  In dealing with high-dimensional data sets, factor models are often useful
for dimension reduction. The estimation of factor models has been actively
studied in various fields. In the first part of this paper, we present a new
approach to estimate high-dimensional factor models, using the empirical
spectral density of residuals. The spectrum of covariance matrices from
financial data typically exhibits two characteristic aspects: a few spikes and
bulk. The former represent factors that mainly drive the features and the
latter arises from idiosyncratic noise. Motivated by these two aspects, we
consider a minimum distance between two spectrums; one from a covariance
structure model and the other from real residuals of financial data that are
obtained by subtracting principal components. Our method simultaneously
provides estimators of the number of factors and information about correlation
structures in residuals. Using free random variable techniques, the proposed
algorithm can be implemented and controlled effectively. Monte Carlo
simulations confirm that our method is robust to noise or the presence of weak
factors. Furthermore, the application to financial time-series shows that our
estimators capture essential aspects of market dynamics.
