Topics in Advanced Econometrics: Volume II Linear and Nonlinear Simultaneous Equations

410 Pages·1994·25.78 MB·English

by  Phoebus J. Dhrymes (auth.)

Topics in Advanced Econometrics
Volume II
Phoebus J. Dhrymes
Topics in Advanced
Econometrics
Volume II
Linear and Nonlinear
Simultaneous Equations
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Phoebus J. Dhrymes
Department of Economics
Columbia University
New York, NY 10027
USA
Library of Congress Cataloging-in-Publication Data
Dhryrnes, Phoebus J.
Topics in advanced econometrics.
(v. 2: Linear and nonlinear simultaneous equations)
Includes bibliographical references
and index.
Contents: [II Probability foundations-v. 2. Linear
and nonlinear simultaneous equations.
\. Econometrics. 2. Probabilities. I. Title.
HB139.D49 1989 330' .01 '5195 89-27330
ISBN-13: 978-1-4612-8731-5 e-ISBN-13: 978-1-4612-4302-1
001: 10.1007/978-1-4612-4302-1
Printed on acid-free paper.
© 1994 Springer-Verlag New York, Inc.
Softcover reprint of the hardcover 1s t edition 1994
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New
York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereaf
ter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even
if the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
by anyone.
Production managed by Natalie Johnson; manufacturing supervised by Genieve Shaw.
Photocomposed copy produced using the author's LaTeX files.
987654321
ISBN-13: 978-1-4612-8731-5
To IngraIn Olkin and Henri Theil, who stimulated
my early interest in econometrics
Preface
This book is intended for second year graduate students and professionals
who have an interest in linear and nonlinear simultaneous equations mod
els. It basically traces the evolution of econometrics beyond the general
linear model (GLM), beginning with the general linear structural econo
metric model (GLSEM) and ending with the generalized method of mo
ments (GMM). Thus, it covers the identification problem (Chapter 3),
maximum likelihood (ML) methods (Chapters 3 and 4), two and three
stage least squares (2SLS, 3SLS) (Chapters 1 and 2), the general nonlinear
model (GNLM) (Chapter 5), the general nonlinear simultaneous equations
model (GNLSEM), the special ca'3e of GNLSEM with additive errors, non
linear two and three stage least squares (NL2SLS, NL3SLS), the GMM
for GNLSEIVl, and finally ends with a brief overview of causality and re
lated issues, (Chapter 6). There is no discussion either of limited dependent
variables, or of unit root related topics.
It also contains a number of significant innovations. In a departure from
the custom of the literature, identification and consistency for nonlinear
models is handled through the Kullback information apparatus, as well
as the theory of minimum contrast (MC) estimators. In fact, nearly all
estimation problems handled in this volume can be approached through
the theory of MC estimators. The power of this approach is demonstrated
in Chapter 5, where the entire set of identification requirements for the
GLSEM, in an ML context, is obtained almost effortlessly, through the
apparatus of Kullback information.
The limiting distribution of dynamic GLSEM is handled through various
convergence theorems for dependent sequences and a martingale difference
Vlll Preface
central limit theorem on a step by step basis, so that the reader may ap
preciate the complexity of the problems and the manner in which such
problems are resolved.
A simplified (two step) FIML estimator is derived whose computational
complexity is quite analogous to that of 3SLS; this enables the reader to
sec precisely why the two estimators need not be numerically identical even
if ~)SLS is iterated.
The method of generalized momcnts (GMM) estimator is presented as a
variant of a 3SLS-like estimator in the context of the GLSEM with additive
errors.
Because notation has been a problem in this subject, 1 I have maintained
a consistent notation throughout the volume, so that one can read about
FI'\IL, LIML. 2SLS, 3SLS, and GMM in the same notation and mutatis
lmdand~8 with the same conventions and formulations. This facilitates the
teaching of the subject, and reduces the unproductive time devoted to
reconciliation of alternative notations and conventions.
The material in this volume can be llsed as the basis for a variety of
one semester or quarter courses, depending on the level of preparation of
the class. If students are conversant with a modicum of modern probability
theory, the rTl'atcrial may be covered for the most part in a semester course.
