Bayesian prediction mean squared error for state space models with estimated parameters

Hamilton (A standard error for the estimated state vector of a state-space model. 1 Economet. 33 (1986), 387-97) and Ansley and Kohn (Prediction mean squared error for state space models with estimated parameters. Biometrika 73 (1986), 467-73) have both proposed corrections to the naive approximation (obtained via substitution of the maximum likelihood estimates for the unknown parameters) of the Bayesian prediction mean squared error (MSE) for state space models, when the model's parameters are estimated from the data. Our work extends theirs in that we propose enhancements by identifying missing terms of the same order as that in their corrections. Because the approximations to the MSE are often subject to a frequentist interpretation, we compare our proposed enhancements with their original versions and with the naive approximation through a simulation study. For simplicity, we use the random walk plus noise model to develop the theory and to get our empirical results in the main body of the text. We also illustrate the differences between the various approximations with the Purse Snatching in Chicago series. Our empirical results show that (i) as expected, the underestimation in the naive approximation decreases as the sample size increases; (ii) the improved Ansley-Kohn approximation is the best compromise considering theoretical exactness, bias, precision and computational requirements, though the original Ansley-Kohn method performs quite well; finally, (iii) both the original and the improved Hamilton methods marginally improve the naive approximation. These conclusions also hold true with the Purse Snatching series.