Fama-MacBeth Regression

The Fama-MacBeth (1973) two-pass cross-sectional regression is the standard methodology for testing whether a variable explains the cross-section of expected stock returns. Nearly every empirical factor paper since 1973 uses this procedure or its variants.

The procedure

First pass (time series): For each asset $i$, estimate factor loadings by regressing returns on factor returns over a rolling or full-sample window:

$R_{it}-R_{ft}={\alpha }_i+{\beta }_i{F}_t+{ϵ}_{it}$

Second pass (cross-sectional): Each month $t$, run a cross-sectional regression of returns on the estimated betas from the first pass:

$R_{it}-R_{ft}={\gamma }_{0t}+{\gamma }_{1t}{\hat{\beta }}_i+{\eta }_{it}$

Inference: Average the monthly cross-sectional slopes ${\gamma }_{1t}$ across time and use the time-series standard deviation of these slopes to compute t-statistics. This accounts for cross-sectional correlation in returns without requiring an explicit covariance model.

Original findings (Fama and MacBeth 1973)

Using NYSE stocks from 1935-1968:

• The market beta-return relation is positive and significant (t=2.57), supporting the capm
• Squared beta and residual variance do not explain returns beyond beta, consistent with CAPM
• The results support the “fair game” efficient market hypothesis: coefficients are serially uncorrelated

Why the method endures

• Automatically corrects for cross-sectional correlation in residuals (a major problem for pooled OLS)
• Allows time-varying risk premia (each month produces a separate estimate)
• Intuitive: the average slope is the average risk premium
• Easily extended to multiple factors and characteristics

Role in the factor investing canon

Fama-MacBeth regressions are the backbone of:

• Fama and French (1992): showed beta’s slope became insignificant (t=0.46) while size (t=-2.58) and book-to-market (t=5.71) dominated
• Lakonishok, Shleifer, and Vishny (1994): portfolio sorts supplemented by FM regressions
• Harvey, Liu, and Zhu (2016): the t > 3.0 threshold applies to FM regression t-statistics

The original 1973 paper is also the first published evidence that market-beta was priced, a finding that Fama himself later overturned in Fama and French (1992).

Sources

• Risk, Return, and Equilibrium: Empirical Tests (File, DOI)