Nonparametric cointegration analysis



Note that, since Fk is real valued, we can also represent Fk by

(A.35)


Fk(x) α α0,k + Σ (αj,kcos(2 π jχ) + βj,ksin(2 π jχ)),

where

co,k = αo,k ;  for j 1:   cj


α ., - i β ,
J,k           J,k

-------ɔ-------, cj, k


α ., + i β ,

J, k         ,i, k

2


(A.36)


Since


exp(2iπjχ) III(■=0),


(A.37)


it follows that


F(x)dx O 0 implies c0,k - α0,k - 0,


(A.38)


hence


Fk(x ) = Σ cj, kp(2 i π jx )∙

Next, observe that


(A∙39)


x

0


i π jy ) dy =


p(2 iπ) - 11(j.≠0)
2
i j


÷ xI(j'=0)


(A∙40)


and

x

Jy exp(2 i π jy ) dy =
0


λx exp(2 i π jx)

ч   2iπj


exp(2 i π jx) - 1
(2
i π j )2


ʌ

I(j'≠0)


÷ -2x21(j' 0),


(A∙41)


hence, for j1 0, j2 0,

44



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