Statistical Inference in Random Coefficient Regression Models

218 Pages·1971·12.22 MB·English

by  P. A. V. B. Swamy (auth.)

Lectu re Notes in
Operations Research and
Mathematical Systems
Economics, Computer Science, Information and Control
Edited by M. Beckmann, Providence and H. P. Kunzi, Zurich
55
P. A. V. B. Swamy
Ohio State University, Department of Economics,
Columbus, Ohio/USA
Statistical Inference in Random
Coefficient Regression Models
Springer-Verlag
Berl in . Heidelberg . New York 1971
Advisory Board
H. Albach . A. V. Balakrishnan . F. Ferschl . R. E. Kalman . W. Krelle . N. Wirth
AMS Subject Classifications (1970): 62J05, 62MlO, 62P20, 90A20
ISBN -13:978-3-540-05603-4 e-ISBN -13:978-3-642-80653-7
001: 10.1007/978-3-642-80653-7
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
Under § 54 of the German Copyright Law whe.re copies are made for other than private use, a fee is payable to the publisher,
the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 75-173909.
To my parents and sister
PREFACE
This short monograph which presents a unified treatment of the theory of
estimating an economic relationship from a time series of cross-sections, is based
on my Ph.D. dissertation submitted to the University of Wisconsin, Madison. To the
material developed for that purpose, I have added the substance of two subsequent
papers: "Efficient methods of estimating a regression equation with equi-correlated
disturbances", and "The exact finite sample properties of estimators of coefficients
in error components regression models" (with Arora) which form the basis for
Chapters 11 and III respectively. One way of increasing the amount of statistical
information is to assemble the cross-sections of successive years. To analyze such
a body of data the traditional linear regression model is not appropriate and we
have to introduce some additional complications and assumptions due to the hetero
geneity of behavior among individuals. These complications have been discussed in
this monograph. Limitations of economic data, particularly their non-experimental
nature, do not permit us to know a priori the correct specification of a model. I
have considered several different sets of assumptionR about the stability of coeffi
cients and error variances across individuals and developed appropriate inference
procedures. I have considered only those sets of assumptions which lead to opera
tional procedures. Following the suggestions of Kuh, Klein and Zellner, I have
adopted the linear regression models with some or all of their coefficients varying
randomly across individuals. The random coefficient approach considered in this
study provides a simple and elegant procedure for handling inter-individual hetero
geneity in a cross-section. The traditional fixed coefficient models can be treated
as special cases of random coefficient models.
It is presumed that the reader has a reasonable familiarity with matrix
notation and operations, with concepts of quadratic forms and statistical estimation.
My thesis on random coefficient regression models grew from Zellner's article: "On
the aggregation problem: A new approach to a troublesome problem" and Rao's article:
"The theory of least squares when the parameters are stochastic and its application
to the analysis of growth curves".
VI
I owe a debt of gratitude to Professor A. S. Go1dberger, my thesis advisor,
for his criticism and helpful suggestions at various stages of my work. His
searching criticism has been of great value in sh~ping my thoughts on this problem.
Despite his busy work he carefully went through my thesis and improved my presen-
tation. He also helped me in getting free computer time. I am indebted to
Professor A. Ze11ner who offered me Research Assistantship for two years from 1964
to 1966. During this period, though I did not do any research on my own, my associ-
ation with him gave me deeper understanding of economic behavior and the problems
of incorporating them in model construction and econometric analysis. He was kind
enough to comment on various parts of my thesis. In Summer 1968 he offered me
Research Associateship and extended an excellent opportunity to prepare my thesis
material for publication. It was entirely due to his efforts Springer Ver1ag,
New York; agreed to publish my thesis. My gratitude goes to him. Responsibility
of any possible remaining errors is mine.
My thanks also go to Mr. Richard D. Porter for programming assistance.
For expert typing of difficult material I am grateful indeed to Mrs. Shir1ey Black
of the Graduate School of Business, University of Chicago and to Mrs. Lily Shu1tz.
My work in this thesis is financed by National Science Foundation under
Grants GS-1S1 and GS-13S0.
Finally, I express my sincere thanks to Professor J.N.K. Rao and
Dr. T. R. Rao who were instrumental in bringing me to the United States -- a
venture whose main product is this thesis.
