Cl: For any δ > 0 and some mo ∈ M there is an 6> 0 such that
sup P (zn(mo) > e) ≤ δ.
n
C2: For any δ > 0 and e > 0 there is a ξ > 0 such that
sup P
n
sup ∖zn(m1)
∖J∣m1-m2∣∣<ξ
Zn(m2)∖ > 6
)≤δ
Verifying Cl and is C2 is what we set out to do. Note that Cl holds trivially as
zn(mo) = 0p(l) by Theorem 3.2 of Park and Phillips (2001). Next choose ξ < 1
and let Hj (m1j,m2j,Xj,t) = Wj(xj,t) [exp (m1Φ(xjιt)) — exp (m2Φ(xj∙,t))]. Condition
C2 follows from the continuity of A(ao,m) and the fact that
< sup n 1/4
I∣m1-m2∣∣<ξ
n J
ΣΣHj (m1,j ,m2,j ,Xj,t) ut
t=1 j=1
—
≤ ξC,
(A7)
for some C < ∞. The last result can be established as follows:
< sup n 1Z4
I∣m1-m2∣∣<ξ
n J
ΣΣHj (m1,j ,m2,j ,xj,t} ut
t=1 j=1
ξexp {(1 + sup ∣∣m∣∣) (sup ∖φ^∖)}∑ E |-
1/4
∑ wj (xjt)Ut f
t=1 J
(A8)
ξ exp J (1 + sup ∣∣m∣∣ ) (sup ∣Φ(s)∣
I ∖ тем J ∖ S
)}∑ ∣e ^-1/4 ∑ wj (xj’t)u
1/2
ξ eχp {(1+ sup ∣∣m∣∣) (sup ∖Φ(s)∖)⅛ (e∣:-■• ∑
u2 ∖ ʃt-ɪ)^l /
{(1+ sup ∖∖m∖∖) (sup ∖Φ(s)∖)}∑ (e(V∙∑ «x,,,)}'■'
where the first inequality is due to equation (B9) in Bierens (1990), and second one
is due to Liapunov’s inequality. The last term above is:
∑ E (n-1'2 ∑ w,fe,)A
j=1 ∖ t=1 J
j n Γ∞
= ∑--*'2∑ /
j=1 t=1 ∙7-∞
wj (t1'2 x)dj,t(x)dx
j
= Σ--1'2
j=1
n ∞o
^2-ll∖
t=1 j-∞o
w2(s)djιt(s∕t1'2)ds
j
2 У sup sup
. 1 t>1 X
J=1 -
z∞
-∞
w2(s)ds + o(1) < ∞,
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