Nonparametric cointegration analysis

corresponding to the r zero eigenvalues. Then:

Lemma 2. Under Assumption 3 and the conditions of Lemma 1,

■          ^ï

RrM(!) ft

RM z (Fk) n

V ^k 7

jointly in k = 1,...,m, where the Y_k ’s andZ are independent q-variate standard normally
distributed, with Y_k defined by (13). Moreover, Z does not depend on F_k .

Such weight functions F_k do exist. In particular,

Lemma 3. If F_k(x) = cos(2kπx), then the conditions (6) through (10) hold. Moreover, we then

have F_k(1) = 1, ʃʃF_k(x)Fk(y)min(xy) dxdy = 1(kπ y², ʃF_k(x)²dx = 1.

There are many ways to choose these functions F_k, but as will be shown in section 5, the
above choice is optimal in some sense.

Denoting

it follows now easily from Lemmas 1-2:

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