Corrected portmanteau tests for VAR models with time-varying variance

Abstract

The problem of test of fit for Vector AutoRegressive (VAR) processes with unconditionally heteroscedastic errors is studied. This problem is motivated by numerous examples of series presenting such a pattern. Our analysis is based on the residual autocorrelations obtained from Ordinary Least Squares (OLS), Generalized Least Squares (GLS) and Adaptive Least Squares (ALS) estimation of the autoregressive parameters. The GLS approach requires the knowledge of the variance structure while the ALS method is adapted to the unknown time-varying unconditional covariance which is estimated by kernel smoothing. It is shown that the ALS and GLS residual autocorrelations are asymptotically equivalent. It is also found that the asymptotic distribution of the OLS residual autocorrelations can be quite different from the standard chi-square asymptotic distribution obtained in a correctly specified VAR model with i.i.d. innovations. As a consequence the standard portmanteau tests which are available in routinely used software are unreliable in our framework. Using our results modified portmanteau tests based on the OLS and ALS residual autocorrelations and which take into account time varying covariance are proposed. The finite sample properties of the goodness-of-fit tests we consider are investigated by Monte Carlo experiments. The theoretical results are also illustrated using a U.S. economic data set.