Fama–MacBeth Regressions for Testing the CAPM

Summary

The document outlines a two-pass Fama–MacBeth procedure for evaluating the Capital Asset Pricing Model. First, time-series regressions estimate each asset’s market beta from excess returns. Then, for each period, a cross-sectional regression relates asset excess returns to those estimated betas and a time-varying market price of risk. The resulting pricing errors can be averaged and tested for whether they are jointly zero.

The question asks why this second-stage regression omits asset-specific intercepts, and whether including them would provide an equally valid joint test. It observes that adding intercepts changes the estimation of the period-specific risk price, so the resulting pricing errors and test are not automatically equivalent. The document gives the model setup and identifies the specification issue, but includes no answer, empirical results, or discussion of test-statistic assumptions. It is therefore useful as a statement of the CAPM testing framework and a methodological question, rather than a resolved comparison of the two specifications.

Key ideas

• The Fama–MacBeth procedure first estimates asset betas using time-series regressions.
• A second-stage cross-sectional regression estimates the period-specific market price of risk.
• Average pricing errors can be tested for whether they are jointly zero.
• Adding asset-specific intercepts changes the second-stage estimation and raises a specification question.
• The document poses this comparison but does not resolve it or provide empirical evidence.

Tags

# CAPM test methodology


# 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.

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