The name is absent

where x_t is a vector unit root process, c_o is a

constant and f (x) =    ^fj^(xj) ^{with f}j(∙)

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 x_t, u_t satisfy the following assumption:

Assumption A:

(i) Let x_t = x_t~₁ + v_t with x₀ = O_p(I) and

Vt = Ψ(L)η_f = ^£ *._%_„

s=1

with Ψ(∕) = 0 and ∑!=₁ ^s ∣∣Φ_s∣∣ < ∞. The sequence η_t is iid with mean zero and
E ∣∣η_t∣∣^r < ∞ with r > 4.

(ii) η_t 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 x_t is adapted to some filtration T_t-₁∙

(iv⁾ {^t = (^ut,^ηt₊1) , ʃt = ^σ (ζ_s, -∞ ≤ 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(∣u_t∣⁷ ∣ ʃt-ɪ) < ∞ a.s. for some
y > 2.

Define the partial sum processes V_n (r) and U_n (r) as:
_

^(Vn^{(r) U}n^(r)) =  ,  £ (^vt^{, u}t⁾ ∙

√n
^v t=1

The processes V_n(r) and U_n(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), ∙∙∙, V_j(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   f^t   Z, ZX

^l<^t'^s>=b^mo2; f ^{v (r)}

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