The asymptotic size and power of the augmented Dickey–Fuller test for a unit root

ABSTRACT

It is shown that the limiting distribution of the augmented Dickey–Fuller (ADF) test under the null hypothesis of a unit root is valid under a very general set of assumptions that goes far beyond the linear AR(∞) process assumption typically imposed. In essence, all that is required is that the error process driving the random walk possesses a continuous spectral density that is strictly positive. Furthermore, under the same weak assumptions, the limiting distribution of the ADF test is derived under the alternative of stationarity, and a theoretical explanation is given for the well-known empirical fact that the test's power is a decreasing function of the chosen autoregressive order p. The intuitive reason for the reduced power of the ADF test is that, as p tends to infinity, the p regressors become asymptotically collinear.

KEYWORDS:

• Autoregressive representation
• hypothesis testing
• random walk

JEL CLASSIFICATION:

• C21
• C12

Acknowledgments

The authors are grateful to Essie Maasoumi, Peter Phillips, David Stoffer and Adrian Trapletti for helpful suggestions, and to Nan Zou for computational support. Many thanks are also due to Li Pan and to the students of the MATH181E class at UCSD in Fall 2012 for their valuable input on the real data example of Section 3.