where xt is a vector unit root process, co is a
constant and f (x) = fj(xj) with fj(∙)
j=1
being an I-regular function. The I-regular class, comprises integrable transformations
that are piecewise Lipschitz (see Park and Phillips, 2001, for more details). For a
statistical analysis of these models, the reader is referred to P&P and Chang et. al.
(2001). The variables xt, ut satisfy the following assumption:
Assumption A:
(i) Let xt = xt~1 + vt with x0 = Op(I) and
Vt = Ψ(L)ηf = £ *.%_„
s=1
with Ψ(∕) = 0 and ∑!=1 s ∣∣Φs∣∣ < ∞. The sequence ηt is iid with mean zero and
E ∣∣ηt∣∣r < ∞ with r > 4.
(ii) ηt has distribution absolutely continuous with respect to Lebesgue measure and
has characteristic function φ(X) = o(∣∣ A∣∣~δ) as A → ∞, for some δ > 0.
(iii) The random vector xt is adapted to some filtration Tt-1∙
(iv) {^t = (ut,ηt+1) , ʃt = σ (ζs, -∞ ≤ s ≤ t)} is a martingale difference sequence
with E [ξtξt I ^t-1] = ∑.
(v) E (u2 I ʃt-ɪ) = σ2 < ∞ a.s. and sup1<t<n E(∣ut∣7 ∣ ʃt-ɪ) < ∞ a.s. for some
y > 2.
Define the partial sum processes Vn (r) and Un (r) as:
_
(Vn(r) Un(r)) = , £ (vt, ut) ∙
√n
v t=1
The processes Vn(r) and Un(r) take values in the set of cadlag functions on the interval
[0,1].
Under Assumption A, the following strong approximation result holds (see P&P):
sup ∣∣(Un (r),Vn (r)) - (U (r), V (r))∣∣ = Op(1),
re [0,1]
where (U(r), V(r)) is a (J+1)-dimensional Brownian motion, comfortably partitioned
as (U(r), V1 (r), ∙∙∙, Vj(r)). In addition, for the purpose of the subsequent analysis, we
need to introduce the (chronological) local time process of the Brownian motion Vj
up to time t defined as
z x 1 ft Z, ZX
— s∣ ≤ e}dr∙
l<t's>=bmo2; f v (r)