Nonparametric cointegration analysis

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

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

K ~ n,                                                                  (A.58)

s

1 - cos(Kx) ~ 2^k2^π2^τ2,                                                             (A.59)

n²

sin(Kx) ~ -²k^πτ ,                                                             (A.60)

n

sin(0.5x) ~ k^π s , cos(0.5 x) ~ 1,                                                   (A.61)

n

sin(y + 0.5x) ~ ²k^π (^{τ + 0.5s}___°J2, cos(y + 0.5x) ~ 1,                        (A.62)

n

hence

[( n-τ )/s ]

∑ cos(2kπ (js + τ - 0.5)/n)

^j=1                                                                                             (A.63)

-2kπτ /n - (2kπ (τ +0.5s-0.5)/n2kk²π ²τ²/n²)    _ τ

4 k π s / n                        2 s

and consequently

lim       cos[2kπ (t - 0.5)/n] = -ɪ.                                            (A.6⁴)

‘                                2 s

Next, observe that

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