The Population-Sample Decomposition Method: A Distribution-Free Estimation Technique for Minimum Distance Parameters

249 Pages·1987·8.697 MB·English

by  A. M. Wesselman (auth.)

THE POPULATION-SAMPLE DECOMPOSITION METHOD
INTERNATIONAL STUDIES IN ECONOMICS AND
ECONOMETRICS
Volume 19
1. Harder T: Introduction to Mathematical Models in Market and Opinion Research With
Practical Applications, Computing Procedures, and Estimates of Computing Require
ments. Translated from the German by P.H. Friedlander and E.H. Friedlander. 1969.
2. Heesterman ARG: Forecasting Models for National Economic Planning. 1972.
3. Heesterman ARG: Allocation Models and their Use in Economic Planning. 1971.
4. Durda~ M: Some Problems of Development Financing. A case Study of the Turkish
First Five-Year Plan, 1963-1967. 1973.
5. Blin JM: Patterns and Configurations in Economic Science. A Study of Social Deci
sion Processes. 1973.
6. Merkies AHQM: Selection of Models by Forecasting Intervals. Translated from the
Dutch by M. van Holten-De Wolff. 1973.
7. Bos HC, Sanders M and Secchi C: Private Foreign Investment in Developing Coun
tries. A Quantitative Study on the Evaluation of its Macro-Economic Impact. 1974.
8. Frisch R: Economic Planning Studies Selected and Introduced by Frank Long. Pref
ace by Jan Tinbergen. 1976.
9. Gupta KL: Foreign Capital, Savings and Growth. An International Crosssection Study.
1983.
10 Bochove CA van: Imports and Economic Growth. 1982.
11. Bjerkholt 0, Offerdal E (eds.): Macroeconomic Prospects for a Small Oil Exporting
Country. 1985.
12. Weiserbs D (ed.): Industrial Investment in Europe: Economic Theory and Measure
ment. 1985.
13. Graf von der Schulenburg J-M, Skogh G (eds.): Law and Economics & The Econom
ics of Legal Regulation. 1986.
14. Svetozar Pejovich (ed.): Socialism: Institutional, Philosophical and Economic Issues.
1987.
15. Heijmans RDH, Neudecker H (eds.): The Practice of Econometrics. 1987.
16. Steinherr A, Weiserbs D (eds.): Employment and Growth: Issues for the 1980s. 1987.
17. Holler MJ (ed.): The Logic of Multiparty Systems. 1987.
18. Brabant JM van: Regional Price Formation in Eastern Europe. Theory and Practice
of Trade Pricing. 1987.
19. Wesselman AM: The Population-Sample Decomposition Method. 1987.
The Population-Sample
Decomposition Method
A Distribution-Free Estimation Technique for
Minimum Distance Parameters
by
A.M. Wesselman
4-
1987 KLUWER ACADEMIC PUBLISHERS ....
DORDRECHT I BOSTON I LANCASTER "
Dislribulors
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Library of Congress Cataloging in Publication D.la
w.. ..d ..n . A. H. • 1956-
The population-aa~le decoapo,ition method.
(IntemaUoTl,lll atudiea in ec:ono..ic. and
econoaetric. ; 19)
BiblioB. ... phy, p.
Include. indexea.
1. Multivariate aTl,lllyala. 2. Sa.pling (Stati.tic.)
3. !.ti~tion theory. I. Title. II. Serie.:
International atudies in ec:ono.lc: • • nd ec:ona.etrici
v. 19.
QA27!.W47 1987 519.5'35 81-21446
ISBN-l3: 978-94-010-8147-4 c-ISBN-13: 978-94-009-3679-9
001: 10.10071978-94-009-3679-9
Copyright
© 1987 by Martinus Nijhoff Publishers, Dordrecht.
Softcover reprint of the hardcover 1st edition 1987
All rights reserved. No part of this publication may be reproduced, stored in a
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The Netherlands.
