Fama–MacBeth Second-Pass Regression for Beta Pricing

Summary

The note explains the second pass of a linear beta-pricing model. First, estimate each asset’s market beta from a time-series regression of its excess returns on the market’s excess return. Then use the estimated betas as independent variables in a cross-sectional regression, with each asset’s average excess return as the dependent variable; the slope estimates the risk premium associated with beta.

It cautions that second-pass standard errors need adjustment, naming Fama–MacBeth and Shanken approaches. The model implies that expected returns are linear in the market factor and that first-pass alphas are jointly zero. The response suggests testing linearity by adding nonlinear beta terms and testing the joint-alpha condition with the GRS statistic. These are model implications and diagnostics, not evidence that the model fits any particular asset sample; the question’s additional residual-variance term is not addressed in the accepted answer.

Key ideas

• Estimate each asset’s beta in a time-series regression before the cross-sectional second pass.
• Use average excess return as the second-pass dependent variable and estimated beta as an explanatory variable.
• The second-pass slope represents the estimated risk premium for beta exposure.
• Adjust second-pass standard errors using approaches such as Fama–MacBeth or Shanken.
• Test linearity with nonlinear terms and test whether first-pass alphas are jointly zero with the GRS statistic.

Tags

# Linear Model setup for Second-pass Regression


# Linear Model setup for Second-pass Regression












I'm confused on modeling the second pass regression given the beta's from the first pass.

First-pass regression :

$r_{it} - r_{ft} = a_{i}+b_{i}(r_{Mt}-r_{ft})+e_{it}$

For estimating this model (9 diff models), I regressed the annual returns for different stocks on the index variable and obtained 9 different betas.

Second-pass regression :

$\overline{r_{i}-r_{f}} = \gamma_{0}+\gamma_{1}b_{i} + \gamma_{2}\sigma^{2}(e_{i})$

Now that I have the betas, I need to regression the average return on the reported betas, but I'm not sure what the dependent variable is in this regression?

What is the dependent variable in this case and how would you run this regression?

## Answer by fni (score 1, accepted)

https://quant.stackexchange.com/a/17030

In the second pass, the independent variables are the first pass estimated betas. That is, you estimate $\hat{\beta_i}$ in time series for every stock i

$$r_{i,t} - r_{f,t} = \alpha_i + \beta_i(r_{M,t}-r_{f,t}) + \epsilon_t$$

and then you estimate risk premia $\hat{\lambda}$ according to the following regression: $$\overline{r_{i,t} - r_{f,t}} = a_0 + \lambda \hat{\beta_i} + u_i$$ Do not forget to adjust second pass standard errors according to Fama&MacBeth or Shanken

This type of (linear beta pricing) models have two implications:

- Expected returns are linear in the market factor

- First pass alphas are jointly equal to zero

To check the first you can add nonlinear terms to the second pass and check if they are significant, to check the second use the GRS statistic.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.