  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.
