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



.—.                                               .

αi, are replaced by estimates with β = β + Op(T-1/2) and αbi, = αi, +
.                                                                  .—.                                                                                         .

Op(T-1/2). It follows that βb0 yi,t-1 = ΓSi,t-1 + Op(T 1/2) and

TT

T-1 X αb0i,yityi0,t-1βb = T-1 X α0i,yityi0,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

(yi0,t-1Φ1) 0 .        0        (yi0,t-1Ψ1) 0

.—.

zit =             .             . .             .                     .              .

0        0 . (yi0,t-1Φr)        0        0


0

(yi0,t-1 Ψr)


θr
φ ι


+v


*
it


ψr

From Theorem 1 it follows that the vector [θ01,... ,θ0r,1,..., φ'r]0 is asymp-
totically normal and, therefore,
Wψ = 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

W k-r


= Wk-r


—  Wk-r


17




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