The name is absent

Cl: For any δ > 0 and some m_o ∈ M there is an 6> 0 such that

sup P (z_n(m_o) > e) ≤ δ.

n

C2: For any δ > 0 and e > 0 there is a ξ > 0 such that

Verifying Cl and is C2 is what we set out to do. Note that Cl holds trivially as
z_n(m_o) = 0_p(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(xj_ιt)) — exp (m₂Φ(x_j∙,_t))]. Condition
C2 follows from the continuity of A(a_o,m) and the fact that

for some C < ∞. The last result can be established as follows:

ξ ^eχp {(¹+ sup ∣∣m∣∣) (sup ∖Φ(s)∖)⅛ (^e∣:-■• ∑

{(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

= Σ-^-1'²

j=1

ⁿ       ∞o

^2-^ll∖

t=1       ^j-∞^o

j

2 У sup sup
. ₁ t>1 X

J=1 -

20

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