Pivotal Statistics for Testing Subsets of Structural Parameters in the IV Regression Model

We construct a novel statistic to test hypothezes on subsets of the structural parameters in an Instrumental Variables (IV) regression model. We derive the chi squared limiting distribution of the statistic and show that it has a degrees of freedom parameter that is equal to the number of structural parameters on which the hypothesis is specified. The statistic has this limiting distribution regardless of the quality of the instruments for the endogenous variables associated with these structural parameters. The instruments have to be valid for the endogenous variables associated with the remaining structural parameters. We analyze the relationship of the novel statistic with the Lagrange Multiplier, the Likelihood Ratio and the GMM over-identification statistic from Stock and Wright (2000). Chi squared limiting distributions for the first two statistics only hold when the instruments are valid for all endogenous variables. A chi squared limiting distribution for the GMM over-identification statistic is obtained under the same conditions as for our novel statistic but has a larger degrees of freedom parameter. For some artificial datasets, we compute power curves and p-value plots that result from the different statistics. We apply the statistic to an IV regression of education on earnings from Card (1995).