This paper develops a first-stage linear regression representation for an 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, 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 procedures to evaluate the validity of instruments by evaluating their statistical significance using the first-stage representation. Monte Carlo experiments provide numerical evidence that the proposed tests work as expected in terms of empirical size and power. An empirical application illustrates the methods.

中文翻译：

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工具变量分位数回归的第一阶段表示

本文开发了工具变量 (IV) 分位数回归 (QR) 模型的第一阶段线性回归表示。分位数第一阶段类似于最小二乘情况，即内生变量在工具和其他外生协变量上的线性投影，不同之处在于 QR 情况是加权投影。权重由 QR 结构模型中给定分位数的新息项的条件密度函数给出。我们还表明，IVQR 模型所需的雅可比识别条件嵌入在分位数第一阶段中。然后，我们建议通过使用第一阶段表示评估工具的统计显着性来评估工具有效性的程序。Monte Carlo 实验提供了数值证据，表明所提出的测试在经验规模和功效方面按预期工作。一个实证应用说明了这些方法。