The optimal level is calculated by minimizing expected loss from hypothesis testing

Under the assumption of equal prior and identical losses from Type I and II errors

1 | ```
GRS.optimal(T, N, K, theta, ratio, Graph = "TRUE")
``` |

`T` |
sample size |

`N` |
the number of portfolio returns |

`K` |
the number of risk factors |

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

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

`Graph` |
show graph if TRUE. No graph otherwise |

Based on the power calculation of the GRS test, as in GRS (1989) <DOI:10.2307/1913625>.

The blue square is the point where the expected loss is mimimized.

The red horizontal line indicate the point of the covnentional level of significance (alpha = 0.05).

`opt.sig ` |
Optimal level of significance |

`opt.crit ` |
Critical value corresponding to opt.sig |

`opt.beta ` |
Type II error probability corresponding to opt.sig |

ratio = theta/thetas

thetas = maximum Sharpe ratio of the K factor portfolios: GRS (1989) <DOI:10.2307/1913625>

Jae H. Kim

Leamer, E. 1978, Specification Searches: Ad Hoc Inference with Nonexperimental Data, Wiley, New York.

Kim, JH and Ji, P. 2015, Significance Testing in Empirical Finance: A Critical Review and Assessment, Journal of Empirical Finance 34, 1-14. <DOI:http://dx.doi.org/10.1016/j.jempfin.2015.08.006>

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

Kim and Shamsuddin, 2016, Reapparaising Empirical Validity of Asset-Pricing Models with consideration of Statistical Power. Working Paper

Kim and Ji (2015)

1 | ```
GRS.optimal(T=90, N=25, K=3, theta=0.25, ratio=0.4) # Figure 3 of Kim and Shamsuddin (2016)
``` |

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