# Understanding the GRS test of the Capital Asset Pricing Model

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?

• You are correct. Testing the CAPM is fraught with difficulties, largely due to it being a one-period model. However, if we don't model as in 12.1 when applying as a multiperiod model, then we are assuming all assets' expected returns are constant through time, and that is clearly false. Feb 14, 2023 at 20:45
• The one-period capm assumes investors adjust starting prices so that the capm will hold in the future single period. Such price adjustments are not included in the (one-period) capm period. (In effect, the one-period capm is untestable.) Yet multi-period testing is forced to include all price movements by necessity, contradicting the capm. Hence, the reason why the capm frequently fails when tested this way. Feb 14, 2023 at 21:06
• @GrahamBornholt, thanks for you input! Your points make sense in general. However, I wonder if they help solve this exact issue. It seems to me the CAPM + the GRS test amount to a well defined set of assumptions (the CAPM) and a well defined procedure (the GRS test) that purports to achieve a specific goal (test the validity of the CAPM). It is my concern that the assumptions + the procedure do not actually facilitate goal achievement for reasons outlined above. Could you point out exactly which part of my argument is mistaken? Feb 15, 2023 at 7:43
• I don't think you are mistaken. When testing the Capm, it is normal to use what used to be called the 'market model' (your 12.1) Feb 15, 2023 at 9:34
• The market model is used in a variety of tests of the capm, not just the GRS test. Feb 15, 2023 at 9:38

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