Weighted L₁-estimates for a VAR(p) time series model

Abstract

The most common method of estimating the parameters of a vector-valued autoregressive time series model is the method of least squares (LS). However, since LS estimates are sensitive to the presence of outliers, more robust techniques are often useful. This paper investigates one such technique, weighted-L ₁ estimates. Following traditional methods of proof, asymptotic uniform linearity and asymptotic uniform quadricity results are established. Additionally, the gradient of the objective function is shown to be asymptotically normal. These results imply that the weighted-L ₁ parameter estimates for this model are asymptotically normal at rate n ^−1/2. The results rely heavily on covariance inequalities for geometric absolutely regular processes and a Martingale central limit theorem. Estimates for the asymptotic variance–covariance matrix are also discussed. A finite-sample efficiency study is presented to examine the performance of the weighted-L ₁ estimate in the presence of both innovation and additive outliers. Specifically, the classical LS estimate is compared with three versions of the weighted- L ₁ estimate. Finally, a quadravariate financial time series is used to demonstrate the estimation procedure. A brief residual analysis is also presented.

Keywords:

• asymptotic normality
• L ₁-estimates
• robust
• vector autoregressive time series
• weights

Acknowledgements

We would like to thank the reviewers of this paper, who provided many helpful suggestions. Their comments were extremely useful in preparing the revised manuscript.