A Specification Test based on the MCMC Output

A test statistic is proposed to assess the model specification after the model is estimated by Bayesian MCMC methods. The new test is motivated from the power enhancement technique of Fan, Liao and Yao (2015). It combines a component (J1) that tests a null point hypothesis in an expanded model and a power enhancement component (J0) obtained from the null model. It is shown that J0 converges to zero when the null model is correctly specified and diverges when the null model is misspecified. Also shown is that J1 is asymptotically X2-distributed, suggesting that the proposed test is asymptotically pivotal, when the null model is correctly specified. The proposed test has several properties. First, its size distortion is small and hence bootstrap methods can be avoided. Second, it is easy to compute from the MCMC output and hence is applicable to a wide range of models, including latent variable models for which frequentist methods are difficult to use. Third, when the test statistic rejects the specification of the null model and J1 takes a large value, the test suggests the source of misspecification of the null model. The finite sample performance is investigated using simulated data. The method is illustrated in a linear regression model, a linear state-space model, and a stochastic volatility model using real data.