Second-order Asymptotics On The Number Of Factors

In statistical inference on large-dimensional factor model,a key ingredient is esti-mating the number of common factors r.Existing studies only consider the consistency of r,but the second-order asymptotics beyond consistency,such as convergence rate and central limit theorem were still unsolved.Therefore,this paper is concerned with second-order asymptotics on the estimated factor number for a large-dimensional ap-proximate factor model.Via a hybrid use of the sample splitting technique and the principal component analysis,we transfer estimating the number of factors of a large-dimensional approximate factor model to estimating the rank of a low-dimensional volatility matrix.Using simple matrix perturbation technique gives a pseudo point es-timate of the rank.We prove that this estimated pseudo rank has asymptotic normal distribution,which can be used to inference on the factor number.Extensive simu-lation studies show that the normal approximation has accurate covering probability Application to hypothesis testing can effectively control the first type of error and have good size and power performance.Empirical studies on two financial real data sets find evidence that the number of factors is around 3.