Unit Root Testing in Heteroskedastic Panels using the Cauchy Estimator

The so-called Cauchy estimator uses the sign as instrument for the first lag in autoregressions, and the resulting t-type statistic has a standard normal distribution even in the unit root case. Thus, nonstandard asymptotics of the usual unit root tests such as the augmented Dickey-Fuller [ADF] test can be avoided. Moreover, the ADF test is affected by unconditional heteroskedasticity asymptotically; but the paper shows that, by using as instruments nonlinear transformations asymptotically homogenous of order 0 (of which the sign is one), asymptotic normality of the IV t-type statistic is maintained under unconditional heteroskedasticity when the series to be tested has no deterministic trends. No estimation of the so-called variance profile and no bootstrap application are required to this end, unlike for the ADF test. When adjusting the differences for deterministic components, however, the null distribution of the Cauchy test for a unit root becomes non-standard, reminiscent of the ADF test. In fact, an examination of the asymptotic distribution of the Cauchy test under a sequence of local alternatives reveals that it has power in the same 1/T neighborhoods as the the ADF test, irrespective of whether a deterministic trend is present in the data or not. The standard normality of the examined Cauchy test can be exploited to build a panel unit root test under cross-sectional dependence with an orthogonalization procedure. The panel test does not require any N asymptotics to establish the limiting distribution, but the paper's analysis of the joint N,T asymptotics for the panel statistic suggests that N should be smaller than T. To render the test applicable when the number of cross-sectional units is larger than the number of time observations, shrinkage estimators of the involved covariance matrix are used. The performance of the discussed procedures is found to be satisfactory in finite samples.