What are the main methods for estimating a factor model?

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AcadiFi · Accepted Answer

Three main approaches to factor model estimation:

1. Time-series regression (Macro factors known in advance):
Factors like market return, GDP, inflation are observable. For each asset, regress its return on factor time series:
Rᵢ,ₜ = αᵢ + Σⱼ βᵢⱼ Fⱼ,ₜ + εᵢ,ₜ
OLS gives betas and residuals. Example: regress 60 months of stock returns on market and SMB factors. The slope coefficients are the asset's factor exposures.

2. Cross-sectional regression (Characteristics known, factor returns unknown):
Asset characteristics (P/B, size, momentum) are observed. For each date, regress cross-section of returns on characteristics:
Rᵢ,ₜ = λ₀,ₜ + Σⱼ λⱼ,ₜ Xᵢⱼ,ₜ + eᵢ,ₜ
The λⱼ,ₜ are estimated factor returns. Fama-MacBeth takes averages across time. Example: each month, regress 500 stock returns on their value and size ranks to extract that month's value premium and size premium.

3. Statistical factor models (No prior factor specification):
Use PCA or maximum likelihood factor analysis on the return covariance matrix to extract latent factors. Neither factors nor loadings are specified — both estimated from data. Example: APT factor analysis — economic interpretation assigned after factors are extracted.

Comparison table:

| Method | Inputs | Estimates | Use case |
|--------|--------|-----------|----------|
| Time-series | Factor returns | Loadings | Macro models, hedging |
| Cross-sectional | Characteristics | Factor returns | Smart-beta, Fama-French |
| Statistical | Returns only | Both | APT, dimensionality reduction |

Tradeoffs: time-series requires ex-ante factor identification but directly gives hedge ratios. Cross-sectional captures characteristic premia but treats characteristics as constants. Statistical is data-driven but factors lack economic meaning until you interpret them.

Barra and MSCI commercial risk models blend approaches — use cross-sectional for style factors and time-series for country/currency.

What are the main methods for estimating a factor model?

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