The details of the test results involved are presented in the separate appendix to this
paper.
8.4. Monte Carlo comparison
In order to check whether the above results are typical for this data set or not, and
which test performs better, we have conducted our nonparametric tests and Johansen’s tests
on 500 replications of ln(wages) and ln(GNP) with sample size n = 80, on the basis of the
estimated ECM (24) with p = 8, and an intercept plus linear trend π00 + π01t, where π01 is
proportional to γ. The first eight observations were taken from the actual data set, and the
errors et were drawn independently from the bi-variate normal distribution with zero mean
vector and variance matrix equal to the estimated variance matrix. All tests are conducted at
the 10% significance level. Johansen’s tests are conducted for p = 6, 8 and 10, in order to
check the sensitivity of these tests for the VAR order p, with an intercept and a linear trend
included in the ECM, and cointegration restrictions on the trend parameters imposed.
The Monte Carlo results, presented in Table 5, indicate that for this data-generating
process our nonparametric test for testing the number of cointegrating vectors performs better
than Johansen’s lambda-max test, even for the correct VAR order p = 8. The nonparametric
test gives in about 79% of all cases the correct answer r = 1, whereas the corresponding
percentages for Johansen’s lambda-max test are only 58% if p = 6, 70% if p = 8 and 56% if
p = 10. This illustrates once more the importance of finding the correct order p of the ECM:
under as well as over-specification of p seem to have quite a damaging effect on the test
results for r. Moreover, Johansen’s test of linear restrictions on the cointegrating vectors
suffers more from size distortions than the nonparametric test, although if we would correct
for size Johansen’s test seems more powerful. At first sight these results seem odd, because
Johansen’s approach is a full ML approach. However, as admitted by Johansen, the optimality
properties of ML may not apply to the nonstandard case involved. The results in Table 5
confirm this conjecture.
<Insert Table 5 about here>
The above Monte Carlo analysis, of course, does not provide sufficient evidence that
29
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