Statistical Inference For Semiparametric Time Series Models Driven By Covariates

Linear time series models are popular in analyzing records that are collected over time.However,they are not useful in dealing with time series data with nonlinear features due to the limitation of linear time series models.To model such features objectively and effectively,statisticians appeal to nonlinear time series models.In this dissertation,two types of semiparametric nonlinear time series models are discussed,including single-index-driven autoregressive model with linear explanatory variables and single-index-driven autoregressive model with varying-coefficient explanatory variables.And the main contents of this dissertation are presented as follows:Single-index varying-coefficient model is a class of significant semiparametric model,which can avoid the "curse of dimensionality",and is easily to be interpreted in practical applications because of the features in model structure.The combination of single-index varying-coefficient model and autoregressive model with explanatory variables becomes single-index-driven autoregressive model with linear explanatory variables.The model can be seen as an extension of partially linear single-index varyingcoefficient model in the context of time series analysis.In this model,the autoregressive component is driven by a single-index structure showing the impact of covariate on the observable time series,while the linear part represents that of another covariate.Therefore,the model is a powerful tool to fit time series data influenced by large amounts of covariates.Estimators for the unknown nonparametric part and the unknown parametric part of the model are obtained based on the local linear smoothing method and the least squares method.Meanwhile,the iterative algorithm to implement the estimation procedure is introduced,and the asymptotic normality of the estimators are proved.Simulation studies and a real data example illustrate the efficiency of the estimation method.Single-index-driven autoregressive model with linear explanatory variables is flexible and adaptable,in which the linear effect of covariate on the observable time series is revealed intuitively by the part of linear explanatory variables.Nevertheless,in practical situations,the rate for this covariate to influence the observable time series may be varying.That is to say,this covariate has a special kind of interaction with some other covariates.Hence,single-index-driven autoregressive model with varyingcoefficient explanatory variables is proposed in this dissertation7 generalizing the part of linear explanatory variables to varying-coefficient explanatory variables.In order to attain the estimation of unknown nonparametric part and the unknown parametric part in the new model,the link function and the coefficient function are approximated locally by linear functions at first.Then,the quasi-initial estimators for unknown functions can be derived by using the local linear method and the average method.Finally,the local linear method and the least squares method,together with the backfitting algorithm are employed to get the estimators for unknown functions and unknown parameter.The iterative algorithm is described in detail to perform the estimation procedure,and the asymptotic normality of the estimators are established.Numerical studies verify that the estimation method is feasible.Although single-index-driven autoregressive model with linear explanatory variables has an advantage in characterizing the relationship between covariates and the observable time series,some components of covariates may be unimportant in actual applications,and they are not significant statistically.Variable selection procedure can make the model more concise and accurate by selecting significant variables and deleting nonsignificant variables in the components of covariates,thus improving the validity of the model.So,for single-index-driven autoregressive model with linear explanatory variables,variable selection for parametric part is conducted.Combining the local linear method and the penalized least squares method,the selection of significant parametric components and the estimation of unknown parameter can be accomplished simultaneously.The iterative algorithm to construct the local estimator for unknown function and the penalized estimator for unknown parameter is summarized.Besides,the consistency and oracle property of the penalized least squares estimator are provided.Simulation studies demonstrate that the variable selection procedure is workable.