How to compute a Fama-Macbeth R-Squared (R2)?

Question

I'm reaching out regarding the R-Squared of a Fama-Macbeth regression. This is often reported in econometric results but I have yet to find a good explanation of how it is computed.

Specifically, if I consider the second stage of a Fama-Macbeth regression, where we are potentially running hundreds of regressions, how are the R-Squareds of these hundreds of regressions aggregated into a final R-Squared for the entire procedure? I understand that the coefficients are aggregated by a simple averaging, but was unclear about the R-Squareds.

I understand that there are codes to do this, but am trying to understand what's under the hood.

Thanks!

EDIT:

From the Fama-Macbeth regression we specify a model where each return $y_{i,t}$ of portfolio $i$ in time period $t$ can be priced by: $$ y_{i,t}=\gamma_0 + \gamma_1 \beta_{1, i}+ \gamma_2 \beta_{2, i} + \dots + \gamma_j \beta_{N,i} $$

where $N$ represents the total number of factors.

When calculating the predicted values to calculate our $R^2$, do we take the residuals of each time period or the mean portfolio returns, i.e. are our residuals $y_{i,t}-\hat{y}_{i,t}$ ($i \times t$ number of resids) or only $y_{i}-\hat{y}_{i}$ (i number of resids).

Answer 1

There's nothing different here. To compute $R^2$, you need the actual values $y_i$ and the fitted (i.e. model predicted) values $\hat{y}_i$. Think of the Fama-Macbeth procedure as just another way to get fitted values $\hat{y}_i$.

Once you have your coefficient estimate $\hat{\mathbf{b}}$ from running Fama-Macbeth. Calculate $R^2$ the usual way: calculate the total sum of squares, obtain the fitted values $\hat{y}_i = \mathbf{x}_i \cdot \hat{\mathbf{b}}$, calculate the explained sum of squares, and then compute $R^2$.

Quick econometrics review

Imagine you have the following panel regression.

$$ y_{it} = \mathbf{x}_{it} \cdot \mathbf{b} + \epsilon_{it} $$

Now let's imagine that the error terms are cross-sectionally correlated (i.e. $\operatorname{E}[\epsilon_{it}\epsilon_{jt}] \neq 0$) but across time, the error-terms are independent (and we have $\operatorname{E}[\epsilon_{it_1}\epsilon_{j t_2}] = 0$ for $t_1 \neq t_2$). Because of the cross-sectional correlation, the typical OLS standard errors are going to be understated. In typical finance settings, they will be massively understated because cross-sectional correlation is big.

What to do?

• Option 1: Compute clustered standard errors, clustering on the time variable. (This is arguably a more modern approach.)
• Option 2: Fama-Macbeth procedure

(Note: in typical situations, you should obtain similar results but the two approaches involve different weighting.)

Back in 1973, cluster robust, Rogers standard errors weren't around yet, and instead Fama and Macbeth developed their immensely intuitive procedure. The basic intuition is that:

1. Each time period is independent, so we can use our regular Stats 1 techniques to estimate the mean and t-stat for a stationary time series.

2. We can run period by period cross-sectional regressions to obtain a time-series of estimates $\{\hat{\mathbf{b}}_t\}$

Combine (1) and (2) and you have the Fama-Macbeth procedure.