Nonparametric cointegration analysis

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

∞

(A.35)

Fk(^x) ^α^α0,k ⁺ Σ (^αj,k^co^s(²^π j^χ) ^{+ β}_j,k^sin(²^π j^χ)),

where

co,k = ^αo,k ; for ^j ≥ ¹: c_j

α ., - i β ,
J,k J,k

-------ɔ-------^, cj, k

α ., + i β ,

J, k ^,i, k

(A.36)

Since

∫^ex^p(²ⁱ^πj^χ)^dχ 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, k^eχp(²ⁱ^π^jx )∙

Next, observe that

(A∙39)

ⁱ^π jy ) dy =

^eχp(2 i^π^jχ) - ¹1(_j.≠₀)
2 i ∏ j

÷ xI(j'=0)

(A∙40)

and

Jy exp(2 i π jy ) dy =
0

^λx exp(2 i π jx)

_ч 2iπj

exp(2 i π jx) - 1
(²ⁱ^π j )²

I(j'≠0)

√

÷ -2x²1(j' 0),

(A∙41)

hence, for j₁ ≠ 0, j₂ ≠ 0,