If not, one has the option of concentrating on Chapters 1, 3, and 4 and
those parts of Chapter 2 that do not delve too deeply into asymptotic the
ory. Alternatively, one might devote a number of lectures on the probability
background and let Topics in Advanced Econometrics: Probability Founda
tions (Volume 1) serve as a reference for various convergence and central
limit theorems needed in the development of asymptotic theory. Thus, a
sernester course may be based on Chapter 1, parts of Chapter 2, and parts
of Chapters 5 and 6. This basically leaves out the classic identification dis
cussion and .t\IL estimation, but covers nonlinear methods in the context
of the general linear model as well as the GNLSEM with additive errors.
In my own tcaching, I devote approximately two weeks to various conver
gence results from Topic8 'in Advanced Econometrics: Probability Founda
tions (Volume I) and, by and large, let this as well as my other book Afath
ematic8 fOT Econmnetrics serve as reference material. Normally, Chapter
6 is never reached, and is covered in the follow-up course on Time Series,
the dit;CllSsion of GMM serving as a natural interface between these two
strands of t.he literature.
I have devdoped the contents of this volume over several years. and
nearly every part has been utilized, at one time or another, as class notes
at Columbia University. I wish to record here my appreciation for the many
suggestions I have received from successive generations of students and
I It would not be an exaggeration to [-my that in reading the literature on thi~
subject, perhaps more than half the effort involved is devoted to deciphering the
particular notation and convention~ of the material being studied.
Preface ix
hope that their advice has made the presentation smoother and more easily
comprehensible.
Finally, the general tenor of the presentation, as well as the selection of
topics, invariably reflects in part the author's conceptual framework and
the role envisioned for the subject in scientific pursuits. It has always been
my view that good empirical econometrics has to be informed by economic
theory and, equally so, by econometric theory. This requires practitioners
to have a thorough grounding in the techniques employed for the purpose
of empirical inference. I deplore the employment of complex or opaque pro
cedures when this is clearly not required by the problem at hand. Equally
important, when writing on theoretical issues it is highly desirable to be
sufficiently well aware of first principles. This enables the investigator to
bring to bear the appropriate tools in the analysis of the issues under dis
cussion and reduces excessive reliance on broad and general theorems to
solve relatively straightforward problems, a feature not uncommon in the
literature of econometric theory. These concerns have led me, on one hand,
to give perhaps too extensive a discussion of thc underlying conceptual
framework, notational conventions, and the motivation and rationalization
of the assumptions made, and on the other, they have led me to pursue
most proofs a.', explicitly as I could manage.
I hope I have succeeded in setting forth the richness of the literature
on the subject as it was developed in the past fifty years or so, and that
this volume will be equally useful to the advanced student, as well as the
interested professional both in economics and in other disciplines as well.
