Table Of ContentLectu re Notes
in Economics and
Mathematical Systems
Edited by M. Beckmann, Providence, and H. P. Kunzi, Zurich
Econometrics
90
J.-F. Richard
Posterior and Predictive
Densities for Simultaneous
Eq uation Models
Springer-Verlag
Berlin· Heidelberg· New York 1973
Editorial Board
J.
H. Albach· A. V. Balakrishnan' P. Dhrymes . Green' D. W. Hildenbrand
R. E. Kalman' W. Krelle . U. P. Ritter' R. Sato . P. Schonfeld
Dr. Jean-Fran~ois Richard
Center for Operations Research
and Econometrics
Universite Catholique de Louvain
de Croylaan 54
B-3030 Heverlee/Belgium
AMS Subject Classifications (1970): 33-04, 33A30, 62-02, 62F15, 62)05, 62P20
ISBN-13: 978-3-540-06525-8 e-ISBN-13: 978-3-642-65749-8
001: 10.1007/978-3-642-65749-8
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 where copies are made for other than private use, a fee is payable to the
publisher, the amount of the fee be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1973. Library of Congress Catalog Card Number 73-16617.
Offsetprinting and bookbinding: Julius Beltz, Hemsbach/Bergstr.
Acknowledgements
I would like to thank, first and foremost, Professor J. Dreze, who originally
interested me in economet~ics in general, and in Bayesian econometrics in parti
cular. During the years spent at CORE, I have benefited greatly from his constant
suggestions, advice and encouragement. This monograph is in the spirit of research
which he himself has undertaken since 1962, and which, thanks to the new perspec
tives he has opened up, lends itself to considerable future development.
I am indebted to Professor J. Meinguet for his unfailing help in the numerous
problems of numerical analysis which confronted me in this monograph. Professor
J.F. Mertens frequently gave me the benefit of his experience in statistics, in
many discussions which were particularly stimulating. I would also like to thank
Professors A. Barten, L. Phlips and T. Rothenberg for their many helpful comments
on an earlier version, and for their suggestions for improving this present one.
My friends and colleagues H. Diaz, J.P. Florens, J.A. Morales, M. Mouchart and H.
Tompa have given me much help at all stages of this work, in numerous discussions
which were very fruitful. Although not directly concerned with this monograph,
I would like to thank Professor A. Zellner, whom I met in Chicago, and who stimu
lated my interest in Bayesian econometrics.
This work could not have been realised without the help of various institu
tions. While W. Hallam Tuck Fellow at the University of Chicago, my work was sup
ported by the Belgian American Educational Foundation. After this, I was supported
by the "Fonds National de la Recherche Scientifique" until nominated assistant at
the University of Louvain. Part of my research in informatic was supported by the
a
"Programme National df Impulsion la Recherche en In format i que ". I have also bene
fited greatly from the stimulating environment at CORE. Lastly, I would like to
thank the IBM company which allowed me to put some of my programs into runing
order on their computers, and the Louvain Computing Center where the necessary
calculations were undertaken.
Last but not least, my wife Colette deserves all praises for her patience.
Contents
I. The Simultaneous Equation Model,
I.' The statistical mOdel; Notations
1.2 The identification problem 3
1.3 Some previous contributions 8
1.4 Review of the contents "
1.5 Technical abstract '3
II. Full Information Analysis of the Two-equation Model. 20
II.' Notations 20
II.2 The Dreze and Morales approach. 23
I
11.3 An alternative approach: structural analysis 29
11.4 Reduced form analysis and prediction. 40
11.5 Some additional remarks 44
III. Limited Information Analysis of the Simultaneous Equation Model. 49
III.' Some posterior joint conditional densities on TI : non-
informative prior density
111.2 Posterior moments of the reduced form parameters limited
information prior density 59
111.2.' Natural conjugate prior density (m,=') 60
111.2.2 Exact prior restrictions (m,=')' 68
III. 2.3 Some extensions for m, > , 76
II1.3 A "limited information" prediction. 8,
VI
Appendix I : The matrix-t density 85
Appendix II : The row-diagonal conditional matricvariate-t density 89
Appendix III : The technicalities of Chapter II 93
Appendix IV : Hypergeometric series ; Their computations 118
Appendix V : The programs of Chapter II 141
Appendix VI : An integral identity (Dickey) 147
IV. Empirical Illustration The Belgian Beef Market 152
IV.1 Morales' model. 152
IV.2 The information content of the identifYing restrictions_ 154
IV.3 Informative approach full information analysis 166
IV.4 Informative approach limi ted information analysis 187
V. Conclusions 196
Appendix VII : The data 201
Appendix VIII : The results 203
References 223
I THE SIMULTANEOUS EQUATION MODEL
1., The statistical model Notations
We consider the following system of structural equations
( '.1) By rz + U
where y is an m-dimensional (column) vector of endogenous variables
z is an n-dimens ional vector of exogenous variables
B is an m x m non singular matrix of coefficients
r is an m x n matrix of coefficients
u is an m-dimensional vector of disturbances
For normalisation purposes, one element of each row of B is arbitrarily set
equal to one.
