Robust inference in conditionally heteroskedastic autoregressions

We consider robust inference for an autoregressive parameter in a stationary linear autoregressive model with GARCH innovations. As the innovations exhibit GARCH, they are by construction heavy-tailed with some tail index κ. This implies that the rate of convergence as well as the limiting distribution of the least squares estimator depend on κ. 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 based on least squares estimates for a fixed number of groups of the original sample. The merit of this approach is that no knowledge about the value of κ nor about the rate of convergence and the limiting distribution of the least squares estimator is required. We verify that the two-sided t-test is asymptotically a level α test whenever α≤5% for any κ≥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.