The Bayesian Score Statistic
We propose a novel Bayesian test under a (noninformative) Jeﬀreys’ prior speciﬁca- tion. We check whether the ﬁxed scalar value of the so-called Bayesian Score Statistic (BSS) under the null hypothesis is a plausible realization from its known and standard- ized distribution under the alternative. Unlike highest posterior density regions the BSS is invariant to reparameterizations. The BSS equals the posterior expectation of the classical score statistic and it provides an exact test procedure, whereas classical tests often rely on asymptotic results. Since the statistic is evaluated under the null hypothe- sis it provides the Bayesian counterpart of diagnostic checking. This result extends the similarity of classical sampling densities of maximum likelihood estimators and Bayesian posterior distributions based on Jeﬀreys’ priors, towards score statistics. We illustrate the BSS as a diagnostic to test for misspeciﬁcation in linear and cointegration models.
Kleibergen, F.R., Kleijn, R.H., & Paap, R.. (2000). The Bayesian Score Statistic (No. EI 2000-16/A). Retrieved from http://hdl.handle.net/1765/18192
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