Optimal Fractional Dickey–Fuller tests

Abstract

This article analyzes the fractional Dickey–Fuller (FDF) test for unit roots recently introduced by Dolado, Gonzalo and Mayoral (2002 Econometrica 70, 1963–2006) within a more general setup. These authors motivate their test with a particular analogy with the Dickey–Fuller test, whereas we interpret the FDF test as a class of tests indexed by an auxiliary parameter, which can be chosen to maximize the power of the test.Within this framework, we investigate optimality aspects of the FDF test and show that the version of the test proposed by these authors is not optimal. For the white noise case, we derive simple optimal FDF tests based on consistent estimators of the true degree of integration. For the serial correlation case, optimal augmented FDF (AFDF) tests are difficult to implement since they depend on the short-term component. Hence, we propose a feasible procedure that automatically optimizes a prewhitened version of the AFDF test and avoids this problem.