  This paper studies new tests for the number of latent factors in a large
cross-sectional factor model with small time dimension. These tests are based
on the eigenvalues of variance-covariance matrices of (possibly weighted) asset
returns, and rely on either the assumption of spherical errors, or instrumental
variables for factor betas. We establish the asymptotic distributional results
using expansion theorems based on perturbation theory for symmetric matrices.
Our framework accommodates semi-strong factors in the systematic components. We
propose a novel statistical test for weak factors against strong or semi-strong
factors. We provide an empirical application to US equity data. Evidence for a
different number of latent factors according to market downturns and market
upturns, is statistically ambiguous in the considered subperiods. In
particular, our results contradicts the common wisdom of a single factor model
in bear markets.
