An approximate factor model of high dimension has two key features. First, the idiosyncratic errors are correlated and heteroskedastic over both the cross-section and time dimensions; the correlations and heteroskedasticities are of unknown forms. Second, the number of variables is comparable or even greater than the sample size. Thus, a large number of parameters exist under a high-dimensional approximate factor model. Most widely used approaches to estimation are principal component based. This paper considers the maximum likelihood–based estimation of the model. Consistency, rate of convergence, and limiting distributions are obtained under various identification restrictions. Monte Carlo simulations show that the likelihood method is easy to implement and has good finite sample properties.

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