GMM Moments for Testing a Constrained Factor Pricing Model

Summary

The document poses an estimation problem for a single-factor asset pricing model when the factor is not itself a return. It specifies time-series regressions of asset excess returns on the factor, with intercepts constrained by factor loadings, the factor risk premium, and the factor mean. The error terms are assumed independent and identically distributed over time, homoskedastic, and independent of the factor.

The suggested route is to begin with the moment conditions associated with unrestricted ordinary least squares regressions, then impose the intercept restriction and estimate the restricted model using generalized method of moments. The task also asks how to construct a test of that restriction, referring readers to standard GMM results. This is a useful statement of the modeling setup and estimation strategy, but the document is posed as a question and supplies no actual moment vector, estimator derivation, or test statistic. It therefore identifies the ingredients and assumptions rather than providing a complete worked procedure; the setup is attributed to a textbook exercise.

Key ideas

• The model regresses each asset's excess return on a single factor in a time-series specification.
• The intercepts are restricted to depend on the factor loadings, risk premium, and factor mean.
• The proposed estimation starts from unrestricted OLS moments and then imposes the pricing restriction.
• GMM can be used to estimate the restricted parameters and test whether the intercept constraints hold.
• The document states the question and assumptions but does not present a worked solution.

Tags

# GMM time-series regression factor model with factors that are not returns


# GMM time-series regression factor model with factors that are not returns












Factor models with factors that are not returns are usually estimated and tested by cross-sectional regressions. However, there is a way to use time-series regression to estimate and test the model. The time-series regression is given by: $$R_t^{ei} = a_i + \beta_i f_t + \varepsilon_t^i \ \ ; \ \ t = 1, 2, \cdots, T \ \text{for each } i$$ where $R_t^{ei} = R_t^i - R_t^f$, i.e., the excess return of asset $i$, $f_t$ is a single factor, and the error term $\varepsilon$ is i.i.d. over time, homoskedastic, and independent of the factors. Furthermore, the above asset pricing model does not leave $a_i$ free, instead they must satisfy $a_i = \beta_i(\lambda - E(f))$ where $\lambda$ is the factor risk premium.

Question: Using GMM, write down a set of moment conditions that you can use to estimate this model and test the restriction on the $a_i$. How would you estimate the parameters and what formula would you use to compute a test? [You can use the already known results of the famous GMM formulas].

The hint is to start with the moments for unrestricted OLS time-series regressions, then impose the constraint between $a$ and $\lambda$, then estimate the restricted model.

For anyone that's interested, this question comes from John Cochrane's Asset Pricing (2005 revised edition) book, chapter 12 question 5.

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