# How can I perform a familywise error rate correction for Johansen cointegration tests?

I want to test multiple possible cointegration relationships with the Johansen cointegration test. I'm currently using the urca package in R with the ca.jo test. I was going to use the Bonferroni correction to correct the familywise error rate. However, ca.jo doesn't report p-values. It only reports test statistics and critical values:

######################
# Johansen-Procedure #
######################

Test type: maximal eigenvalue statistic (lambda max) , with linear trend

Eigenvalues (lambda):
[1] 0.335639191 0.001256000

Values of teststatistic and critical values of test:

test 10pct  5pct  1pct
r <= 1 |   1.26  6.50  8.18 11.65
r = 0  | 408.52 12.91 14.90 19.19
[More ca.jo output snipped]


How can I correct for rejecting the null incorrectly when I perform multiple cointegration tests?

You could use alternatively the function rank.test() from package tsDyn, which provides the p-values for the Johansen test, based on the gamma approximation of Doornik (1998, 1999)

Compare:

> library(tsDyn)
> library(urca)
>
> data(denmark)
> sjd <- denmark[, c("LRM", "LRY", "IBO", "IDE")]
> summary(ca.jo(sjd, type="eigen", K=2))

######################
# Johansen-Procedure #
######################

Test type: maximal eigenvalue statistic (lambda max) , with linear trend

Values of teststatistic and critical values of test:

test 10pct  5pct  1pct
r <= 3 |  0.56  6.50  8.18 11.65
r <= 2 |  6.59 12.91 14.90 19.19
r <= 1 | 10.15 18.90 21.07 25.75
r = 0  | 31.51 24.78 27.14 32.14

> ve <- VECM(sjd, lag=1, estim="ML")
> summary(rank.test(ve))[,c(1,5,6)]
r      eigen eigen_pval
1 0 31.5135590    0.01196
2 1 10.1452836    0.73451
3 2  6.5888726    0.54673
4 3  0.5560158    0.45588


Refs:

• Doornik, J. A. (1998) Approximations to the Asymptotic Distributions of Cointegration Tests, Journal of Economic Surveys, 12, 573-93

• Doornik, J. A. (1999) Erratum [Approximations to the Asymptotic Distribution of Cointegration Tests], Journal of Economic Surveys, 13, i