Understanding the GRS test of the Capital Asset Pricing Model

Question

I posted a similar question on Quantitative Finance Stack Exchange a while ago. It has not received any answers, thus I am reposting a version of it here in the hope of finding a larger pool of experts.

Suppose we are interested in testing the Capital Asset Pricing Model (CAPM) using data on $N$ assets observed for $T$ time periods. The CAPM states that $$ E(R^{ei})=\beta_i E(f) \tag{*} $$ where $R^{ei}$ is the excess return on an asset $i$, $f$ is the excess return on the market portfolio and $\beta_i\equiv\frac{\text{Cov}(R^{ei},f)}{\text{Var}(f)}$. While the CAPM is a single-period model, let us assume it works in $T$ time periods so that $E(R^{ei}_t)=\beta_i E(f_t)$ where $E(f_t)\equiv E(f)\equiv\lambda$. Let us also assume the relevant covariances and variances and thus also $\beta$s are constant over time, too.

According to Cochrane "Asset Pricing" (2005) Chapter 12, the Gibbons, Ross and Shanken (GRS) test of the CAPM (GRS, 1989) 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 $H_0\colon \alpha_1=\dots=\alpha_N=0$.

Now, I do see how setting $\alpha_1=\dots=\alpha_N=0$ in $(12.1)$ and integrating w.r.t. $f_t$ implies $(*)$. However, I do not think that $(*)$ implies $\alpha_1=\dots=\alpha_N=0$ in $(12.1)$. I am only able to get the following:
$$ R^{ei}_t = 0+\beta_i E(f)+\varepsilon^i_t. \tag{12.1'} $$ It is as if we have measurement error in $(12.1)$ compared to $(12.1')$, as $f_t$ gets in place of $E(f)$.

How do I reconcile $(12.1)$ with $(12.1')$ so that the GRS test makes sense?

Answer 1

For each asset $i$, consider the time-series regression to estimate $\alpha_i$ and $\beta_i$: $$ R^e_{it} = \alpha_i + \beta_i f_t + \epsilon_{it} $$

Take expectations to get rid of the error term and get:

$$ \operatorname{E}[R^e_{i}] = \alpha_i + \beta_i \operatorname{E}[f]$$

Meanwhile (*) says $\operatorname{E}[R^e_{i}] = \beta_i \operatorname{E}[f]$. This can only be true if $\alpha_i = 0$. If $\alpha_i \neq 0$ then $\operatorname{E}[R^e_i] \neq \beta_i \operatorname{E}[f]$. If you can reject $\forall_i \alpha_i = 0$ then you can reject the factor model that you're testing.

As a practical matter, the way these tests work out is that if you have portfolios or assets with any interesting and significant spread in expected returns then you can typically reject any asset pricing model. You can statistically reject the Fama-French three factor model. The more practical but fuzzy test is whether the model is telling you something useful about the data! Even though you can reject the Fama-French 3 Factor Model with the right test assets, that model captures something about the cross-section of expected returns that's empirically there while the CAPM is just entirely in tension with the data.