2. The data-generating process
Consider the q-variate unit root process with drift zt = μ + zt-1 + ut, where ut is a zero
mean stationary process, and μ is a vector of drift parameters. We assume that zt is
observable for t = 0,1,2,..,n. Due to the Wold decomposition theorem, we can write (under
some mild regularity conditions),
∞
ut = Cv C,vt-j = C(L ) V, (1)
J=0
where vt is a q-variate stationary white noise process, and C(L) is a q×q matrix of lag
polynomials in the lag operator L. For convenience we assume that C(L) is a rational lag
polynomial, and that the vt ’s are Gaussian white noise, so that ut is a Gaussian VARMA
process:
Assumption 1. The process ut can be written as (1), with vt i.i.d. Nq(0,Iq) and C(L) =
1(L)-1C2(L), where C1(L) and C2(L) are finite-order lag polynomials, with all the roots of
det(C1(L)) lying outside the complex unit circle.
This assumption is more restrictive than necessary, but it will keep the argument below
transparent, and focussed on the main issues. See Phillips and Solo (1992) for weaker
conditions in the case of linear processes. Also, we could assume instead of Assumption 1
that ut is stationary and ergodic, so that we can write ut = εt + wt - wt-1, where εt is a
martingale difference process with variance matrix comparable with C(1)C(1)T. Cf. Hall and
Heyde (1980, p.136), and equation (2) below. Note that we do not restrict the lag polynomial
2(L), except for the implicit restrictions imposed by Assumption 2 below.
Since by construction the lag polynomial C(L) - C(1) is zero at L = 1, we can write
ut = C(L)Vt = C(1)Vt ÷ (C(L)-C(1))Vt = C(1)Vt ÷ (1-L)D(L)Vt
(2)
C C(1)Vt + wt - wt_ 1