Fractional Integration and the Phillips-Perron Test

Abstract

This paper derives the asymptotic distribution of the Phillips-Perron unit root tests statistics and some of their variants under a general non-stationary fractionally-integrated I (1+d) process, for Є (-0.5,0.5). By using the Newey-West estimator of long-run variance, we show that both the Phillips-Perron’s t statistics and standardized coefficients estimator are consistent against a non-stationary but mean-reverting alternative, such as the I (1+d) process for d Є (-0.5,0). However, only the t statistic from a no-drift and no-time trend regression is consistent against a non-stationary and non-mean-reverting alternative, such as the I(1+d) process for d Є (0,0.5). Simulation results also confirm that the power of these test statistics in large samples will decrease as the lag number increases in the construction of a Newey-West estimator of the long-run variance.