2010/07/07 by Valentin Patilea, Patilea, Valentin, Hamdi Raïssi +1
Economics, Econometrics and Finance · Mathematics · #Advanced Statistical Methods and Models #Complex Systems and Time Series Analysis #FOS: Computer and information sciences #Financial Risk and Volatility Modeling #Methodology (stat.ME) #stat.ME
paper · pdf · doi:10.48550/arxiv.1007.1193
openalex publication_date 2010/07/07 · arxiv created 2010/07/08 · arxiv updated 2010/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.