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.
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 conditional mean regression of the endogenous variables on the instruments, with the difference that 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. In addition, we show that the required Jacobian identification conditions for IVQR models are embedded in the quantile first-stage. The first-stage regression is a natural framework to evaluate the validity of instruments , and in particular, the validity of the Jacobian identification conditions. Hence, we suggest testing procedures to evaluate the adequacy of instruments by evaluating their statistical significance using the first-stage result. This procedure may be specially useful in QR since the instruments may be relevant at some quantiles but not at others , which indicates the use of weak-identification robust inference . 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.
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quantile
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analogue
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for
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In addition, we show that the required Jacobian identification conditions for IVQR models are embedded in the quantile first-stage.
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. Thus, we are able to use the first-stage result and
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in IVQR models
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. In the QR case,
using the first-stage result. This procedure may be specially useful in QR since
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or at the mean
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