Fama–MacBeth Pricing Errors Across Assets and Time
Summary
The document clarifies the meaning of pricing errors in the Fama–MacBeth asset-pricing procedure. At each time period, a cross-sectional regression relates assets’ excess returns to their factor exposures and estimates period-specific factor premia. The residual for each asset in each period is that observation’s pricing error, so the full set of errors has an asset-by-time structure.
The reported pricing error for an asset is obtained by averaging its period-specific residuals over time; factor-premium estimates are likewise averaged across periods. Thus the intercept-like term should not be confused with a single common alpha for each month: the notation tracks an error for each asset and period before forming asset-level averages. The answer also notes that variation in the period-by-period estimates is used to obtain sampling errors. This is a notation-focused explanation and does not address choices such as beta estimation, weighting, or inference adjustments in particular applications.
Key ideas
- Fama–MacBeth estimation runs a separate cross-sectional return regression in each period.
- Each asset-period regression residual represents a pricing error for that asset at that time.
- The residuals form an asset-by-time collection before they are averaged over time.
- Average factor premia and average asset pricing errors summarize the period-specific estimates.
- Variation across period estimates is used to calculate sampling errors.
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# Residuals Fama MacBeth Regression
# 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
## Answer by skoestlmeier (score 3, accepted)
https://quant.stackexchange.com/a/44033
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.