  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.
