The possibility that a structural equation may not be identified
casts doubt on measures of estimator precision that are usually used.
Using the Fieller–Creasy problem for illustration, we argue that
an observed identifiability test statistic is directly relevant to the
precision with which the structural parameters can be estimated, and
hence we argue that inference in such models should be conditioned on
the observed value of that statistic (or statistics).
We examine in detail the effects of such conditioning on the
properties of the ordinary least squares (OLS) and two-stage least
squares (TSLS) estimators for the coefficients of the endogenous
variables in a single structural equation. We show that (a)
conditioning has very little impact on the properties of the OLS
estimator but a substantial impact on those of the TSLS estimator; (b)
the conditional variance of the TSLS estimator can be very much larger
than its unconditional variance (when the identifiability statistic is
small) or very much smaller (when the identifiability statistic is
large); and (c) conditional mean-square-error comparisons of the two
estimators favor the OLS estimator when the sample evidence only weakly
supports the identifiability hypothesis but favor TSLS when that
evidence moderately supports identifiability.
Finally, we note that another consequence of our argument is that the
statistic upon which Anderson–Rubin confidence sets are based is
in fact nonpivotal.We are grateful for the
constructive comments offered by Peter Phillips and three anonymous referees
that greatly improved the paper. Giovanni Forchini acknowledges support from
ESRC grant NR00429424115.