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