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Factor models used for Asset Pricing, Business Cycle Analysis, Monitoring/Forecasting, Consumer Theory, etc.

Classical FA

• \(T \to \infty\), \(N\) fixed
• \(e_{it}\) are iid over \(t\) and ind. over \(i\), i.e. \(\Omega := \E(e_te_t')\) is a diagonal matrix
• \(F_t\) are iid and ind. of \(e_{it}\)
• \(\widehat{\Sigma}\) is assumed to be \(\sqrt{T}\)-consistent for \(\Sigma\)
• \(\lambda_i\) can be consistently estimated, but not \(F_t\) (Anderson 1984)

Approximate FA (Chamberlain & Rothschild 1983)

• nondiagonal \(\Omega := \E(e_te_t')\) allowed
• \(\Omega := \E(e_te_t')\) has bounded eigenvalues
• Hence largest eigenvalue bounded by \(\boxed{\underset{i}{\max} \sum_{j=1}^N\abs{\Omega_{ij}}}\)
• PCA ~ FA, when \(N\to\infty\) (assuming \(\Sigma\) known)
• Connor & Korajczyk: unknown \(\Sigma\) and \(N\gg T\)

Strict FA (APT of Ross 1976)

• \(e_{it}\) is uncorrelated across \(i\)

HD FA (Bai 2003)

• \(N,T \to \infty\) (also \(N\to \infty, T\) fixed)
• serial and cross-section dependence for the idiosyncratic errors
• heteroskedasticity in both dimensions

POET (Fan 2013)

\(\boxed{y_{it} = b_i'f_t + u_{it}} \quad i\le p, \; t\le T.\)
\(\Sigma = B cov(f_t)B' + \Sigma_u\)