Nonparametric cointegration analysis

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 H_r that there are r cointegrating vectors against the alternative H_r+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 z_t :

p-1

∆z = ∏ Π∆z + γβTz  + e.,                                                  (²⁴)

t                j t-j       ^{1 ,}     t-p         t^,

j=1

where the Π_j, 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 e_t’s are i.i.d. N_q(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(S_poS_o,1S_op λS_pp) = 0 = 0, where S_ij = (1/n)Σn=₁ R_iβ_j^τ_t, i,j =
o,p, with R_o,t the residual vector of the regression of ∆z_t on ∆z_t-1 ,...,∆z_t-p+1 , and R_p,t the
residual vector of the regression of z_t-p on ∆z_t-1 ,...,∆z_t-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 H_r
against Hr₊₁, is nλ_r+1. The trace test tests Hr against Hq, which is equivalent to the alternative
that z_t is stationary. Johansen proves that (λ₁,...,λq ) converges in distribution to (c ₁,...,cr
,0,...,0), where the c_j^,s are positive constants, and n(λr₊₁ ,....,λq) converges in distribution to (ε₁
,...,εq_-r), where the ε_j.^,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 λ_j,_m by μ_j,_m = 1/(n√λq_+1-j,_m) and replacing Johansen’s
eigenvalues in his lambda-max and trace tests by these μ_jm ’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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