Functional Central Limit Theorem approximations and the distribution of the Dickey–Fuller test with strongly heteroskedastic data

We simulate the small-sample distribution of the Dickey–Fuller (DF) test with data generated from various GARCH(1,1) processes where the parameters α and β are close to the boundary of integration. As the length of the sample increases, the small-sample distributions of the DF test converge slowly to the asymptotic one, and the convergence is even slower as α+β approaches unity. This suggests that, with strongly heteroskedastic data, we must use caution when relying on asymptotic tools that use the Functional Central Limit Theorem (FCLT). Indeed, with close-to-integrated GARCH(1,1) data, the asymptotic DF critical values lead to grossly oversized tests.