In addition, define zn (m) as
zn{m) =
n 1/4 ∑n=ι [A(a,m)C 1(α)g(¾α) — W(¾m)] ut
y/ s2(m)
The process zn(m) is asymptotically equal to B(m). Moreover, by virtue of Assump-
tion C, zn(m) is tight. This is formally stated in the subsequent lemma:
Lemma 5: Suppose that H0 holds.
(i) Under Assumptions A and B', we have supmeM B(m)
— zn(m)
= 0p(ɪ),
(ii) Under Assumptions A and C, zn is tight.
Next, we report our main result. Let C(M) be the space of all continuous functions
on M equipped with the metric p(c1,c2) = supmeM ∣c1(m) — c2(m)∣. The following
theorem shows the limit distribution of our test statistic under the null hypothesis.
Moreover, it establishes that the sup-statistic is unbounded in probability against any
I-regular alternative.
Theorem 3: Let Assumptions A, B' and C hold. Then, as n → ∞, we have:
(i) Under H0, B(m) converges to z(m)2, where z(m) is a mixed Gaussian element
of C(M) with covariance function
Γ(mbm2 ) = E
∑∫=ι ʃɪ Gj(mi,s)Gjm -s)ds
√^s2(mι)s2(m2)
where m1,m2 ∈ M and for i = ʃɪ, 2},
Gj(mi, s) = L1/2(G 0) [Aj(aod,mij)Cj~1(aod)gj(s, aoj) — Wj(s) exp (mijΦ(s))] and
s2(mi) = ∑j=1 Lj(ɪ, 0) ʃɪ [Aj(ao,j,mi,jc)gj(s,ao,j) — wj(s)exp(mi,j$(s))]2 ds.
In addition,
(ii) UnderH 1,
sup B(m) → sup z(m)2.
mzM mzM
sup B(m,yvn → sup c(m),
mzM mzM
with supmeM c(m) > 0 a.s.
For stationary models (e.g. Bierens (1990), de Jong (1996)), the limit distribution of
B(m) under H0 is Gaussian. In our case it is mixed Gaussian. In addition, under
stationarity the limit distribution of the sup-statistic is data dependent (see Bierens
(1990) and de Jong (1996)). This is true under the present framework as long as the
fitted model involves multiple covariates. The limit distribution depends on the local
11
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