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?


1 Answer 1


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.


Cochrane, John, Asset Pricing, 2005, p. 230

  • 1
    $\begingroup$ Apologies for the wrong formatting of the table, I will upload the correct one. a~a" is the mean absolute alpha The number of portfolio returns that are being examined.16175 The number of time periods used to produce the estimates: 647 observation Whether we're working with monthly returns (I assume so). Monthly $\endgroup$
    – rahaa
    Aug 25, 2017 at 16:05
  • $\begingroup$ @rahaa That makes more sense, let me update my answer. $\endgroup$ Aug 25, 2017 at 19:14

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