Palgrave Handbook of Econometrics: Applied Econometrics

David F. Hendry 35

occurs to changes across different “regimes” (see, e.g., Heckman, 2000; Hendry,
2004; and the references therein).

1.4.5.3 Weak exogeneity and economic theory

Much economic theory concerns relationships between means such as:

μy=β′μz. (1.25)

A famous example is the permanent income hypothesis (PIH), whereμyis per-
manent consumption andμzis permanent income, so the income elasticity of
consumption is unity∀β. Most demand and supply functions relate to expected
plans of agents; expectations and Euler equation models involve conditional first
moments, as do GMM approaches; policy relates planned instruments to expected
targets, etc. Since constructs likeμyandμzare inherently unobservable, additional
assumptions are needed to complete the model. For example, Friedman (1957) uses:

yt=μy+ (^) y,t andzt=μz+ (^) z,t whereE
[
(^) y,t (^) z,t
]
=0, (1.26)
which precludes weak exogeneity ofztforβgiven the dependence between the
means in (1.25). Allowingμyto also depend on second moments would not alter
the thrust of the following analysis.
Econometrics, however, depends on second moments of observables. Consider
the regression:
yt=γ′zt+vtwhere vt∼IN
[
0,σv^2
]

. (1.27)

Forzt∼INn

[

μz,zz

]

withE

[

ztvt

]

= 0 ∀t:

E

[

yt|zt

]

=γ′zt, (1.28)

then, fory′=

(

y 1 ...yT

)

andZ′=

(

z 1 ...zT

)

:

̂γ=

(

Z′Z

)− 1

Z′y, (1.29)

so that second moments are used to estimateγ. Here, (1.27) entailsE[yt]=γ′E[zt],
and from (1.28):

E

[

ztyt

]

=E

[

ztz′t

]

γ orσyz=zzγ, (1.30)

both of which involveγ. Thus, there seems to be no difference between how means
and variances are related, which is why second moments can be used to infer about
links between first moments. However, when any relation like (1.25) holds, then
σyzandzzin (1.30) must be connected byβ, notγ, if valid inferences are to result
about the parameters of interestβ. Weak exogeneity is needed, either directly in
(1.27), or indirectly for “instrumental variables.” This is more easily seen from the
joint distribution:
(
yt
zt

)

∼INn+ 1

[(

μy

μz

)

,

(

σyy σyz′

σzy zz

)]

, (1.31)