# Testing asset pricing models with Roll's critique in mind

Roll's critique (Roll, 1977) can be summarized as follows (quoting Wikipedia):

1. Mean-variance tautology: Any mean-variance efficient portfolio $$R_{p}$$ satisfies the CAPM equation exactly: $$E(R_{i})-R_{f}=\beta_{{ip}}[E(R_{p})-R_{f}].$$ Mean-variance efficiency of the market portfolio is equivalent to the CAPM equation holding. This statement is a mathematical fact, requiring no model assumptions.
Given a proxy for the market portfolio, testing the CAPM equation is equivalent to testing mean-variance efficiency of the portfolio. The CAPM is tautological if the market is assumed to be mean-variance efficient. (And here is a question about how to interpret this, exactly.)
2. The market portfolio is unobservable: The market portfolio in practice would necessarily include every single possible available asset, including real estate, precious metals, stamp collections, jewelry, and anything with any worth. The returns on all possible investments opportunities are unobservable.
From statement 1, validity of the CAPM is equivalent to the market being mean-variance efficient with respect to all investment opportunities. Without observing all investment opportunities, it is not possible to test whether this portfolio, or indeed any portfolio, is mean-variance efficient. Consequently, it is not possible to test the CAPM.

The critique sounds quite devastating, thus my questions:

1. Given Roll's critique, should we drop empirical tests of the CAPM?
If not, what can be concluded from an empirical test of the CAPM?
2. Does the same or analogous critique apply to multifactor models?

Here is what I was able to find in the literature:

1. Cochrane "Asset Pricing" (revised edition, 2005) and his lecture series on YouTube – nothing.
2. Bodie, Kane and Marcus "Investments" (12th edition, 2021) present Roll's critique and then just state the following: Given the impossibility of testing the CAPM directly, we can retreat to testing the APT.... I take this as giving up on testing the CAPM because of the critique. But researchers have continued testing asset pricing models (including the CAPM, I think) also after 1977, so there must be something to that...
3. Campbell, Lo & MacKinlay "The Econometrics of Financial Markets" (1996) mention on p. 214-215 that
(i) Stambaugh (1982) found the results to be insensitive to alternative versions of the benchmark portfolio and
(ii) Kandel and Stambaugh (1987) and Shanken (1987a) find that as long as the correlation between the actual market portfolio and the proxy is above 0.70, rejection based on the proxy implies rejection based on the actual market portfolio.
These points address the second point of Roll's critique (unobservability) but not the first one (tautology).

This is a repost from Quantitative Finance Stack Exchange where the question was well received and got quite a bit of attention due to a bounty, but still got no answers.