The name is absent



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, -∞ ≤ st)} 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(ut7 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

se}dr∙


l<t's>=bmo2; f v (r)



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