An Empirical Likelihood Goodness-of-Fit Test For Time Series

The goodness-of-fit test for time series is vital to statistics. The problem of testing a parametric mean regression against a nonparametric alternative is not new for an independent and identically distributed setting. Yet for dependent setting more efforts arc needed to complete relevant theory. Owen(1998) produced Empirical Likelihood as a nonparametric inference method for complete samples. It has sample property like bootstrap. Compared to classical statistics, this method has many prominent ad-vantages such as Wilk's property and Barlett correctability. Based on the predecessor, Chen, Hardlc and Li(2003) made full use of the advantages of empirical likelihood and improved the resolution of the goodness-of-fit test.Due to the prominent properties of the local polynomial method for estimating the conditional mean function m(x)=E(Y|X=x), in this thesis, we improved the method of Chen, Hardle and Li(2010) by replacing NW estimation with local polyno-mial estimation for the conditional mean function. Moreover, we produced the empir-ical likelihood of estimators and proved a lemma to estimate the empirical likelihood ratio. Based on the ratio we built a test statistics and compared it with HM statistics. Thus some advantages of the empirical likelihood statistics could be found. In the end, we adopted theoretical asymptotic distribution and bootstrap method respectively to estimate the distribution of the empirical likelihood statistics and thus get the rejection region. Consequently, one can expect to have a better resolution of the goodness-of-fit test.