Nonparametric cointegration analysis



Cov


6                    x ʃ

(jF (x ) W(x ) dx), (]Fj (y ) W(y ) d'


(A.10)


^Fι(x)Fj(y)min(xjy)dxdy] ■ Iq = O for i j,


Cov


6 . x ,                   - xï

Fi (x ) W(x ) dx], (Fj(1) W(1) -ʃfj (y ) W(y ) dy] ,


F   (1FxFi(x ) dx ʃ ʃʃF (x )fj (y)min(x ,y ) dxdy] ’ 1q


(A.11)


Cov


ʃ Fi(y)dy


ï ï

dx

J J


O for ij ,


6

jF1 (1) W(1) - ʃfi(x) W(x)a⅛j, (Fj(1) W(1) - ʃf (y) W(y)dyj ^


= (f∕1)Fj(1) - F1 (ɪ)ʃf(x)dx F Fj(1)∫f (x)dx


F (-Fi(1)Fj(1) + F1 (1)∫Fj(x)dx +


y × 1q

Fj(1)Fi (x) dx

y × Iq


(A.12)


Q.E.D.


= (ʃʃk(x)fj(y)min(xX')dxdy f Fi(1)Fj(I)J × Iq
ʃ ʃF.(x)F.(x)dx × Iq= O for i j.


Proof of Lemma 2: Let F be a typical function Fk, with derivative f, and let ξ be a cointegrating
vector. Using Lemma 9.6.3 in Bierens (1994, p.200) it follows now that


39




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