Bayesian Inference in Econometric Models Using Monte Carlo Integration

Methods for the systematic application of Monte Carlo integration with importance
sampling to Bayesian inference in econometric models are developed. Conditions under
which the numerical approximation of a posterior moment converges almost surely to the
true value as the number of Monte Carlo replications increases, and the numerical accuracy
of this approximation may be assessed reliably, are set forth. Methods for the analytical
verification of these conditions are discussed. Importance sampling densities are derived
from multivariate normal or Student t approximations to local behavior of the posterior
density at its mode. These densities are modified by automatic rescaling along each axis.
The concept of relative numerical efficiency is introduced to evaluate the adequacy of a
chosen importance sampling density. The practical procedures based on these innovations
are illustrated in two different models.
KEYWORDISm:portance sampling, numerical integration, Markov chain model, ARCH
linear model.

1. INTRODUCTION
ECONOMETRMICODELS are usually expressed in terms of an unknown vector of
parameters e E O G Rk, which fully specifies the joint probability distribution of
the observations X = {g,, . . .,g,). In most cases there exists a probability
density function f(Xle), and classical inference then often proceeds from the
likelihood function L(e) =f(Xle). The asymptotic behavior of the likelihood
function is well understood, and as a consequence there is a well developed set of
tools with which problems of computation and inference can be approached;
Quandt (1983) and Engle (1984) provide useful surveys. The analytical problems
in a new model are often far from trivial, but there are typically several
approaches that can be explored systematically with the realistic anticipation that
one or more will lead to classical inference procedures with an asymptotic
justification.