where wt = D(L)vt and D(L) = (C(L)-C(1))/(1 -L) = ∑∞0DkL k. Thus:
t
zt =z о- w o+ μ t + wt + C (1)∑ vj. (3)
j=1
The process zt is cointegrated with r linear independent cointegrating vectors ξj ,j= 1,..,r,
say, if rank(C(1)) = q - r < q. Then ξTTC(1) = 0T for j = 1,..,r, hence it follows from (3) that
ξTTzt is trend stationary, with trend function ξTT(z0 - w0) + ξJμt.
Note that Assumption 1 guarantees that C(L)vt and D(L)vt are well-defined stationary
processes and that ∑Ck, ∑CkCTk, ∑Dk and ∑DkDTk converge. Cf. Engle (1987). For later
reference it will be convenient to write the latter matrix as:
∞
DkD DkDkr- D.D.r. (4)
k=0
Assumption 1 will be our maintained hypothesis, together with the following
assumption:
Assumption 2. Let Rr be the matrix of the eigenvectors of C(1)C(1)T corresponding to the r
zero eigenvalues. Then the matrix RTrD(1)D(1)TRr is nonsingular.
Moreover, for the time being we shall assume that the cointegration relations RTrzt are
stationary about a possible intercept but not about a trend. Thus:
Assumption 3. RTμ = 0.
This assumption will be dropped in due course, but is maintained temporary in order to stay
focused on the main issues.