),0) = zt(yt - x't(3). The Method of Moments There are q equations here since there are q instruments, so we require q > p for identification. 1.1.2 Method of Moments Estimation We now consider how to estimate a parameter vector 9 using moment conditions as given in Definition 1.1. Consider first the case where q = p, that is, where 9 is exactly identified by the moment conditions. Then the moment conditions E(f(xu9)) = 0 represent a set of p equations for p unknowns. Solving these equations would give the value of 9 which satisfies the moment conditions, and this would be the true value 90. However, we cannot observe E(f(.,.)), only f(xt,9). The obvious way to proceed is to define the sample moments of f(xu 9) t=l which is the Method of Moments (MM) estimator of E(f(xu9)). If the sample moments provide good estimates of the population moments, then we might expect that the estimator 6T that solves the sample moment conditions fr(9) = 0 would provide a good estimate of the true value 90 that solves the population moment conditions E(f(xt,9)) = 0. In each of Examples 1.1-1.3, the parameters are exactly identified by the moment conditions. We consider each in turn. EXAMPLE 1.5 For the gamma distribution in Example 1.1, fr(9) = 0 implies and t=l

Author Laszlo Matyas Isbn 9780521660136 File size 7.22MB Year 1999 Pages 332 Language English File format PDF Category Mathematics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare The generalized method of moments (GMM) estimation has emerged over the past decade as providing a ready to use, flexible tool of application to a large number of econometric and economic models by relying on mild, plausible assumptions. The principal objective of this volume, the first devoted entirely to the GMM methodology, is to offer a complete and up to date presentation of the theory of GMM estimation as well as insights into the use of these methods in empirical studies. It is also designed to serve as a unified framework for teaching estimation theory in econometrics. Contributors to the volume include well-known authorities in the field based in North America, the UK/Europe, and Australia.     Download (7.22MB) Stochastic versus Deterministic Systems of Differential Equations Mathematical Modelling of Inelastic Deformation Theory Of Factorial Design: Single- And Multi-stratum Experiments Degradation Processes in Reliability Nonlinear Partial Differential Equations and Free Boundaries. Volume 1: Elliptic Equations Load more posts

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