Discriminating among Linear Models with Interdependent Disturbances

PROBLEMS OF COMPARING or choosing among models of a stochastic process are frequently encountered in empirical research. In many such situations, conventional statistical procedures offer little guidance since they assume that the model is given. If the alternative models can be nested in a more general model, standard estimation and testing procedures can be employed. Often, however, such general models are not readily available and other considerations may dictate against their use. Recently, there has been considerable progress in the development of methods for comparing alternative non-rested models. A review of this work both Bayesian and non-Bayesian, is given in Gaver and Geisel [1]. Discussions of the Bayesian approach to the comparison of linear regression models are given in Zellner [3, Ch. 10] and Lempers [2], among others. In this paper we consider Bayesian comparison of linear models in which the disturbances have non-scalar covariance matrices. General posterior odds expressions are given and specialized to the case of first order autoregressive disturbances. We also consider a specification error problem in this context; that is, we examine the effect of ignoring the non-scalar covariance structure on the posterior odds ratio. For the first order autoregressive disturbance case we give an approximate expression indicating the magnitude of the error involved in computing the posterior odds ignoring the serial correlation. The accuracy of this approximation is investigated via a small sampling experiment. We use the following notatiori: Let the ith model, Mi (i = 1, 2,. .., N), be y = Xi/3i + yi where y is a T x 1 vector of observations on the random (dependent) variable of interest, Xi is a T x ki matrix of observations on the explanatory variables of Mi (Xi is assumed to be non-stochastic with rank ki), p3i is a ki x 1 vector of unknown parameters of Mi, and ui is a T x 1 vector of disturbances of Mi (yi is assumed to have a normal distribution with E(yi) = 0, and E(yiyii) = U22i where 2Ji is an unknown T x T positive definite symmetric matrix with trace (i) = T).2 Probability functions for the models are denoted by P( ), densities for parameters by 7r( ), and densities for observations by p( ).