Nonlinear estimation using estimated cointegrating relations

The Granger–Engle procedure consists of two steps. In the first step, a long-run cointegrating relationship is estimated, and in the second stage, this estimated long-run relationship is used to estimate a distributed lag model. This paper establishes the limit distribution of the second-stage estimator if the model estimated in the second stage is other than linear. One may expect that the estimation of the cointegrating relationship does not affect the limit distribution of the second-stage estimator; however, it is shown that unless a regularity condition holds, this intuition is false. Clearly 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 that if a constant is included in the long-run least-squares regression, the (possibly nonlinear) second-stage estimator will be asymptotically normally distributed.