Co1umbus, Ohio P.A.V.B. Swamy
June 1971.
TABLE OF CONTENTS
Chapter I -- Introduction 1
1.1 Purpose and Outline of the Study 1
1.2 Review of the Literature on Regression Models with Random
and Fixed Coefficients 2
1.3 Conclusions 23
Chapter 11 -- Efficient Methods of Estimating a Regression Equation
with Equicorre1ated Disturbances 24
2.1 Introduction 24
2.2 Some Useful Lemmas 25
2.3 A Regression Model with Equicorre1ated Disturbances 26
2.4 Analysis of Time Series of Cross-Sections 32
2.5 Estimation When the Variance-Covariance Matrix of
Disturbances is Singular 54
2.6 Estimation When the Remaining Effects are Heteroskedastic 54
2.7 Conclusions 59
Appendix 60
Chapter III -- Efficient Methods of Estimating the Error Components
Regression Models 63
3.1 Introduction 63
3.2 Some Matrix Results 63
3.3 Covariance Estimators 64
3;4 Estimation of Error Components Models 66
3.5 A Class of Asymptotically Efficient Estimators 72
3.6 Small Sample Properties of the Pooled Estimator 75
3.7 A Comparison of the Efficiencies of Pooled and OLS Estimators 77
3.8 A Comparison of the Efficiency of Pooled Estimator with Those
of its Components 80
3.9 Alternative Estimators of Slope Coefficients and the Regression
on Lagged Values of the Dependent Variables 82
3.10 Analysis of an Error Components Model Under Alternative
Assumptions 84
3.11 Maximum Likelihood Method of Estimating Error Components Model 85
3.12 Departures from the Basic Assumptions Underlying the Error
Components Model 88
3.13 Conclusions 93
Appendix 95
nn
Page
Chapter IV -- Statistical Inference in Random Coefficient Regression
Models Using Panel Data 97
4.1 Introduction 97
4.2 Setting the Problem 97
4.3 Efficient Methods of Estimating the Parameters of RCR Models 99
4.4 Estimation of Parameters in RCR Models when Disturbances are
Serially Correlated 127
4.5 Problems Associated with the Estimation of RCR Models Using
Aggregate Data 131
4.6 Forecasting with RCR Models 133
4.7 Relaxation of Assumptions Underlying RCR Models 138
4.8 Similarities Between RCR and Bayesian Assumpti9ns 140
4.9 Empirical CES Production Function Free of Management Bias 142
4.10 Analysis of Mixed Models 143
4.11 Conclusions 155
Chapter V -- A Random Coefficient Investment Model 156
5.1 Introduction 156
5.2 Grunfe1d's Hypothesis of Micro Investment Behavior 156
5.3 Estimation and Testing of Random Coefficient Investment Model 159
5.4 Aggregate Investment Function 170
5.5 Comparison of Random Coefficient Model with Fixed Coefficient
Macro Model 171
5.6 Comparison of Random Coefficient Model with Fixed Coefficient
Micro Model 172
5.7 Conclusions 175
Chapter VI -- Aggregate Consumption Function with Coefficients Random
Across Countries 176
6.1 Introduction 176
6.2 Aggregate Consumption Model 177
6.3 Source and Nature of Data 178
6.4 Fixed Coefficient Approach 178
6.5 Random Coefficient Approach 182
6.6 Conclusions 190
Chapter VII -- Miscellaneous Topics 191
7.1 Introduction 191
7.2 Identification 191
7.3 Incorporation of Prior Information in the Estimation of
RCR Models 194
7.4 Conclusions 200
Appendix 201
Bibliography
CHAPTER I
INTRODUCTION
1.1 Purpose and Outline of the Study
The purpose of this study is to examine various approaches which are
useful in estimating a relation from the combined time series and cross-section
data. The study in this thesis can be divided into two parts. In the first part
fc>'1lowing Kuh (1959, 1963) we consider a regression equation with additional
effects associated with both the time direction and cross-sectional units and
develop efficient methods of estimating the parameters of that model under
different sets of assumptions. In the second part we examine critically the
specifying assumptions traditionally entertained while analyzing cross-section
behavior. In doing so, we will consider Klein's (1953, pp. 216-218) suggestion to
treat the coefficients of a regression model as random in cross-section analyses to
account for spacial or interindividual heterogeneity. Under alternative sets of
assumptions, we will study random coefficient regression (RCR) models which treat
both intercept and slope coefficients as random variables and will develop appro
priate statistical inference procedures. Following Zellner (1966), we discuss the
implications of RCR models for the aggregation problem. Further, we discuss the
problems of identification, forecasting, and of using stochastic prior information
within the framework of RCR models.
