Adaptive Estimation of Vector Autoregressive Models with Time-Varying Variance: Application to Testing Linear Causality in Mean

Mardi | 2011-05-17

Valentin PATILEA – Hamdi RAÏSSI – libre

Linear Vector AutoRegressive (VAR) models where the innovations could beunconditionally heteroscedastic and serially dependent are considered. Thevolatility structure is deterministic and quite general, including breaks ortrending variances as special cases. In this framework we propose OrdinaryLeast Squares (OLS), Generalized Least Squares (GLS) and Adaptive LeastSquares (ALS) procedures. The GLS estimator requires the knowledge ofthe time-varying variance structure while in the ALS approach the unknownvariance is estimated by kernel smoothing with the outer product of the OLSresiduals vectors. Different bandwidths for the different cells of the timevaryingvariance matrix are also allowed. We derive the asymptotic distributionof the proposed estimators for the VAR model coefficients and compare theirproperties. In particular we show that the ALS estimator is asymptoticallyequivalent to the infeasible GLS estimator. This asymptotic equivalence isobtained uniformly with respect to the bandwidth(s) in a given range andhence justifies data-driven bandwidth rules. Using these results we build Waldtests for the linear Granger causality in mean which are adapted to VARprocesses driven by errors with a non stationary volatility. It is also shownthat the commonly used standard Wald test for the linear Granger causalityin mean is potentially unreliable in our framework (incorrect level and lowerasymptotic power). Monte Carlo and real-data experiments illustrate the useof the different estimation approaches for the analysis of VAR models withtime-varying variance innovations.