On Conﬁdence Intervals for Autoregressive Roots and Predictive Regression

Abstract

A prominent use of local to unity limit theory in applied work is the construction of conﬁdence intervals for autogressive roots through inversion of the ADF t statistic associated with a unit root test, as suggested in Stock (1991). Such conﬁdence intervals are valid when the true model has an autoregressive root that is local to unity (τ = 1 + (c/n)) but are invalid at the limits of the domain of deﬁnition of the localizing coeﬀicient c because of a failure in tightness and the escape of probability mass. Consideration of the boundary case shows that these conﬁdence intervals are invalid for stationary autoregression where they manifest locational bias and width distortion.

In particular, the coverage probability of these intervals tends to zero as c approaches -∞, and the width of the intervals exceeds the width of intervals constructed in the usual way under stationarity. Some implications of these results for predictive regression tests are explored. It is shown that when the regressor has autoregressive coeﬀicient |τ| < 1 and the sample size n approaches inﬁnity, the Campbell and Yogo (2006) conﬁdence intervals for the regression coeﬀicient have zero coverage probability asymptotically and their predictive test statistic Q erroneously indicates predictability with probability approaching unity when the null of no predictability holds. These results have obvious implications for empirical practice.

Keywords: Autoregressive root, Conﬁdence belt, Conﬁdence interval, Coverage probability, Local to unity, Localizing coeﬀicient, Predictive regression, Tightness