Nonparametric cointegration analysis



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 - rq. 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.



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