# Why estimate the (known) market return in the cross-sectional regression of Fama-MacBeth?

Suppose we are given a dataset with $$T$$ time periods and $$N$$ assets or portfolios. We are interested in estimating and testing the CAPM. Using Fama-MacBeth style analysis, we first estimate $$N$$ time series regressions of the form $$r^*_{i,t}=\alpha_i+\beta_i r^*_{m,t}+u_{i,t} \tag{1}$$ for $$i=1,\dots,N$$ where $$r^*_i:=(r_{i,t}-r_{f,t})$$ is asset's $$i$$ excess return and $$r^*_m:=(r_{m,t}-r_{f,t})$$ is the market's excess return, $$r_{f,t}$$ being the risk-free rate. This way we obtain the fitted betas $$\hat\beta_i$$ for $$i=1,\dots,N$$. We then estimate $$T$$ cross-sectional regressions $$r^*_{i,t}=\alpha_i+\lambda_t\hat\beta_i+v_{i,t} \tag{2}$$ for $$t=1,\dots,T$$.

We then obtain $$\hat{\lambda} = \frac{1}{T} \sum^{T}_{t=1} \hat{\lambda}_t$$ and its estimated asymptotic variance as $$\widehat{\text{AVar}}(\hat{\lambda}) = \frac{1}{T^2} \sum^{T}_{t=1} (\hat{\lambda}_t - \hat{\lambda} )(\hat{\lambda}_t - \hat{\lambda} )'$$. However, we could skip this bit if we are not directly interested in inference about the true $$\lambda$$.

The CAPM implies $$\alpha_i=0 \ \forall i$$, and an assessment of $$H_0\colon \alpha_i=0 \ \forall i$$ is a common test of the CAPM.

Update: The following is based on a mistaken thought about what the CAPM implies. I realized that with the help of Matthew Gunn. I leave it as is for historical consistency.

If only $$(2)$$ contained $$\beta_i$$s in place of $$\hat\beta_i$$s, the CAPM would also imply $$\lambda_t=r^*_{m,t}$$. Thus, I would be tempted to impose $$\lambda_t=r^*_{m,t}$$ in $$(2)$$ to yield $$r^*_{i,t}=\alpha_i+\hat\beta_i r^*_{m,t}+w_{i,t} \tag{3}.$$ Since the systematic risk component $$\hat\beta_i r^*_{m,t}$$ is given, what would have to be estimated would be the $$\alpha_i$$ in $$(r^*_{i,t}-\hat\beta_i r^*_{m,t})=\alpha_i+w_{i,t} \tag{3'},$$ allowing us to test $$H_0\colon \alpha_i=0 \ \forall i$$ later on. The intuition behind wanting to impose $$\lambda_t=r^*_{m,t}$$ is that estimating something that we actually observe sounds like a way to introduce unnecessary noise in the model.

Now, I do see the subtle difference (which may be not so subtle numerically) between $$\hat\beta_i$$ and $$\beta_i$$ that does not allow us to impose $$\lambda_t=r^*_{m,t}$$. (That would make $$\alpha_i$$ in $$(3)$$ include the $$\alpha_i$$ from $$(1)$$ plus $$(\beta_i-\hat\beta_i) r^*_{m,t}$$.) Is this the reason for why we estimate $$\lambda_t$$ in $$(2)$$? Or is there some other reason for (roughly speaking) estimating the market return when we actually observe it?

If you're just interested in statistically testing the CAPM, I don't see much of a reason to go beyond testing the joint significance of the hypothesis $$H_0: \forall_i \;\alpha_i= 0$$. Run the time-series regressions then run Gibbons Ross Shanken (fancy name for F-test) that all $$\alpha_i = 0$$ or run a $$\chi^2$$ test. Done!

There's the famous aphorism of Box though that, "all models are wrong but some are useful." Let's say we statistically reject the CAPM, hence the CAPM is wrong in some strict sense.

Might it still be a useful theory? The testable implication of the CAPM is that expected returns are linearly increasing in a security's (or portfolio's) market beta. Is that a useful statement? It turns out the answer is no, and you can see it with a cross-sectional regression.

