Lemma 4: Under Assumption B, the set Mt = {m ∈ Rj : ç(m)2 = θ} has Lebesgue
measure zero a.s.
Theorem 2 next, demonstrates the limit properties of the B (m) test statistic under
the null and the alternative hypothesis.
Theorem 2: Suppose that Assumptions A-B hold. Then, for any m ∈ Rj∖ (M U M,),
as n → ∞ :
(i) UnderH0:
B(m) → y2,
(ii) UnderH 1:
B(m)∕^n → c(m),
with c(m) > 0 a.s.
In view of Theorem 2, the function c(m) can be zero only on sets of Lebesgue measure
zero. Therefore, consistency can be achieved by choosing m from a continuous distri-
bution. A consistent test of functional form based on randomised m is proposed by
Bierens (1987). Alternatively, a consistent test can be based on an appropriate func-
tional of B(m). By virtue of Theorem 1, in the limit the numerator of the I?(m) can
be zero only on null sets. Hence, any compact non-trivial subset of Rj contains some
m* such that c(m*) > 0. An obvious choice for m* is the maximiser of B(m) over some
compact set of positive Lebesgue measure. This is exactly the approach advocated
by Bierens (1990). Following Bierens (1990), we consider the Kolmogorov-Smirnov
functional of ]3(m):
sup B(m), (6)
mEM
where M is a compact subset of Rj.
Next, the limit properties of the sup-statistic are explored. Assumption B ensures
that a test statistic based on randomised m is well defined in the limit. Nonethe-
less, to ensure that the test statistic of (6) is well defined asymptotically, a stronger
assumption is required:
Assumption B': infto∈m ç(m)2 > 0 a.s.
Theorem 2 essentially follows from the asymptotic theory of Park and Phillips (2001).
To obtain the limit distribution of the sup-statistic however, further limit results are
required. First, we need some additional assumption about the covariates of the
model:
Assumption C: The process, t 1∕2xj∙,t, 1 ≤ f ≤ J has density function dj,t(x) that
is uniformly bounded2 i.e. supt>1 supæ dj,t(x) < ∞.
2By Lemma 3.1 in Potscher (2004), the following requirement is sufficient for Assumption C: η^t
has characteristic function φ^(r) such that lim,. ..ɪ |r|Æ φ^(r) = 0, for some ð > 1.
10