  Linear Vector AutoRegressive (VAR) models where the innovations could be
unconditionally heteroscedastic and serially dependent are considered. The
volatility structure is deterministic and quite general, including breaks or
trending variances as special cases. In this framework we propose Ordinary
Least Squares (OLS), Generalized Least Squares (GLS) and Adaptive Least Squares
(ALS) procedures. The GLS estimator requires the knowledge of the time-varying
variance structure while in the ALS approach the unknown variance is estimated
by kernel smoothing with the outer product of the OLS residuals vectors.
Different bandwidths for the different cells of the time-varying variance
matrix are also allowed. We derive the asymptotic distribution of the proposed
estimators for the VAR model coefficients and compare their properties. In
particular we show that the ALS estimator is asymptotically equivalent to the
infeasible GLS estimator. This asymptotic equivalence is obtained uniformly
with respect to the bandwidth(s) in a given range and hence justifies
data-driven bandwidth rules. Using these results we build Wald tests for the
linear Granger causality in mean which are adapted to VAR processes driven by
errors with a non stationary volatility. It is also shown that the commonly
used standard Wald test for the linear Granger causality in mean is potentially
unreliable in our framework. Monte Carlo experiments illustrate the use of the
different estimation approaches for the analysis of VAR models with stable
innovations.
