Nonparametric cointegration analysis

Cov

6                    x ʃ

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

(A.10)

^F_ι(x)F_j(y)min(x_jy)dxdy] ■ I_q = O for i ≠ j,

Cov

6 . x ,                   - xï

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

^{F   (1}Fx^Fi^(x ) ^dx ʃ ʃʃF ^(x )^fj (y^{)min(x ,}y ) ^dxdy] ’ ¹q

(A.11)

Cov

_ʃ F_i(y)dy

ï ï

dx

J J

O for i ≠ j ,

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)∫F_j(x)dx +

y × ¹q

Fj(1)∫Fi (x) dx

^y × Iq

(A.12)

Q.E.D.

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

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

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