Acknowledgements
This volume has been written in a time span of four years. From May 1983
until January 1985 I was affiliated to the Centre for Research in Public
Economics of the Leyden University and during the remaining period
(until May 1987) I was affiliated to the Econometric Institute of the
Erasmus University in Rotterdam. The research has been made possible by
financial support from the Netherlands Organization for Advancement of
Pure Research (ZWO).
I am grateful to my colleagues, who worked with me during these four
"PSD years"; especially the "Leyden-group", that moved with me from
Leyden to Rotterdam. With their advices they contributed directly or
indirectly to the final results of the research.
I am especially indebted to Bernard van Praag, who initialized the PSD
approach and encouraged and advised me in many stages of the project.
Finally I would like to thank Ellen Biesheuvel and Aagske Sporry at the
Erasmus University for their careful typing of the manuscript.
THE POPULATION-SAMPLE DECOMPOSITION METHOD:
A DISTRIBUTION-FREE ESTIMATION TECHNIQUE FOR MINIMUM DISTANCE PARAMETERS
Table of contents
I. Introduction to the Population-Sample Decomposition Approach
1.1 The linear statistical model
1.2 Minimum distance parameters subject to minimal model assumptions 9
II. The Estimation of Linear Relations; The Sample Part of PSD
11.1 Method of moments and asymptotic distribution theory 15
11.2 Asymptotic estimation of covariance functions 20
III. Principal Relations
111.1 Basic formulation of the principal relations 31
111.2 The distance matrix Q 46
111.3 Simultaneous equations systems 52
111.4 Seemingly unrelated regressions 69
111.5 Restricted seemingly unrelated regressions 79
111.6 Canonical correlation analysis 94
IV. Principal Factors
IV.l Basic formulation of principal factors 110
IV.2 Principal relations versus principal factors 125
IV.3 Principal components analysis 132
V. Goodness-of-Fit Measures
V.l Coefficients of multiple correlation and angles between 140
random vectors
V.2 Coefficients of linear association for principal relations 149
and principal factors
V.3 Coefficients of linear association for simultaneous 162
equations systems
V.4 Coefficients of linear associat~on for seemingly unrelated 170
regressions
VI. Review
VI.1 A schematic representation of the parameters 180
VI.2 List of notation and summary of results 184
VII. Computational Aspects of the Population-Sample Decomposition
VII.1 Fourth-order central moments 191
VII.2 Pre- and post-multiplication of V by the gradient matrix 199
VII.3 The PSD method in practice 205
Appendix
Preliminaries on matrix algebra 214
References 225
Author Index 237
Subject Index 239
I. Introduction to the Population-Sample Decomposition Approach
1.1. The linear statistical model.
One of the basic objectives of the social sciences is the drive to discover
regularities within a set of objects, usually called the population. If we
were able to obtain a complete set of information about those phenomena which
are of interest to us, we would be in a position to make deterministic
statements. However, as the population under consideration is often too large
to be observed in its entirety, it is impossible to obtain the necessary
information from all the objects of the population. It is for this reason that
the phenomena are only observed for a subset of objects, that are randomly
sampled from the population. Let us assume that the resulting set of
observations can be described by a set of vectors. The observed vectors are
collected in a dataset, called the sample. The assumed regularities in the
population must be a reflection of corresponding relations in the sample
observations, as the sample represents the phenomenon to be observed. In this
way, the real-world regularities can be estimated by means of the sample
observations.
The aim of this study is to provide a solution to the problem of how to use
sample information subject to minimal assumptions with respect to linear
statistical models, that are of interest in multivariate statistical analysis.
