7. Cointegrating systems with unconstrained drift
7.1. Non-seasonal drift
Until so far all our derivations were based on the assumption that the drift parameter
vector μ is orthogonal to the cointegrating vectors. Cf. Assumption 3. The problem is that
without Assumption 3 the result of Lemma 2 no longer holds, due to the fact that
Σn=11cos(2kπt/n) = n/2, although Σnt=1cos(2kπt/n) = 0, so that, with Fk(x) = cos(2kπx), the result
of Lemma 2 now becomes:
RM(Ft√n - 1rμ.-'
RM (Fk) n
∖ k √
( __________'
1CD (1) Yk ftFk(x )2dx
` Fk(1)R,TD,Z
and consequently, part (15) of Lemma 4 becomes:
D
RC-A R →
m
(
ɪ _
RqT-rC(DE XkXkTC(1)TR,-t
k- 1
m _
RrT μ∑ XkTC (1)R
F k1
'
RTCd)∑ Xt μ tr
-,
Rr μμtR
√
Clearly, this will render all our test results invalid. However, a minor change of the functions
Fk will cure the problem:
Lemma 8. Let Fn,k(x) = cos[2kπ(nx - ½)/n] and Fk(x) = cos(2kπx). Then
nn
Fnk (t (t/n ι = tLk(τ/.)( t/n 1 = 0,
t=1 t-1
(38)
and
lim n sup F^nk(x) - Fk(x)∣ = kπ .
n -∞∖ 0≤x≤1 ’
(39)
23
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