ADAPTIVE ESTIMATION OF VECTOR AUTOREGRESSIVE MODELS WITH TIME VARYING VARIANCE: APPLICATION

Niveau: Supérieur, Doctorat, Bac+8ADAPTIVE ESTIMATION OF VECTOR AUTOREGRESSIVE MODELS WITH TIME-VARYING VARIANCE: APPLICATION TO TESTING LINEAR CAUSALITY IN MEAN Valentin Patileaa and Hamdi Raïssib? a Université européenne de Bretagne, IRMAR-INSA & CREST (Ensai) b Université européenne de Bretagne, IRMAR-INSA First version January 2010 This version December 2010 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 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.square estimation innovation x˜t?1x˜ ?t?1 square causality tests als estimator wald tests stationary volatility linear causality