  It is known that the common factors in a large panel of data can be
consistently estimated by the method of principal components, and principal
components can be constructed by iterative least squares regressions. Replacing
least squares with ridge regressions turns out to have the effect of shrinking
the singular values of the common component and possibly reducing its rank. The
method is used in the machine learning literature to recover low-rank matrices.
We study the procedure from the perspective of estimating a minimum-rank
approximate factor model. We show that the constrained factor estimates are
biased but can be more efficient in terms of mean-squared errors. Rank
consideration suggests a data-dependent penalty for selecting the number of
factors. The new criterion is more conservative in cases when the nominal
number of factors is inflated by the presence of weak factors or large
measurement noise. The framework is extended to incorporate a priori linear
constraints on the loadings. We provide asymptotic results that can be used to
test economic hypotheses.