The corresponding reduced form is given by
( '.2) y JIz + v
-,
where JI v = B u
Assuming that T-joint observations on y and z are available, we represent them
= =
by the T x m matrix Y (Y, ... Ym) and the T x n matrix Z (z, ... zn)'
= ( =
U u, .,. um) and V (v, •.. vm) being the corresponding T x m matrices of
unobserved disturbances.
(1. 3) YB' zr' + U
2
(,.4) Y ZIT' + V
As usual, we shall assume that, given Z, the rows of V are jointly normally
distributed, identically and independently for all t : , .... T.
or, equivalently
where I: BrIE'
The likelihood is then
-1.
( 1. 7a) l(ylz,IT,n) ~ Inl 2 exp - 1 tr n-'(Y-ZIT')'(Y-ZIT')
2
or
T
l(ylz,B,r,l:) ~ 11:1 2 IIBIIT exp _1 tr I:-'(B -r)(Y'Y Y'Z)( B')
2 Z'Y Z'Z -r'
3
I.2 The identification problem
The fact that the likelihood (1.7) may be expressed either as a function
m+1) 3m-1
of the (n + ~ .m parameters (n,n), or as a function of the (n + --2--).m
parameters (B,r,l:), is known as the "identification problem". The likelihood
is stationary on any set of values (B,r,l:) implying the same (n,n) matrices.
As a consequence, maximum likelihood methods do not yield unique estimates of
B, r and l:. Classically this problem has been solved by imposing additional
exact prior restrictions on (B,r.l:), such that the transformation from (n,n)
to (B,r,l:) is one-to-one. The model is then said to be "identified". It may
even be "overidentified" if the structural constraints imply some restrictions
on the parameters (n,n). (See Fisher [10])
In a Bayesian framework we shall consider instead stochastic prior infor-
mation. The counterpart of the identification problem is then most conveniently
expressed in a theorem discussed by Dr~ze in [5] :
( 1.8)
where h denotes any prior conditional density on B, and h* the corresponding
posterior density. ConditionallY on n and n, the data are non-informative
about the B coefficients.
More generally our prior information will be described by means of a prior
density function for some of the parameters, as well as by means of exact cons-
traints on some parameters. This clearly calls for an extension of the classical
concept of identification. Such an extension could be based on some comments
4
we made in a joint paper (J.P. Florens, M. Mouchart and J.F. Richard [11 l)
about identification in the error-in-variables mOdels(1).
Let 5 be the set of structures, i.e. the set of possible values for
(B,r ,r) :
(1. 9) 5 Rm(m-1) x Rmn x C
m
m(m+1)
where Cm {r Ir £ R 2 r is P.D.S.}
Two structures (B1 ,r 1 ,r 1) and (B2 ,r 2' r2) are said to be "observationally equi
valent" if, and only if:
(1.10)
Such will be the case if, and only if
(1.11)
and we shall use the notation
Let then S/~ denote the quotient class of 5 with respect to the equivalence
relation'" .
( 1) We also present l. n that paper an extensl.o n 0 f the concept 0 f II 0 b servat l. ona1 1y
equivalent" structures which is more oriented toward a decision framework. We
are presently investigating that topic.
5
In such a framework, the classical concept of identification may be pre-
sented as follows. We introduce a set of constraints ~(B,r,l:) = 0 and define
T, a subset of 5 by :
( 1.13) T= {(B,r,l:)I(B,r,l:) e:S; ~(B,r,E) O}
The model is then identified if, and only if, the elements of T/~ are all
singletons.
A poss~'b Ie Bayes'~a n concept (1) would call ~. nstead for a pr.~or d,ens~t y
fo(B,r,E) which would induce a probability measure on the elements of any
equivalence class in T/-;;. • On the basis of Dreze's theorem, we should then
consider prior densities fo(B,r,E) which, through the integrand transforma-
tion from (B,r,E) to (B,rr,n), would result in a proper conditional prior den
sity fo(Blrr,n). It is clear that any proper prior density on B would be suffi-
cient, but even if we selected a "non-informative" marginal prior density on
B, we could still rely on some prior information on (r,E).
A lack of identification essentially results in the fact that the data
are non informative on some of the aspects of the model. On the other hand :
" Yet a key distinction is made between decision and nuisance parame-
ters and an eventual lack of identification is considered as consequen-
tial only if, for a given prior information, it results in a marginal
("Dreze is presently working on a concept of "identification in probability"
which should be the Bayesian analogue of the classical theory of identifica
tion. We have just received a preliminary version [7] of a paper on that topic.
