Diebold-Rudebusch (1991) On the Power of Dickey-Fuller Tests Against Fractional Alternatives

Summary

A short note showing that Dickey-Fuller unit-root tests have low power against fractionally-integrated alternatives. When the true process is an ARFIMA with non-integer $d$ that is mean-reverting but not a unit root, DF tests rarely reject the $I(1)$ null, so the widespread failure to reject unit roots in macroeconomic data may partly reflect this lack of power rather than genuine unit roots.

Key Claims

• Fractional nests the unit root. The ARFIMA model $\phi(L)(1-L)^d x_t=\theta(L)\varepsilon_t$ allows any real $d$ and contains the unit root ($d=1$) as a special, potentially restrictive, case; it flexibly characterizes persistent processes.
• Low power. Applying DF tests to fractionally-integrated data ($1/2<d<3/2$), the tests have quite low power — they seldom reject the unit-root null even when the series is fractionally integrated and mean-reverting.
• Convergence-rate dependence (Sowell 1990). In the White (1958) AR(1) regression with a fractionally-integrated innovation of order $\delta=d-1$, the estimator satisfies $(\hat\beta-1)=O(T^{-1-2\delta})$; convergence is faster or slower than the standard $O(T^{-1})$ as $d\gtrless 3/4$, and the asymptotic fractional-unit-root distribution is severely misleading in all but very large samples.
• Interpretation. The pervasive non-rejection of unit roots (e.g. Nelson-Plosser 1982) may partly stem from low power against fractional alternatives; such series could be long-memory rather than $I(1)$.

Concepts Introduced or Extended

• Unit Root Tests
• Long Memory and Fractional Integration

Entities Mentioned

• Francis X. Diebold
• Glenn D. Rudebusch

Quotes

"We examine the properties of Dickey-Fuller unit root tests under fractionally-integrated alternatives and find that these tests have quite low power."

My Take

A compact but consequential note: it reframes the "unit roots everywhere" empirical regularity as possibly an artifact of a test that cannot tell $I(1)$ from mean-reverting fractional integration. Together with Diebold-Inoue (2001) on regime switching mimicking long memory, it brackets the identification problem from both sides — DF tests miss fractional alternatives, and long-memory estimators can be fooled by breaks. The practical upshot for this wiki's unit-root and long-memory pages is the same discipline: don't read a non-rejection of the unit root as evidence of an exact unit root, and consider the full fractional range.