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

Appendix A

Proof of Theorem 1

First assume that α_i and Σ_i are known so that z_it = (α_i⁰Σ_i^-1α_i)^-1α⁰_iΣ_i^-1∆y_it
and z_i⁺_t = zit - y_i⁽_,¹_t⁾_-1 .

From Johansen (1991) it is known that in a cointegrated VAR(1) model
we have
t

yit = β⊥ (α_i⁰_,⊥β⊥)^-1α⁰_i,⊥     εis + uit ,

s=1

where u_it is (asymptotically) stationary. From T ^{-1/2 Pt}_s=1 ε_is = T ^{-1/2 P[}_t^a₌^T₁^] ε_it ⇒
B_i (a), where B_i(a) ≡ B_i is a Brownian motion with covariance matrix Σ_i. It
follows that

T

T ^{- X} yi_ty_i⁰_t ⇒ β⊥(α⁰_i,⊥β⊥)^- α⁰_i,⊥
t=1

Using E(f B_iBi) = (1 /2)∑_i we obtain

Furthermore,

^yi⁽,²t⁾-1 ^vi⁰t  ^{=  β}2,⊥(^α0i,⊥^β⊥)^-1α0i,⊥

where v_i⁰_t = (α⁰_iΣ_i^-1αi)^-1α⁰_iΣ_i^-1εit. Since εi,t-j is independent of vit for j ≥ 1
we have

NT

ΣΣvec(y_i⁽_,²_t⁾_-1v_i⁰_t)vec(y_i⁽_,²_t⁾_-1v_i⁰_t)⁰

i=1 t=1

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

T√₇N ∑∑*c ( B -1 V0t )

^{T N} i=1 t=1

15

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