Discussion of “Feature Matching in Time Series Modeling” by Y. Xia and H. Tong
Edward L. Ionides · Statistical Science · 2011
Xia and Tong have made a novel contribution to the debate on whether and how to carry out some sort of feature matching in preference to a statistically efficient alternative such as the maximum likelihood estimate (MLE).They show that an estimation criterion emphasizing long-term predictions has some advantages over the MLE on some misspecified time series models.However, emphasizing long-term predictions must lead to a down-weighting of higher-frequency information in the data.In particular, Xia and Tong's catch-all approach does not typically share the statistical efficiency of MLE when the model fits the data adequately.Further, it is necessarily the case (whatever fitting method is used) that some scientific inferences one might wish to conclude from fitting a misspecified model are statistically invalid.Scientific interpretation of fitted parameter values and predictions using a model that is a statistically poor match to the data therefore requires considerable care.One seeks models that are simultaneously scientifically relevant and provide an adequate statistical description of the data, and then statistical efficiency becomes an important consideration for drawing scientific conclusions from limited data.Flexible modern inference methods facilitate the development and statistical analysis of such models.I will discuss these issues in the context of Xia and Tong's analysis of Nicholson's blowfly data.Similar considerations arise in their measles example, and have been investigated by He, Ionides and King (2010).Xia and Tong's APE(≤1) estimate is equivalent to the MLE only for a specific choice of stochastic model.From their equation (3.12), we see that APE(≤1) corresponds to the MLE for additive, Gaussian, constantvariance process noise with no measurement error.For Xia and Tong's blowfly model, the log-likelihood at the APE(≤1) point estimate is -1568.5 whereas the log-likelihood at the APE(≤T ) point estimate is -1569.5.A chi-squared approximation indicates that