Interval Estimation Of The Tail Index Of A Unit Root Model With GARCH Errors

In generalized autoregressive conditional heteroskedasticity (GARCH) model, relevant samples quantities such as the sample autocorrelation function and extreme value theory contain a lot of useful information about the time series. These quantities usually depend on the tail index of the related time series. Therefore, it is an important and meaningful task of estimating the tail index. Since the tail index of such a model is generally determined by the sample moment equations with the basic unknown parameters. For the case of a unit root model with GARCH errors, its tail index is also based on the sample moment equations. Tail index is estimated in two steps:first, we will estimate the basic unknown parameters by the quasi maximum likelihood method; secondly, to get into the parameter estimator and solve the sample moment equations. Tail index estimates has good convergence rate, but its asymptotic variance has a complex structure. In order to avoid the above problems and construct a confidence interval of the tail index, this paper proposes the profile empirical likelihood method. On the aspect of theoretical properties, we prove the asymptotic normality of the tail index estimator, show the existence and consistency of the solution to the profile empirical likelihood function reaching the minimum, and establish Wilkes Theorem for the proposed method. In addition, the simulation study shows that the profile empirical likelihood method works good in terms of the convergence accuracy.