Nonparametric cointegration analysis

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 F_k replaced by F_n,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 F_n,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 z_t 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 ^cτ^dτ,t ⁺ "^,
τ =0

where the d_τ,_t’s are seasonal dummy variables, i.e., d_τ,_t = 1 if t = js + τ for some integer j
and d_τ,_t = 0 if not, and the c_T’s are q-vectors of coefficients. However, the modified weight
function F_n,k do not sufficiently filter out the seasonal drift:

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