Nonparametric cointegration analysis

n²T (H) →trace(W*_mV* ζ¹J,                                                          (³⁵)

m∖                       s, m s,q r,ml

whereas if this null hypothesis is false, T_m(H) converges in distribution to the sum, T₁,_m(H),
say, of the s ₁ solutions of (34), hence plim_n→∞n²T_m(H) = ∞. Thus, denoting the critical value
of the trace test at the α×100% significance level by M_α,s,q-r,m, we reject the null if n 2 ^rTm (H) ≥
^Mα,s,q-r,m ^.

6.3. The choice of "m"

A Monte Carlo simulation of the limiting distribution (35), based on 10,000
replications of the random vectors Y^*_k^* and X^*_k for k = 1,...,m, with m = max(s,q-r),...,20,
reveals that M_α,s,q-r,m is decreasing in m for m ≥ q-r+s, and infinite for m < q-r+s, due to
(near-) singularity of the matrix (32). See the separate appendix to this paper. Using the
approximation

P ( n ² Tm (H) ≥ Mα_m ) ≈ PT1m (H) ≥ n 2Mas,j

and

Lemma 7.

n ∖M

α    α, s, q-r, m

λ R (R_a^τ_rC(1)C(1)^tR )

mιn∖ q ^{r v}∙ ^{7 v}∙ q q-rj

where the right-hand side probability is an increasing function of m, and λ_min(.) stands for the
minimum eigenvalue of the matrix involved,
it follows that in order to boost the power of the test we should choose m "large". On the
other hand, m should not be too large, as otherwise m acts as being dependent on the sample
size n, which may distort the size of the test. Since the critical values M_α,s,q-r,m hardly change
anymore for m > 2(q-r+s), and since we have to choose m ≥ q as otherwise the matrix Â_m

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