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

.—.                                               .

α_i,⊥ are replaced by estimates with β_⊥ = β_⊥ + O_p(T^-1/2) and αb_i,⊥ = α_i,⊥ +
.                                                                  .—.                                                                                         .

O_p(T^-1/2). It follows that β^b_⊥⁰ y_i,t-1 = ΓS_i,t-1 + O_p(T ^1/2) and

TT

T^{-1 X} αb⁰_i,⊥∆yity_i⁰_,t-1β^b⊥ = T^{-1 X} α⁰_i,⊥∆yity_i⁰_,t-1β⊥ + op(1) .

t=1                              t=1

Consequently, replacing β⊥ and αi,⊥ by consistent estimates does not change
the asymptotic distribution.

Proof of Theorem 3

Since the null distributions of the LR and the Wald statistics are asymptot-
ically identical (e.g. Engle 1984), we first consider the Wald statistic of the
null hypothesis. To this end we write the second step regression as

biitt = θ'j(^φjyi,t-1) + φj(^ψjyi,t-1) + ^v*   ^j = ^{1 ,}...^,r^,

where Ψj is a k × (k-r-qj) dimensional matrix such that the matrix [Φj, Ψj]
has full column rank. The null hypothesis is equivalent to <pj = 0. The set
of equations can be written as

θ1

ψr

From Theorem 1 it follows that the vector [θ⁰₁,... ,θ⁰_r,φ^,₁,..., φ'_r]⁰ is asymp-
totically normal and, therefore, W_ψ = φ'Var(<b)^- 1b is asymptotically χ²
distributed with r (n — r) — V^=₁ Qj degrees of freedom, where b is the least-
squares estimate of φ = [φ⁰₁,..., φ'_r]⁰.

Appendix B

For the model with a constant or a linear time trend, the Brownian motions
in Theorem 2 are replaced by the expressions

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