On Bayesian Structural Inference in a Simultaneous Equation Model

Econometric issues that are considered fundamental in the development of Bayesian structural inference within a Simultaneous Equation Model are surveyed. The difficulty of specifying prior information which is of interest to economists and which yields tractable posterior and predictive distributions has started this line of research. A major issue is the nonstandard shape of the likelihood due to reduced rank restrictions. It implies that existence of structural posterior moments under vague prior information is a nontrivial issue. The problem is illustrated through simple examples using artificially generated data in a so-called limited information framework where the connection with the problem of weak instruments in classical econometrics is also described. A positive development is Bayesian inference of implied characteristics, in particular, dynamic features of a Simultaneous Equation Model. The potential of Bayesian structural inference, using a predictive approach for prior specification and using Monte Carlo simulation techniques for computational purposes, is illustrated by means of a prior and posterior analysis of the US business cycle in the period of the depression. A structural prior is elicited through investigation of the implied predictive features. Some connections with modern time series econometrics are emphasized, in particular, the formal mathematical equivalence of overidentification in a SEM and cointegration in an vector autoregressive model. It is argued that Bayesian structural inference is like a Phoenix. It was almost a dead topic in the late eighties and early nineties but it has become of renewed importance in models where reduced rank analysis occurs. These models include structural vector autoregressive models, ∗I am indebted to Eric Zivot, Rodney Strachan and, in particular, to Frank Kleibergen for stimulating discussions on the topic of this paper. I am also grateful to Lennart Hoogerheide, Bernt Stigum, Tore Schwert and Nils Hjort for helpful comments on an earlier version of this paper. Responsibility for any errors is entirely mine.