Nonlinear estimation using estimated cointegrating relations

The Granger-Engle procedure consists of two steps. In the first step, a lon g-run cointegrating relationship is estimated, and in the second stage, thi s estimated long-run relationship is used to estimate a distributed lag mod el. This paper establishes the limit distribution of the second-stage estim ator if the model estimated in the second stage is other than linear. One m ay expect that the estimation of the cointegrating relationship does not af fect the limit distribution of the second-stage estimator; however, it is s hown that unless a regularity condition holds, this intuition is false. Cle arly this regularity condition holds in the standard linear case. A simple example where the limit distribution changes is the addition of the square of the cointegrating relationship to the second stage distributed lag model that is estimated by least squares. Surprisingly however, it turns out tha t if a constant is included in the long-run least-squares regression, the ( possibly nonlinear) second-stage estimator will be asymptotically normally distributed. (C) 2001 Elsevier Science S.A. All rights reserved. JEL classi fication. C22; C32.