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

\[\boxed{X_{it} = \lambda_i'F_t + e_{it}} \quad i\le N, \; t\le T.\]

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)

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