Proof of Theorem 1: The proof is similar to that of Bierens (1990). Suppose mo is
suchthat Jr g(s)em°φ(s)ds = 0. Noticethat
A [{s : φ)em°φ^ = 0}] = A [{s : g(s) = 0}] > 0.
By Lemma A, there is δ > 0 such that Jr [g(s)em°φ(s)] emφs',ds = 0, for 0 < ∖m∖ < δ.
Therefore, Jr q(s)emφs'1 ds = 0, for 0 < ∖m — mo∖ < δ. Hence, infmeM,m=mo ∖m — mo∖ >
0. This implies that M is a set of isolated points of the real line, and therefore is
countable. It is also straightforward to show that is non-dense in R.B
Proof of Lemma 4: By Lemma 3.1 of Chang et. al. (2001) the result can be proved
along the lines of Lemma 2 in Bierens (1990). ■
Proof of Theorem 2: We start with the limit result under H0. By Lemma 3.1
of Chang et. al. (2001), the terms An(a.m) and Cn(a) have well defined limits,
A(a.m), C (a) say. Set D(a.m) = A'(a.m)C 1 (a. m) and notice that D(a.m) can be
partitioned as D(a.m) = [D1(a1.m1)...... Dj(aj.mj)]. Let an, an be mean values of
à and ao. By Lemma 2, Lemma 3.1 of Chang et. al. (2001) and the mean value
theorem, the numerator of B(m) rescaled by n 1/2 is
n
n-1/4£ (yt
t=1
- 2
c — g(xt.a)') W(xt,m)
n n
1/4 1 Σ [An(an.m)C-1(fin.m)(∕(xt.ao) — W(xt,m)] ut — (C —
11=1
n
n
co) yyw(xt.m)
t=1
Il2
n^jn [Dj (mj .aoj )gP(xj,t .ao,j )
-∣ 2
Wj (xj,t) exp (mjΦ(xj,t))] ut + Op(n-1/2)
Lj=ι
t=1
Lj=ι
j f ∕*∞
(1. 0) / [Dj (mj .ao,j )gj (s.ao,j)
4_1 I 7-∞
— Wj(s) exp (mjΦ(s))]2 ds ^ Wj(1)
where (W1(1)..... Wj(1)) ~ N(0.σ2Ij) and independent of Lj(1. 0),s. In addition,
by Lemma 3.1 of Chang et. al. (2001), the denominator of B(m) rescaled by n
1/2 is
1/2ç2(m) → ς2(m.ao)
σ2
∑ Uj(1.0) ∞ [Dj (m.ao,j )gj (s.ao,j) — Wj (s)exp(mj T(s))]2 Ls l.(A2)
j=1 I 9-∞ J
The result follows from (A1) and (A2).
18
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