Conditional Inference for Possibly Unidentified Structural Equations

Abstract

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.

Confidence sets based on inverting Anderson–Rubin tests

“…Even when they are strong, it never has the correct conditional coverage. Forchini and Hillier (2003) have argued that the AR statistic is not in fact pivotal, because it does not depend on the parameter of interest everywhere in the parameter space, and that confidence sets based on it are therefore invalid. Our paper is concerned with the more detailed properties of AR confidence sets, but some of the issues that arise below are related to this important point.…”

Further results on projection-based inference in IV regressions with weak, collinear or missing instruments

“…2 See, for example, Nelson and Startz (1990a, b), Buse (1992), Hall et al (1996), Dufour (1997), Staiger and Stock (1997), Wang and Zivot (1998), Zivot et al (1998), Startz et al (1999), Chao and Swanson (2000), Stock and Wright (2000), Dufour and Jasiak (2001), Hahn and Hausman (2002a, b), Kleibergen (2002Kleibergen ( , 2004Kleibergen ( , 2005, Moreira (2003a, b), Moreira and Poi (2001), Yogo (2002, 2003), , Perron (2003), Wright (2003Wright ( , 2002, Bekker and Kleibergen (2003), Hall and Peixe (2003), Forchini and Hillier (2003), Andrews et al (2004), Dufour and Taamouti (2004), and the reviews of and Dufour (2003). 3 We borrow the terminology ''robust to weak instruments'' from Stock et al (2002, p. 518).…”

Conceptual frameworks and experimental design in simultaneous equations

“…Although it has not been completely ignored (see, for example, Forchini and Hillier, 2003), we believe that the literature has dramatically underestimated the practical consequences of this observation. In the absence of compelling arguments in favour of one conceptual framework, and its associated experimental design, over the other our results suggest that it would be prudent to explore both designs when analysing the behaviour of tests such as those considered here.…”