A first-stage representation for instrumental variables quantile regression

This paper develops a first-stage linear regression representation for the instrumental variables (IV) quantile regression (QR) model. The quantile first-stage is analogous to the least squares case, i.e., a linear projection of the endogenous variables on the instruments and other exogenous covariates, with the difference that the QR case is a weighted projection. The weights are given by the conditional density function of the innovation term in the QR structural model, conditional on the endogeneous and exogenous covariates, and the instruments as well, at a given quantile. We also show that the required Jacobian identification conditions for IVQR models are embedded in the quantile first-stage. We then suggest inference procedures to evaluate the adequacy of instruments by evaluating their statistical significance using the first-stage result. The test is developed in an over-identification context, since consistent estimation of the weights for implementation of the first-stage requires at least one valid instrument to be available. Monte Carlo experiments provide numerical evidence that the proposed tests work as expected in terms of empirical size and power in finite samples. An empirical application illustrates that checking for the statistical significance of the instruments at different quantiles is important. The proposed procedures may be specially useful in QR since the instruments may be relevant at some quantiles but not at others.