Nonparametric cointegration analysis

Separate Appendix to:
NONPARAMETRIC COINTEGRATION ANALYSIS
by Herman J.Bierens

Following Phillips (1987), we use throughout this appendix the symbol "⇒" to indicate weak
convergence (cf. Billingsley 1968), convergence in distribution, or convergence in probability.
From the context it will be clear which mode of convergence applies.

Proof of Lemma 1: Denoting the partial sums associated with v_t and w_t by

[^χn ]

Snv(^χ) ⁼ 0 i^fχ ∈ [0^,n ^¹); Snv(^χ) ⁼ ∑^v_tif ^χ ∈ ^[n ^¹,1L
t= 1

^{[xn ]}

Snw(^χ) = 0 i^fχ ∈ [0^{,n l}) Sn^w(^χ) = ∑^f_t if^χ ∈ ^[n^¹,ι].

t=1

respectively, it follows easily that

where W is a q-variate standard Wiener process. Next, denote the partial sums associated with
z_t and ∆z_t by

[^χn ]

Sn(^χ) = 0 i^{f χ} ∈ [0^,n ¹); Sn(^χ) = ∑-^χ i^{f χ} ∈ ^[n ^¹,1],
t= 1
^[χn^]

^s^^δ(^χ) ⁼ 0 i^{f χ} ∈ [0^,n ¹); Sn^z(^χ) ⁼ ∑∆zt^{if χ} ∈ ^[η ^^1,1L
t=1

respectively. Then it follows from (3) and (A.2) that

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