Next, we specify the test hypothesis. Under the null hypothesis, all the fitted
components gj (xj ,aj), 1 ≤ j ≤ J are correctly specified. In particular, we say that
a fitted component is correctly specified, if the function gj∙(.,aj∙) differs from its true
counterpart (i.e. fj(.)) on a set of Lebesgue measure zero at most. This is formally
stated below:
Definition 1.
H0 : (correct specification) For all 1 ≤ j ≤ J,
ʌ [{∙s : fj(s) = gj(s,ao,j)}] = o,
for a some αθιj∙ ∈ Aj.
H1 : (incorrect specification) For some 1 ≤ j ≤ J,
ʌ[{s : fj(s) = gj(s,aj)}] >o,
for all αj∙ ∈ Aj∙.
Clearly, our formulation of the test hypothesis is in general different than that
of Bierens (1990). Nevertheless, if the covariate xt has certain properties, the two
approaches are equivalent. First, notice that under (1), the null hypothesis that
appears in (4) can written as
H0 : P [f (xt) = g(xt, a)] = 1, for a some ao ∈ A.
The following lemma shows that the two approaches are equivalent, under certain
conditions.
Lemma 1. Let q(.) : Rj → R be Borel measurable.
(i) If the random variable x has absolutely continuous distribution with respect to
the Lebesgue measure, then:
ʌ[s ∈ Rj : q(s) = 0] = O ^ P[q(x) = 0] = 1.
(ii) If P[x ∈ D] > 0 for all Borel sets D of positive Lebesgue measure, then:
P[q(x) = 0] = 1 ^ ʌ[s ∈ Rj : q(s) = 0] = 0.
In view of Lemma 1 and setting q(.) = f (.) — g(., ao), the result follows. Notice that
condition (i) of Lemma 1 requires that the covariates are continuously distributed.
This is one of the maintained assumptions of the paper (Assumption A above), which
is also a standard assumption in the literature e.g. Park and Phillips (1999, 2001),
Jeganathan (2003), Pbtscher (2004), de Jong (2004), de Jong and Wang (2005). On
the other hand, condition (ii) essentially requires that the covariates are unrestricted.
Clearly, it rules out bounded random variables. Condition (ii) can be easily checked