Statistical Inference For Semiparametric GARCH-M Models

Measurement of the risk aversion has always been a hot topic in Financial Econometric.The GARCH-M model can effectively describe the relationship between conditional variance and conditional expectation,its volatility coefficient can be used to explain risk aversion.Empirical studies show that the risk aversion may be affected by some exogenous variables,so the functional coefficient GARCH-M model is constructed to explain this effect.These models can not only evaluate the market's relative risk aversion,but also quantify the impact of exogenous variables on risk aversion,so it has attracted much attention.However,the error series of the model is not observable,and the variance series is difficult to observe,so the statistical inference of this kinds of models are very difficult.By replacing the unobservable error sequence in the variance equation with the observable time-delay series,some alternative functional coefficient GARCH-M models can be constructed.Some empirical studies show that the alternative models can also measure risk aversion well,and can assess slao the time-varying characteristics of risk aversion.Therefore,the statistical inference of functional coefficient GARCH-M model is gradually becoming the focus of the current research.Empirical likelihood approach can be used for both parameter inference and nonparameter inference,and has many excellent properties,so it attracts many scholars' interest.Empirical likelihood inference does not require estimating variance,and does not assume that the datas come from a particular distribution family,so it is highly adaptable.Through various improvements and developments in recent years,empirical likelihood approach has been successfully applied to the statistical inference of various models,such as linear and generalized linear models,partial linear models,variable coefficient models and single index models.Empirical likelihood inference of the time series models and the models with dependent errors is an important research direction at present.On this basis of these studies,we studied the empirical likelihood inference of semi-parametric GARCH-M model.The main studies contents include the following aspects:(1)Empirical likelihood approach is used to test the nonparametric components of functional coefficient ARCH-M model.That is,how exogenous variables affect risk aversion.The quasi-maximum likelihood estimators of the parametric parts is given firstly,and we gain the consistency and asymptotic normality of the estimator.And then,the empirical log-likelihood ratio test statistic is constructed by using the estimation equation.It is proved to be asymptotically Chi-square distribution,and then the asymptotic confidence region of the test problem is obtained.Numerical simulation shows that the empirical likelihood test is sensitive to the alternative hypothesis.(2)Empirical likelihood approach is developed to estimate parametric and nonparametric parts in functional coeficient ARCH-M models.Firstly,The empirical log-likelihood ratio function for parameters are constructed by the least square method,and it proved to be asymptotic convexity and to be asymptotically standard Chi-squre distribution.At the same time,the maximum empirical likelihood estimation(MELE)for parameters are shown to be asymptotically Normal.Simulation study shows that the empirical likelihood method works better than the normal approximation method in terms of average areas of confidence regions for parameters.Secondly,based on the MELE of parameters,the empirical likelihood approach is again applied to estimate nonparametric parts.With the profile likelihood method,the empirical log-likelohood ratio for functional coeficient is constructed,and it is proved to be also asymptotically standard Chi-squre distribution.Thus the empirical likelihood confidence band of functional coefficient is obtained.Numerical simulation results show that the confidence band is robust,and the MELE performs well.(3)Empirical likelihood approach is used to test the risk aversion of functional coefficient GARCH-M model.That is,whether exogenous variables affect risk aversion.Based on the quasi maximum likelihood estimators of parameters,empirical likelihood ratio test statistics are constructed with the estimation equations.It is proved that this function is asymptotical standard chi-square distribution,and a rejection region with the level of significa is obtained.Simulation studies show that the empirical likelihood test is quite sensitive to the alternative hypothesis.In this dissertation,empirical likelihood approach first studies the statistical inference of semiparametric GARCH-M model,which enriches the estimation and test methods of the models and broadens the application scope of empirical likelihood method,so it has certain theoretical and applied value.