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.
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