and using the same arguments as Bierens (1982, Theorem I(I)) it can be shown that
A [{s : q(s) = 0}] > 0 implies Jr q(s)em"φ(s)ds = 0 for some m ∈ R. In addition,
because Φ(s) is bounded, the requisite result follows along the lines of Bierens (1982,
Theorem 1(II)).H
Proof of Lemma 1: The proof for part (i) is trivial (see for example Halmos (1950)
p. 104 and 128). For (ii) set D = Js ∈ Rj : q(s) = 0} and suppose that A[D] > 0.
Then by assumption, P[x ∈ D] > 0. Clearly this implies that P[q(x) = 0] < 1, which
is a contradiction. Therefore, A[D] = 0.И
Proof of Lemma 2: We first prove consistency. As in Marmer (2005), we consider
the concentrated objective function:
n
Qn(a) = ∑(yt - C(a) - g(xt, a))2 ,
t=1
where C(a) = n 1 ɪɪɪ (yt - g(xt, a)) = n 1 ɪɪɪ (f (xt) - g(xt, a))+n 1 ɪɪɪ ut+co.
Next,
sup ∣C(a)
a∈A
- Co∣ ≤ sup
a∈A
n
n 1 ∑tt(xt) -g(xt,a))
t=1
n
n 1 Ut
t=1
= °P(1),
where the last equality above follows from Park and Phillips (2001, Theorem 3.2)
and Chang et. al. (2001, Lemma 3.1). This establishes consistency of C. For â notice
that if xj∙,t has absolutely continuous distribution with respect to Lebesgue measure,
then fj(xj∙,t) = g<j(xjt, aoj) a.s. Given this, Park and Phillips (2001, Theorem 3.2) and
Chang et. al. (2001, Lemma 3.1), it is easy to show that:
n 1/2 (Qn(a) - Qn(ao))
j f∞
∑⅛ (0.1) J∞
(gj (s,ao,j)
- gj∙(s. aj∙))2 ds
uniformly in a. In view of the above, condition b(ii) of Lemma 2 and condition CN1
of Park and Phillips (2001, p. 133) we get â → ao.
Given the consistency of the LS estimator, the limit distribution result follows
easily along the lines of Park and Phillips (2001, Theorem 5.1).■
Proof of Lemma 3: The consistency of C can be established as above. Also, using
the same arguments as those in the previous proof, we get
uniformly in a and the result follows. ■
n 1/2 (Qn(a) - Qn(a*))
j ∞oo
→ ∑ij (0.1) /
j=1 j∞o
(fj(s) - gj(s. aj))2 - (fj(s) - gj(s. aj}}
ds,
17
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