32 Methodology of Empirical Econometric Modeling
1.4.5.1 Weak exogeneity
The notion of exogeneity, or synonyms thereof, in relation to econometric
modeling dates back to the origins of the discipline (see, e.g., Morgan, 1990;
Hendry and Morgan, 1995), with key contributions by Koopmans (1950) and
Phillips (1957). Weak exogeneity was formalized by Engle, Hendry and Richard
(1983), building on Richard (1980) (see Ericsson, 1992, for an exposition), and is a
fundamental requirement for efficient conditional inference, which transpires to
be at least as important in integrated systems as in stationary processes (see Phillips
and Loretan, 1991). Weak exogeneity is equally relevant to instrumental variables
estimation, since the marginal density ofztthen relates to the distribution of the
claimed instruments: asserting orthogonality to the error term is often inadequate,
as shown by the counter-examples in Hendry (1995a).
Further,ztis strongly exogenous forθifztis weakly exogenous forθ, and:
Dzt(
zt|Xtt−− 1 s,X^10 −s,qt,κ 2)
=Dzt(
zt|Ztt−− 1 s,X^10 −s,qt,κ 2). (1.14)
When (1.14) is satisfied,ztdoes not depend uponYt− 1 soydoes not Granger-
cause z, following Granger (1969). This requirement sustains marginalizing
Dzt(zt|Xtt−− 1 s,X^10 −s,qt,κ 2 )with respect toY^1 t− 1 , but does not concern condition-
ing. Consequently, Granger causality alone is neither necessary nor sufficient for
weak exogeneity, and cannot validate inference procedures (see Hendry and Mizon,
1999).
The consequences of failures of weak exogeneity can vary from just a loss of
estimation efficiency through to a loss of parameter constancy, depending on the
source of the problem (see Hendry, 1995a, Ch. 5). We now illustrate both extreme
cases and one intermediate example.
Outperforming Gauss–Markov. First, consider a standard regression setting where
Gauss–Markov conditions seem satisfied:
y=Zβ+ with ∼NT[
0 ,σ
2 I]
, (1.15)whenZ=(z 1 ...zT)′is aT×kmatrix, rank(Z)=k, and ′=(
1 ...
T), with:
E[y|Z]=Zβ,and henceE[Z′ ]= 0. OLS estimates ofβ, the parameter of interest here, are:
̂β=β+(
Z′Z)− 1
Z′ ∼Nk[
β,σ
2(
Z′Z)− 1 ]
.However, ordinary least squares (OLS) need not be the most efficient unbiased
estimator ofβ, and an explicit weak exogeneity condition is required to preclude
that possibility whenZis stochastic. For example, let:
zt=β+νtwhere νt∼INk[ 0 ,],estimated by the mean vector:
β=β+ν∼Nk[
β,T−^1 ]
,