Cross-validation criteria for SETAR model selection (Q2740035)

A central issue in the analysis of discrete-time self-exciting threshold autoregressive (SETAR) models is the choice of a suitable model order. For linear regression and linear time series model selection a variety of criteria has been proposed for this purpose. For identifying the order of a SETAR model, Akaike's information criterion (AIC) is commonly applied in practice. However, one should be cautious in using this criterion since the exact likelihood function of a time series generated by a SETAR process does not exist. This is a basic assumption underlying the derivation of AIC and ``improved'' versions of AIC, like BIC of \textit{H. Akaike} [Ann. Inst. Stat. Math. 30, 9-14 (1978; Zbl 0441.62007)]. In fact, for linear time series models only order selection criteria based on sample reuse such as cross-validation (CV) or bootstrapping can be strictly justified.NEWLINENEWLINENEWLINEIn this paper, three CV-criteria, denoted, respectively, by \(C\), \(C_c\) and \(C_u\), are proposed for general SETAR model selection. These criteria are tied to the out-of-sample prediction performance of SETAR models which is often an important goal in practice. The derivation of \(C\) is within a natural CV-framework. The criterion \(C_c\) is similar in spirit to \(AIC_c,\) i.e., a ``corrected'' version of \(AIC\) for linear models. The criterion \(C_u\) is a variant of \(C_c\) having a similar property as \(AIC_u,\) an approximately unbiased estimator of the Kullback-Leibler information in linear time series models. The simulation studies indicate that, in small-sample situations, \(C_u\) performs much better than \(AIC_c\) and various other SETAR model selection criteria. Further, it is shown that an analogue of \(AIC_u\) for SETAR model selection outperforms \(AIC,\) \(AIC_c\) and \(BIC\), all defined within a SETAR framework, except when the sample size is large. In fact, the author recommends the use of \(C_u\) in practice.