Nonparametric cointegration analysis

6 Sn(x ) '	/ ʌ x
n↑[n Sn& z (x )	⇒	c(ɪ)ʃwty )dy к c (1) W(x ) ,
√n vv >

(A.4)

It follows from Lemma 9.6.3 in Bierens (1994, p.200) that

JL                     . c .

FX-(t/n)z, F F(1)S∙(1) - f(x)'∙.(x)dx,
t= 1                                             ^j

(A.5)

where f_k is the derivative of F_k, and similarly for ∆z_t. Using (A.4), (A.5), and the straightforward
qualities

F_k(1)Jw(x)dx - ʃfk(x)ʃW(y)dydx = ʃF_k(x) W(x)dx,

0

Var(jF_k(x)W(x)dx∣ ʃ ʃʃF_k(x)F_k(y)min(x,y)dxdy ∙ I_q,

(A.6)

(A.7)

Var Fk(1) W(1) - ʃf (x) W(x)dj

(A.8)

⁼ [^{f (1)2} - ^2Fk⁽¹⁾ ʃ^xfk^(x)^{dx +} ʃʃfk^(x)^fk(y^)min(x,y)^dxdy∣ ■ ¹_q = (ʃ^Fk^(x)^2dx_l

Cov

6.             ` _l                           Ô

(ʃFk(x) W(x)dx∖, (Fk(1) W(1)-ʃfk(x) W(x)dx}^

ʃʃ^Fk^(x )^fk(y^{)min(x y} ) ^dxdy^

(A.9)