Statistical Analysis Of Some Semi-parametric Factor Models

In the past few years,semi-parametric models have developed greatly.It not only has the characteristic of modeling flexibility of non-parametric models,but also has the characteristic of intuitive and easy interpretation of parametric models.On the one hand,motivated by the extensive application of the Fama–French factor model in the financial field,this paper aims to study it with semi-parametric models,so as to make it more flexible.The models studied in this paper are not only applicable to the Fama–French factor data,but also to other panel data,such as real estate data,environmental data and medical data.With the evolvement of time,the data studied by scholars generally have a large time span.For example,Fama–French factor data can be traced back to 1926.Hence,it is crucial to summarize timevariant characteristics of this type of dataset.If we assume parameters in models are time-invariant,the models may exist modeling error.Or,you may use data spanning a smaller time to fit the model,the di culty you may face with is what criterion you choose to determine the number of time.In this paper,we apply the t T rescaled time technique to investigate time-varying behaviors of the Fama–French factor data with a semi-parametric model.Moreover,a long historic time series may be nonstationary.The traditional method is to detrend or di?erence the series.However,it is di cult to make a statistical inference on the original data.One way to solve this problem is provided by the concept of local stationarity,according to which the series are stationary over a short time span but non-stationary over a long time span.It is convenient to deduce statistical properties of the estimates of unknown parameters and functions in a model.On the other hand,this paper extends the research of traditional parametric factor models to a semi-parametric framework.The factor model can exhibit more information by taking auxiliary information into consideration.This paper constructs factor loadings as single-index functions,which have many advantages of single index forms,in particular,to avoid the curse of dimensionality.Generally speaking,this paper is divided into two parts.The first part is to use semi-parametric models to study the dynamic characteristics of the Fama–French factor data.In this paper,we will study the estimation method,asymptotic property and hypothesis test of the proposed model.The second part is a semi-parametric factor model with single-index loadings.The estimation procedure and its asymptotic properties are studied.The article is arranged as follows:The first chapter introduces the research background,research methods(local stationary series,factor model,single-index model,nonparametric estimation method,etc.)and the structure of the paper.The second chapter introduces the estimation and inference for a dynamic semiparametric Fama–French factor model.We assume that the factor of the traditional Fama–French factor model is a time-varying bivariate unknown function.In this paper,we creatively propose a four-step estimation method,which combines the Bspline and the local linear method to estimate unknown parameters and functions in the model.This method has the advantage that it not only inherits the computational e ciency of spline estimation but also helps to construct the asymptotic distribution of the estimate by the local linear method.Moreover,this paper studies the asymptotic properties of the estimates,especially their oracle properties,in a setting of(N,T→ ∞).In addition,this paper proposes the generalized likelihood ratio test to test whether the model is dynamic,and studies its asymptotic property.Numerical simulation is conducted to evaluate the performance of the proposed estimation method and hypothesis test.Finally,this paper proposes a method to estimate the sample covariance matrix based on the proposed model,and applies it to Markowitz’s mean-variance optimal portfolio theory to construct the portfolio.Compared with other models,the proposed method has excellent return.In the third chapter,we introduce the estimation and inference for a time-varying additive semiparametric model.It is the further development of the second chapter.This chapter assumes that the bivariate function in the model in Chapter 2 is the product of two univariate functions about covariates and tT.This simplifies the form of the model in Chapter 2 and avoids overfitting.We use B-spline and least square method to estimate unknown parameters and functions.We propose to test the su ciency of the multiplicative assumption by using L2 test statistics.We study the asymptotic properties of the proposed estimation methods and hypothesis tests,and use numerical simulation to evaluate their performances.The forth chapter introduces a semi-parametric factor model,which decomposes the dynamic behaviour of a high dimensional data into a few low dimensional time series factors.The factor loadings are assumed to be unknown functions of covariates and have a form of single-index.It is di?erent from traditional factor models in which the factor loadings are parametric.An estimation procedure based on Damped Newton algorithm is proposed to iteratively estimate factors and factor loadings.The algorithm combines the steepest decent algorithm and the Newton algorithm.It not only has the advantage of high speed of the Newton algorithm,but also inherits the high stability advantage of the steepest decent algorithm.The asymptotic properties of estimated parameters,factors and factor loadings are derived.Simulation studies are conducted to evaluate the performance of the estimation procedure.We apply our method to investigate the trajectories of global COVID-19 cumulative confirmed cases.The fifth chapter is the summary of this dissertation,and the future work is prospected.