Nonparametric cointegration analysis

Var If(xx )w [ _nx ] d^x

(A.19)

→fjʃ(Xfy)I(x=y)dxdy Var(w0) = 0.

Furthermore, denoting

S ^w(1) S ^w(x)
s„(F) F F(1)  „ - ʃfx)   „.lx,

and using the easy equality

∑E(w,wnT) = ∑' (w0'w) - £E(w0w^τ),
t= 1                         t= 1                          j=0

(A.20)

(A.21)

Assumption 1 implies that s_n(F) and w_n are jointly normally distributed with covariance

n-[ nx ]-1

∑ ^e(w 0w^T)

Covars„⁽F) ^, w„) = ʃf(^x) — —-----d^x O О^(1/√ⁿ ).

„П

(A.22)

Finally, (A.2) implies that

s„(F) ⇒ D(1)(F(1) W(1) - ʃf(x) W(x)dx),
whereas

(A.23)

w_n ~ N_q(0,DD_tT)                                                            (A.2⁴)

cf. (4). Lemma 2 now easily follows from these results. Q.E.D.

Proof of Lemma 3 : Let z = exp(2 ikπ/n ) = cos(2 kπ/n ) + i .sin(2 kπ/n ), and observe that zⁿ = 1. Then

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