Long–range Dependent Time Series Specification

Model specification of short–range dependent stationary time series has become a very active research field in both econometrics and statistics since about two decades ago. In the meantime, estimation of long–range dependent stationary time series models has also been quite active. To the best of our knowledge, however, model specification of stationary time series with long–range dependence (LRD) has not been discussed in the literature. This is probably due to unavailability of certain central limit theorems for weighted quadratic forms of stationary time series with LRD. In this paper we try to tackle such difficult issues by establishing a nonparametric model specification test for parametric time series with LRD. In order 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. In order to implement the 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 show that both the proposed theory and the simulation procedure work well and that the proposed test has little size distortion and reasonable power.