Efficient Detrending in the Presence of Fractional Errors∗

This paper examines the efficiency gains of GLS detrending over OLS detrending in the model yt = z′ tγ + ut, where z ′ t = (1, t, ..., t k−1) and ut ∼ I(d), −1/2 < d < 3/2. A famous theorem on trend removal by OLS regression (usually attributed to Grenander and Rosenblatt (1957) gave conditions for the asymptotic equivalence of GLS and OLS in deterministic trend estimation with I(0) errors. When the errors are fractionally integrated of order d 6= 0, this asymptotic equivalence no longer holds. In this case, the asymptotic relative efficiency depends initialization. Under infinite past initialization, the GLS estimator, when consistent, is not only asymptotically more efficient for any value of d but also converges faster by a rate of √ log(T ) when d = k − 1/2, k = 0, 1, 2, .... Under the origin initialization, i.e. ut = 0, for t ≤ 0, GLS still achieves substantial efficiency gain, but in general, it does not enjoy a faster convergence rate for any d. Two exceptions are (1) a trending variable is of order td−1/2 and the GLS of this specific trend converges faster than OLS by a rate of √ log(T ). (2) d = −0.5 and GLS estimators of all coefficients converge faster than the OLS by a rate of √ log(T ) JEL Classification: C13; C22