Empirical Likelihood For Some Time Series Models

Empirical likelihood has been studied extensively in the literature because of its general-ity and effectiveness. In this paper, different types of mixing data, namely, the m-dependent. martingale, p-mixing and a-mixing sequences are discussed. By the different properties of mixing, the asymptotic normality and the convergence speed of different estimating function are also discussed. In particular, we apply empirical likelihood to two classes of nonlinear time series models, that is TAR models and bilinear models, different estimating equations for both models are structured. By the properties of the estimating functions, the empirical likelihood estimations are asymptotically normal distributed, and the corresponding empir-ical likelihood ratio statistic are asymptotically chi-square distributed. Furthermore, time dependency of data is a significant characteristics of time series, the parameters of the model are affected by historical data, forming a certain relationship of size, then we should consider the situation when the parameters satisfy order restraint.In the second chapter, empirical likelihood are used for moving average models. Using the properties of m-dependent, we structure a suitable function for the model, and then get the asymptotic normality and the convergence speed of the estimating function. Under certain conditions, the empirical likelihood estimation is asymptotically normal distributed, and the corresponding empirical likelihood ratio statistic is asymptotically chi-square distributed.In the third chapter, autoregressive moving average models are considered by construct-ing an estimating equation. The asymptotic normality and the convergence speed of the estimating function are given by the central limit theorem and the law of the iterated loga-rithm of p-mixing sequences, separately. The asymptotic normality of the empirical likelihood estimation and the asymptotic chi-square distribution of the test statistics are also proved.In the fourth chapter, empirical likelihood are applied to threshold autoregressive mod-el. By constructing estimating equations, the estimation and test for the parameters of the model are given. Using the central limit theorem of martingale and the law of the iterated logarithm of a-mixing sequences, the asymptotic distribution and convergence speed of the estimating function are discussed. Under certain assumptions, we prove that the estima-tion is asymptotically normal distributed and the test statistic is asymptotically chi-square distributed.In the fifth chapter, the empirical likelihood methods are used for simple bilinear mod-els. Both estimating and testing problem of the parameters are discussed by constructing estimating equations. Using the properties of α-mixing sequences, we get the asymptotic properties and convergence speed of the estimating function. Then the limit properties of both empirical likelihood estimation of parameters and empirical likelihood ratio statistics are discussed.In the sixth chapter, we discuss the empirical likelihood when parameters with some order restraint. When using time series, the parameters of models are always affected by the time. We consider the model with weakly dependent processes. When the parameters are under some order restraint, the empirical likelihood estimating and testing problems are discussed. The expression of the constrained empirical likelihood estimator is given. The asymptotic properties of the estimator are also given. In the end, we prove that the corresponding testing statistic is asymptotically weighted mixture chi-square distributed.