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 an

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.