If you run the cross-sectional regression: $$\operatorname{E}[r^*_{i}] = \gamma_0 + \gamma_1 \beta_i + v_{i}$$

you get also get a more qualitative take on whether the model is doing something useful. Is $$\gamma_1$$ even positive? You also estimate $$\gamma_0$$, the excess return on the 0 beta portfolio, which is kind of an interesting concept. (Is it indeed 0 or really quite different?) This cross-sectional regression ins a kind of debugging regression: why are the CAPM alphas so big and significant? What's going wrong?

For each asset $$i$$, you first ran the time-series regression to estimate $$\alpha$$ and $$\beta$$ (where $$r^*_i = r_i - r_f$$):

$$$$r^*_{i,t} = \alpha_i + \beta_i r^*_{m,t} + \epsilon_{i,t}$$$$

where $$r^*$$ denotes an excess return (eg.$$r_{i,t} - r_{f,t}$$). Hence: $$$$\operatorname{E}[r^*_{i}] = \alpha_i + \beta_i \operatorname{E}[r^*_{m}]$$$$

Let me slightly rewrite that what we're ultimately interested in is the cross-sectional regression:

$$\operatorname{E}[r^*_{i}] = \gamma_0 + \gamma_1 \beta_i + v_{i}$$

If the CAPM is true, then $$\gamma_0 = 0$$ and $$\gamma_1 = \operatorname{E}[r^*_m]$$. Your question in some sense, why estimate, can't we just impose that and see what happens? Sure!

Then you can equate those two lines:

\begin{align*} \alpha_i + \beta_i \operatorname{E}[r^*_{m,t}] &= \gamma_0 + \beta_i \gamma_1 + v_{i} \\ &= \beta_i \operatorname{E}[r^*_m] + v_i \end{align*} Hence if you impose $$\gamma_0 = 0$$ and $$\gamma_1 = \operatorname{E}[r^*_m]$$ you get: $$v_i = \alpha_i$$

The error terms $$v_i$$ in the cross-sectional regression are just the $$\alpha_i$$ estimated in the time-series regression. This is what John Cochrane likes to call the cross-sectional implications of the time-series regression (I also go into it here).

So you can just impose $$\gamma_0 = 0$$ etc... and jump straight to testing $$\alpha_i$$. You can also do the intermediate, in some sense debugging exercise of running the cross-sectional regression.

### If your factor isn't a return...

Let's imagine you had a theory that expected returns were linearly increasing in their covariance with aggregate consumption. Consumption isn't the return on a financial return, and you'll see that the substitution trick which implied $$\alpha_i = 0$$ no longer works.

In this case, you'd have to run the cross-sectional regression to estimate the factor premium etc....

• Thank you! This is interesting and helpful. You say if the CAPM is true, then $\gamma_1=E[r_m]$. I was actually thinking to impose $\gamma_{1,t}=r^*_{m,t}$, i.e. different realized market returns in different periods. I wonder if that makes any sense. I suppose it would if we only had a single period $t$, as it would be a direct implication of the CAPM. It becomes a bit confusing with multiple periods. After all, we are testing a single-period model on multiple periods of data, so what gets to vary over time and what must stay constant? Commented Jan 31, 2023 at 20:07
• I also thought the test of $H_0\colon \alpha_i=0 \ \forall i$ is done on the basis of cross-sectional regressions, not time series ones, as the CAPM is about the cross section. Does no one do that? Commented Jan 31, 2023 at 20:15
• @RichardHardy I had a typo; you're correct $\gamma_1 = \operatorname{E}[r^*_m]$ with that formulation. Commented Jan 31, 2023 at 21:41
• @Richard Hardy The cross-sectional regression asset pricing want to run is $\operatorname{E}[r^*_i] = \gamma_0 + \gamma_1 \beta_i + \epsilon_i$. They want to say expected returns are linear in covariance of the return with some magical SOMETHING. In the simple, typical model people estimate, the $\gamma_1$ parameter isn't time-varying. To get consistent standard errors, a common thing to do is the Fama-Macbeth procedure which gives you different $\hat{\gamma}_{1,t}$ estimates for each period, but you're trying to estimate 1 time invariant parameter $\gamma_1$. Commented Jan 31, 2023 at 21:54
• Imposing $\gamma_{1,t} = r^*_{m,t}$ isn't the conceptually right thing to do... there really isn't a $\gamma_{1,t}$ unless you have a much more complicated model with time-varying risk premia (which is probably right btw... but that's another topic). Commented Jan 31, 2023 at 21:55