Why the GRS Test Can Test the CAPM Through Joint Alpha Restrictions
Summary
The document examines whether replacing the CAPM’s constant expected factor return with its realized time-varying return in time-series regressions creates an errors-in-variables problem that invalidates the GRS test. The proposed test runs asset-return regressions on the factor and jointly tests whether all intercepts are zero, treating nonzero intercepts as pricing errors. The questioner worries that regression estimates may not consistently recover parameters from a formulation using the factor’s expected return.
The accepted response cites a related analysis and concludes that a nonzero intercept in the regression implies the CAPM cannot hold. On that reasoning, the joint zero-alpha hypothesis remains a valid test of the CAPM. The document does not develop the derivation in detail or discuss finite-sample behavior, test assumptions, or alternative specifications, so its conclusion is best read as a focused clarification of the stated concern rather than a full treatment of GRS inference.
Key ideas
- The GRS procedure tests whether the intercepts across asset time-series regressions are jointly zero.
- The concern is that realized factor returns replace the factor’s expected return in the regression specification.
- The accepted response argues that a nonzero regression intercept is incompatible with the CAPM.
- The document concludes that the joint alpha test is valid for testing the CAPM, without detailing broader inference conditions.
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# Testing the CAPM: does GRS account for errors in variables (measurement error)?
# Testing the CAPM: does GRS account for errors in variables (measurement error)?
Suppose we are interested in testing the CAPM using the GRS test. Consider $N$ assets observed for $T$ time periods. Using the notation of Cochrane "Asset Pricing" (2005), the GRS test amounts to running $N$ time series regressions of the form $$ R^{ei}_t=\alpha_i+\beta_i f_t+\varepsilon^i_t \tag{12.1} $$ and testing the joint hypothesis $H_0\colon \alpha_1=\dots=\alpha_N=0$. The $\alpha$s are treated as pricing errors, so they better be zero if the CAPM is an adequate model.
However, I have a quibble with using $(12.1)$ for testing the CAPM. Consider the following. The CAPM states that $$ E(R^{ei})=\beta_i E(f). $$ While it is a single-period model, let us assume it works in all $T$ periods so that $E(R^{ei}_t)=\beta_i E(f_t)$ where $E(f_t)\equiv E(f)=:\lambda$. Let us further assume the relevant covariances and variances and thus $\beta$s are constant over time, too. This implies $$ R^{ei}_t = \tilde\alpha_i+\tilde\beta_i \lambda+\varepsilon^i_t. \tag{12.1'} $$ with $\tilde\alpha_1=\dots=\tilde\alpha_N=0$.
Comparing $(12.1)$ to $(12.1')$, we see that the former replaces $\lambda$ with $f_t$ thus introducing a measurement error (a.k.a. errors in variables). Therefore, the OLS point estimates $\hat\alpha^{\text{12.1 by OLS}}_i$ and $\hat\beta^{\text{12.1 by OLS}}_i$ are not even consistent$\color{red}{^*}$ for the true values $\tilde\alpha_i$ and $\tilde\beta_i$ corresponding to $(12.1')$.
Question: Does the GRS test take care of $\lambda$ being replaced by $f_t$?
$\color{red}{^*}$I have now realized the statement about consistency may be wrong, since we have a rather special case of $\lambda$ being a constant rather than a variable. I will check and come back to this later...
Due to lack of answers, a version of this question has been reposted on Cross Validated Stack Exchange.
## Answer by Richard Hardy (score 1, accepted)
https://quant.stackexchange.com/a/74657
See Matthew Gunn's answer to the linked thread on Cross Validated. It turns out that $\alpha_i\neq 0$ in $(12.1)$ does imply the CAPM cannot hold. To see that, take the expectations of both sides of $(12.1)$ under $\alpha_i\neq 0$. Therefore, a test of $H_0\colon \alpha_1=\dots=\alpha_N=0$ is a valid test of the CAPM.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.