*! version 4.3 22jul2020
help grsftest

Title

grsftest -- Module to perform the Gibbons, Ross, Shanken (1989, GRS) test of mean-variance efficiency of asset returns in empirical asset pricing models.

Syntax

grsftest
varlist
[if]
, factor(varlist)
[details]

Required
factor(varlist) one or more factor portfolios with excess returns (e.g. xmkt smb hml)

Optional
details displays the estimation results of factor model. It reports the estimated intercepts (on average and by individual asset) and summary statistics of the asset and factor portfolio returns.

Description

grsftest calculates the Gibbons, Ross, Shanken (1989, GRS) F-test statistic for the test assets in varlist and the factor portfolios in factor(). When grsftest runs the time-series regression, it assumes the returns for both the test assets and factor portfolios are excess returns (in excess of the riskless rate of return). grsftest uses the following formula to calculate GRS F-test statistic:

GRS F-test, H0 representation:

F = [T/N][(T-N-K)/(T-K-1)] * (â' Sigma^-1 â) / (1 + E[rp]' Lambda^-1 E[rp]) ~ F(N,T-N-K)

where, T denotes the number of observations in terms of time series. N denotes the number of test assets. K denotes the number of factor portfolios. E[p] is the sample means of the factor portfolios. â is the estimated intercepts. Sigma is the estimated covariance matrix of residuals. Lambda is the estimated covariance matrix of factor portfolios without degrees of freedom adjustment: Lambda = (1/T)(rp'*rp) - E[rp]E[rp]'

Remarks

grsftest does not adjust for the degrees of freedom, when calculating estimator of the sample covariance matrix of the factor portfolios. This approach avoids a common misrepresentation of the GRS paper. See Kamstra and Shi (2020) for more.

Stored Results

grsftest stores the following in r(), regardless [details] is specified or not:

Matrices

r(summary) estimated intercepts on average and by individual asset

r(alphas) summary statistics of the asset and factor portfolio returns

Examples

. grsftest excess_ret1 excess_ret2, factor(xmkt smb hml) d

References

Gibbons, M.R., S. Ross, and J. Shanken, 1989. "A test of the efficiency of a given portfolio" Econometrica, 57(5), 1121-1152.

Kamstra, M.J., R. Shi, 2020. "A Note on the GRS Test" Working Paper

Author

Mengnan(Cliff) Zhu

Brandeis International Business School

cliffzhu@brandeis.edu