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**This section is under development!**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)

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

\(\Sigma = B cov(f_t)B' + \Sigma_u\)