# 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).

• Yes, Mr. Hsueh, please clarify the meaning of the graph in your original post. As it is, with no labels, it is quite unclear. Jan 8 at 21:09
• I've edited my post to clarify my question. Sorry for the confusion. Jan 9 at 5:49
• Thank you, my interpretation of the chart was not correct. Jan 9 at 12:09

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.
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.