  This paper develops a first-stage linear regression representation for the
instrumental variables (IV) quantile regression (QR) model. The first-stage is
analogue to the least squares case, i.e., a conditional mean regression of the
endogenous variables on the instruments, with the difference that for the QR
case is a weighted regression. 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. The first-stage regression is a natural framework to evaluate the
validity of instruments. Thus, we are able to use the first-stage result and
suggest testing procedures to evaluate the adequacy of instruments in IVQR
models by evaluating their statistical significance. In the QR case, the
instruments may be relevant at some quantiles but not at others or at the mean.
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.
