CAPM test methodology

Setting

If we test the CAPM using Fama-MacBeth regressions we do the following:

First, run cross-sectional regressions to determine the beta loading

$$R^i_t- r^f_t = a_i +\beta_i (R^M_t- r^f_t) + \epsilon_t^i$$

Afterwards, we determin the market price of risk for each $t$ and calculate the model errors

$$R^i_t- r^f_t = \hat{\beta}_i \lambda^M_t + u_t^i \quad (1) \\ \alpha_{i,t} = R^i_t- r^f_t - \hat{\beta}_i \lambda^M_t$$ Then we can test if the $\bar{\alpha}_i = mean(\alpha_{i,t})$ are jointly zero or not using certain test statistics.

Question

I want to understand choice of regression model in equation $(1)$. Why don't we use the following $$R^i_t- r^f_t = \alpha_i + \hat{\beta}_i \lambda^M_t + u_t^i \quad (2)$$ and test if the $\alpha_i$ are jointly zero?

The only difference between $(1)$ and $(2)$ is the inclusion of $\alpha$ into the $\lambda_t^M$ estimation. For both approaches it is possible to come up with simple test statistics. So both approaches can be used equally easy.

• "The only difference between (1) and (2) is the inclusion of α into the β estimation". I don't understand this statement. In both (1) and (2) $\hat{\beta}$ is fixed, it is not being estimated, it came out of the time series regressions of the first part. – Alex C Oct 16 '16 at 17:42
• I mixed up $\lambda$ and $\beta$ – Phun Oct 16 '16 at 17:56