# 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.

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.