Identification-Robust Inference for Endogeneity Parameters in Linear Structural Models

We provide a generalization of the Anderson-Rubin (AR) procedurefor inference on parameters which represent the dependence between possiblyendogenous explanatory variables and disturbances in a linear structuralequation (endogeneity parameters). We focus on second-orderdependence and stress the distinction between regression and covariance endogeneity parameters. Such parameters have intrinsic interest(because they measure the effect of "common factors" which induce simultaneity) and play a centralrole in selecting an estimation method (because they determine "simultaneity biases" associated withleast-squares methods). We observe that endogeneity parameters may not beidentifiable and we give the relevant identification conditions. We developidentification-robust finite-sample tests for joint hypotheses involvingstructural and regression endogeneity parameters, as well as marginalhypotheses on regression endogeneity parameters. For Gaussian errors, weprovide tests and confidence sets based on standard-type Fisher criticalvalues. For a wide class of parametric non-Gaussian errors (possiblyheavy-tailed), we also show that exact Monte Carlo procedures can be appliedusing the statistics considered. As a special case, this result also holdsfor usual AR-type tests on structural coefficients. For covarianceendogeneity parameters, we supply an asymptotic (identification-robust)distributional theory. Tests for partial exogeneity hypotheses (forindividual potentially endogenous explanatory variables) are covered asinstances of the class of proposed procedures. The proposed procedures areapplied to two empirical examples: the relation between trade and economicgrowth, and the widely studied problem of returns to education.