Replication data for: Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure

Quantile regression (QR) fits a linear model for conditional quantiles, just as ordinary least squares (OLS) fits a linear model for conditional means. An attractive feature of OLS is that it gives the minimum mean square error linear approximation to the conditional expectation function even when the linear model is misspecified. Empirical research using quantile regression with discrete covariates suggests that QR may have a similar property, but the exact nature of the linear approximation has remained elusive. In this paper, we show that QR minimizes a weighted mean-squared error loss function for specification error. The weighting function is an average density of the dependent variable near the true conditional quantile. The weighted least squares interpretation of QR is used to derive an omitted variables bias formula and a partial quantile regression concept, similar to the relationship between partial regression and OLS. We also present asymptotic theory for the QR process under misspecification of the conditional quantile function. The approximation properties of QR are illustrated using wage data from the US census. These results point to major changes in inequality from 1990-2000.分位数回归（Quantile Regression，QR）为条件分位数拟合线性模型，正如普通最小二乘法（Ordinary Least Squares，OLS）为条件均值拟合线性模型。普通最小二乘法的一项显著优势在于，即便线性模型存在设定误差，它仍能为条件期望函数提供最小均方误差的线性近似。针对离散协变量开展的分位数回归实证研究显示，QR或许具备类似特性，但此类线性近似的确切本质仍未明确。本文证明，QR可最小化设定误差下的加权均方误差损失函数，其中加权函数为因变量在真实条件分位数附近的平均密度。基于QR的加权最小二乘解释，本文推导了遗漏变量偏误公式与部分分位数回归概念，其逻辑与偏回归和OLS之间的关系相仿。此外，本文还提出了条件分位数函数设定有误情况下，QR过程的渐近理论。本文利用美国人口普查的工资数据，对QR的近似性质进行了实证例证。上述结果揭示了1990年至2000年间收入不平等状况的重大变化。