The name is absent

Lemma 4: Under Assumption B, the set M^t = {m ∈ R^j : 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 ∈ R^j∖ (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 R^j 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 R^j.

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': inf_to∈_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∕²x_j∙,t, 1 ≤ f ≤ J has density function dj,_t(x) that
is uniformly bounded² i.e. sup_t>₁ sup_æ 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.

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