  In this paper, a new and convenient $\chi^2$ wald test based on MCMC outputs
is proposed for hypothesis testing. The new statistic can be explained as MCMC
version of Wald test and has several important advantages that make it very
convenient in practical applications. First, it is well-defined under improper
prior distributions and avoids Jeffrey-Lindley's paradox. Second, it's
asymptotic distribution can be proved to follow the $\chi^2$ distribution so
that the threshold values can be easily calibrated from this distribution.
Third, it's statistical error can be derived using the Markov chain Monte Carlo
(MCMC) approach. Fourth, most importantly, it is only based on the posterior
MCMC random samples drawn from the posterior distribution. Hence, it is only
the by-product of the posterior outputs and very easy to compute. In addition,
when the prior information is available, the finite sample theory is derived
for the proposed test statistic. At last, the usefulness of the test is
illustrated with several applications to latent variable models widely used in
economics and finance.
