Autoregressive Order Identification for VAR Models with Non Constant Variance

The identification of the lag length for vector autoregressive (VAR) models by mean of Akaike information criterion (AIC${\textit{AIC}}$), partial autoregressive, and correlation matrices (PAM and PCM hereafter) is studied in the framework of processes with time-varying variance. It is highlighted that the use of the standard tools is not justified in such a case. As a consequence considering the adequate expression of the likelihood of the model and the adaptive estimator of the autoregressive parameters, we propose new identifying tools. More precisely we approximate the correct AIC${\textit{AIC}}$ for VAR models with time-varying variance using an adaptive AIC${\textit{AIC}}$. It is found that the adaptive AIC${\textit{AIC}}$ is robust to the presence of unconditional heteroscedasticity. Corrected confidence bounds are proposed for the usual PAM and PCM obtained from the ordinary least squares (OLS) estimation. We also use the unconditional variance structure of the innovations to develop adaptive PAM and PCM. Noting that the adaptive estimator is more accurate than the usual OLS estimator, it is underlined that the adaptive PAM and PCM have a greater ability than the OLS PAM and PCM for detecting unnecessary large lag length for VAR models. Monte Carlo experiments show that the adaptive AIC${\textit{AIC}}$ selects satisfactorily the correct autoregressive order of VAR processes. An illustrative application using US international finance data is presented.