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Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
232
Luc Bauwens
Bayesian Full Information
Analysis of Simultaneous
Equation Models Using
Integration by Monte Carlo
Springer-Verlag
Berlin HeidelberQ New York Tokyo 1984
Editorial Board
H. Albach M. Beckmann (Managing Editor) P. Ohrymes
G. Fandel J. Green W. Hildenbrand W. Krelle (Managing Editor) H.P. KGnzi
G.L. Nemhauser K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fUr Gesellschafts- und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Author
Dr. Luc Bauwens
C.O.R.E.
34, Voie du Roman Pays
B-1348 Louvain-La-Neuve, Belgium
ISBN-13: 978-3-540-13384-1 e-ISBN-13: 978-3-642-45578-0
001: 10.1007/978-3-642-45578-0
Library of Congress Cataloging in Publication Data. Bauwens, Luc, 1952-. Bayesian full infor
mation analysis of simultaneous equation models using integration by Monte Carlo. (Lecture
notes in economics and mathematical systems; 232) Bibliography: p. 1. Linear models
(Statistics). 2. Equations, Simultaneous. 3. Monte Carlo method. I. Title. II. Series.
QA276.B33 1984 519.5 84-14137
ISBN-13: 978-3-540-13384-1
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
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© by Springer-Verlag Berlin Heidelberg 1984
2142/3140-543210
ACKNOWLEDGMENTS
I want to thank Professor J.-F. Richard for his many fruitful suggestions in
the course of this research. There is no doubt that this monograph would not have
been written if I had not benefited from his enlightening advises during the years
I spent at CORE.
My thanks are also due to Professors J. Dreze and H. van Dijk (of Erasmus Univer
sity at Rotterdam), and to my colleague and friend M. Lubrano, for occasional discus
sions on the topic.
All these persons, as well as Professors A. Barten and D. Weiserbs, made also
many useful comments on the manuscript, and thereby helped me to improve the presen
tation of this monograph.
E. Pecquereau deserves my gratitude for her prompt and efficient typing.
I am indebted to H. Tompa, although he was not directly concerned with this
research, for transmitting and sharing generously his incomparable experience in
numerical analysis and computer programming during his long period of activity at
CORE.
Financial support of the "Projet d'Action Concertee" of the government of Belgium
under oontract"SO/S5-12 is gratefully acknowledged. This contract has also financed
the numerous computations I had to make. It has been a pleasure to work on CORE
computer.
My wife Bernadette and my children Ariane and Ronald have always comforted me
by their agreeable company; let they be praised for their unfailing patience.
TABLE OF CONTENTS
Page
INTRODUCTION
I. THE STATISTICAL r~ODEL 4
1.1 Notation 4
1.2 Interpretation 5
1.3 Likelihood function 6
II. BAYESIAN INFERENCE: THE EXTENDED NATURAL-CONJUGATE APPROACH 8
11.1 Two reformulations of the likelihood function 8
11.2 The extended natural-conjugate prior density 9
11.3 Posterior densities 14
11.4 Predictive moments 15
11.5 Numerical integration by importance sampling 16
III. SELECTION OF IMPORTANCE FUNCTIONS 21
IIL1 General criteria 21
IIL2 The AI(E) approach 22
III.2.1. Properties of the posterior density of 6 22
III.2.2. Student importance function (STUD) 25
III.2.3. Poly-t based importance function Case I (PTFC) 26
III.2.4. Poly-t based importance function Case II (PTDC) 27
III.2.5. Poly-t based importance function Case III (PTST) 28
III.2.6. Conclusion 32
IIL3 The AI(y) approach 32
IV. REPORT AND DISCUSSION OF EXPERIMENTS 35
IV.1 Report 35
IV. I. I. BBM 36
IV.I.2. Johnston 43
IV.I.3. Klein 47
IV.I.4. EX 55
IV.I.5. W 62
IV.2 Conclusions 65
VI
V. EXTENSIONS 67
V.I Prior density 67
V.2 Nonlinear Models 67
CONCLUSION 70
APPENDIX A DENSITY FUNCTIONS : DEFINITIONS, PROPERTIES AND ALGORITHMS
FOR GENERATING RANDOM DRAWINGS 71
A.I The matricvariate normal (MN) distribution 71
A.II The inverted-Wishart (iW) distribution 72
A.IlI The multivariate Student distribution 75
A.IV The 2-0 poly-t distribution 76
A.V The m-I (0 <m ~ 2) poly-t distribution 78
APPENDIX B THE TECHNICALITIES OF CHAPTER III 82
B.I Definition of the parameters of (3.3) and (3.6) 82
B.II Computation of the posterior mode of 0 83
B.IlI Computation of (3.15) 84
APPENDIX C PLOTS OF POSTERIOR MARGINAL DENSITIES AND OF IMPORTANCE
FUNCTIONS 86
APPENDIX D THE COMPUTER PROGRAM 108
FOOTNOTES 110
REFERENCES 112
INTRODUCTION
In their review of the "Bayesian analysis of simultaneous equation systems",
Dr~ze and Richard (1983) - hereafter DR - express the following viewpoint about the
present state of development of the Bayesian full information analysis of such sys
tems
i) the method allows "a flexible specification of the prior density, including well
defined noninformative prior measures";
ii) it yields "exact finite sample posterior and predictive densities".
However, they call for further developments so that these densities can be eval
uated through 'numerical methods, using an integrated software packa~e. To that end,
they recommend the use of a Monte Carlo technique, since van Dijk and Kloek (1980)
have demonstrated that "the integrations can be done and how they are done".
In this monograph, we explain how we contribute to achieve the developments
suggested by Dr~ze and Richard.
