Two Independent Pivotal Statistics that test Location and Misspecification and add up to the Anderson-Rubin Statistic

We show that the Anderson-Rubin (AR) statistic is the sum of two independent piv- otal statistics. One statistic is a score statistic that tests location and the other statistic tests misspecification. The chi-squared distribution of the location statistic has a degrees of freedom parameter that is equal to the number of parameters of interest while the degrees of freedom parameter of the misspecification statistic equals the degree of over- identification. We show that statistics with good power properties, like the likelihood ratio statistic, are a weighted average of these two statistics. The location statistic is also a Bartlett-corrected likelihood ratio statistic. We obtain the limit expressions of the location and misspecification statistics, when the parameter of interest converges to infinity, to obtain a set of statistics that indicate whether the parameter of interest is identified in a specific direction. We show that all exact distribution results straight- forwardly extend to limiting distributions, that do not depend on nuisance parameters, under mild conditions. For expository purposes, we briefly mention a few statistical models for which our results are of interest, i.e. the instrument al variables regression and the observed factor model.