The marginal likelihood of dynamic mixture models

Analytical results for reducing the parameter space dimension when computing the marginal likelihood are given for the broad class of dynamic mixture models. These results allow the integration of scale parameters out of the likelihood by Kalman filtering and Gaussian quadrature. The method is simple and improves the accuracy of four marginal likelihood estimators, namely, the Laplace method, the Chib estimator, reciprocal importance sampling, and bridge sampling. For some empirically relevant cases like the local level and the local linear models, the marginal likelihood can be obtained directly without any posterior sampling. Implementation details are given in some examples. Two empirical applications illustrate the gain in accuracy achieved.