Residuals Fama MacBeth Regression

I am still asking myself what the pricing error terms in the Fama-MacBeth regression are.

Are they the intercept I regress across all assets in each month, once? Or are they the residuals of each asset in each month?

To clarify this I attach the picture of the formula I am referring to:

Also consider this example:

I have returns of 100 stocks over 120 months.

If the alphas were the residuals, I would have a 120x100 matrix.

If the alphas were an intercept I regress (just like beta) it would be a 120 alpha values vector.

The post I am referring to:

Calculating the pricing error in Fama-Macbeth Regression for Fama/French 5 Factor model

Skoestlmeier says it is an intercept. But, from the above image (source: Cochrane) it seems to me, that alphas are the residuals for every asset i=1,...,N over each month t.

I would be very grateful for clarification.

Best Regards

It's all about the notation - so i try to be very precise now.

The Fama-MacBeth approach is a cross-sectional regression at each period of time: $$R_{t}^{ei}= \beta_{i}^{'}\lambda_t+a_{it}$$

where $$R_{t}^{ei}$$ is the excess-return of asset $$i$$ at time $$t$$ and $$\beta_{i}^{'}$$ denotes the estimated beta-factor of the stock. As stated in Cochrane (Asset Pricing, rev. edition, 2004, p. 235):

[...], $$\beta$$ are the right-hand variables, $$\lambda$$ are the regression coefficients, and the cross-sectional regression residuals $$\alpha_i$$ are the pricing errors.

You are right, that for $$n$$ assets over $$T$$ periods of time, this would result in a $$T \times n$$ matrix of pricing errors $$\alpha_{it}$$ (hence the double subscript).

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

As described in my answer here, we use the standard deviations of these (averaged) cross-sectional estimates to generate the sampling errors for these estimates.

• Thank you for your time and efforts to help us! I appreciate it so much. Thank you :) – J.Pop Feb 13 at 17:37
• another question came to my mind and I hope it is okay if I try to ask you this: If I have alphas statistically different from zero, would it be correct to conclude, that there are different/additional sources of risk, that the factor beta does not capture? As economic interpretation/conclusion. – J.Pop Feb 17 at 20:02
• also, should the chi-squared values for the alphas be <= X^2 (test statistic Chi-Squared). – J.Pop Feb 17 at 22:04
• If the alpha is significantly different from zero, then the model is not able to fully explain the expected return, so your interpretation is right. On how to calculate the standard error (which you need for calculating the significance) for $\alpha$, you may look at the last section of my answer here. – skoestlmeier Feb 18 at 15:30