Some Research On Functional Regression Model

In past decades,with the rapid development of technology and improvement of data collection,big data especially functional data(FD)that are in the form of curves have emerged in the fields of finance,economy,biological information,medicine,meteorological environment,human body kinematics and so on.Because FD is a continuous function in time,space or other dimensions,the traditional multivariate analysis method is no longer applicable.Therefore,functional data analysis(FDA)has become one of the international hotspots and frontier issues in current academic research.Functional linear regression is an important tool in FDA and has attracted the attention of researchers.Its results not only has important theoretical significance,but also has a wide range of applications.We focus on statistical inference problems and the application for some functional regression models.The main content is as follows:(1)We study the application of functional linear model in financial field.Through examining the implied volatility curves of the individual stocks in US market,we find some of the curves possess very bizarre shapes,which are not linear or quadratic.This motivates us to use some nonparametric smoothing methods to fit the implied volatility curve based on the observed option price data,and then study its economic determinants from the perspective of functional data.We propose the functional Fama-Mac Beth regression method,which contains a cross-sectional functional regression first and then a point to point -test for the regression coefficient function.Then,we use the option and stock data from Jan 1996 to Dec 2015 for empirical analysis to illustrate advantages of our proposed method via comparing the results in the current literatures.(2)We investigates the series correlation test of error term in the partially functional linear regression model.Financial and economic data often have serial correlation,which will reduce the fitting effect and prediction accuracy of the model if we model for them directly.First,we reduce the infinite dimensional functional covariate to the finite dimensional space by the functional principal component analysis(FPCA).Then,we extend the series correlation test method of scalar data linear model to functional linear model and obtain the large sample properties.Finally,we evaluate the good finite sample properties of proposed statistics through simulation studies,and illustrate their utility via an analysis of the residential electricity consumption data.(3)We study the variable selection of multiple functional regression models with autoregressive errors.The construction and estimation of the model is very challenging with more than one functional covariates and dependent response variable.We reduce the infinite dimensional functional independent variable to the finite dimensional space by FPCA,and then select a relevant set of functional variables and determine the order of the autoregressive error term simultaneously based on the Group Smoothly Clipped Absolute Deviation(Group SCAD)criterion.Under some regularity conditions,we establish the consistency of model selection and asymptotic normality of autocorrelation parameter estimators.In addition,simulation studies show an excellent finite sample performance.(4)We focus on the global hypothesis test of the linearity in high dimensional partially functional linear model.In practical application,we often encounter mixed data of functional data and high-dimensional data.Therefore,We construct U-type test statistics under the null hypothesis and local alternative hypothesis after dimension reduction of functional covariates by FPCA.Under certain regularity conditions,we obtain their the asymptotic normality.Simulation results and an empirical analysis of air pollution data show that proposed test statistics have good finite sample properties.