  This article develops nonparametric cointegrating regression models with
endogeneity and semi-long memory. We assume 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 that are 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 convergence rate of the proposed test is
parameter dependent, and its limit theory involves the local time of fractional
Brownian motion. The present paper modifies the test statistic proposed for the
long memory case by Wang and Phillips (2016) to be suitable for the semi-long
memory case. With this modification, the limit theory for the test involves the
local time of standard Brownian motion. Through simulation studies, we
investigate properties of nonparametric regression function estimation with
semi-long memory regressors as well as long memory regressors.
