The name is absent



Proof of Theorem 1: The proof is similar to that of Bierens (1990). Suppose mo is
suchthat J
r 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 δ0 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


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(02Ij) 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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