function resolves this problem, because Wj (xjff and ut are asymptotically orthogonal.
The second difference between W(xt, m) and the Bierens (1990) weighting function is
that the former is additively separable in the regression variables. Clearly, this con-
forms with the structure of the models under consideration. Whether a non-additive
separable weighing function could form the basis of a consistent test, is an open
question, as no limit theory exists for multivariate integrable functions.
Our test statistic is a functional of:
Ê(m) = e<=i ~ ⅞-a)> w<xt'm>f, m ∈ m.
ξ (m)
where:
Ç2(m) = (n~1 ∑"=1 U2) ∑"=1 [An(ffm)Cff(a)g(xt,a) - W(xt,m)]2 and ς2(m) its
distribution limit.
An(a,m) = n ш J2n=1 9'(xt, a)W(xt,m) and A(a,m) its distribution limit.
Cn(a) = n ш J2n=1 5,(xt, α)<7 '(xt, a) and C (a) is its distribution limit.
Cn 1 (a) and C 1 (a) are the inverses of Cn(a) and C (a), when they exist.
The following result is analogous to Theorem 1 of Bierens (1990) and is essential
for the development of a fully consistent test for the I-regular family.
Theorem 1: Let q : R → R integrable with q(s) = 0 on a set of positive Lebesgue
measure. Assume Φ : R → C is bijective and continuously differentiable with C being
an open and bounded subset of R. Then, the set
has Lebesgue measure zero and is non-dense in R.
<^m ∈ R :
q q(s)
Λ
exp (mΦ(s)) ds
=0}
The integral of Theorem 1 corresponds to the numerator of the Ê(m) statistic, in
the limit. Under H1, the numerator of Ê(m) can be zero only on a set of Lebesgue
measure zero, in large samples. In fact, M is a set of isolated points of the real line.
The subsequent assumption is similar to one of the regularity conditions of Bierens
(1990). Its purpose is to ensure that the denominator of Ê(m) is non-zero asymptot-
ically.
Assumption B: There are integrable and Borel measurable functions μj∙(s), 1 ≤
j ≤ J on R, such that the matrix Jr kj(s)kj(s)ds is non-singular, with kj(s) =
[Tj (s),9j(s)]'.
The following lemma shows that the Ê (m) statistic has a well defined limit unless m
belongs in a null set.
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