.—. .
αi,⊥ are replaced by estimates with β⊥ = β⊥ + Op(T-1/2) and αbi,⊥ = αi,⊥ +
. .—. .
Op(T-1/2). It follows that βb⊥0 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,φ,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