The name is absent

and using the same arguments as Bierens (1982, Theorem I(I)) it can be shown that
A [{s : q(s) = 0}] > 0 implies J_r q(s)e^m"^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 ∈ R^j : 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(x_t, a))² ,

t=1

where C(a) = n ¹ ɪɪɪ (y_t - g(x_t, a)) = n ¹ ɪɪɪ (f (x_t) - g(x_t, a))+n ¹ ɪɪɪ u_t+c_o.
Next,

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 x_j∙,_t has absolutely continuous distribution with respect to Lebesgue measure,
then fj(x_j∙,_t) = g<j(xj_t, a_oj) 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:

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 → a_o.

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. ■

17

<<   15   16   17   18   19   20   21   >>

More intriguing information