Conditional Inference for Possibly Unidenti...ed Structural Equations

The possibility that a structural equation may not be identi...ed casts doubt on the measures of estimator precision that are normally used. We argue that the observed identi...ability test statistic is directly relevant to the precision with which the structural parameters can be estimated, and hence argue that inference in such models should be conditioned on the observed value of that statistic (or statistics). We examine in detail the e¤ects of conditioning on the properties of the ordinary least squares (OLS) and two-stage least squares (TSLS) estimators for the coe¢cients 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 identi...ability statistic is small), or very much smaller (when the identi...ability statistic is large); and (c) conditional mean-square-error comparisons of the two estimators favour the OLS estimator when the sample evidence only weakly supports the identi...ablity hypothesis, can favour TSLS slightly when that evidence is moderately favourable, but there is nothing to choose between the two estimators when the data strongly supports the identi...cation hypothesis.