Adaptive Estimation of Var with Time-Varying Variance : Application to Testing Linear Causality in Mean and Var Order

Abstract

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 two applications are proposed. First, 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. Second, we investigate the properties of the classical portmanteau tests when the innovations are unconditionally heteroscedastic. We find that the asymptotic distributions of the OLS residual autocorrelations can be quite different from the standard chi-square asymptotic distribution. Corrected portmanteau tests which take into account changes in the volatility using data driven adjustments for the critical values are proposed. Monte Carlo experiments and real data examples illustrate the theoretical results. The talk is based on joint work with Hamdi Raïssi.