with intercept, and with intercept and time trend. The nonparametric estimate of a
corresponds to the OLS coefficient of ln[GNP] in the regression with intercept and time trend,
whereas Johansen’s estimate of a corresponds to the regression with intercept only. Therefore
we now include an intercept plus linear time trend in the ECM (24), say π00 + π01t. However,
it seems reasonable to impose cointegration restrictions on π01, i.e., we assume that π01 is
proportional to γ, as otherwise there would be a quadratic trend in zt, which seems unlikely.
In view of the previous result we first specified p = 6, but for that case the test results for r
were inconclusive. Therefore we next specified p = 8, which yields conclusive test results: r =
1. In both cases the LR test of the restriction that π01 is proportional to γ, given r =1,is
accepted.
The estimation of the cointegrating vector, and the tests of linear restrictions on the
cointegrating vector has been based on the ECM with p = 8 without imposing the restriction
that π01 is proportional to γ, because otherwise we have to test these linear restrictions jointly
with linear restriction on π01. Cf. Johansen (1994). The estimate involved of the standardized
cointegrating vector is now (1,-.7)T, which is in tune with our nonparametric estimate (the
difference is only from the third decimal digit onwards). Again we have conducted the same
series of LR tests as before. Now only the value a = -.7 is accepted at the 10% significance
level and the values -.7 and -.75 are accepted at the 5% level. Thus the previous estimate of
a, -.75, is now rejected at the 10% significance level! This demonstrates the sensitivity of
this LR test w.r.t. to the specification of the ECM. However, once the correct specification of
the ECM has been found Johansen’s test of linear restriction on the cointegrating vector
seems more powerful than the corresponding nonparametric test.
The above empirical comparison of our nonparametric cointegration analysis with
Johansen’s approach demonstrates that our approach is capable of giving the same answers
regarding the number of cointegrating vectors and the cointegrating vectors themselves as
Johansen’s ML method, but with much less effort. Our approach gives clear answers, using
only one set of tables, regardless whether or not the cointegrated system has drift and/or the
cointegration relations contain a linear trend, and there is no ambiguity in interpreting the test
results.
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