4.1. The lambda-min test, and a comparison with Johansen’s tests
The results in Theorems 1-2 suggest to use the test statistic λq-rm for testing the null
hypothesis Hr that there are r cointegrating vectors against the alternative Hr+1. We shall call
this test the lambda-min test, which (as will be shown below) is in the same spirit as
Johansen’s lambda-max test.
Johansen’s (1988) original approach is based on the following ECM of the q-variate
unit root process zt :
p-1
∆z = ∏ Π∆z + γβTz + e., (24)
t j t-j 1 , t-p t,
j=1
where the Πj, j >0,areq×q and β and γ are q×r parameter matrices with r the number of
cointegrating vectors (the columns of β), and the et’s are i.i.d. Nq(0,Σ) errors. By stepwise
concentrating all the parameter matrices in the likelihood function out, except the matrix β,
Johansen shows that the ML estimator of β can be derived from the eigenvectors of the
generalized eigenvalue problem det(SpoSo,1Sop λSpp) = 0 = 0, where Sij = (1/n)Σn=1 Riβjτt, i,j =
o,p, with Ro,t the residual vector of the regression of ∆zt on ∆zt-1 ,...,∆zt-p+1 , and Rp,t the
residual vector of the regression of zt-p on ∆zt-1 ,...,∆zt-p+1 . Moreover, the ordered eigenvalues
λ1 ≥ .... ≥ λq involved can be used for testing hypotheses about the number of cointegrating
vectors. In particular, Johansen proposes two LR tests for the number of cointegrating vectors,
the trace test and the lambda-max test. The test statistic of the latter test, for testing Hr
against Hr+1, is nλr+1. The trace test tests Hr against Hq, which is equivalent to the alternative
that zt is stationary. Johansen proves that (λ1,...,λq ) converges in distribution to (c 1,...,cr
,0,...,0), where the cj,s are positive constants, and n(λr+1 ,....,λq) converges in distribution to (ε1
,...,εq-r), where the εj.,s are positive random variables. Comparing Johansen’s generalized
eigenvalue results with Theorems 1-2 we see that we can mimic Johansen’s tests by
transforming our generalized eigenvalues λj,m by μj,m = 1/(n√λq+1-j,m) and replacing Johansen’s
eigenvalues in his lambda-max and trace tests by these μjm ’s. Then Johansen’s lambda(mu)-
max test becomes our lambda-min test. In this paper we shall focus on the lambda-min test
only, because for this test it is possible to optimize the power of the test to m, and the order
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