Least squares estimates the conditional mean; quantile regression estimates any conditional quantile (τ = 0.5 is the median) by minimising the asymmetric check loss, so a few outliers or a skewed error distribution tilt the fit far less.
The fit uses Schlossmacher's iteratively reweighted least squares with an annealed threshold: each round solves a weighted least squares problem, and the threshold shrinks tenfold so the answer converges to the exact check-loss minimiser. Pseudo-R² is Koenker–Machado's local fit measure.
Frequently asked questions
Why compare τ = 0.1 with τ = 0.9?
If the slopes differ across quantiles, the effect of x is not just shifting the distribution but changing its spread — something mean regression averages away.
Does it give standard errors?
This tool reports point estimates and pseudo-R². Inference for quantile regression needs bootstrap or kernel methods, which are a different machinery from the closed-form t tests of OLS.
中文说明
最小二乘估计条件均值；分位数回归通过最小化不对称检查损失估计任意条件分位数（τ = 0.5 即中位数），因此少数离群值或偏态误差分布对拟合的影响小得多。
拟合采用 Schlossmacher 迭代重加权最小二乘并退火阈值：每轮解一个加权最小二乘问题，阈值逐轮缩小十倍，使解收敛到检查损失的精确最小化点。伪 R² 采用 Koenker–Machado 局部拟合指标。
常见问题
为什么要对比 τ = 0.1 与 τ = 0.9？
若不同分位数的斜率不同，说明 x 不只是平移分布、还在改变离散程度——这是均值回归会平均掉的信息。
为什么没有标准误？
本工具给出点估计与伪 R²。分位数回归的推断需要自助法或核方法，与 OLS 的闭式 t 检验是两套机制。