Research On Semiparametric Models

High dimensional data modelling and analysis have been a large literature in statistics and econometrics.Semiparametric models,as an important regression analysis tool,generally involves estimating unknown smooth functions and unknown finite dimensional parameters.Specifically,semiparametric models,often projecting the unknown functions on lower dimensional parametric space,on the one hand,can tackle the model misspecification problem suffered by parametric models and avoid the ”curse of dimensionality”problem encountered by nonparametric models,and on the other hand,are capable of retaining high estimation efficiencies and possessing good modelling interpretations.During the last decades,single index models(multi-index models)and varying coefficient models,two important semiparametric types,have received extensive developments in theory as well as in fertitle applications covering various science fields.This dissertation focuses on modelling extensions,metheodology improvements and theoretical developments of these two models.The second chapter concerns estimating the asymptotic distribution of the parameter index estimator in the single index conditional mean model.It is well known that the asymptotic variance of many parameter index estimators,such as the minimum average variance estimator(MAVE),admits a sandwitch formula which involves many unknown quantities.Thus,to estimate the variance,one has to plug-in estimate all the nuisance quantities,which however is cumbersome and not prefered.We propose the MAVE based wild bootstrap method to deal with this problem.Theoretically,we prove that the wild bootstrap estimator of the parameter index shares the same asymptotical normality of the original MAVE estimator.Thus,as long as enough bootstraps are conducted,the asymptotical distribution can be accurately estimated.As an application of the consistency result,we propose a Wald type test for the parameter index,and show that the bootstrap based test is more powerful than the same test which is based on the traditional plug-in variance estimator through simulated experiments.A real horse mussel data analysis also demonstrates our bootstrap strategy.Variance function modelling and estimation are another critical and challenging issues.The remaining of this thesis is devoted to modelling the variance function with single index(multi-index)and varying coefficient structures,and providing their corresponding valid estimation methods.The following three chapters are summerized.The third chapter investigates estimation methods for the single index conditional variance function.We introduce two estimators of the single index parameter vector through maximizing quasi-likelihood functions.The resulting parameter index estimators are respectively called the quasi-likelihood OPG(LOPG)estimator and the quasilikelihood MAVE(LMAVE)estimator,both of which can achieve root-n consistency.Our proposed methods can also guarantee the positivity of the variance function estimator.We show that the proposed methods can estimate the conditional variance with the same asymptotical efficiency as if the conditional mean function is given.Asymptotical distributions of the proposed estimators are also derived.Simulation experiments on comparisons with some exisiting estimatin methods and an application to hitters’ salary data demonstrate our proposed estimation approaches.The fourth chaper extends the OPG based method proposed in the previous chapter to estimating the multi-index conditional variance model.Specifically,we also maximize the quasi-likelihood function,which is equivelent to minimizing a penalized loss function,and obtain the multi-index matrix estimator via OPG.Moreover,a version of the ”leave-one-out” cross-validation method is employed to estimate the structure dimensionality.Simulation studies assess the sample performance of our proposed methods.In the fifth chapter,we put forward a noval varying coefficient volatility model and give its estimation method.The proposed volatility model links the varying coefficient part with an unknown smooth variance function.Based on the model’s identification conditions,the estimation procedures which are also based on maximizing quasi-likelihood functions include estimating the unknown direction and norm of the varying coefficient function and estimating the unknown conditional variance function.The resulting variance function estimator can also guarantee nonnegativity.Asymptotical distributions of all the functional estimators are derived.Simulation studies indicate that our proposed estimation methods are valid.An application to stock returns also illustrates our proposed model and methods.