Quantile Cointegration in the Autoregressive Distributed-Lag ...

(i) Let D G := diag([ √ nι ′ 1+qk , nι′ k ]′ ) and G := [G 1 , . . . , G n ] ′ . Then,
(ii) Furthermore,
⎡
⎤
n
n∑
−1 n −1 ¯W′ t n −3/2 X ′ t
D −1
G G′ GD −1
G = ⎢ n
⎣
−1 ¯Wt n −1 ¯Wt ¯W′ t n −3/2 ¯Wt X ′ t
⎥
⎦
t=1
n −3/2 X t n −3/2 X t ¯Wt n −2 X t X ′ t
⎡
1 0 ′ ∫ 1 ¯B 0 W (r) ′ dr
⇒ ⎢ 0 E [ ] ¯Wt ¯W′
⎣
t 0 ′
∫ 1 ¯B
∫ 1
0 W (r)dr 0 ¯B 0 W (r) ¯B W (r) ′ dr
⎤
⎥
⎦ ;
D −1
G G′ Ψ τ (U) =
n∑
⎢
⎣
t=1
⎡
n −1/2 ψ τ [U t (τ)]
n −1/2 ψ τ [U t (τ)] ¯W t
n −1 ψ τ [U t (τ)]X t
⎤ ⎡
⎥
⎦ ⇒ ⎢
⎣
⎤
B ψ (1, τ)
B ψ·W (1, τ) ⎥
∫
⎦ ;
1 ¯B 0 W (r)dB ψ (r, τ)
(iii) D −1
H G′ K(τ) = O P (1), where D H := diag([nι ′ 1+qk , n3/2 ι ′ k ]′ );
(iv) M := n −2 X ′ [I − ˜W(˜W ′ ˜W)
−1 ˜W′ ]X ⇒ ∫ 1
0 ˜B W (r)˜B W (r) ′ dr; and
(v) n −1 X ′ [I − ˜W(˜W ′ ˜W)
−1 ˜W′ ]Ψ τ (U) ⇒ ∫ 1
0 ˜B W (r)dB ψ (r, τ). □
Proof of Corollary A1: (i) Lemmas A1(i), A2(i) and A2(ii) imply that
{
n −1
n∑
t=1
¯W t , n −1
n∑
t=1
¯W t ¯W′ t , n −3/2
n∑
t=1
¯W t X ′ t
}
→ P
{
0, E[ ¯Wt ¯W′ t ], 0 } .
Next, Lemma A3 implies that
} {∫ 1
{n −3/2 X t , n −2 X ′ tX t ⇒
0
∫ 1
}
¯B W (r)dr, ¯B W (r) ¯B W (r)dr .
0
Combining these two results we obtain the desired result in Corollary A1(i).
(ii) Assumption 1(vi) implies that
{
n −1/2
n∑
ψ τ [U t (τ)], n −1/2
t=1
}
n∑
ψ τ [U t (τ)] ¯W t ⇒ {B ψ (1, τ), B ψ·W (1, τ)} .
t=1
Moreover, Lemma A3 implies that n −1 ∑ n
1 ψ τ [U t (τ)]X t ⇒ ∫ 1
0 ¯B W (r)dB ψ (r, τ). By combining these
results, we show that the asymptotic limit of D −1
G G′ Ψ τ (U) is equal to Corollary A1(ii).
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