EM Algorithm for Factor Models (因子模型的 EM 算法)

Definition. The Expectation–Maximization (EM) algorithm applied to factor models with latent factor returns. When the factor returns f_t are unobserved, direct maximization of the marginal log-likelihood p(r_t \mid \theta) is intractable, but the complete-data log-likelihood p(r_t, f_t \mid \theta) is Gaussian and admits closed-form conditional expectations. Iterating an E-step (compute posterior of f_t) and an M-step (maximize expected complete-data log-likelihood) gives monotone improvement in the marginal likelihood.

General recipe

Given observed returns r_t and latent factor returns f_t, model:

r_t \mid f_t \sim \mathcal{N}(B f_t, D), \quad f_t \sim \mathcal{N}(0, F)

Parameters \theta = \{B, F, D\}. E-step:

p(f_t \mid r_t, \theta^{\text{old}}) = \mathcal{N}(\mu_{f \mid r}, \Sigma_{f \mid r})

with

\Sigma_{f \mid r}^{-1} = F^{-1} + B^\top D^{-1} B, \quad \mu_{f \mid r} = \Sigma_{f \mid r} B^\top D^{-1} r_t

M-step: maximize

Q(\theta, \theta^{\text{old}}) = \mathbb{E}_{f_t \mid r_t, \theta^{\text{old}}}[\log p(r_t, f_t \mid \theta)]

For Gaussian models, this gives closed-form updates for B, F, D using the weighted sufficient statistics E[r_t r_t^\top], E[r_t f_t^\top], E[f_t f_t^\top].

Why factor models love EM

• Latent variables f_t are natural (factor returns not directly observed except via regression).
• Complete-data likelihood is simple: Gaussian \Rightarrow closed form.
• Convergence guarantee: marginal log-likelihood is non-decreasing.
• Missing data handled for free: marginalize missing r_{i,t} via the same posterior.

Closed-form M-step in the augmented risk model (Tzikas et al., 2026)

For the augmented model with fixed base loadings B and learned transient loadings X, the M-step for X reduces to a block matrix formula involving:

X = (S_{rg}^\top - B S_{fg}^\top) (S_{gg} - S_{fg} \Sigma_F^{-1} S_{fg}^\top + \Sigma_F^{-1})^{-1}

where S_{rg}, S_{fg}, S_{gg} are weighted cross-moment matrices from the E-step. The full update for \Sigma_B and D is similarly closed form. The paper reports 20–30 iterations to convergence on the Barra USE4 + 870-asset problem.

Initialization strategies

For the transient-factor augmentation:

1. Regress r_t on B under EWMA weights → residual e_t.
2. Eigen-decompose EWMA covariance of e_t → top-k eigenvectors form initial X.
3. Initialize D to match the EWMA sample covariance's diagonal.

Failure modes and remedies

Connections

• Risk Model Enhancement via Transient Statistical Factors — primary application.
• Gaussian latent variable model — the family EM applies to here.
• Maximum likelihood estimation — the underlying objective EM optimizes.

Reference

Tzikas et al. (2026), see summaries/202606-mle-transient-factor-mining/202606-mle-transient-factor-mining.