Bayesian simultaneous equations analysis using reduced rank structures

Diffuse priors lead to pathological posterior behavior when used in Bayesia n analyses of simultaneous equation models (SEM's), This results from the l ocal nonidentification of certain parameters in SEM's. When this a priori k nown feature is not captured appropriately, it results in an a posteriori f avoring of certain specific parameter values that is not the consequence of strong data information but of local nonidentification. We show that a pro per consistent Bayesian analysis of a SEM explicitly has to consider the re duced form of the SEM as a standard linear model on which nonlinear (reduce d rank) restrictions are imposed, which result from a singular value decomp osition. The priors/posteriors of the parameters of the SEM are therefore p roportional to the priors/posteriors of the parameters of the linear model under the condition that the restrictions hold. This leads to a framework f or constructing priors and posteriors for the parameters of SEM's. The fram ework is used to construct priors and posteriors for one, two, and three st ructural equation SEM's, These examples together with a theorem, showing th at the reduced forms of SEM's accord with sets of reduced rank restrictions on standard linear models, show how Bayesian analyses of generally specifi ed SEM's can be conducted.