Empirical Likelihood For Partial Linear Models With ARCH Errors

Empirical likelihood method is a nonparametric statistical inference method by Owen and has well asymptotical properties. How to use this method for the statistical inference of the partially linear models with the ARCH errors is a hot issue. Although some literatures using maximum likelihood estimation method to structure the empirical likelihood estimation, the fourth order moments of the error term is limited. This requirement is too harsh for the financial time series. Thus, the literature use LAD estimation method to structure the empirical likelihood estimation and respectively deduce asymptotic properties of the LAD estimator and the empirical likelihood ratio statistic under the condition of the error term is fat-tailed distribution. Finally, this article has carried on the monte carlo simulation and simulate the coverage probabilities of confidence regions for the two methods. According to the simulation results compare the advantages of the two methods. This thesis will mainly discuss the following several aspects:Firstly, we structure parameters objective function of the parameters for the partially linear models with the ARCH errors. Then, we calculate the minimum value of the objective function and get the least absolute deviation estimator.Secondly, according to the martingale central limit theorem and the ergodic theorem, we prove the asymptotic normality of the LAD estimation and give asymptotic normal confidence region.Thirdly, we structure the empirical likelihood ratio statistic based on the LAD estimation. Then, we prove the asymptotic normality of the empirical likelihood ratio statistic and give empirical likelihood confidence region.Fourthly, we simulate the data and calculate the coverage probabilities of confidence regions. According to compare, it is concluded that the empirical likelihood method has very well the advantages.