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