# GRS Test Statistic and p-value based on Maximum Likelihood Estimator for Covariance matrix

### Description

W statistic given in (7) of GRS (1989) <DOI:10.2307/1913625>

### Usage

1 | ```
GRS.MLtest(ret.mat, factor.mat)
``` |

### Arguments

`ret.mat` |
portfolio return matrix, T by N |

`factor.mat` |
matrix of risk factors, T by K |

### Details

T: sample size, N: number of portfolio returns, K: number of risk factors

### Value

`GRS.stat` |
GRS test statistic |

`GRS.pval` |
its p-value |

`theta` |
maximum Sharpe ratio of the K factor portfolios |

`thetas` |
slope of the efficient frontier based on all assets |

`ratio` |
theta/thetas, proportion of the potential efficiency |

### Note

Applicable to CAPM as well as a multi-factor model

### Author(s)

Jae H. Kim

### References

Gibbons, Ross, Shanken, 1989. A test of the efficiency of a given portfolio, Econometrica, 57,1121-1152. <DOI:10.2307/1913625>

### See Also

Fama and French, 1993, Common risk factors in the returns on stocks and bonds, Journal of Financial Economics, 33, 3-56. <DOI:10.1016/0304-405X(93)90023-5>

Fama and French, 2015, A five-factor asset-pricing model, Journal of Financial Economics, 116-1-22. <DOI:http://dx.doi.org/10.1016/j.jfineco.2014.10.010>

### Examples

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