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.


1 Answer 1


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.


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