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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).

The factor zig zag around zero

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).

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  • $\begingroup$ Yes, Mr. Hsueh, please clarify the meaning of the graph in your original post. As it is, with no labels, it is quite unclear. $\endgroup$
    – nbbo2
    Jan 8 at 21:09
  • $\begingroup$ I've edited my post to clarify my question. Sorry for the confusion. $\endgroup$ Jan 9 at 5:49
  • $\begingroup$ Thank you, my interpretation of the chart was not correct. $\endgroup$
    – nbbo2
    Jan 9 at 12:09

1 Answer 1

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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.

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  • $\begingroup$ Not sure if I am answering your question, but I hope this will be helpful. $\endgroup$ Jan 7 at 17:10
  • $\begingroup$ Ah, so the significance of a factor in Fama-MacBeth regression depends on the time-horizon. In my example, the expected stock returns on a short time horizon would definitely not be zero, and the zig-zag factor would be significant $\endgroup$ Jan 8 at 19:18
  • $\begingroup$ @ShawnHsueh, consider editing your post to clarify the issues questioned in the comments under your post. Thank you. $\endgroup$ Jan 8 at 19:51
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    $\begingroup$ @ShawnHsueh, I had understood you correctly. My answer still applies after your edit. $\endgroup$ Jan 9 at 6:52

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