Nonparametric cointegration analysis

n

t= 1

∑ dj

I^j-¹ '

Γ ..    .         ..1 -ʌ

cos[2kπ (t - 0.5)/n] = ∑
t=1

cos[2kπ (t - 0.5)/n∖

[( n-τ )/s ]

= ∑ j cos[(2kπ s/n)j + 2kπ (τ - 0.5)/n]

j=1

(A.65)

= 1 h_t. ..₁(2 k π s / n ,2 k π (τ - 0.5)/n ) + 1 h..  .,₁(-2 k π s / n ,-2 k π (τ - 0.5)/n ),

2 [( n-τ )/s p∙               ^,         ×              ^{7 7}       2 [( n^-τ )/s ]^v                  ’ ^{v              7    75}

where

K

h_κ(x,У) ⁼ e'^y∑ te'^xt - e^{i(y + 0}5^x)gK(x),

(A.66)

with g_K defined by (A.54). Thus

^hK^(x, У) ^{+ h}κ⁽ -^χ, -У)

cos(y+0.5 x )

cos(0.5x)(1 - cos(Kx) - (2K- 1)sin(0.5x)sin(Kx)

-2sin²(0.5 x )

(A.67)

- sin(y+0.5x)

(2K- 1)sin(0.5x)cos(Kx) - cos(0.5x)sin(Kx)

+ sin(0.5 x )

-2sin²(0.5 x )

Again substituting K = [(n-τ)/s], x = 2kπs/n, y = 2kπ(τ-0.5)/n it follows that

^hK^(x,У) ^{+ h}K⁽-x^, “У) ~ 2k²π²τ²/n² - (2K-1)(kπ s/n)(2kπτ/n)
ⁿ                             -(2 k ²π² s²/n²) n

_ 2kπ (τ +0.5s-0.5) × (2K-1)(kπs/n) -(2kπτ/n) + (kπs/n)
ⁿ                            -2( k ²π² s²/ n²) n

(A.68)

and thus

50