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

-→ N(0, Ω^-1 ® Σv ) .

! ■---^ !

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

NT        NT

T   ∑ ∑ yg - 1 V0t = _t√_n ∑∑ УЗ-1 vit + op (1)

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

and, thus, replacing α_i and Σ_i 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

β⊥y_z,t-1 = β⊥β⊥(α)^-1 αi,⊥ XX . + Op(T¹ /²)
s=1

ΓSi,t-1 + op(T^1/2) ,

where Sit = ^Pt_s=1 α⁰_i,⊥εis and Γ = β_⊥⁰ β⊥ (α⁰_i,⊥β⊥)^-1 . Accordingly, under the
null hypothesis α⁰_i,⊥∆y_it = α_i⁰_,⊥ε_it the respective Wald statistic can be written

as

×  ^XT ΓSi,t-1∆Si⁰t

t=1

where Σ ¹ /² is a symmetric matrix with the property Σ ¹ /²Σ ¹ /² = α'_i ⊥Σα_i,⊥.
It remains to show that the limiting distribution is not affected if β_⊥ and
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