Economics Letters 45 (1994) 267-271 016%1765/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved 267 Efficiency of least-squares-estimation of polynomial trend when residuals are autocorrelated Ralf Busse, Roland Jeske, Walter Kriimer* zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM Department of Statistics, University of Dortmund, P.O. Box 50 05 00, D-46@ Dortmund 50, Germany Received 17 February 1993 Accepted 18 September 1993 We derive lower bounds for the efficiency of OLS relative to GLS in the polynomial trend model where disturbances are stationary AR( 1). JEL classification : C22 1. Introduction Regressing data on polynomial trends is common practice economics [see, for example, Shiller (1981) and the subsequent in various branches of empirical variance bounds literature, where logs of share prices are often regressed on time, and many other applications in empirical finance and economics]. More often than not, the disturbances in such regressions are acknowledged to be serially dependent, but estimation is still done by OLS. Therefore there appears to be interest in the relative efficiency of OLS in polynomial regressions, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK yr=Po+P*t+...+PKfK+U, (t=l,. ..,T), (I) where the disturbances U, are stationary but not necessarily serially independent. It has long been known [see, for example, Grenander (1954)] that in this model OLS is in general asymptotically as efficient as GLS when sample size zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ T tends to infinity, but lower bounds for the relative efficiency of OLS in finite samples are only known for very special cases. Below we assume stationary AR(l) disturbances u,, where U,=P%l+eI> lPl<l, (2) and where the disturbance covariance matrix V is given by ui, = IJ:~‘~-“. The covariance matrix of the OLS coefficient estimator b = (X’X))‘Xy (where x,; = t’, t = 1,. . . , T; i = 0, . . , K) is then cov(& = (X’X)_‘x’vx(x’x)-‘, and the covariance- matrix of (x’v-‘x)-‘x’v-’ y is cov(p”) = (X’V-‘X)-l. The relative efficiency * Corresponding author. SSDI 0165-1765(94)00427-4 the GLS estimator j!?= of OLS, defined as