A basic idea is to use known properties of the porterior density of the param
eters of the structural form to design the importance functions, i.e. approximations
of the posterior density, that are needed for organizing the integrations. If in
particular the prior density is in the class of extended natural-conjugate densities,
which has been proposed by Morales (1971) and Dr~ze and Morales (1976), the main
properties we use are the following :
i) the posterior density for the coefficients of a single equation, conditional on
the coefficients of the other equations, belongs to the class of 1-1 poly-t den
sities;
ii) there exist sufficient conditions for the existence of moments of the posterior
density of the coefficients of the structural form.
We concentrate on this case, which is reviewed in DR (section 6.3).
In order to give an intuitive view of the usefulness of these properties, let us
discuss briefly the method of importance sampling, which is the Monte Carlo technique
of numerical integration used throughout this monograph. Let c(e) be a kernel of
the posterior density of the parameters e (E e) and let gee) be an integrable func
tion of e. The expected value of gee) can be written as
Ie
gee) wee) fee) de
(I)
Ie
wee) fee) de
where wee) = ..:(e) / fee) is the weight function, and fee) is a so-called importance
function, i.e. a density function with the following two properties :
i) fee) must be a good approximation of K(8), so that wee) varies as little as
2
possible;
ii) it must be possible to generate random drawings from f(6).
It is then possible to estimate (1) by a ratio of two sample averages
N
-I
N L g(6k) w(6k)
k=1
(2)
N
N-1 :E w(6k)
k=1
where {6k, k = 1,2, ••• ,N} is a sample of N independent random drawings from f(6).
The knowledge of the conditional densities of the posterior can be exploited in
several ways to build an importance function; for example, the importance function can
be defined as the product of the posterior conditional 1-1 poly-t density for the
coefficients of a single equation, and a suitably chosen marginal density for the
other coefficients. Also, if an upper bound on the order of existence of moments of
the posterior can be imposed upon the importance function,' the two density functions
are likely to have similar tails; this can prevent the occurence of extreme values
(relative to the average) of the weight function w (6) when 6 is drawn in the tails
of the posterior.
The use of 1-1 poly-t densities as (conditional) importance functions requires
the availability of an efficient algorithm for generating random drawings from such
distributions. Such an algorithm has been proposed by Bauwens and Richard (1982).
An important criterion that has guided this research is a requirement for autom
atized procedures. The importance functions we propose are automatic in the sense
that their parameters are the same functions of the parameters of the posterior den
sity whatever the modeZ. Automatized procedures are needed for implementing a user
friendly computer package, thanks to which interested econometricians could estimate
a simultaneous equation model in the Bayesian framework, without programming effort.
However, such procedures must be carefully thought out, since generality could entail
inefficiencYj this is particularly true for Monte Carlo methods, where it is important
to use as much as possible all the information one has about a problem. We consider
therefore several classes of importance functions which are suited to specific cases,
so that for a particular model, it should be relative Iv easy to identify to which case
it corresponds.
CONTENTS
The book contains five chapters and four appendices.
Chapter I introduces the notations of the model, discusses its interpretation
and presents its likelihood function.
3
Chapter II considers Bayesian inference with an extended natural-conjugate prior
density; two reformulations of the likelihood function are used in order to integrate
analytically different subsets of the parameters of the structural form and the sec
tion concludes with a discussion of the Monte Carlo technique (importance samp1inp,)
that is used to integrate numerically the remaining parameters.
Chapter III proposes a selection of importance functions for the two approaches
distinguished in the previous chapter.
Chapter IV reports some practical experiments and draws some general conclusions.
Chapter V indicates various extensions.
Appendix A gives the notations used for the following probability density func
tions : the matricvariate normal, the multivariate Student, the inverted-Wishart,
the 2-0 po1y-t and them-l po1y-t. It discusses also the use of some properties of
these distributions for generating random drawings from them.
Appendix B contains some technicalities of Chapter III.
Appendix C contains a few plots of posterior marginal densities and of the cor
responding importance functions, for some parameters of the models used in Chapter IV.
Appendix D describes briefly the computer program that has been written to im
plement the methods described in this monograph.
CHAPTER I THE STATISTICAL MODEL
1.1. NOTATION
We adopt the following formulation of the structural form of the dynamic linear
simultaneous equation model
(I. 1) XA = YB + zr = U
where X=[y z];
y = [Y YI ] is a T)( m matrix of observed endo~enous variables; the sub
matrix Y is of dimensions T)( m ;
z is a T x n matrix of observed predetermined variab les;
(m m
A = [;] is an + n) )( matrix of constants and unknown distinct coef
ficients; B is a nonsingu1ar matrix of order iii', whose diagonal ele
ments aU' for i = 1, ••• , m, are equal to 1 (normalisation rule) 1 ;
U = [U 0] is a T x mm atrix; U is a T x m matrix of unobserved distur
m- U
bances; the last m columns of are equal to zero, in which case
the corresponding columns of A do not contain unknown parameters
(linear identities) •
The following assumptions are also made :
,
i) predete,r minedness : V t <; T, V i ~ 0 such that t + i <; T, the row vectors Zt
and ut+i are linearly independent;
ii) normality: U has a matricvariate normal density (as defined in Appendix A),
Le.
Txm I
(I.2) P (U) = f MN (U 0, L ~ IT) •
The i-th equation (V i <; m) of the system (1.1) can be written as
(1.3)
where Yi is the i-th column of Y;
X( i) = - [ Y( i) z (i)] is the T x ti submatrix of X consisting of the
observations on the ti explanatory variables appearing in the i-th
equation;
<5. is the t. x 1 vector of unrestricted coefficients in the i-th equation;
1 1
u. is the i-th column of U.
1
