Nonparametric cointegration analysis



corresponding to the r zero eigenvalues. Then:

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

         ^ï

RrTD (1) Y1 ^F1(x)2dx


Fk(1)RrτD,Z


RrM(!) ft

RM z (Fk) n

V k 7

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

Such weight functions Fk do exist. In particular,

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

have Fk(1) = 1, ʃʃFk(x)Fk(y)min(xy) dxdy = 1(kπ y2, ʃFk(x)2dx = 1.

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

Denoting

Fk(x )2 dx


γ k =            j

y ^^I^k(x )Fk((y )min(x, y ) dxdy


, δk


Fk (1)

ʃFk (x )2 dx


(14)


it follows now easily from Lemmas 1-2:



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