How the GRS Test Responds to Rescaling a Market Factor

# GRS test does not reject a scalar multiple of the market factor


# GRS test does not reject a scalar multiple of the market factor












I have been playing with the GRS test (see my R script below) in relation to Why not use a time series regression when the factor is not a return?. I generated a $10,000\times 26$ matrix of returns on 25 "portfolios" and the market portfolio according to the CAPM. I used the function `GRS.test::GRS.test` in R to test the CAPM, and I could not reject the $H_0$. Then I added idiosyncratic $\alpha$s to each "portfolio", tested the CAPM again and now could easily reject the $H_0$. So far so good.

Then I generated data according to the CAPM again and ran the GRS test with the market's excess return replaced by an unrelated random variable (a random factor). The test easily rejected $H_0$. For completeness, I added idiosyncratic $\alpha$s to each "portfolio" and tested the CAPM again with the random factor. As in the case with the true factor, I could again easily reject the $H_0$. So far so good.

Then I generated data according to the CAPM again and ran the GRS test with the market's excess return replaced by a scalar multiple of itself with a scaling factor $c=100$. To my surprise, the test could not reject the $H_0$! The $p$-value distribution over the simulation runs was close to Uniform[0,1], as we would expect when testing the true factor. For completeness, I added idiosyncratic $\alpha$s to each "portfolio" and tested the CAPM again with the random factor. As in the case with the true factor, I could again easily reject the $H_0$.

Question: What is going on? Should the GRS test not reject a model with a scalar multiple of the true factor?

(You can run the script online and see the results for yourself at https://rdrr.io/snippets/. Just paste the script there and click "Run". You may deleted the # symbol before the commented lines to vary between the specifications of $\alpha$s and the factor that is supplied to the GRS test.)

```
library(MASS)
library(GRS.test)
data("data")  # Fama-French data: market's excess return and 25 portfolios (5x5, sorted on SMB and HML) 
data=data/100 # because original data was in percent

M=1e2 # number of simulation runs
T=1e4 # sample size (length of time series)
N=25  # number of test assets/portfolios
#c=1   # scaling factor
c=100 # scaling factor

Sigma=cov(data[,c(8:32,2)])    # empirical covariance matrix; the last column is the market, the other 25 columns are the portfolios
alpha =rep(0,N+1)              # Jensen's alpha is set to zero for all assets
#set.seed(-1); alpha=runif(n=N+1,min=-0.01,max=0.01) # Jensen's alpha, in this case nonzero for all assets
beta_m=rep(NA,N+1); for(i in 1:(N+1)) beta_m[i]=Sigma[i,N+1]/Sigma[N+1,N+1] # actual betas from Fama-French data
mu_mte=rep(mean(data[,2]),T)   # expected value of market excess return, in this case time-constant and in line with Fama-French data

results=list(); pvals=rep(NA,M)
for(j in 1:M){
 # Generate a matrix of returns according to a 1 factor model with a scaling constant c for the factor:
 set.seed(j); r_matrix=mvrnorm(n=T, mu=alpha+beta_m*mean(mu_mte), Sigma=Sigma)
 f_matrix=cbind(r_matrix[,N+1])*c               # factor return matrix for the function GRS.test()
 #set.seed(-j); f_matrix=cbind(rnorm(T,mean=5)) # random, unrelated factor returns
 results[[j]]=GRS.test(r_matrix[,1:N],f_matrix) # GRS test
 pvals[j]=results[[j]]$GRS.pval
}

# Plot p-values of all simulation runs of the GRS test
plot(sort(pvals),xlim=c(0,M),ylim=c(0,1)); abline(a=0,b=1/M,lwd=2,col="red")
```

P.S. At first, there was also a puzzle about not being able to reject a random factor, but that was solved by increasing the sample size and thereby the power of the test. The related question and answer can be found here.