Subset hypotheses testing and instrument exclusion in the linear IV regression

This paper explores the sensitivity of plug-in based subset tests to instrument exclusion in linear IV regression. Recently, identification-robust statistics based on plug-in principle have been developed for testing hypotheses specified on subsets of the structural parameters. However, their robustness to instrument exclusion has not been investigated. Instrument exclusion is an important problem in econometrics and there are at least two reasons to be concerned. Firstly, it is difficult in practice to assess whether an instrument has been omitted. Secondly, in many instrumental variable (IV) applications, an infinite number of instruments are available for use in large sample estimation. This is particularly the case with most time series models. If a given variable, say X(t), is a legitimate instrument, so too are its lags X(t-1), X(t-2), ... Hence, instrument exclusion seems highly likely in most practical situations. In this paper, we stress that the usual high level assumption of the identification may be misleading when potential relevant instruments are omitted. We propose an analysis of the asymptotic distributions of the LIML estimator and the plug-in based statistics when potential instrument are omitted. Our results provides several new insights and extensions of earlier studies. We show that even when partial identification holds, the asymptotic distribution of the LIML estimator of the identified linear combination is no longer a Gaussian mixture, even though it is still consistent. This contrasts with the usual IV estimator of the identified linear combination, which is still asymptotically a Gaussian mixture despite the exclusion of relevant instruments. As a result, the asymptotic distributions of the plug-in based subset statistics that exploit the LIML estimator are modified in a way that could lead to size distortions. We provide an empirical illustration using a widely considered returns to education example, which clearly shows that the confidence sets of the returns to education resulting from the plug-in principle are highly sensitive to instrument exclusion.