# GRS.optimal: Optimal Level of Significance for the GRS test: Normality... In GRS.test: GRS Test for Portfolio Efficiency, Its Statistical Power Analysis, and Optimal Significance Level Calculation

## Description

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

The F-distributions are used to calculate the power, under the normality assumption

## Usage

 `1` ```GRS.optimal(T, N, K, theta, ratio, p = 0.5, k = 1, Graph = TRUE) ```

## Arguments

 `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 `p` prior probability for H0, default is p = 0.5 `k` relative loss, k = L2/L1, default is k = 1 `Graph` show graph if TRUE. No graph otherwise

## Details

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

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

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

## Value

 `opt.sig ` Optimal level of significance `opt.crit ` Critical value corresponding to opt.sig `opt.beta ` Type II error probability corresponding to opt.sig

## Note

ratio = theta/thetas

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

Jae H. Kim

## References

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, 2017, Empirical Validity of Asset-pricing Models: Application of Optimal Significance Level and Equal Probability Test

Kim and Choi, 2017, Choosing the Level of Significance: A Decision-theoretic Approach

## Examples

 `1` ```GRS.optimal(T=90, N=25, K=3, theta=0.25, ratio=0.4) # Figure 3 of Kim and Shamsuddin (2017) ```

### Example output

```\$opt.sig
[1] 0.11914

\$opt.crit
[1] 1.451265

\$opt.beta
[1] 0.1182896
```

GRS.test documentation built on Dec. 4, 2017, 9:03 a.m.