On Confidence Intervals for Autoregressive Roots and Predictive Regression

Abstract

Local to unity limit theory is used in applications to construct confidence intervals (CIs) for autoregressive roots through inversion of a unit root test (Stock (1991)). Such CIs are asymptotically valid when the true model has an autoregressive root that is local to unity (ρ = 1 + c n ), but are shown here to be 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. Failure at the boundary implies that these CIs have zero asymptotic coverage probability in the stationary case and vicinities of unity that are wider than O(n −1/3 ). The inversion methods of Hansen (1999) and Mikusheva (2007) are asymptotically valid in such cases. Implications of these results for predictive regression tests are explored. When the predictive regressor is stationary, the popular Campbell and Yogo (2006) CIs 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 cautionary implications for the use of the procedures in empirical practice.

The Uniform Validity of Impulse Response Inference in Autoregressions

“…For example, Stock (1991) proposes constructing confidence intervals for the dominant autoregressive root by inverting unit root tests. Phillips (2014), however, proves that inference about the AR(1) slope parameter based on Stock's (1991) confidence interval, while asymptotically valid when the root is local to unity, has zero coverage asymptotically when the root is far enough from unity. 3 The lack of a uniform asymptotic approximation across the parameter space has undermined the profession's confidence in the accuracy of standard confidence intervals in applied work and has created interest in confidence intervals that remain asymptotically valid whether the AR (1) slope parameter is unity, close to unity or far from unity.…”