Question about Fama Macbeth Regression (Confusion about paper)

Question

I'm reading this paper Zura Kakushadze: 4-Factor Model for Overnight Returns https://arxiv.org/pdf/1410.5513.pdf and I am slightly confused about the methodology of the regressions.

It says it uses Fama MacBeth to use the residuals for a mean reversion strategy. Now, I am also somewhat confused about Fama MacBeth as well.

From what I understand, first you run cross sectional regressions to get your betas, then you run a time series regression to get the risk premiums.

So, if we had say four factors, and let's say we had 10 years of daily data (2,520 days) and let's say we had 5,000 stocks. So if my understanding is clear, first we would use the cross sectional regression to estimate 4*5,000=20,000 betas? And then you get the risk premia, which you would have a value for each day for each factor (2,520 per factor)?

Given I understand that correctly (PLEASE tell me if I am mistaken), I am confused what this paper is doing. It looks like it calculates the betas directly. But it's saying it is running the cross sectional regressions to get the factor returns. Is that the same as the risk premium?

Any clarification would be much appreciated. I am trying to replicate this but am having major confusion.

Answer 1

Fama-MacBeth procedure (Step 1):

So if my understanding is clear, first we would use the cross sectional regression to estimate 4*5,000=20,000 betas?

That is not right, because betas (and other risk-factor loadings) are estimated by a time-series regression:

$$ R_{i,t}^e = \alpha_i + \beta_{i, MktRf} MktRf_t + \beta_{i, SMB} SMB_t + \beta_{i, HML} HML_t + \epsilon_{i, t}$$

where $R_{i,t}^e$ is the excess return of stock $i$ (i.e. in excess of a risk-free rate of return) and the right-hand variables are the well known risk-factor premiums of the Fama/French 3-factor model.

For your example with 5,000 stocks, you would run the above regression a minimum of 5,000 times if you consider full-sample betas, i.e. your factor exposures are constant over the full period of time. Fama/MacBeth (1973) however use a rolling 5-year regression to estimate the factor loadings which increases the amount of time series regressions a lot.

Fama-MacBeth procedure (Step 2):

Step 2 starts with the estimated risk-exposures $\hat{\beta}_i \equiv [\beta_{i, MktRf}, \beta_{i, SMB}, \beta_{i, HML}]'$ from the above first step (i consider the simple case of full-sample betas, so $\hat{\beta}_i$ is constant over time).

Now you apply a cross-sectional regression at each period of time $T$ over all stocks $i$: $$R_{i,t}^{e}= \hat{\beta}_{i}^{'}\lambda_t+a_{it}$$

This results in estimates for $\lambda_t$ and $a_{it}$ for each period of time $T$.

What Fama/MacBeth (1976) suggest is, that we estimate $\lambda$ and $\alpha_i$ as the average of these cross-sectional regression estimates, i.e. $$\hat{\lambda} = \frac{1}{T} \sum_{t=1}^{T}{\hat{\lambda}}_{t}$$ $$\hat{a}_i = \frac{1}{T} \sum_{t=1}^{T}{\hat{a}}_{it}$$

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Additional remarks:

• For the statistical significance of your estimated risk-premiums see my answers [1] or [2].

• A carefully described video from John Cochrane on the Fama/MacBeth procedure is available here.