For empirical implementation of RCR models considered in this thesis,
panel data provide the base. By panel data we mean temporal cross-section data
obtained by assembling cross-sections of several years, with the same cross-section
units appearing in all years. The cross-section units could be households, firms
or any micro economic units. Data on a number of units for a single year with one
observation per unit constitute a single cross-section. We apply our statistical
procedures in estimating the parameters of a random coefficient investment model
using panel data on certain corporations in the U.S.A, We also use aggregate time
series data on each of twenty-four countrie_s to estimate the aggregate consumption
function with coefficients varying randomly across countries.
2
The plan of the thesis is as follows: We devote the remaining sections
of this chapter to a review of the literature, to an explicit statement of the
assumptions which are appropriate for a cross-section or a temporal cross-section
analysis, and to an indication of the statistical problems connected with such
analyses. In Chapter 11 we discuss efficient methods of estimating a regression
equation with equicorrelated disturbances. In Chapter 111 we develop efficient
methods of estimating the parameters of an equation with additional random-effects
associated with both the time direction and cross-sectional units. In Chapter IV
we specify a linear regression model with random coeffiqients and develop the
relevant statistical inference and forecasting procedures. We apply these proce-
dures to analyze certain investment and consumption functions in Chapters V and VI,
respectively. We consider the problems of identification and incorporating prior
information in Chapter VII.
1.2 Review of the Literature on Regression Models
with Random and Fixed Coefficients
(a) Pooling Cross-Section and Time Series Data
in the Estimation of a Regression Model
The discussion in this section is based on Kuh (1959), Mundlak (1961),
Nerlove (1965, pp. 157-187), and Balestra and Nerlove (1966). Nerlove considered
the following dynamic model to analyze a time series of cross-sections.
K-l G
Yit 130 + k~l 6kzkit + g~l 'YgYit_g + uit
(i 1,2, ••• ,n t 1,2, ... , T) (1.2.1)
We have observations on y's, and z's for n individual units. The disturbance term
uit is decomposed into two statistically independent parts: an individual effect,
and a remainder. Thus,
(1.2.2)
Nerlove assumes that the random variables ~. and v. are independent such that
~ :t-t-
3
o
for all i and t (1.2.3a)
E\-LiU it = o for all i and t (1.2.3b)
if i=i' and t=t'
(1.2.3c)
otherwise;
,i
if i=i'
= { \-L (1.2.3d)
o otherwise •
He further assumes that the U it and the \.Lit are independent of themselves for
different individuals.
In his important contribution on empirical production function free of
management bias, Mund1ak (1961) specifies a log linear production function with
nonstochastic regressors and includes a time invariant individual firm effect \-Li'
n
He treats the \-L.'s as fixed parameters obeying the restriction ~ \.L. = O. In this
L i=l L
case the coefficients can be estimated by applying ordinary least squares (OLS) to
observations expressed as deviations from their respective individual firm means.
Kuh (1959) also specifies a linear regression model with nonstochastic regressors
and includes an individual time invariant random effect \-Li' Unlike Ner1ove, Kuh
assumes that I-Li is correlated with U it'
If we combine (1.2.1) and (1.2.2) and write
K-1 G
«(30 + I-Li) + k~l °kzkit + g~l Ygy it-g + Ui t (1.2.4)
it is clear that the dynamic model as described in (1.2.1)-(1.2.3d) has an inter-
cept varying randomly across units and fixed slopes. Under the assumptions
(1.2.3a)-(1.2.3d) the variance-covariance matrix of ·u. 's is a block diagonal
Lt
matrix and Aitken's (1934-35) generalized least squares estimator of coefficients
in (1.2.1) is a best linear unbiased estimator (BLUE). The Aitken estimator is not
operational if the variances of error terms are unknown.
Ner10ve presented an interesting discussion on the application of the
maximum likelihood (ML) method to estimate the parameters in the model (1.2.1).
He concluded that the ML method was inapplicable to the above situation if it