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
