Table Of ContentECONOMETRICS
economics
ECONOMETRICS
STATISTICAL FOUNDATIONS
AND APPLICATIONS
PHOEBUS J. DHRYMES
Professor of Economics
Columbia University
Springer-Verlag New York· Heidelberg· Berlin
1974
Library of Congress Cataloging in Publication Data
Dhrymes, Phoebus J 1932-
Econometrics: statistical foundations and applica
tions.
Corrected reprint of the 1970 ed. published by
Harper & Row, New York.
1. Econometrics. I. Title.
[HB139.D48 1974] 330'.01'8 74-10898
Second printing: July, 1974.
First published 1970, by Harper & Row, Publishers, Inc.
Design: Peter Klemke, Berlin.
All rights reserved.
No part of this book may be translated or reproduced in any form without
written permission from Springer-Verlag.
© 1970 by Phoebus J. Dhrymes and 1974 by Springer-Verlag New York Inc.
ISBN-13:978-0-387-90095-7 e-ISBN-13:978-1-4613-9383-2
001: 10.1007/978-1-4613-9383-2
PREFACE TO SECOND PRINTING
T
he main difference between this edition by Springer-Verlag and the
earlier one by Harper & Row lies in the elimination of the inordinately
high number of misprints found in the latter. A few minor errors of
exposition have also been eliminated. The material, however, is
essentially similar to that found in the earlier version.
I wish to take this opportunity to express my thanks to all those who
pointed out misprints to me and especially to H. Tsurumi, Warren
Dent and J. D. Khazzoom.
New York PHOEBUS J. DHRYMES
February, 1974
v
PREFACE TO FIRST PRINTING
T
his book was written, primarily, for the graduate student in econo
metrics. Its purpose is to provide a reasonably complete and rigorous
exposition of the techniques frequently employed in econometric
research, beyond what one is likely to encounter in an introductory
mathematical statistics course. It does not aim at teaching how one
can do successful original empirical research. Unfortunately, no one
has yet discovered how to communicate this skill impersonally.
Practicing econometricians may also find the integrated presentation
of simultaneous equations estimation theory and spectral analysis a
convenient reference.
I have tried, as far as possible, to begin the discussion of the various
topics from an elementary stage so that little prior knowledge of the
subject will be necessitated. It is assumed that the potential reader is
familiar with the elementary aspects of calculus and linear algebra.
Additional mathematical material is to be found in the Appendix.
Statistical competence, approximately at the level of a first-year
course in elementary mathematical statistics is also assumed on the
part of the reader.
The discussion, then,. develops certain elementary aspects of multi
variate analysis, the theory of estimation of simultaneous equations
systems, elementary aspects of spectral and cross-spectral analysis,
and shows how such techniques may be applied, by a number of
examples.
It is often said that econometrics deals with the quantification of
economic relationships, perhaps as postulated by an abstract model.
vii
As such, it is a blend of economics and statistics, both presupposing a sub
stantial degree of mathematical sophistication. Thus, to practice econometrics
compentently, one has to be well-versed in both economic and statistical theory.
Pursuant to this, I have attempted in all presentations to point out clearly the
assumptions underlying the discussion, their role in establishing the conclusions,
and hence the consequence of departures from such assumptions. Indeed, this
is a most crucial aspect of the student's training and one that is rather frequently
neglected. This is unfortunate since competence in econometrics entails, inter
alia, a very clear perception of the limitations of the conclusions one may obtain
from empirical analysis.
A number of specialized results from probability theory that are crucial for
establishing, rigorously, the properties of simultaneous equations estimators
have been collected in Chapter 3. This is included only as a convenient reference,
and its detailed study is not essential in understanding the remainder of the
book. It is sufficient that the reader be familiar with the salient results presented
in Chapter 3, but it is not essential that he master their proof in detail. I have
used various parts of the book, in the form of mimeographed notes, as the basis
of discussion for graduate courses in econometrics at Harvard University and,
more recently, at the University of Pennsylvania.
The material in Chapters I through 6 could easily constitute a one-semester
course, and the remainder may be used in the second semester. The instructor
who may not wish to delve into spectral analysis quite so extensively may include
alternative material, e.g., the theory of forecasting.
