Testing of symmetry of innovations in autoregression

Abstract: We consider a stationary linear $AR(p)$ model with zero mean. The autoregression parameters as well as the distribution function (d.f.) $G(x)$ of innovations are unknown. We consider two situations. In the first situation the observations are a sample from a stationary solution of $AR(p)$. Interesting and essential problem is to test symmetry of $G(x)$ with respect to zero. If hypothesis of symmetry is valid then it is possible to construct nonparametric estimators of $AR(p)$ parameters, for example, GM-estimators, minimum distance estimators and others. First of all we estimate unknown parameters of autoregression and find residuals. Based on them we construct a kind of empirical d.f., which is a counterpart of empirical d.f of the unobservable innovations. Our test statistic is the functional of omega-square type from this residual empirical d.f. Its asymptotic d.f. under the hypothesis and the local alternatives are found. In the second situation the observations subject to gross errors (outliers). The distribution of outliers is unknown, their intensity is $O(n^{-1/2})$, $n$ is the sample size. We test the symmetry of innovations again but by constructing the Pearson's type statistic. Its asymptotic d.f. under the hypothesis and the local alternatives are found. We establish the asymptotic robustness of this test as well.