Nonparametric cointegrating regression models have been extensively used in financial markets, stock prices, heavy traffic, climate data sets, and energy markets. Models with parametric regression functions can be more appealing in practice compared to non-parametric forms, but do result in potential functional misspecification. Thus, there exists a vast literature on developing a model specification test for parametric forms of regression functions. In this paper, we develop two test statistics which are applicable for the endogenous regressors driven by long memory and semi-long memory input shocks in the regression model. The limit distributions of the test statistics under these two scenarios are complicated and cannot be effectively used in practice. To overcome this difficulty, we use the subsampling method and compute the test statistics on smaller blocks of the data to construct their empirical distributions. Throughout, Monte Carlo simulation studies are used to illustrate the properties of test statistics. We also provide an empirical example of relating gross domestic product to total output of carbon dioxide in two European countries.

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