Nonparametric cointegration analysis



n

gn(x) = e "5ixteixt

1 e -0.5iχdy,(eix)t = 1
i         dx t= 1              i

( ʌ

ixd e,χ 1 eβ∙xx

dx     1 -eix

\          /


(       . ■                „                         \ A

1 -e         +e -°-i ix( 1-( x+1) einx)

e -0.5ix_e 0.5ix}2        e ~o∙5ix-e 0'5ix


_ cos(0.5x)-isin(0.5x_etnx^ + 2insin(0.5x)eιxx

-4sin2(0.5 x )                   -4sin2(0.5 x )


(A.54)


Thus,


cos(0.5x)(1 - cos(xx) - (2x— 1)sin(0.5x)sin(xx)

-4sin2(0.5 x )

i (2x— 1)sin(0.5x)cos(xx) - cos(0.5x)sin(xx) + sin(0.5x)

-4sin2(0.5 x )


gx(x ) + gx(x ) =


cos(0.5x)(1 - cos(xx) - (2x-1)sin(0.5x)sin(xx)


-2sin2(0.5 x )


(A.55)


Since cos(2kπ) = 1 and sin(2kπ) = 0, the second equality in (38) follows. The proof of the first
equality goes similarly, and (39) is trivial. Q.E.D.


Proof of Lemma 9: Observe that


K

∑ expi (xt + У) =

t≡ 1


exp( i (y + 0.5 x ))-------1----exp( iKx )------

exp(-0.5 ix) - exp(0.5ix)


_ (cos(y + 0.5x) + i sin(y + 0.5 x))(1 - cos(Kx) - i sin(Kx))
-2
i sin(0.5 x )

_ i cos(y + 0.5x)(1 - cos(Kx)) + sin(y + 0.5 x)sin(Kx)
2sin(0.5
x )

cos(y + 0.5x)sin(Kx) - sin(y + 0.5x)(1 - cos(Kx))

2sin(0.5 x )                       ’


(A.56)


hence


48




More intriguing information

1. The name is absent
2. The name is absent
3. A THEORETICAL FRAMEWORK FOR EVALUATING SOCIAL WELFARE EFFECTS OF NEW AGRICULTURAL TECHNOLOGY
4. Dendritic Inhibition Enhances Neural Coding Properties
5. The name is absent
6. The name is absent
7. Migration and Technological Change in Rural Households: Complements or Substitutes?
8. The name is absent
9. The name is absent
10. Dynamiques des Entreprises Agroalimentaires (EAA) du Languedoc-Roussillon : évolutions 1998-2003. Programme de recherche PSDR 2001-2006 financé par l'Inra et la Région Languedoc-Roussillon