Adaptive estimation of vector autoregressive models with time-varying variance: application to testing linear causality in mean

TL;DR

This paper develops adaptive estimation methods for vector autoregressive models with time-varying variance, enabling more reliable testing of linear causality in mean under heteroscedasticity.

Contribution

It introduces ALS estimators that adaptively estimate time-varying variance, and demonstrates their asymptotic equivalence to infeasible GLS estimators, improving causality testing.

Findings

ALS estimator is asymptotically equivalent to GLS

Standard Wald tests may be unreliable under heteroscedasticity

Monte Carlo simulations validate the proposed methods

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. 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.