  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.
