Specification Testing For Time Series Models With Long-range Dependent Heteroscedastic Errors

In recent three decades,various statistics based on semiparametric and nonpara-metric techniques have been,widely applied in specifica.tion testing for independent and short-range dependent time series.Nevertheless,recent studies on economics,en-vironment and finance applications have shown that many real data sets may display long-range dependence(LRD).Consequently,in this paper we focus on the specification testing for time series models with long-range dependent heteroscedastic errors.Under cert,ain assumpt,ions,we generalize the result of time series models with homoscedastic errors.For the hypothesis testing of the unknown function m(·)in the nonlinear time series model Yt =m(Xt)+ ?(Xt)et,under the conditions that Xt?i.i.d.and {et} is a sequence of strictly stationary linear process with LR.D,we propose a general test,statistic and establish its asymptotic distribution.To implement our proposed test in practice,we develop a parametric bootstrap simulation procedure for estimating the value l? which denotes the 1-? quantile of the exact finite-sample distribution of the proposed test statistic.In order to study the asymptotic properties of the bootstrap estimation,we compare the empirical size and power function of the test statistic under different parameter settings.Finally,our finite-sample studies demonstrate that both the proposed asymptotic theory and the estimated critical value work well.