Fama–MacBeth Testing with Firm-Specific Characteristics

Summary

The document asks whether the second stage of a Fama–MacBeth procedure remains valid when the first-stage asset returns use characteristics specific to each firm, such as size and book-to-market. It outlines the familiar two-stage setup: estimate asset exposures from time-series regressions, then run repeated cross-sectional regressions of returns on those estimated exposures.

The central concern is that firm-specific characteristics vary across both firms and time, unlike common factor returns, so substituting them into the first-stage equation may alter what the estimated coefficients represent and whether the second-stage regressors are appropriate. The document only states the question and provides no answers, formal argument, assumptions, or empirical evidence. It is therefore useful as a methodological prompt, but it does not resolve identification, estimation, or inference issues for this specification.

Key ideas

• The standard procedure first estimates time-series factor exposures and then prices those exposures in cross sections.
• The question substitutes firm-level size and book-to-market characteristics into the first-stage return equation.
• Firm-specific, time-varying characteristics raise questions about the interpretation of the estimated coefficients.
• The document supplies no formal justification or conclusion about validity or inference.

Tags

# Is Fama-MacBeth an appropriate testing methodology when factors are firm-specific?


# Is Fama-MacBeth an appropriate testing methodology when factors are firm-specific?












Recently I saw a paper that used a Fama-MacBeth procedure with firm-specific factors. I am a bit confused about this.

In a standard Fama-MacBeth procedure there's two steps (let's use the FF-3 factor model):

- For each asset $i$, run the following time-series regression over $T$ time periods:

$$R_{it} - R_{ft} = \alpha_i + \beta_{iM}(R_{Mt} - R_{ft}) + \beta_{iSMB} \cdot SMB_t + \beta_{iHML} \cdot HML_t + \epsilon_{it}$$

- At each time $t$, run a cross-sectional regression across $N$ assets:

$$R_{it} - R_{ft} = \gamma_{0t} + \gamma_{1t} \beta_{iM} + \gamma_{2t} \beta_{iSMB} + \gamma_{3t} \beta_{iHML} + \eta_{it}$$

Where the betas are the ones estimated in step 1.

Now let's say I replace equation (1) with firm-specific factors, say size and book to market:

$$R_{it} - R_{ft} = \alpha_i + \beta_{iM}(R_{Mt} - R_{ft}) + \beta_{i,size} \cdot Size_{i,t} + \beta_{i,b2m} \cdot B2M_{i,t}+ \epsilon_{it}$$

Now this should invalidate step 2 of the procedure, if my intuition is correct.

Is there a formal argument for this?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.