Bayesian Simultaneous Equations Analysis Using Reduced Rank Structures

Diffuse priors lead to pathological posterior behavior when used in Bayesian analyses of simultaneous equation models (SEM's). This results from the local nonidentification of certain parameters in SEM's. When this a priori known feature is not captured appropriately, it results in an a posteriori favoring of certain specific parameter values that is not the consequence of strong data information but of local nonidentification. We show that a proper consistent Bayesian analysis of a SEM explicitly has to consider the reduced form of the SEM as a standard linear model on which nonlinear (reduced rank) restrictions are imposed, which result from a singular value decomposition. The priors/posteriors of the parameters of the SEM are therefore proportional to the priors/posteriors of the parameters of the linear model under the condition that the restrictions hold. This leads to a framework for constructing priors and posteriors for the parameters of SEM's. The framework is used to construct priors and posteriors for one, two, and three structural equation SEM's. These examples together with a theorem, showing that the reduced forms of SEM's accord with sets of reduced rank restrictions on standard linear models, show how Bayesian analyses of generally specified SEM's can be conducted.