Hamdi Raïssi

CREST-Ensai \& Irmar, France


Adaptive Inference for Linear AutoRegressive Models with Time-Varying Variance

Linear Vector AutoRegressive (VAR) models where the innovations could be unconditionally heteroscedastic and serially dependent are considered. 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 allowed. We derive the asymptotic distribution of the proposed estimators for the VAR coefficients and compare their properties. The ALS estimator is shown to be asymptotically equivalent to the infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) and hence justifies data-driven bandwidth rules. Using these results we investigate the classical Wald tests for Granger causality and the portmanteau tests when the innovations have time-varying variance and we propose new corrected versions. Finally, we address the problem of testing for the presence of ARCH effects in the univariate AR models with unconditionally heteroscedastic innovations. Noting that the squared residuals are uncorrelated when a deterministic function is used to describe the unconditional volatility, adaptive McLeod and Li's portmanteau and ARCH-LM tests for checking for second order dynamics are provided.
Moreover, the failure of the standard versions of these two tests for checking the presence of second order dynamics in series with time-varying unconditional variance is underlined.
The theoretical results are illustrated by mean of Monte Carlo experiments and considering real US economic and financial data sets.
This survey talk is based on joint work with Valentin Patilea.