A COMPANION TO Theoretical Econometrics

Bayesian methods

While increased computational power has had important implications for many areas of econometrics, the impact has probably been most dramatic in the area of Bayesian econometrics. It has long been accepted that the implementation of Bayesian methods by practitioners has been hindered by the unavailability of flexible prior densities that admit analytical treatment of exact posterior and predictive densities. For the SUR model, the problem is that the joint posterior distribution, /(p, X-11 y, X), has complicated marginal posteriors. Approximate inferences can be based on a conditional posterior, /(P | X-1 = X-1, y, X), but exact inferences using the marginal posterior distribution, /(P |y, X), are problematic.

Richard and Steel (1988) and Steel (1992) have been somewhat successful in extending the exact analytical results that are available for SUR models. Steel (1992) admits, these extensions fall short of providing models that would be of interest to practitioners, but suggests ways in which their analytical results may be effectively used in conjunction with numerical methods.

A better understanding and availability of computational approaches has meant that there are fewer impediments to the routine use of Bayesian methods amongst practitioners. Percy (1992) demonstrated how Gibbs sampling could be used to approximate the predictive density for a basic SUR model. This and other work on Bayesian approaches to SUR estimation is briefly reviewed in Percy (1996).

More recently, Markov chain Monte Carlo methods have enabled Bayesian analyses of even more complex SUR models. Chib and Greenberg (1995) con­sider a Bayesian hierarchical SUR model and allow the errors to follow a vector autoregressive or vector moving average process. Their contribution aptly illus­trates the power of Markov chain Monte Carlo methods in evaluating marginal posterior distributions, which previously have been intractable.

Joint estimation of a SUR model is typically motivated by the presence of disturbance covariation. Blattberg and George (1991) suggest that joint estima­tion may also be justified in the absence of such dependence if one feels that there are similarities between the regression parameters. When individual estima­tion leads to nonsensical parameter estimates, they suggest the use of a Bayesian hierarchical model to shrink estimates across equations toward each other thereby producing less estimator variation. They refer to seemingly unrelated equations, SUE rather than SUR.

With these kinds of developments it is not surprising to see more Bayesian applications of SUR models. Examples include Bauwens, Fiebig, and Steel (1994), Griffiths and Chotikapanich (1997) and Griffiths and Valenzuela (1998).