  Pervasive cross-section dependence is increasingly recognized as a
characteristic of economic data and the approximate factor model provides a
useful framework for analysis. Assuming a strong factor structure where
$\Lop\Lo/N^\alpha$ is positive definite in the limit when $\alpha=1$, early
work established convergence of the principal component estimates of the
factors and loadings up to a rotation matrix. This paper shows that the
estimates are still consistent and asymptotically normal when $\alpha\in(0,1]$
albeit at slower rates and under additional assumptions on the sample size. The
results hold whether $\alpha$ is constant or varies across factor loadings. The
framework developed for heterogeneous loadings and the simplified proofs that
can be also used in strong factor analysis are of independent interest.
