The impact of fat-tailed distributions on some leading unit roots tests

Abstract

There is substantial evidence that many time series associated with financial and insurance claim data are fat-tailed, with a (much) higher probability of " outliers' compared with the normal distribution. However, standard tests, or variants of them, for the presence of unit roots assume a normal distribution for the innovations driving the series. Application of the former to the latter therefore involves an inconsistency. We assess the impact of this inconsistency and provide information on its impact on inference when innovations are drawn from the Cauchy and sequence of t(v) distributions. A simple prediction that fat tails will uniformly lead to over-sizing of standard tests (because the fatness in the tail translates to the test distribution) turns out to be incorrect: we find that some tests are over-sized but some are under-sized. We also consider size retention and the power of the Dickey-Fuller pivotal and normalized bias test statistics and weighted symmetric versions of these tests. To make the unit root testing procedure feasible, we develop an entropy-based test for some fat-tailed distributions and apply it to share prices from the FTSE100.