Weighted L_1-estimates for a VAR(p) time series model

摘要

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 quadric-ity 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.