Semiparametric Methods in Econometrics

210 Pages·1998·8.662 MB·English

by  Joel L. Horowitz (auth.)

Lecture Notes in Statistics 131
Edited by P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,
1. Olkin, N. Wermuth, S. Zeger
Springer Science+Business Media, LLC
Joel L. Horowitz
Semiparametric Methods in
Econometrics
Springer
Ioel L. Horowitz
Department of Economics
University ofIowa
Iowa City, IA 52242
Llbrary of Congress Cataloglng-ln-Publlcatlon Data
Harawltz. 0oel.
Semlparametric methods In econometrlcs I Joel L. Horowitz.
p. cm. -- (Lecture notes In statistlcs ; 131)
Includes blb11ograph1cal references and index.
ISBN 978-0-387-98477-3 ISBN 978-1-4612-0621-7 (eBook)
DOI 10.1007/978-1-4612-0621-7
1. Econometrlcs. 2. Estlmatlon theory. I. Tltle. II. Series,
Lecture notes In statlstlcs (Sprlnger-Verlag) ; v. 131.
HB139.H65 1998
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© 1998 Springer Science+Business Media New York
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ISBN 978-0-387-98477-3
ToN, S, andK
Preface
This book is based on a series of lectures on semi parametric estimation that I
gave at the Paris-Berlin Seminar in Garchy, France, in October 1996. The
topics that are covered in the book are the same as those of the lectures, but the
presentation in the book is much more detailed. Some of the material is new
and is presented here for the first time.
Several people deserve special thanks for helping to make this book
possible. I thank Wolfgang HardIe for arranging the invitation to the Paris
Berlin Seminar that led to the book. Wolfgang Hardie, Ruud Konning, George
Neumann, and Harry Paarsch read drafts of the manuscript. I am deeply
indebted to them for comments and suggestions that greatly improved the
presentation. I also thank John Kimmel for all his help and advice on the
details of production. Partial financial support was provided by the National
Science Foundation under grant SBR-9617925. Finally, I thank my wife,
Susan, for her support and patience during both the writing of the book and the
research that made it possible.
Contents
Preface vii
1. Introduction
2. Single-Index Models 5
2.1 Definition of a Single-Index Model 5
2.2 Why Single-Index Models Are Useful 6
2.3 Other Approaches to Dimension Reduction 10
2.4 Identification of Single-Index Models 14
2.5 Estimating G in a Single-Index Modei 20
2.6 Optimization Estimators of f3 23
2.7 Direct Semiparametric Estimators 35
2.8 Bandwidth Selection 49
2.9 An Empirical Example 52
3. Binary Response Models 55
3.1 Random-Coefficients Models 56
3.2 Identification 57
3.3 Estimation 65
3.4 Extensions of the Maximum Score and Smoothed
Maximum Score Estimators 84
3.5 An Empirical Example 100
4. Deconvolution Problems 103
4.1 A Model of Measurement Error 104
4.2 Models for Panel Data 113
4.3 Extensions 125
4.4 An Empirical Example 137
x Contents
5. Transformation Models 141
5.1 Estimation with Parametric T and Nonparametric F 142
5.2 Estimation with Nonparametric T and Parametric F 152
5.3 Estimation when Both T and Fare Nonparametric 168
5.4 Predicting Y Conditional onX 176
5.5 An Empirical Example 177
Appendix: Nonparametric Estimation 179
A.l Nonparametric Density Estimation 179
A.2 Nonparametric Mean Regression 187
References 191
Index 201
Chapter 1
Introduction
Many estimation problems in econometrics involve an unknown function as
well as an unknown finite-dimensional parameter. Models and estimation
problems that involve an unknown function and an unknown finite
dimensional parameter are called semiparametric.
There are many simple and familiar examples of semi parametric estimation
problems. One is estimating the vector of coefficients j3 in the linear model
Y=Xj3+U,
where Y is an observed dependent variable, X is an observed (row) vector of
explanatory variables, and U is an unobserved random variable whose mean
conditional on X is zero. If the distribution of U is known up to finitely many
parameters, then the method of maximum likelihood provides an
asymptotically efficient estimator of j3 and the parameters of the distribution of
U. Examples of finite-dimensional families of distributions are the normal, the
exponential, and the Poisson. Each of these distributions is completely
determined by the values of one or two constants (e.g., the mean and the
standard deviation in the case of the normal distribution). If the distribution of
U is not known up to finitely many parameters, the problem of estimating j3 is
semi parametric. The most familiar semiparametric estimator is ordinary least
squares, which is consistent under mild assumptions regardless of the
distribution of U. By contrast, a parametric estimator of j3 need not be
consistent. For example, the maximum likelihood estimator is inconsistent if X
is exponentially distributed but the analyst erroneously assumes it to be
lognormal.
The problem of estimating the coefficient vector in a linear model is so
simple and familiar that labeling it with a term as fancy as semiparametric may
seem excessive. A more difficult problem that has received much attention in
econometrics is semi parametric estimation of a binary-response model. Let Y
be a random variable whose only possible values are 0 and 1, and let X be a
vector of covariates of Y. Consider the problem of estimating the probability
that Y = 1 conditional on X. Suppose that the true conditional probability is
J. L. Horowitz, Semiparametric Methods in Econometrics
© Springer Science+Business Media New York 1998
2 Semiparametric Methods in Econometrics
P(Y = llX = x) = F(xP),
where F is a distribution function and P is a vector of constant parameters that
is conformable with X. If F is assumed to be known a priori, as in a binary
probit model, where F is the standard normal distribution function, the only
problem is to estimate p. This can be done by maximum likelihood. F is rarely
known in applications, however. If F is misspecified, then the maximum
likelihood estimators of p and P( Y = llX= x) are inconsistent except in special
cases, and inferences based on them can be highly misleading. In contrast to
estimation of a mean, where the simplest and most familiar estimator is
automatically semiparametric, the distribution function F has a non-trivial
influence on the most familiar estimator of a binary-response model.
Many other important estimation problems involve unknown functions in
non-trivial ways. Often, as is the case in the foregoing examples, the unknown
function is the distribution function of an unobserved random variable that
influences the relation between observed variables. As will be discussed later
in this book, however, the unknown function may also describe other features of
a model. The methods needed to estimate models that include both an
unknown function and an unknown finite-dimensional parameter
(semiparametric models) are different from those needed to estimate models
that contain one or more unknown functions but no finite-dimensional
parameters. Thus, it is important to distinguish between the two types of
models. Models or the latter type are called nonparametric. This book is
concerned with estimation of semi parametric models.
Semi parametric estimation problems have generated large literatures in
both econometrics and statistics. Most of this literature is highly technical.
Moreover, much of it is divorced from applications, so even technically
sophisticated readers can have difficulty judging whether a particular technique
is likely to be useful in applied research.
This book aims at mitigating these problems. I have tried to present the
main ideas underlying a variety of semiparametric methods in a way that will
be accessible to graduate students and applied researchers who are familiar
with econometric theory at the level found (for example) in the textbooks by
Amemiya (1985) and Davidson and MacKinnon (1993). To this end, I have
emphasized ideas rather than technical details and have provided as intuitive
an exposition as possible. I have given heuristic explanations of how important
results are proved, rather than formal proofs. Many results are stated without
any kind of proof, heuristic or otherwise. In all cases, however, I have given
references to sources that provide complete, formal proofs.
I have also tried to establish links to applications and to illustrate the ability
of semi parametric methods to provide insights about data that are not readily
available using more familiar parametric methods. To this end, each chapter
contains a real-data application as well as examples without data of applied
problems in which semi parametric methods can be useful.