David F. Hendry 37and sufficient in simultaneous systems when the restrictions are given by subject-
matter theory: we could call this a technical issue. Cowles Commission researchers
showed that the “reduced form” (or statistical system) was always identified in their
formulation, and that all just-identified models were isomorphic to that statistical
system, hence tests of overidentified “structural forms” could be derived by com-
paring their two likelihoods. Their analysis, therefore, entailed that the “structural
form” is actually areductionof the statistical system, so logically can be obtained
from it without any prior knowledge of the relevant restrictions. Thus, when a
model is identified relative to an identified system, the identification restrictions
in question do not have to be knowna priori, but can be found by a suitable algo-
rithm (see, e.g., Hendry and Krolzig, 2005) – indeed, several overidentified, but
distinct, representations can coexist (see, e.g., Hendry, Lu and Mizon, 2008). Such
a conclusion is predicated on the statistical system itself being identified, which
requires sufficient explanatory variables – the vexed topic of “exogeneity” discussed
in section 1.4.5 above. Moreover, for an overidentification test to be valid, the sta-
tistical system must be well specified, so needs to be modeled and evaluated first
(see, e.g., Spanos, 1990), after which it can be reduced to a “structural form.” Never-
theless, that prior identification restrictions must be known in advance remains the
dominant belief, which if true, would preclude empirical modeling not preceded
by a rigorous theory derivation that entailed sufficient restrictions.
Second, interpretation is a regular seminar question along the lines: “How do
you know you have identified the demand curve” (as opposed to some other
entity)? This is essentially an economic theory issue, and only substantive theory
can resolve such a debate. It is separate from uniqueness: a regression of price on
quantity is always unique, but hardly qualifies as a demand curve just because the
regression coefficient is negative.
Third, even if both uniqueness and interpretation are confirmed, the result still
need not correspond to reality, which is an empirical issue (and related to the usage
of the word “identification” in, say, Box and Jenkins’, 1976, analysis, as well as the
quote above). An estimated equation may be unique and interpretable but not the
relevant relation. Thus, all aspects of model building are involved in establishing
satisfactory identification.
Recently, problems of weak instruments, and the resulting issue of identification,
have become salient (see, among others, Staiger and Stock, 1997; Stock and Wright,
2000; Stock, Wright and Yogo, 2002; Kleibergen, 2002; Mavroeidis, 2004).
1.4.8 Parameter constancy
“Yes, all his horses and all his men,” Humpty Dumpty went on. “They’d
pick me up again in a minute, they would!” (Lewis Carroll, 1899)Parameters are the entities which must be constant if the specified model is to
be a useful characterization of reality. However, that does not preclude the coef-
ficients in any model formulation from changing, as in “random coefficients”
models or “structural time series” (see, e.g., Hildreth and Houck, 1968; Harvey,