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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.

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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. $$\hat{\lambda} = \frac{1}{T} \sum_{t=1}^{T}{\hat{\lambda}}_{t}$$ $$\hat{a}_i = \frac{1}{T} \sum_{t=1}^{T}{\hat{a}}_{it}$$


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.

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  • $\begingroup$ Thank you for the reply. I believe that makes sense in terms of what Fama MacBeth is. In that case, can you help me explain what that paper I linked is doing? Are the "betas" it talks about actually the factor exposures, or is he just already starting with beta hats and moving to step 2? $\endgroup$ – user9592573 Mar 2 at 14:58
  • $\begingroup$ Section 2.9 is confusing because they state to run a cross-sectional regression (exp. 2) over the period of five years. In fact, this is a time-series regression. Section 2.4-2.6 describes a procedure to estimate the betas (factor loadings) which they use to finally estimate the factor return series (usually it is the other way around: factor returns are calculated given a certain strategy [see Fama/French] and then the factor loadings are estimated). I assume they use exp. 2 again for the Fama/MacBeth procedure, but in my opinion, the paper is poorly written and unclear in many ways. $\endgroup$ – skoestlmeier Mar 7 at 9:35

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