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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)

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