A parametric approach to the estimation of cointegration vectors in panel data



It follows that


TN~Nv(b(Bb - B)


= T Nvec


NT

XXyi(,2t)-1yi(,2t)-10
i=1 t=1


-1


NT

X X (2) v0
yi,t-1vit

i=1 t=1


-→ N(0, Ω-1 ® Σv ) .

! ■---^ !

Finally, it is easy to verify that if αbi-αi = Op(T-1/2) and Σb i-Σi = Op(T-1/2)
we have

NT        NT

T   ∑ ∑ yg - 1 V0t = tn ∑∑ УЗ-1 vit + op (1)

i=1 t=1 i=1 t=1

and, thus, replacing αi and Σi by a consistent estimator does not affect the
asymptotic distribution.

Proof of Theorem 2

First, assume that αi and β (and therefore αi, and β) are known. The
vector of regressors results as

wi


βyz,t-1 = ββ(α)-1 αi, XX . + Op(T1 /2)
s=1

ΓSi,t-1 + op(T1/2) ,

where Sit = Pts=1 α0i,εis and Γ = β0 β (α0i,β)-1 . Accordingly, under the
null hypothesis
α0i,yit = αi0,εit the respective Wald statistic can be written

as

λiw(r)


×  XT ΓSi,t-1Si0t

t=1


Sit Si0,t-1Γ0   Xt=T1 ΓSi,t-1Si0,t-1Γ0


-1


-1 / 2} + Op (1)


tr½-1 /2 (∑∆SitSit-ə (∑Sit-1 S0,t-1


-1


X
=1


Si,-1Si0t


-1 / 2} + Op (1)


where Σ 1 /2 is a symmetric matrix with the property Σ 1 /2Σ 1 /2 = α'i Σαi,.
It remains to show that the limiting distribution is not affected if
β and
16



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