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.
BAYESIAN SIMULTANEOUS EQUATIONS ANALYSIS USING REDUCED RANK
STRUCTURES