Nonparametric Hypothesis Testing For Heavy-tailed Time Series

The phenomena of heavy tails exit in many area of science,such as insurance and economics.This thesis mainly focuses on the nonparametric hypothesis testing for heavytailed time series.We study the hypothesis testing for errors using empirical likelihood method in the procedure of fitting an ARMA-GARCH model when the errors have infinite fourth moment,as well as the Portmanteau-type test for unit root when errors have infinite variance.This thesis consists of following parts.First,in the procedure of fitting an ARMA-GARCH model,the self-weighted quasi maximum exponential likelihood estimation(SWQMELE)is often employed instead of the quasi maximum likelihood estimation(QMLE)to reduce the moment constraints.However,SWQMELE requires the errors to have zero median instead of zero mean which is assumed by QMLE.Because changing zero mean to zero median destroys the ARMAGARCH structure and has a serious effect on skewed data,we consider an efficient empirical likelihood test for zero mean of errors in the application of SWQMELE to ensure that the model still concerns conditional mean.We derive the limit distributions of the test under both null hypothesis and local alternative hypothesis under mild assumptions.A simulation study confirms the good finite sample performance before applying the test to the US housing price indexes and financial returns for the study of comovement.Second,to improve the robustness against heteroscedasticity and heavy tails,we employ the empirical likelihood test for testing zero median of errors by using the GARCH structure to reduce the moment effect without estimating the GARCH model.The limit distribution of the test under null hypothesis and local alternative hypothesis are developed as well.The effectiveness of the proposed test is confirmed by simulation study.The data analysis shows that some financial returns do not have zero median of errors,which cautions the use of the SWQMELE.Third,to improve the robustness of the test for zero mean of the errors,we also consider to use a random weighted bootstrap method combined with the idea of using the GARCH structure to reduce the moment effect without estimating the GARCH model.Asymptotic theory for our proposed test has been developed under both null hypothesis and local alternative hypothesis.Empirical analysis on several exchange rates shows the hypothesis testing before fitting the model is necassary.Finally,the above study focuses on ARMA-GARCH model whose errors have infinite fourth moment and finite variance.To further study the time series with infinite-variance errors,we focus on the test for the existence of the unit-root of an autoregressive model under a heavy-tailed linear noise,where the noise ?t=??j=0dj?t-j and {?t} is a sequence of i.i.d.innovations belonging to the domain of attraction of an ?-stable distribution for some ? ?(0,2).We propose a Portmanteau-type statistic based on the sample covariance which is nonparametric and easy to implement.Under certain regular assumptions,the limit distribution of the statistics is shown to be a functional of a standard stable distribution.The good finite sample studies show that the new test outperforms conventional unit root tests in size and power.Applications to the daily world crude oil price and 3-month AA financial commercial paper rate are also given to illustrate the performance of the new test.