Estimation Of Approximate Factor Models Via Penalized Maximum Likelihood

In many applications of economics,finance,and other scientific fields,re-searchers often face a large panel dataset.Factor models have been popular for large sets of data since they provide an effective way of summarizing information from these data sets.The estimation of parameters is the primary problem to study factor models.This paper study the penalized maximum likelihood estimation of the approximate factor models and establish consistency of estimators.The key assumption we make is that idiosyncratic component have a sparse co-variance matrix.This enables us to introduce the penalty function which is used to penalize the elements of the idiosyncratic covariance matrix.The penalty function is used in the form of weighted l1.We suggest three specific choices for wij,meanwhile naming the penalty function Lasso,Adaptive-lasso,SCAD,respectively.We obtain the factor loadings,common factors,and idiosyncratic covariance matrix through minimizing the sum of negative gaussian quasi likelihood and penalty function.While the principal components based methods estimate the covariance matrices and individual factors and loadings separately,in contrast,the penalized maximum likelihood method estimates the factor loading parameters and the idiosyncratic co-variance matrix jointly.In the numerical studies,we compare penalized maximum likelihood method with the regular principal components method,the generalized principal components method,the maximum likelihood method on their estimation performances.Our numerical studies show that the proposed method performs superior than the other methods.The structure of this paper is as follows.In chapter 1,we discuss the back-ground,significance and status of the research.In chapter 2,we introduce the model,the relevant assumptions and the main results of this paper and its proof.In chapter 3,we discusses the problem of computation and simulation.The last chapter is about the summarize of the paper and we put forward some unresolved problems and the future research direction.