Estimation Of Error Distribution And Factor Number Of High Dimensional Approximate Factor Model

The study of error distribution plays an important role in the financial field.In this paper,we present the residual empirical distribution of the high dimensional approx-imate factor model and establish a large sample property.It is shown that empirical procedure of the residual sequence has the oracle property when T = o(p),where p and T represent the dimension and the sample size respectively.The residual empirical distribution function is consistent when the condition is relaxed to(?)T = o(p).The si-multaneous confidence bands of the error distribution function which were constructed based on oracle property has accurate coverage rate.Extensive simulations show that the estimated simultaneous confidence band has good coverage rate.Actual data anal-ysis shows that the idiosyncratic return distributions of the three stocks about State Agriculture Technology(000004),Polaroid(000008)and Shenhuaxin(000010)have no structural changes while the idiosyncratic return distribution of the stock about China Baoan(000009)has obvious structural changes before and after the 2008 economic cri-sis.In addition,this paper proposes a new method to estimate the number of factors.This method reduces the dimension by constructing the sequence of the principal com-ponent,and obtains the estimator of the number of factors by the method of matrix perturbation.Results show that the consistency of the estimated number of factors only requires the parametric ? satisfy p1-?/n2 = o(1).The data simulation shows that the estimated number of factors is highly accurate when the dimensions are greater than 200.