Robust inference in conditionally heteroskedastic autoregressions

We consider robust inference for an autoregressive parameter in a stationary autoregressive model with GARCH innovations when estimation is based on least squares estimation. As the innovations exhibit GARCH, they are by construction heavy-tailed with some tail index $\kappa$. The rate of consistency as well as the limiting distribution of the least squares estimator depend on $\kappa$. In the spirit of Ibragimov and Müller (“t-statistic based correlation and heterogeneity robust inference”, Journal of Business & Economic Statistics, 2010, vol. 28, pp. 453-468), we consider testing a hypothesis about a parameter based on a Student’s t-statistic for a fixed number of subsamples of the original sample. The merit of this approach is that no knowledge about the value of $\kappa$ nor about the rate of consistency and the limiting distribution of the least squares estimator is required. We verify that the one-sided t-test is asymptotically a level $\alpha$ test whenever $\alpha \le $ 5% uniformly over $\kappa \ge 2$, which includes cases where the innovations have infinite variance. A simulation experiment suggests that the finite-sample properties of the test are quite good.