Testing the number of common factors by bootstrap in high-dimensional factor models

This paper proposes to test the number of common factors in high-dimensional factor models by bootstrap. We provide asymptotic distributions for the eigenvalues of bootstrapped sample covariance matrix under mild conditions. The spiked eigenvalues converge weakly to Gaussian limits after proper scaling and centralization. The limiting distribution of the largest non-spiked eigenvalue is mainly determined by order statistics of bootstrap resampling weights, and follows extreme value distribution. We propose two testing schemes based on the disparate behavior of the spiked and non-spiked eigenvalues. The testing procedures can perform reliably with weak factors, cross-sectionally and serially correlated errors. Our technical proofs contribute to random matrix theory with convexly decaying density and unbounded support, or with general elliptical distributions.