Generally, I felt that empirical work is easily accessible in journals and
similar publications, and for this reason, the number of empirical examples is
small. By now, the instructor has at his disposal a number of pUblications on
econometric models and books of readings in empirical econometric research,
from which he can easily draw in illustrating the possible application of various
techniques.
J have tried to write this book in a uniform style and notation and preserve
maximal continuity of presentation. For this reason explicit references to
individual contributions are minimized; on the other hand, the great cleavage
between the Dutch and Cowles Foundation notation is bridged so that one can
follow the discussion of 2SLS, 3SLS, and maximum likelihood estimation in a
unified notational framework. Of course, absence of references from the dis
cussions is not meant to ignore individual contributions, but only to insure the
continuity and unity of exposition that one commonly finds in scientific, mathe
matical, or statistical textbooks.
Original work relevant to the subject covered appears in the references at
the end of each chapter; in several instances a brief comment on the work is
inserted. This is only meant to give the reader an indication of the coverage and
does not pretend to be a review of the contents.
Finally, it is a pleasure for me to acknowledge my debt to a number of
viii PREFACE
individuals who have contributed directly or indirectly in making this book
what it is.
I wish to express my gratitude to H. Theil for first introducing me to the
rigorous study of econometrics, and to I. Olkin from whose lucid lectures I
first learned about multivariate analysis. T. Amemiya, L. R. Klein, J. Kmenta,
B. M. Mitchell, and A. Zellner read various parts of the manuscript and offered
useful suggestions. V. Pandit and A. Basu are chiefly responsible for compiling
the bibliography. Margot Keith and Alix Ryckoff have lightened my burden by
their expert typing.
PHOEBUS J. DHRYMES
January, 1970
PREFACE ix
CONTENTS
1. ELEMENTARY ASPECTS OF
MULTIVARIATE ANALYSIS 1
1.1 Preliminaries
1.2 Joint, Marginal, and Conditional Distributions 5
1.3 A Mathematical Digression 9
1.4 The Multivariate Normal Distribution 12
1.5 Correlation Coefficients and Related Topics 20
1.6 Estimators of the Mean Vector and Covariance Matrix and their
Distribution 25
1.7 Tests of Significance 34
2. APPLICATIONS OF
MULTIVARIATE ANALYSIS 42
2.1 Canonical Correlations and Canonical Variables 42
2.2 Principal Components 53
2.3 Discriminant Analysis 65
2.4 Factor Analysis 77
xi
3. PROBABILITY LIMITS, ASYMPTOTIC
DISTRIBUTIONS, AND PROPERTIES OF
MAXIMUM LIKELIHOOD ESTIMATORS 84
3.1 Introduction 84
3.2 Estimators and Probability Limits 84
3.3 Convergence to a Random Variable: Convergence in Distribution
and Convergence of Moments 90
3.4 Central Limit Theorems and Related Topics 100
3.5 Miscellaneous Useful Convergence Results 110
3.6 Properties of Maximum Likelihood (ML) Estimators 114
3.7 Estimation for Distribution Admitting of Sufficient Statistics 130
3.8 Minimum Variance Estimation and Sufficient Statistics 136
4. ESTIMATION OF
SIMULTANEOUS EQUATIONS SYSTEMS 145
4.1 Review of Classical Methods 145
4.2 Asymptotic Distribution of Aitken Estimators 161
4.3 Two-Stage Least Squares (2SLS) 167
4.4 2SLS as Aitken and as OLS Estimator 183
4.5 Asymptotic Properties of 2SLS Estimators 190
4.6 The General k-Class Estimator 200
4.7 Three-Stage Least Squares (3SLS) 209
5. APPLICATIONS OF CLASSICAL AND
SIMULTANEOUS EQUATIONS TECHNIQUES
AND RELATED PROBLEMS 222
5.1 Estimation of Production and Cost Functions and Specification
Error Analysis 222
5.2 An Example of Efficient Estimation of a Set of General Linear
(Regression) Models 234
5.3 An Example of 2SLS and 3SLS Estimation 236
5.4 Measures of Goodness of Fit in Multiple Equations Systems:
Coeficient of (Vector) Alienation and Correlation 240
5.5 Canonical Correlations and Goodness of Fit in Econometric
Systems 261
5.6 Applications of Principal Component Theory In Econometric
Systems 264
5.7 Alternative Asymptotic Tests of Significance for 2SLS Estimated
Parameters 272
xii CONTENTS
