  In this paper we propose using a nonparametric model specification test for
parametric time series with long-range dependence (LRD). To establish
asymptotic distributions of the proposed test statistic, we develop new central
limit theorems for certain weighted quadratic forms of stationary time series
with LRD. To implement our proposed test in practice, we develop a
computer-intensive parametric bootstrap simulation procedure for finding
simulated critical values. As a result, our finite-sample studies demonstrate
that both the proposed theory and the simulation procedure work well, and that
the proposed test has little size distortion and reasonable power.
