Linear Model setup for Second-pass Regression

Question

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 1

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.