David F. Hendry 33then it is easy to construct scenarios whereβis much more efficient than̂β.
Consequently, even in simple regression, for the Gauss–Markov theorem to be
of operational use one needs the condition thatβcannot be learned from the
marginal distribution.
Weak exogeneity in cointegrated systems. Second, cointegrated systems provide a
major forum for testing one aspect of exogeneity. Formulations of weak exogeneity
conditions and tests for various parameters of interest in cointegrated systems are
discussed in,inter alia, Johansen and Juselius (1990), Phillips and Loretan (1991),
Hunter (1992), Urbain (1992), Johansen (1992), Dolado (1992), Boswijk (1992)
and Paruolo and Rahbek (1999). Equilibrium-correction mechanisms which cross-
link equations violate long-run weak exogeneity, confirming that weak exogeneity
cannot necessarily be obtained merely by choosing the “parameters of interest.”
Conversely, the presence of a given disequilibrium term in more than one equation
is testable. Consider an apparently well-defined setting with the following bivariate
DGP for theI(1) vectorxt=
(
yt:zt)′
from Hendry (1995c):yt=βzt+u1,t (1.16)zt=λyt− 1 +u2,t, (1.17)where:
(
u1,t
u2,t
)
=(
00
ρ 1)(
u1,t− 1
u2,t− 1)
+((^) 1,t
(^) 2,t
)
, (1.18)
and:
(
(^) 1,t
(^) 2,t
)
∼IN 2
[(
0
0
)
,
(
σ 12 γσ 1 σ 2
γσ 1 σ 2 σ 22
)]
=IN 2 [ 0 ,]. (1.19)
The DGP in (1.16)–(1.19) defines a cointegrated vector process in triangular form
(see Phillips and Loretan, 1991) which can be written in many ways, of which the
following equilibrium-correction form is perhaps the most useful:
yt=βzt+ (^) 1,t
zt=λyt− 1 +ρ
(
yt− 1 −βzt− 1
)
- (^) 2,t, (1.20)
where (^) t=
(
(^) 1,t: (^) 2,t
)′
is distributed as in (1.19).
The parameters of the DGP are
(
β,λ,ρ,γ,σ 1 ,σ 2
)
. When cointegration holds,
βandσ 1 can be normalized at unity without loss of generality, and we also set
σ 2 =1. The parameter of interest isβ, which characterizes the long-run relation-
ship betweenytandzt. LetIt− 1 denote available lagged information (theσ-field
generated byXt− 1 ). Then, from (1.19) and (1.20), the conditional expectation of
ytgiven
(
zt,It− 1)
is:E[
yt|zt,It− 1]
=βzt+γzt−γρ(
yt− 1 −βzt− 1)
−γλyt− 1. (1.21)