Nonparametric cointegration analysis

∑v*v* T
XkXk

k= 1

- λRTC(1)C(1) RJ"

= 0,

(A.50)

where the X^*_i’s are i.i.d. N_q-r(0,I_q-r), and the latter minimum solution is equal to, and bounded
from below by

inf

^η η ^tR C(1)C(1)TRqJ"η

inf__lm_______

_η_η T_η

_in_fη TRqT_rC (1) C (1) TR_rrη
η η Tη

(A.51)

^к^k¹

T λ. RTC(1)C(1)TR, ).

mιn∖ q ^r^{v 7} V ^{7 ,}q-rf

This proves the inequality involved. Since

m +1

√

(A.52)

and M_α_,s,q-r,m is decreasing in m, it follows now that the right-hand side lower bound involved
increases with m. Q.E.D.

Proof of Lemma 8: For k > 0 we can write

(A.53)

2J2 t cos(2kπ (t-0.5)/n) = g_n(2kπ /n) + g_n(-2kπ /n),
where