Corrected portmanteau tests for VAR models with time-varying variance
Résumé
The problem of test of fit for Vector AutoRegressive (VAR) processes with unconditionally heteroscedastic errors is studied. The volatility structure is deterministic but time-varying and allows for changes that are commonly observed in economic or financial multivariate series. Our analysis is based on the residual autocovariances and autocorrelations obtained from Ordinary Least Squares (OLS), Generalized Least Squares (GLS)and Adaptive Least Squares (ALS) estimation of the autoregressive parameters. The ALS approach is the GLS approach adapted to the unknown time-varying volatility that is then estimated by kernel smoothing. The properties of the three types of residual autocovariances and autocorrelations are derived. In particular 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 iid innovations. As a consequence the standard portmanteau tests are unreliable in our framework. The correct critical values of the standard portmanteau tests based on the OLS residuals are derived. Moreover, modified portmanteau statistics based on ALS residual autocorrelations are introduced. Portmanteau tests with modified statistics based on OLS and ALS residuals and standard chi-square asymptotic distributions under the null hypothesis are also proposed. An extension of our portmanteau approaches to testing the lag length in a vector error correction type model for co-integrating relations is briefly investigated. 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 two U.S. economic data sets.