Robust Inference in Autoregressions With Multiple Outliers

Abstract

We consider robust methods for estimation and unit root [UR] testing in autoregressions with infrequent outliers whose number, size and location can be random and unknown. We show that in this setting standard inference based on OLS estimation of an augumented Dickey-Fuller [ADF] regression may not be reliable, since (i) clusters of outliers may lead to inconsistent estimation of the autoregressive parameters, and (ii) large outliers induce a jump component in the asymptotic null distribution of UR test statistics. In the benchmark case of known outlier location, we discuss why the augmentation of the ADF regression with appropriate dummy variables not only ensures consistent parameter estimation, but also gives rise to UR tests with significant power gains, growing with the number and the size of the outliers. In the case of unknown outlier location, the dummy based approach is compared with a robust, mixed Gaussian, Quasi Maximum Likelihood [QML] approach, novel in this context. It is proved that, when the ordinary innovations are Gaussian, the QML and the dummy-based approach are asymptotically equivalent, yielding UR tests with the same asymptotic size and power. Moreover, as a by-product of QML the outlier dates can be consistently estimated. When the innovations display tails fatter than Gaussian, the QML approach ensures further power gains over the dummybased method. Simulations show that the QML ADF-type t-test, in conjunction with standard Dickey-Fuller critical values, yields the best combination of finite-sample size and power.

Robust estimation and inference for heavy tailed GARCH

“…This is not surprising since true additive outliers render QML biased (see Mendes [18], Muler and Yohai [47], cf. Cavaliere and Georgiev [13], Muler et al [46]).…”

Exploiting Infinite Variance Through Dummy Variables in Nonstationary Autoregressions

“…Their set up covers both autoregressions with a unit (or local-to-unit) root and stationary autoregressions, and large-sample properties are obtained under the assumption of …nite fourth moments of the innovations. Near-UR autoregressions augmented with dummy variables have also been analysed in Cavaliere and Georgiev (2009), where it is shown that when …nite-variance innovations are contaminated by infrequent, large outliers, the inclusion of dummy variables increases the e¢ ciency of the AR parameter estimator (leaving the consistency rate unchanged) and gives rise to UR tests with signi…cant power gains.…”

Model-based asymptotic inference on the effect of infrequent large shocks on cointegrated variables

“…. , Γ l is affected; see Cavaliere and Georgiev (2009) for a univariate result. Our estimators are still consistent in this case, and Gaussian asymptotic inference applies.…”