  This paper studies the impact of bootstrap procedure on the eigenvalue
distributions of the sample covariance matrix under a high-dimensional factor
structure. We provide asymptotic distributions for the top eigenvalues of
bootstrapped sample covariance matrix under mild conditions. After bootstrap,
the spiked eigenvalues which are driven by common factors will converge weakly
to Gaussian limits after proper scaling and centralization. However, the
largest non-spiked eigenvalue is mainly determined by the order statistics of
the bootstrap resampling weights, and follows extreme value distribution. Based
on the disparate behavior of the spiked and non-spiked eigenvalues, we propose
innovative methods to test the number of common factors. Indicated by extensive
numerical and empirical studies, the proposed methods perform reliably and
convincingly under the existence of both weak factors and cross-sectionally
correlated errors. Our technical details contribute to random matrix theory on
spiked covariance model with convexly decaying density and unbounded support,
or with general elliptical distributions.
