Endogeneity in Nonlinear Regressions with Integrated Time Series

Abstract

This article considers the nonlinear regression with integrated regressors that are contemporaneously correlated with the regression error. We, in particular, establish the consistency and derive the limit distribution of the nonlinear least squares estimator under such endogeneity. For the regressions with various types of regression functions, it is shown that the estimator is consistent and has the same rate of convergence as for the case of the regressions with no endogeneity. Whether or not the limit distribution is affected by the presence of endogeneity, however, depends upon the functional type of the parameter derivative of regression function. If it is asymptotically homogeneous, the limit distribution of the nonlinear least squares estimator has an additional bias term reflecting the presence of endogeneity. On the other hand, the endogeneity does not have any effect on the nonlinear least squares limit theory, if the parameter derivative of regression function is integrable. Regardless of the presence of endogeneity, the least squares estimator has the same limit distribution in this case. To illustrate our theory, we consider the nonlinear regressions with logistic and power regression functions with integrated regressors that have contemporaneous correlations with the regression error.