Why Fama–MacBeth Can Miss a Zero-Mean Factor Risk Premium
Summary
The document examines a toy asset-pricing example in which a factor perfectly explains differences in stock returns through different factor loadings, while the factor itself fluctuates around zero. The apparent puzzle is that a cross-sectional regression can estimate a nonzero factor slope in individual periods, yet a Fama–MacBeth test across periods may find that the average slope is not significantly different from zero.
The reply distinguishes explanatory power from a risk premium. If the factor has zero expected value, the stocks’ expected excess returns are also zero in this setup, even though their period-by-period returns move with their betas. Fama–MacBeth’s overall test of the average price of risk can therefore fail to reject zero without disproving the factor’s role in explaining returns. The interpretation depends on the toy model’s assumptions, including excess returns, risk neutrality, and a zero-mean factor; it is not a general claim that the method cannot detect useful factors.
Key ideas
- Fama–MacBeth tests the average cross-sectional price of risk across periods.
- A factor can explain period-by-period return differences while having a zero average risk premium.
- Failure to reject a zero average slope does not by itself refute a factor’s explanatory role.
- The example assumes excess returns, risk neutrality, and a factor with zero expectation.
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Full text
# Why cannot Fama-MacBeth regression identify a zero-mean factor with explanatory power?
# Why cannot Fama-MacBeth regression identify a zero-mean factor with explanatory power?
Imagine a factor perfectly explain the return of all the stocks in a universe, and the factor has a zig-zag shape around zero (as shown by the image).
Since the factor perfectly explain the return of the stocks, the gamma (fitted from the cross-sectional regression or second-stage regression) would be the same as the factor itself. If we perform t-test on the gamma, it wouldn't have significance. However, shouldn't Fama-MacBeth regression be able to find factors that have explanatory power to the stock returns?
EDIT1
Added x-y labels on the figure.
In this single-factor toy model, all the stocks have zig-zag return with different amplitude (higher beta stocks have higher amplitude).
## Answer by Richard Hardy (score 2, accepted)
https://quant.stackexchange.com/a/77929
Suppose all of the returns are excess returns. (Otherwise, make them.) You are testing $\text{H}_{0}\colon\ \gamma_1=0$ in $r_i^*=\gamma_0+\gamma_1 \beta_i+u_i$. Since the factor perfectly explains the stock returns, you would reject $\text{H}_{0}$ in each period. But since the Fama-MacBeth method looks at multiple periods at once and since $\mathbb{E}(\gamma_1)=0$, you would not reject $\text{H}_{0}$ overall. And that would be correct in the sense that the factor does not command a risk premium.
Fama & MacBeth (1973) write on p. 610:
> Equation (6) has three testable implications <...> (C3) In a market of risk-averse investors, higher risk should be associated with higher expected return; that is, $E(\tilde{R}_m)-E(\tilde{R}_0)>0$.
In your market the investors are risk neutral, as every stock has an expected excess return of zero (the factor has an expectation of zero, a stock $i$ has that times $\beta_i$). Thus the implication C3 that Fama & MacBeth employ in their paper does not apply to your case, so you cannot take the nonrejection of $H_0$ above as evidence against your one-factor model.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.