Remark: (a) Notice that estimator for the intercept is consistent, even if some inte-
grable component is misspecified.
(b) If the component gj (s, αj∙) is misspecified, the NLS estimator âj converges to a
well defined quantity aj that is in general different than the true parameter αoj. It is
obvious from Lemma 3 however, that if a gj∙ (s, αj∙) is correctly specified, αj∙ converges
to the true parameter, even if there are other misspecified or omitted components.
This phenomenon is due to the fact that integrable transformations of different unit
root processes, are asymptotically orthogonal.
3 The test
As discussed earlier, the null hypothesis of our test is determined by the Lebesgue
measure rather some probability measure. Some basic results due to Bierens (1982,
1990) extend naturally to our framework, when the probability measure is replaced
by the Lebesgue measure. The handling of asymptotics however is different in our
case, as it relies largely on the asymptotic theory of Park and Phillips (2001) and
Chang et. al. (2001). In the remaining of this section we present our test statistic,
and derive its limit properties under the null and the alternative hypothesis.
The test statistic under consideration is based on the following sample moment:
n
n^1^4 (yt ~ g(ʃ^,ɑ)) W(xt,rn), m E M
t=ι
with M a compact subset of Rj, and
j
W(xt,m) = ∑wj(^j,t)exp(mjΦ(xjιt)) ∙ (5)
j=1
The function wj∙(∙) > 0 is I-regular, and Φ(∙) is bijective and bounded. Just like
Bierens (1990), the weight function employed is based in the exponential transforma-
tion. Nonetheless, there are two noticeable differences between the Bierens weighting
function and W(xt,m). First, we utilise an additional weighting function, wj∙(∙), that
is chosen to be I-regular. Bierens (1990) points out, that an additional weighting
might be employed to improve power against certain alternatives. Under the current
framework, the use of an integrable weighting function is necessary. The aim of resid-
ual based specification tests is to detect abnormal fluctuation in the residuals that
typically arises when the model is misspecified. Integrable transformations of unit
root process however, exhibit very weak signal. In particular, the intensity of g(xt, a)
is weaker than that of the error term1 (ut). As a result, the functional part of the
model is "obscured" by the error term. The employment of some integrable weighting
1Notice, for instance, that ^"=1 ff2(x-t) = Op{^n) while ^"=1 "'E = Op(n).