Inferential Theory for Factor Models of Large Dimensions

TLDR

The principal components estimator is used because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator under normality. This paper develops an inferential theory for factor models of large dimensions. The authors derive convergence rates and limiting distributions for estimated factors, loadings, and common components, developing the theory for large N and T under general conditions allowing correlations and heteroskedasticities. The estimated common components are asymptotically normal with rate min(√N,√T); factors and loadings are generally normal, sometimes not, and can converge faster; stronger results hold when idiosyncratic errors are serially uncorrelated and homoskedastic, and a necessary and sufficient consistency condition is derived for large N with fixed T.

Abstract

This paper develops an inferential theory for factor models of large dimensions. The principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator (if normality is assumed). We derive the rate of convergence and the limiting distributions of the estimated factors, factor loadings, and common components. The theory is developed within the framework of large cross sections (N) and a large time dimension (T), to which classical factor analysis does not apply. We show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of N and T. The estimated factors and their loadings are generally normal, although not always so. The convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components. These results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions. Stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic. A necessary and sufficient condition for consistency is derived for large N but fixed T.