How do cross-sectional regressions estimate factor returns?

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AcadiFi · Accepted Answer

In cross-sectional regression, at each time t you regress asset returns on asset characteristics (factor exposures or 'loadings'): r_{i,t} = a_t + B_t * x_{i,t-1} + e_{i,t}, where x_{i,t-1} is the vector of characteristics (value, size, momentum, etc.) for asset i known at t-1, and B_t is the cross-sectional slope vector. B_t IS the factor return for period t—the return on a portfolio with unit exposure to that factor and zero exposure to others.

Repeating this regression month by month yields a time series of factor returns {B_t}, which you average and test: if average factor return is significantly positive/negative, the characteristic predicts returns.

For Crestline's 500-stock universe with 5 factors (value, size, momentum, quality, low-vol), monthly cross-sectional regression produces 5 factor return series over 20 years (240 observations each). Mean factor return for value = 0.35%/month, t-stat = 2.4, significant at 5%.

Time-series regression works differently: for each asset, regress r_{i,t} on factor returns F_t (observed, like HML, SMB): r_{i,t} = alpha_i + beta_i * F_t + e_{i,t}. This estimates asset-level betas given observed factor returns. Fama-French uses this approach with pre-formed long-short portfolios as factor proxies.

Cross-sectional regression is more flexible—you can use continuous characteristics rather than constructing long-short portfolios. It's the standard approach for proprietary factor research (BARRA, Axioma) and commonly used in Crestline-style systematic strategies.

How do cross-sectional regressions estimate factor returns?

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