Quantile Regression

What is Quantile Regression?

Quantile regression fits a line to a chosen quantile of the data instead of its mean. Ordinary linear regression minimizes squared errors and delivers the conditional mean; quantile regression, formalized by Koenker and Bassett, minimizes a tilted absolute-error loss (the pinball or check loss) in which, for a target quantile q, points above the fit are weighted by q and points below by 1 minus q. Setting q to 0.5 yields median regression (least absolute deviations); q of 0.05 and 0.95 yield outer lines with roughly 5% of the data beyond each, in-sample.

On charts it shows up mainly as robust fits and channels. A median-slope fit resists the outlier bars that drag least-squares lines around, and outer quantile fits form a channel calibrated to how often price actually closed outside it, rather than assuming normality the way sigma-multiple versions of the linear-regression channel do. The Theil-Sen estimator, the median of the slopes over all pairs of points, is a related robust choice for the slope itself.

How traders use it

• Robust trend measurement: median (q = 0.5) or Theil-Sen fits give slope estimates a single news spike cannot hijack, useful on gappy or thinly traded instruments.
• Quantile channels: outer fits such as the 5th and 95th percentiles frame stretch zones whose in-sample exceedance rate is set by construction; out-of-sample the rate is not guaranteed, so excursions are context, not automatic fades.
• Distribution-aware projection: fitting several quantiles at once sketches the whole conditional distribution of price or returns, which matters when the tails drive the decision more than the average does.