The effect of additive outliers on a fractional unit root test

Abstract

It is well known that additive outliers that occur with a small probability have a bias effect on the asymptotic distribution of classical unit root statistics. This paper shows that such outliers do not affect the asymptotic distribution in the case where the error term is fractionally integrated of order d, where $$0 < d < 1/2$$ , while there is a bias for the case $$-1/2 < d < 0$$ . Convergence to the asymptotic distribution is slow, such that the bias effect of outliers may be important in finite samples, for which numerical evidence is provided. We then show that these results essentially do not change if the unknown d is replaced by a consistent estimator, which may have a slow rate of convergence. Such an estimator can be obtained using first an outlier correction procedure, and then estimate d for the outlier-corrected data. We finally apply our results to a realized volatility series of the S&P 500 for which we find evidence against the unit root hypothesis, as opposed to a procedure which neglects outliers.