The focussed information criterion for generalised linear regression models for time series

The present paper proposes the focussed information criterion (FIC) to tackle the model selection problems pertinent to generalised linear models (GLM) for time series. As a first step towards constructing the FIC, we formally discuss the local asymptotic theory of quasi-maximum likelihood estimation for time series GLM under potential model misspecification. The general FIC formula is derived subsequently that is useful for the simultaneous selection of the order of the autoregressive response as well as a subset of important covariates. We also develop the average FIC (AFIC) that is instrumental in selecting an overall good model for a range of covariates and time regions and establish the equivalence of the AFIC with the classical Akaike's information criterion (AIC). We demonstrate our theory with the analysis of rainfall patterns in Melbourne by means of the logistic and Gamma regression models.