Nonparametric cointegration analysis



n

∑z

t=1


zn-1
z-----

z-1


(A.25)


and

V tz ∙ = z dLŸ ,'..!L .2,/1 - i cos(kπ /n ) 1
t! dz⅛ z-1 2 ^ sin(kπ /n) ,

(A.26)


Thus, taking the real part, we have

nn

(A.27)


COs cos(2kπ t/n) о 0,      t cos(2kπ t/n) = 1 n,

t= 1                                             t-1                                       2

which proves the conditions (6) and (7). The other condition follow from the proof of Lemma
6 below. Q.E.D.

Proof of Lemma 4: We only prove (17); the other parts of Lemma 4 follow straightforwardly
from Lemmas 1-2. It is a standard exercise in linear algebra to verify that

(
Rq-d^ R
q r m q-r


RrTÂ R
rm


q-r


V 1

RqT, R
q r m r

RrTÂ R

r m r ^


л11
m

л21
m


1
<

Â22

m ^


(A.28)


where

22

A
m


n 2(n2RrTÂ R
\ r m r


(nRaTt R )(n2RT R )1(nRrT R )) 1

∖ q r m r-,q     r m r' q r m q-r f

T ^             T 1          , T ^      ∖-1

- ( nRrTAmRq_r )(RqTÂmRq-r )^( nRqTAmRr))


(A.29)


= -n (RTÂ R )1( nRTÂ R )(л22/n2) = (Л21)T

q q m m q-r' q q r m r'q m 7 vm7

Therefore,

42



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