Nonparametric cointegration analysis

with (the log of) real money, the real interest rate and the inflation rate. It seems that only
three out of seventeen tested series are likely (close to) genuine unit root processes, namely
the logs of GNP, wages and velocity of money. For the other series the test results were
either in favor of the trend stationarity hypothesis or inconclusive, in the sense that not all the
tests favored the same hypotheses. The latter results may be due to the presence of trend
breaks, but we did not test for that. Next, inspection of the plots of the remaining three series
(see Schotman and Van Dijk, 1991) revealed possible cointegration of the logs of GNP and
wages. Therefore, we selected these two series for our cointegration analysis. The two-
dimensional vector time series involved has length n = 80 (from 1909 to 1988). Finally, as a
double check we applied our nonparametric cointegration test to each of the two series: the
unit root hypothesis could not be rejected at the 10% significance level.

8.2. Nonparametric cointegration analysis

The result of our nonparametric cointegration analysis is that the null hypothesis of no
cointegration (r = 0) is rejected at the 5% significance level, whereas the null hypothesis r =
1 is accepted at the 10% significance level. Thus we conclude that ln(wages) and ln(GNP) are
cointegrated: r = 1. This result is confirmed by the estimation approach in section 4.4: the
function gm(r) defined by (26), with m = 2, takes the values gm(0) = 1382.966, gm(1) = 3.087,
gm(2) = 28164.158, hence the estimated number of cointegrated vectors is 1. The estimate of
the standardized cointegrating vector is (1, -.70)^T, i.e., ln(wages) - .7ln(GNP) is (trend)
stationary.

In order to see how "significant" the estimated cointegrating vector is, we have
conducted a series of trace tests (which in this case coincide with the lambda-max tests), for 2
× 1 matrices H = (1,a)^T with

a ∈ {-.4,-.5,-.6.-.65,-.7,-.75,-.8,-.9,-1}.

The null hypothesis is accepted at the 10% significance level for a ranging from -.6 to -.8,
and at the 5% level for a ranging from -.5 to -.9.

8.3. Johansen’s approach

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