5% level in most occasions, when W1 is employed. It is obvious from our simulation
experiment that, when a small penalty term is used, the B(m) statistic tends to be
significantly larger B(mo). In this instance the modified Bierens test may suffer from
overrejection of the null hypothesis. Notice however, that in Table (4) B(rn) equals
B(mo) in most cases, even when a small penalty term is employed.
6 Appendix
Lemma A. Let q(s) : R → R be Borel measurable. The function Φ(s) : R → C is
bijective and continuously differentiable with C an open and bounded subset of R. If
q(.) is integrable, then
A [{s : q(s) = 0}] > 0 if and only if [ q(s')emφ^s^ ds = 0
√R
for m E R in an arbitrarily small neighborhood of zero.
Proof of Lemma A: (i) The “if” part is trivial. We show the “only if” part. Consider
the Borel measurable functions
qι(s) = max {q(s), 0} and q2(s) = max {-q(s) 0}
and notice that q = q1 - q2. Assume that
c1 = ʃ q1(s)ds > 0 and c2 = ʃ q2(s)ds > 0.
Define the probability measures6 Fi, i = {1,2} on the Borel field restricted on C (Bc)
as
Fi(B) = ɪ q qi(Φ 1(s)) Φ 1(s) ds, B
ci Jb
EBc ∙
Then
' q(s)emφV>ds = ∣ q√s)emφ¾ - ʃ q2(s)emφ¾
q q1 (φ-1(s))ems
Jc
c1 q emrdF1(r) -
Cc
Φ-1(s) ds - q q2(Φ~1 (s))ems
Φ 1(s) ds
(aI)
Jc
c2 f emrdF2(r)
Jc
= C1η1(m) - c2η2(m),
where the second equality above is due to Billingsley (1979, Theorem 17.2). Notice
that ηi(m) is the moment generating function of Fi, i = {1, 2}. In view of this,
6Noticethat ʃɔ qi(Φ 1(s)) ∣Φ 1(s)∣ ds = Jr qi(s)ds = c⅛.
16
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