Appendix A
Proof of Theorem 1
First assume that αi and Σi are known so that zit = (αi0Σi-1αi)-1α0iΣi-1∆yit
and zi+t = zit - yi(,1t)-1 .
From Johansen (1991) it is known that in a cointegrated VAR(1) model
we have
t
yit = β⊥ (αi0,⊥β⊥)-1α0i,⊥ εis + uit ,
s=1
where uit is (asymptotically) stationary. From T -1/2 Pts=1 εis = T -1/2 P[ta=T1] εit ⇒
Bi (a), where Bi(a) ≡ Bi is a Brownian motion with covariance matrix Σi. It
follows that
T
BiBi0 αi,⊥ (β⊥0 αi,⊥)-1β⊥0
T - X yityi0t ⇒ β⊥(α0i,⊥β⊥)- α0i,⊥
t=1
Using E(f BiBi) = (1 /2)∑i we obtain
1
NT2
NT
XXyi(,2t)-1yi(,2t)-10
i=1 t=1
p
-→
-⅛ .
2 2
Furthermore,
t-1
εisvi0t
s=1
yi(,2t)-1 vi0t = β2,⊥(α0i,⊥β⊥)-1α0i,⊥
where vi0t = (α0iΣi-1αi)-1α0iΣi-1εit. Since εi,t-j is independent of vit for j ≥ 1
we have
1∙ 1 p
lιm —EB
T,N→∞ NT2
NT
ΣΣvec(yi(,2t)-1vi0t)vec(yi(,2t)-1vi0t)0
i=1 t=1
= lim E
T,N→∞
1
NT2
NT
XXyi(,2t)-1yi(,2t)-10
i=1 t=1
NT
NT ΣΣ Vitvit
i=1 t=1
= (1 /2)Ω2 ® ∑υ
and, therefore,
NT
^ N(0, 2Ω2 Θ ∑υ) .
T√7N ∑∑*c ( B -1 V0t )
T N i=1 t=1
15
More intriguing information
1. The name is absent2. The name is absent
3. The name is absent
4. Evaluating the Success of the School Commodity Food Program
5. Should Local Public Employment Services be Merged with the Local Social Benefit Administrations?
6. The name is absent
7. Environmental Regulation, Market Power and Price Discrimination in the Agricultural Chemical Industry
8. On Social and Market Sanctions in Deterring non Compliance in Pollution Standards
9. The name is absent
10. The Trade Effects of MERCOSUR and The Andean Community on U.S. Cotton Exports to CBI countries