Phoebus .J. Dhrymes
Bronxville. NY
.July 1993
Contents
Preface vii
Contents of Volume I xv
1 Extension of Classical Methods I 1
1.1 Introduction.........
1.2 A Brief Historical Review . . . . 4
1.3 The Nature of the GLSEM ... 6
1.4 The GLSEM: Assumptions and Notation. 10
1.4.1 Assumptions and Conventions. 12
1.4.2 Notation... . . . . . .. 1.5
1.5 Inconsistency of OLS Estimators 21
1.6 Two Stage Least Squares (2SLS) 27
1.6.1 The Original Derivation . 27
1.6.2 An Alternative Formulation 28
1. 7 Three Stage Least Squares (3SLS) 32
1.8 Restricted 2SLS and 3SLS Estimators 39
1.9 Tests of Prior Restrictions . . . . . . . 42
1.9.1 Generalities. . . . ..... . . 42
1.9.2 A Restricted Least Squares Interpretation of 2SLS
and 3SLS . . . . 43
Questions and Problems . . . . . . . . . . . . . . . . . . .. 50
Xl! Contents
Appendix to Chapter 1 53
Preliminaries to Hausman's Test 53
Examples , ........... . 57
2 Extension of Classical Methods II 61
2.1 Limiting Distributions . . . . . . . . . ... , .. 61
2.l.1 Preliminaries................ 61
2, l.2 Limiting Distributions for Static GLSEM 63
2.1.:3 Limiting Distributions for Dynamic GLSEM 70
2.2 Forecasting from the GLSEM .. , 83
2,2.1 Generalities ....... . 83
2.2.2 Forecasting from the URF . 84
2.2.:3 Forecasting from the RRF . 9:3
2.~3 The Vector Autoregressive :~'iIodel (VAR) . 102
2.4 Instrumental Variables (IV) .. , ... , , 104
2.4.1 2SLS and :3S1.S as IV Estimators. 105
2.1.2 2SLS and :3SLS &<; Optimal IV Estimators 109
IV and Insufficient Sample Size . , ... , , . 115
2.5.1 The Nature of the Problem .... , . 115
2.5.2 Iterated Instrumental Variables (IIV) 116
2.6 k-class and Double k-class Estimators .. 119
2.7 Distrihution of LM Derived Estimators. 120
2.8 Properties of Specification Tests 122
2,8,1 Single Equation 2SLS ... 122
2.8.2 Systemwide 2SLS and 3S1.S l:n
2.8.:3 Relation to Hausman's Test 1:36
Questions and Problems . . . . . . 139
Appendix to Chapter 2 141
Convergence of Second Moment Matrices 141
Convergence for Dependent Sequences . . 144
Preliminaries and Miscellaneous 144
Convergence of Second Moments of
Final Form Errors . . . . . . . . . . 1:17
3 Maximum Likelihood Methods I 153
:3.1 Introduction.,.,... .. 15:3
a.2 The Identification Problem 154
;3,2.1 Generalities ... 154
3.2.2 The Simple Supply-Demand Model 155
a.2.~~ Identification by Exclusion Restrictions 157
:3.2.4 Identification by Linear Restrictions , . 166
3.2.5 Identification and thc Reduced Form .. 171
3.2.6 C~ovariance and Cross Equation Restrictions. 177
3.2.7 A ).:Iore General Framework. , ....... . 182
Contents xiii
3.2.8 Parametric Nonlinearities and Identification. 194
3.3 ML Estimation of the RF . . . . . . . . 196
3.3.1 General Discussion and ILS . . . 196
3.3.2 Estimation of the Reduced Form 199
3.4 FIML Estimation . . . . . . . . . . 201
:3.5 Simplified FIML Estimators . . . . 203
3.6 Properties of Simplified Estimators 208
3.6.1 Consistency . . . .... 208
:3.7 Limiting Distribution of FIML 212
Questions and Problems . 220
4 LIML Estimation Methods 223
4.1 The "Concentrated" Likelihood Function 223
4.1.1 A Subset of m* Structural Equations 223
4.2 The Single Equation LIML Estimator 230
4.3 Consistency of the LIML Estimator. . 234
4.4 An Interesting Interpretation of LIML 238
4.5 Indirect Least Squares (ILS) . . . .. . 239
4.6 Relation of LIML to Other Estimators 246
4.7 Limiting Distribution of LIML Estimators 248
4.8 Classic Identifiability Tests 249
Questions and Problems 256
Appendix to Chapter 4 257
Limiting Distribution of (T~ - 1) 257
5 Nonlinear ML Methods 263
5.1 Motivation 263
5.2 A Mathematical Digression 264
5.3 Aspects of Likelihood Functions . 268
5.3.1 An Interesting Inequality 269
5.4 Fisher Information 269
5.4.1 Alternative Representation of the
Information Matrix . 273
5.5 The Cramer-Rao Bounds 274
5.6 Martingale Properties of Likelihood Functions . 277
5.7 Kullback Information . 279
5.8 Convergence a.c. of ML Estimators 281
Ei.8.1 Independent Observations 285
5.8.2 Generalizations 291
5.9 The General Nonlinear Model (GNLM) 299
5.9.1 Consistency 299
5.9.2 Identification 303
5.9.3 Asymptotic Normality 304
5.10 The GNLM with Restrictions 310