Interpreting GRS and Chi-Square Tests of Asset Pricing Alphas
Summary
This document explains how to distinguish the finite-sample Gibbons, Ross, and Shanken F-test from an asymptotic chi-square test when assessing whether multiple time-series regression intercepts are jointly zero. The F-test is described under assumptions of normally distributed, homoskedastic errors that are uncorrelated over time. The chi-square alternative is presented as requiring fewer distributional assumptions, with a statistic built from the estimated alpha vector, its covariance matrix, factor moments, sample length, and number of test assets.
The example concerns 25 value-weighted portfolios and reports a chi-square statistic with 25 degrees of freedom, yielding a very small p-value. The key interpretive caution is that a chi-square result should not be labeled as the GRS F-test. The document gives a brief formula and points to asset-pricing references, but does not explain implementation details, finite-sample behavior, or how violations of the stated assumptions affect inference.
Key ideas
- The GRS test jointly tests whether the intercepts in a set of factor regressions are zero.
- The classical GRS F-test relies on normal, homoskedastic errors with no serial correlation.
- An asymptotic chi-square test can be used with fewer distributional assumptions.
- The chi-square reference distribution uses degrees of freedom equal to the number of test assets.
- A chi-square statistic and p-value should not be reported as a GRS F-test.
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Full text
# how to interpret the GRS F test values?
# how to interpret the GRS F test values?
I'm comparing the performance of Fama French three factor and Carhart four factor models. For the regression analysis, I have used the 25 Value Weighted portfolios sorted on size and B/M.
The Table above are the values obtained for the GRS ([Gibbons, Ross and Shanken][3]) test. I'm not sure about the way to analyse this table. Can anyone help me please?
## Answer by Matthew Gunn (score 11, accepted)
https://quant.stackexchange.com/a/35782
You don't have a GRS test there that all the alphas are zero. You have a $\chi^2$ test that all the alphas are zero. (The p-value associated with that test statistic corresponds to a chi-squared distribution with 25 degrees of freedom. `1 - chi2cdf(81.338394, 25) = 7.029276349879154e-08`)
Perhaps examine this answer here.
#### Quick review of the F-test (GRS test)
- Under the assumption of normal error terms, that are homoskedastic and uncorrelated over time, one can apply an F-test that all the alphas are zero.
The Gibbons Ross Shanken (GRS) test is what finance calls a statistical F-test for the hypothesis that all the alphas (from a set of time-series regressions) are zero. Each $\alpha_i$ is the intercept term in a time-series regression of excess returns $r_{it} - r^f_t$ on factors.
Perhaps examine this answer on the meaning of alpha and why a test that all alphas are zero constitutes a joint test of market efficiency and an asset pricing model.
#### Quick review of $\chi^2$ 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 (in your case 25), 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:
$$ J \sim \chi^2\left(n \right)$$
Definition of variables are given here. Cochrane (2005) shows how to derivate the test statistic as a special case of the Sargan-Hansen J test. You might examine Cochrane's notes here.
#### References:
Cochrane, John, Asset Pricing, 2005, p. 230Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.