Optimal tests for autoregressive models based on autoregression rank scores

Abstract

Locally asymptotically optimal tests based on autoregression rank scores are constructed for testing linear constraints on the structural parameters of AR processes. Such tests are asymptotically distribution free and do not require the estimation of nuisance parameters. They constitute robust, flexible and quite powerful alternatives to existing methods such as the classical correlogram-based parametric tests, the Gaussian Lagrange multiplier tests, the optimal non-Gaussian and ranked residual tests described by Kreiss, as well as to the aligned rank tests of Hallin and Puri. Optimality requires a nontrivial extension of existing asymptotic representation results to the case of unbounded score functions (such as the Gaussian quantile function). The problem of testing AR$(p - 1)$ against AR$(p)$ dependence is considered as an illustration. Asymptotic local powers and asymptotic relative efficiencies are explicitly computed. In the special case of van der Waerden scores, the asymptotic relative efficiency with respect to optimal correlogram-based procedures is uniformly larger than one.