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
jʃFi (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 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)∫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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