Subset hypotheses testing and instrument exclusion in the linear IV regression

This paper investigates the asymptotic size properties of robust subset tests when instruments are left out of the analysis. Recently, robust subset procedures have been developed for testing hypotheses which are specified on the subsets of the structural parameters or on the parameters associated with the included exogenous variables. It has been shown that they never over-reject the true parameter values even when nuisance parameters are not identified. 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. For example, some components of the “identifying” instruments that are excluded from the structural equation may be quite uncertain or “left out” of the analysis. 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 Xt, is a legitimate instrument, so too are its lags Xt1; Xt2. Hence, instrument exclusion seems highly likely in most practical situations. After formulating a general asymptotic framework which allows one to study this issue in a convenient way, I consider two main setups: (1) the missing instruments are (possibly) relevant, and, (2) they are asymptotically weak. In both setups, I show that all subset procedures studied are in general consistent against instrument inclusion (hence asymptotically invalid for the subset hypothesis of interest). I characterize cases where consistency may not hold, but the asymptotic distribution is modified in a way that would lead to size distortions in large samples. I propose a “rule of thumb” which allows to practitioners to know whether a missing instrument is detrimental or not to subset procedures. I present a Monte Carlo experiment confirming that the subset procedures are unreliable when instruments are missing.