Violation of Exogeneity : A Joint Test of Structural Parameters and Correlation

In instrumental variable regression, structural parameter estimation relies on knowledge of the correlation between the structural error and the instruments. The current practice is to assume that this correlation is zero. However, this assumption is unlikely to be satisfied in applied work. In this paper, we first show that to make valid inferences about the structural parameters we need to take into account their joint behavior with the correlation between the instruments and the error term. We analyze both the overidentified and just identified cases. We suggest a new test statistic that corrects the bias in the regular t test that is caused by a violation of perfect exogeneity. One can then build joint confidence intervals for the structural parameters and the correlation parameter by inverting the test statistic. We also see that in both the overidentified and just identified cases, the confidence intervals are informative about the structural parameters. Two empirical examples highlight the value of this tool for applied research