36 Methodology of Empirical Econometric Modeling
so given (1.25):
E[
yt|zt]
=μy−γ′μz+γ′zt=(β−γ)′μz+γ′zt, (1.32)which coincides with (1.28) only ifγ=β, so means and variances are then related
by identical parameters.
Note from (1.31):
zt=μz+ut, (1.33)so impulse responses cannot be identified uniquely as originating from perturbing
utorμz. But from (1.32), the response ofytto these perturbations in (1.33) will
differ unlessγ=β, so weak exogeneity is essential for unique impulse responses,
which cannot be based on an arbitrary choice of Cholesky decompositions (only
one variant could coincide with valid conditioning).
1.4.6 Functional form
Alice began to remember that she was a Pawn, and that it would soon be
time to move. (Lewis Carroll, 1899)In practice, one cannot expect every functional form specification to coincide
with that which generated the data, however well-based its logic or theory creden-
tials. There are theories of what various linear and other approximations deliver
(see, e.g., White, 1980b, 2008), but such approximations cannot ensure non-
systematic residuals. Automatic model selection has been extensively applied to
select functional forms from quite large classes using data evidence (see,inter alia,
Perez-Amaral, Gallo and White, 2003, 2005; Castle, 2005; Castle and Hendry, 2005,
and section 1.5). In low-dimensional models, semiparametric and nonparamet-
ric methods are often used to avoid specifying the functional form, but can be
susceptible to unmodeled outliers and breaks.
1.4.7 Identification
...watching one of them that was bustling about among the flowers, pok-
ing its proboscis into them, “just as if it was a regular bee,” thought Alice.
However, this was anything but a regular bee: in fact, it was an elephant.
(Lewis Carroll, 1899)Identification has three attributes of uniqueness, interpretation, and correspon-
dence to reality (see, e.g., Hendry, 1995a), which we discuss in turn. Since
unidentified parameters entail a non-unique model specification – so what is esti-
mated need not match the parameters of the generating process – identification is
a fundamental attribute of a parametric specification.
First, a general understanding of identification as uniqueness has been developed
(see, e.g., Fisher, 1966; Rothenberg, 1971; Sargan, 1983: Hsiao, 1983, provides an
overview), building on the rank and order conditions so well known to be necessary