Factor Empirical Process Of Large-dimensional Aproximate Factor Model

The distribution of factor return for an individual variable are important in the forecasting and applications.Using the recently developed theory for large dimensional approximate factor model,it can be used to estimate the consistent factor return by the principal component analysis method.There is a nature question,is it reliable to use the estimated empirical distribution replace the distribution of real factor return?If so,what are the requirements for dimension and sample size?This paper will answer the above questions by the research on the empirical process of principal component estimator sequence of factor return.The result shows,the difference between the em-pirical process about principal component estimator sequences of factor return and the empirical process about real factor return is very small,the order is op(1),the difference between the empirical distribution about principal component estimator sequences of factor return and the empirical distribution about real factor return is very small,the order is op((?),and T is the sample size.Based on the oracle property of the empirical process,we construct simultaneous confidence bands(SCBs)for the distribution of the factor return,the confidence bands has accurate coverage.Based on the oracle proper-ty,the empirical process of the principal component estimator sequence of factor return requires p and T satisfy T = o(p),and p is the dimension.While the consistency of the estimated factor distributions just need p and T satisfy(?)= 0(p).Simulation stud-ies check that the estimated SCBs have accurate coverage frequencies.Our real data analysis use four stocks:State Agriculture Technology(000004),Baolilai(000008),Chinese Baoan(000009),and Shenhuaxin(000010),result shows that the distributions of factor return of the four stocks have structural changes before and after the 2008 economic crisis.