Nonlinear estimators with integrated regressors but without exogeneity

This paper analyzes nonlinear cointegrating regressions as have been recently analyzed in a paper by Park and Phillips in Econometrica. I analyze the consequences of removing Park and Phillips’ exogeneity assumption, which for the special case of a linear model would imply the asymptotic validity of the least squares estimator for linear cointegrating regressions. For the linear model, the unlikeliness of such an exogeneity assumption to hold in practice has inspired the “fully modified” technique, the “leads and lags” technique, and Park’s “canonical regressions”. In this paper, a “fully modified” type technique is proposed for nonlinear cointegrating regressions. The mathematical tool for proving this result is a new so-called “convergence to stochastic integrals” result. This result is proven for objects that are summations of a stationary random variable times an asymptotically homogeneous function of an integrated process. The increments of the integrated process are allowed to be correlated with the stationary random variable. This result is derived by extending results by Chan and Wei and by de Jong and Davidson. ∗The comments of Peter Phillips and Alex Maynard on an earlier version of this paper are gratefully acknowledged. 1