Nonparametric cointegration analysis

where w_t = D(L)v_t and D(L) = (C(L)-C(1))/(1 -L) = ∑∞₀D_kL ^k. Thus:

t

z_t =^z о- ^w o⁺ μ ^{t +} wt ⁺ C (¹)∑ ^v_j.                                                        ⁽³⁾

j=1

The process z_t is cointegrated with r linear independent cointegrating vectors ξ_j ,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
ξTTz_t is trend stationary, with trend function ξTT(z₀ - w₀) + ξJμt.

Note that Assumption 1 guarantees that C(L)v_t and D(L)v_t are well-defined stationary
processes and that ∑C_k, ∑C_kC^T_k, ∑D_k and ∑D_kD^T_k converge. Cf. Engle (1987). For later
reference it will be convenient to write the latter matrix as:

∞

DkD DkDk^r- D.D.^r.                                                            (4)

k=0

Assumption 1 will be our maintained hypothesis, together with the following
assumption:

Assumption 2. Let R_r be the matrix of the eigenvectors of C(1)C(1)^T corresponding to the r
zero eigenvalues. Then the matrix R^T_rD(1)D(1)^TR_r is nonsingular.

Moreover, for the time being we shall assume that the cointegration relations R^T_rz_t 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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