Nonparametric cointegration analysis

4.4. Estimating the number of cointegrating vectors

Rather than testing for the number of cointegrating vectors, we can also estimate it
consistently, as follows. Denote

1 Y¹

q

∏λ _t
I^k1

q

^{n 2 q}II^λk,m i^frq.

where m is chosen from Table 1 for one of the three significance levels and the test result for
r, provided r < q, and m = q, say, if the test result is r = q . Then g_m(r) converges in
probability to infinity if the true number of cointegrating vectors is unequal to r, and g_m(r) =
Op(1) if the true number of cointegrating vectors is indeed r. Thus, taking r_m =

argmin₀≤_r≤₁{g_m(r)} we have lim_n→∞P(r_m = r) = 1. This approach may be useful as a double-
check on the test results for r.

5. The choice of the weight functions F_k

The best choice of the weight functions F_k is such that the power of the lambda-min
test is maximal, but again this is not feasible because the power depends on nuisance
parameters. However, Lemma 5 suggests that the second best choice is to choose the F_k’s as
to minimize the squared γ_k ’s, subject to the conditions (6) through (10). In doing so, it will
be convenient to replace first the conditions (6) and (7) by the weaker conditions

F_k(x ) dx о 0
and

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