Define excess return $r^x_{it} = r_{it} - r^f_{t}$ as the return $i$ minus the risk free rate, and $f_{jt}$ similarly denotes the excess return of factor $j$ at time $t$. Let's say we have some factor model of returns where:
$$ r^x_{it} = \alpha_i + \sum_j \beta_{i,j} f_{jt} + \epsilon_{it}$$
F-test / GRS Test
If we assume the error terms $\epsilon_{it}$ follow the normal distribution, are uncorrelated over time, and are homoskedastic, a standard F-test can be used for the hypothesis that $\alpha_i = 0$ for all $i$. Finance calls this the GRS test for Gibbons, Ross, and Shanken who originally applied it to asset pricing. A statistician may see it as a form of F-test. Following similar notation as in Cochrane (2005):
- Let $\tau$ be the number of time periods of your data.
- Let $\Sigma_f$ be a $k \times k$ sample covariance matrix of your factors.
- Let $\Sigma$ be an $n \times n$ sample covariance matrix of your residuals $\epsilon_{it}$.
- Let $\boldsymbol{\alpha} = \begin{bmatrix}\alpha_1 \\ \ldots \\ \alpha_n \end{bmatrix}$ be a vector of your alphas.
- Let $\boldsymbol{\mu_f}$ be a $k \times 1$ vector giving the sample mean returns of the factors.
The GRS test statistic is given by:
$$ f_{GRS} = \left( \frac{\tau - n - k}{n}\right) \frac{\boldsymbol{\alpha}' \Sigma^{-1} \boldsymbol{\alpha}}{ 1 + \boldsymbol{\mu_f}' \Sigma_f^{-1} \boldsymbol{\mu_f}} $$
where test statistic $f_{GRS}$ follows the $F$ distribution:
$$ f_{GRS} \sim F\left(n, \tau - n - k \right)$$
$\chi^2$ test
Dropping the assumption of normally distributed error terms, there exists a test-statistic that asymptotically approaches the $\chi^2$ distribution. Let $n$ be the number of test assets, and let $T$ be the number of time periods. Define test statistic $J$ as:
$$ J = T \frac{\boldsymbol{\alpha}' \Sigma^{-1} \boldsymbol{\alpha}}{ 1 + \boldsymbol{\mu_f}' \Sigma_f^{-1} \boldsymbol{\mu_f}} $$
$J$ follows the $\chi^2$ distribution with $n$ degrees of freedom:
$$ J \sim \chi^2\left(n \right)$$
Cochrane (2005) shows how to derivate the test statistic as a special case of the Sargan-Hansen J test. You might also examine Cochrane's notes here.
Collinear residuals
If some of your test returns are essentially the same return, then you're going to have a rank deficient $\Sigma$ matrix. Use the pseudo-inverse $\Sigma^+$ instead of the inverse $\Sigma^{-1}$ and use $n$ as the number of linearly independent assets. This $n$ can be estimated by conducting a singular value decomposition of $\Sigma$ and counting the number of singular values above some threshold. (This is how MATLAB and some other packages estimate matrix rank.)
Discussion:
Generally what you find with any of these F tests or $\mathcal{X}^2$ tests is that the hypothesis that all the alphas are zero is overwhelmingly rejected. What does that mean?
One possibility is that the model of returns is misspecified, that the Carhart four factor model or whatever you're using doesn't perfectly describe the joint distribution of returns. These factor models sort of work (in the sense that they're useful), but they're not that precise either.
If you find that mutual funds have non-zero alpha relative to this asset pricing model does it mean:
- Some mutual funds have positive skill (and/or some have negative
skill)?
- That we have the wrong asset pricing model?
The two aren't really distinguishable.
There's also a famous saying of Box that all models are wrong but some are useful. Generally speaking, you can statistically reject any asset pricing model with the right test assets. Whether the model is useful is a more nuanced discussion.
Testing for betas?
You seem to want to test for a value weight market beta of 1? I don't see how this is at all useful. Mechanically, the value weight market beta across all securities must be one! I'm struggling to see how this would be useful?
Bootstrapping
Another approach to creating consistent test statistics is bootstrapping.
For example, Fama and French (2010) construct bootstrap simulations of what distribution of mutual fund returns would be expected under a hypothesis of zero alphas under the Fama French three factor model.
References
Cochrane, John, Asset Pricing, 2005, p. 230 link
Fama, Eugene and Kenneth French, "Luck versus Skill in the Cross-Section of Mutual Fund Returns", Journal of Finance
