Nonparametric cointegration analysis

Lemma 4. Let rank C(1) = q

- r. Then

O

nI

r

√A

R.A R

^Iqr

O

^v

O

nI
r

( „.

R-A R

q ^r m q-r

nR_r^τA R

V ^r m q-r

, „ A
nR—A R
q ^r m r

n²R_r^τA R

^r m r ^

and

(
^lq-^q

O

^v

Rq^τ,C(1)∑ X₁X^τC (1) R

R- 1

^Rr^-D (1) ∑ γ ^τχχ'(1) R

k-1

JLi_r

R^τB R

Iqr

O

^v

A

O

JLi_r

A

Rq^τ-_rc(1)∑ γ XY D (1) ^τR,

k =1

m

Rr^τD (1)∑ γ² Y_kY_k^τD (1) ^τR

k= 1

(15)

(                  _r- a a a

R-B R  JLrLb R

^q > m q-r ^{q q} ' m r

JnR_r^τB R   nR^τB R

V ^r m q-r        ^r m r ^

Rq^TrC(1)Σ YkYk^τC(1)^-Rqr

m

R_q^τ_rC(1)У δ YZ ^τD-R
q ^r X /Z k k k r _r

(16)

k=1                                          k=1

mm

R,^τD. ∑ δ WX(1) ^τR_rr  R^τD. ∑ δ 2ZZ ^τD.^τR_r

V          k-1                                            k-1                        ^

where the random vectors X_i, Yj and Z are the same as in Lemmas 1-2. Moreover,

R ^,.'W'R
m

n²

D (O
→

^o a

V
r, m
√

(17)

where

m

= R,^τD (1)∑ γ 2 Y_tY^τD (1) ^τR_r

k= 1

(

- Rr^τD(1)∑ γkYkXk^τC(1)^τR_t

(

^m

Rq^τ-rC (1)∑ X_kXk^-C (1) ^τR_l

∖

Y√

k= 1

q-r

√

k= 1

q-r

√

,    JL       ,    _

R,^-C(1)∑ γ„xRd(1)^τR,..

k =1

10