Stochastic local and moderate departures from a unit root and its application to unit root testing

Local-to-unity and moderate-deviations specifications have been popular alternatives to unit root modeling. This article considers another kind of departures from a unit root, of the form $c{v}_t/T^{\beta }$, where $v_t$ is random and $\beta$ determines the distance from a unit root. We classify the stochastic departures into two types: local and moderate. This classification task is completed by investigating the asymptotic behavior of unit root tests that assume the stochastic unit root (STUR) processes as the alternative hypothesis. The stochastic local-to-unity model arises when $\beta =3/4$; in this case, the test statistics have limiting distributions different from those under the unit root null, and their asymptotic powers are greater than size. Moderate deviations emerge when $1/2\le \beta <3/4$, in which case the test statistics diverge. We also propose new tests for a unit root against an STUR, whose construction is based on the limit theory developed in this article. To evaluate the performance of these new tests, we derive the limiting Gaussian power envelope under the local alternative from an approximate model.