The proof of this lemma is straightforward. Note that the functions cos(2kπ(t-.5)/n) are
known as Chebishev time polynomials, of even order. See, e.g., Hamming(1973).
It follows now easily from Lemma 8 that:
Theorem 7. With the weight functions Fk replaced by Fn,k , the results of Theorems 1 through
6 carry over to cointegrated systems with drift, without the need for Assumption 3.
Note that, due to (39), the optimality of the modified weight functions Fn,k is preserved.
Moreover, note that without Assumption 3 we allow the cointegration relations to be trend
stationary. This case is considered only very recently by Johansen (1994) and, in a slightly
different way, by Perron and Campbell (1993). Toda (1994) compares the two approaches
involved by Monte Carlo simulation.
7.2. Seasonal drift
Next, consider the case where zt is a seasonal vector time series process with s
seasons. In that case the drift may differ per season:
s-1
zt = zt-1 + cτd cτdτ,t + ",
τ =0
where the dτ,t’s are seasonal dummy variables, i.e., dτ,t = 1 if t = js + τ for some integer j
and dτ,t = 0 if not, and the cT’s are q-vectors of coefficients. However, the modified weight
function Fn,k do not sufficiently filter out the seasonal drift:
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