Nonparametric Cointegrating Regression Functions with Endogeneity and Semi-Long Memory

This article develops nonparametric cointegrating regression models with endogeneity and semi-long memory. We assume that semi-long memory is produced in the regressor process by tempering of random shock coefficients. The fundamental properties of long memory processes are thus retained in the regressor process. Nonparametric nonlinear cointegrating regressions with serially dependent errors and endogenous regressors driven by long memory innovations have been considered in Wang and Phillips (2016). That work also implemented a statistical specification test for testing whether the regression function follows a parametric form. The limit theory of test statistic involves the local time of fractional Brownian motion. The present paper modifies the test statistic to be suitable for the semi-long memory case. With this modification, the limit theory for the test involves the local time of the standard Brownian motion and is free of the unknown parameter d. Through simulation studies, we investigate the properties of nonparametric regression function estimation as well as test statistic. We also demonstrate the use of test statistic through actual data sets.