Fama–MacBeth Cross-Sectional Tests and the CAPM Risk Premium

Summary

The document discusses how to distinguish a broad test of a positive risk–return relationship from a more specific test of the CAPM. It presents a cross-sectional return model with beta, squared beta, and residual risk as explanatory terms, and identifies the mean slope on beta as the coefficient associated with the positive trade-off implication of a general two-parameter equilibrium model.

It then proposes modifying the dependent variable to use returns in excess of the risk-free rate and testing whether the average beta slope equals the market’s expected excess return. The author asks whether this specification has precedent or whether there is a reason it is uncommon. The text cites Fama and MacBeth’s stated scope but supplies no answer, empirical estimates, or evaluation of the proposed modification. It is therefore a framing of an asset-pricing research question rather than a report of test results; inference would also depend on the model specification and how the time-varying coefficients are estimated.

Key ideas

• The Fama–MacBeth setup described uses cross-sectional returns and beta-related explanatory variables.
• A positive average beta coefficient represents a general positive relationship between risk and expected return.
• The document proposes testing the CAPM’s more specific prediction using excess returns.
• The proposal is posed as a question and the document provides no empirical result or resolution.

Tags

# Testing the CAPM a la Fama & MacBeth: specific trade-off between expected return and risk


# Testing the CAPM a la Fama & MacBeth: specific trade-off between expected return and risk












Fama & MacBeth (1973) test a two-parameter model of market equilibrium by examining whether its implications hold empirically. They work with the following generalization of the model: $$ \tilde R_{i,t} = \tilde\gamma_{0,t} + \tilde\gamma_{1,t}\beta_i + \tilde\gamma_{2,t}\beta_i^2 + \tilde\gamma_{3,t}\sigma_i + \tilde\eta_{i,t} \tag{7}. $$

Implication (C3) of a two-parameter model is $\mathbb{E}(\tilde\gamma_{1,t})>0$, i.e. there is a positive trade-off between expected return and risk.

Fama & MacBeth state explicitly on p. 612-613 that they are not testing the CAPM but a more general two-parameter model. Now, if we were to test the CAPM by their method, we could modify the generalized model to $$ \tilde R_{i,t}-R_{f,t} = \tilde\gamma_{0,t} + \tilde\gamma_{1,t}\beta_i + \tilde\gamma_{2,t}\beta_i^2 + \tilde\gamma_{3,t}\sigma_i + \tilde\eta_{i,t} \tag{7'}. $$ and modify the implication (C3) to (C3'): $\mathbb{E}(\tilde\gamma_{1,t})=\mathbb{E}(R_{m,t})-R_{f,t}$ which is more specific than (C3) in case of the CAPM. (Another option would be to keep (C3) and then add (C3') as an extra implication.) I have not seen this done. Certainly, I do not have a great overview of the asset pricing literature, but I would expect to have run into this by now if this was mainstream. Have I simply overlooked this type of testing, or is there a reason why it is not done?

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