Bayesian Estimates of Equation System Parameters_ An Application of Integration by Monte Carlo

Monte Carlo (MC) is used to draw parameter values from a distribution defined on the
structural parameter space of an equation system. Making use of the prior density, the
likelihood, and Bayes' Theorem it is possible to estimate posterior moments of both
structural and reduced form parameters. The MC method allows a rather liberal choice of
prior distributions. The number of elementary operations to be performed need not be an
explosive function of the number of parameters involved. The method overcomes some
existing difficulties of applying Bayesian methods to medium size models.
The method is applied to a small scale macro model. The prior information used stems
from considerations regarding short and long run behavior of the model and from
extraneous observations on empirical long term ratios of economic variables. Likelihood
contours for several parameter combinations are plotted, and some marginal posterior
densities are assessed by MC.

1. INTRODUCTION
INRECENT YEARS several Bayesian methods of estimating parameters of simultaneous
equation systems have been introduced (see, e.g., Drkze [5], Zellner [21],
Harkema [Ill, Rothenberg [17 and 181, and Richard [16], and the references
cited there). An important motive for research in this area is the analysis of
economic policy problems from a decision theoretic point-of-view. It appears that
in this context Bayesian estimates are more satisfactory than classical ones. The
analysis of these problems requires the use of numerical methods, for, in order to
obtain analytically tractable results, restrictions have to be imposed which are less
attractive from an economic point of view (see Rothenberg [17, pp. 139-1441
and Harkema [ll]).
The application of numerical methods appears to be hampered by the amount
of computational work involved (see Rothenberg [17, p. 1401). However, the
numerical work for several econometric problems is restricted to the computation
of first and second order moments, e.g., analysis of economic policy problems
based on a quadratic loss function (see Zellner [21, Chapter Ill), or MELO
estimators of ratios of parameters (see Zellner [22]).