By using the word "model" a direct reference is made to a second aspect that,
together with· the set of sample observations, plays an important role in
statistical theory. In multivariate statistical analysis one starts by
creating a model on the basis of intuitions about relationships in the
population. The model is assumed to he an approximation of reality and gives a
formal description of the data generating process. The parameters of the model
are estimated by means of the observed sample values and the distributional
properties of the reSUlting estimates are based on the assumption that the
data are generated by this particular model. The question arises as to whether
it is always permissible to postulate that the hypothesized model is a good
approximation of reality. Naturally, if some hypotheses are implied by the
theory, or by empirical evidence from previous research results, then they
should be included in the model. However, one often finds that a part of the
model forms a dubious approximation of reality and is merely postulated for
-2-
the sake of convenient estimation. Before we continue with an outline of an
alternative approach to the estimation of statistical relations based on
minimal model assumptions. attention will be paid to some general ideas behind
data analysis and modeling in multivariate statistical analysis. Firstly. some
ideas about exploratory data analysis. based on non-random sample values. will
be formalized (see also Tukey (1977». Secondly. these ideas will be extended
into a propabilistic context. yielding confirmatory data analysis based on
model assumptions and probability theory. and statistical inference techniques
for the resulting estimators.
Suppose that {xn: nal ••••• ~} is a set of N observations. the values of which
are indicated by the (kxl) vector xn' That is. xn stands for the value of the
nth observation on the vector of interest. If the relevant aspects of the
population are reflected by a (kxl) vector x. then a functional specification
of the regularities in the population can be formalized as
( 1.1) f(x; b) = o.
with f being a function that is assumed to be known up to its parameter vector
b. The problem is how to derive the value of b. by means of the N values of x
given in the sample. The hypothesized functional form of f constitutes. when
relation (1.1) is placed in a probabilistic context. a part of the statistical
model. However. in order to start with a data descriptive technique. we do not
yet assume the existence of any randomness. Consequently. relation (1.1)
should apply to each observation ~. so that we may write
(1.2) for n-I ••••• N.
However. generally it is not possible to find a suitable value for b. and we
hall have to make do with a value for b in the senae that (1.2) will hold "as
ccurately as possible" for all observations. according to some criterion of
:it. If on is defined as the deviation of f(xn;b) from zero. hence
(1.3) for n-l ••••• N
then a well-known criterion of fit is given by the sum of squares
-r
(1.4)
Minimizing (1.4) with respect to b. an optimal value of b. say b. is obtained
as a function of the sample values xl ••••• xN and 5n - f(xn;b) can be seen as
the deviation of xn from the hypothesized functional approximation.
So far. no randomness has been assumed and nothing has been said about the
reliability of the value b. The vector b is merely a descriptive statistic.
However. in the early thirties. the realization dawned that the resulting
A
value b varies. when it is calculated on distinct samples that are dealing
with the same phenomenon. Especially under the influence of R.A. Fisher (much
of modern statistical methodology is founded on Fisher's classics (1925.
1935» these methods have been embedded in a probabilistic context. That is.
the sample can be seen as a realization of a stochastic event and the
resulting estimator b should be considered as a realization of a stochastic
event too. So now we are interested in the statistical properties of the
resulting estimator. In multivariate statistical analysis this problem is
solved by assuming a specific data-generating model. which is specified up to
a number of unknown parameters. These parameters are estimated from the
observed sample and inferences are based on the assumptions about the
stochastic structure of the model. A way of introducing the stochastic concept
into the previously described data analysis. is .by assuming that one or more
of the components of xn are random in character. One may distinguish the
random variables from the fixed values by partitioning ~ into subvectors Yn
and zn' where the random subvector Yn is denoted by a capital in order to
emphasize its stochastic character. As a consequence. the function f(Yn.zn;b)
is random as well and (1.3) is rewritten as
(1.5) for n=l ••••• N
with el ••••• eN a sequence of random error terms. that are usually assumed to
have zero expectations. Assumptions about the functional form of f and about
the distribution of en are called ~odel specifications.
It is worth noting that the data are supposed to comply with the model.
Consequently the estimation methods and the statistical inference about the