Quantile Regression

Quantile regression aims to estimate either the conditional median or other quantiles of a response variable. In this intercept-only model, we find a constant prediction c that minimizes the mean pinball loss for a given quantile tau.

Experiment

You can interact with this model by using the controls on the Quantile Regression interactive module. Try following these steps to see how the optimal interval changes:

1. Set Dataset to Spread, Quantile tau (%) to 50 and Constant prediction to 1. Read mean loss 0.700000 and minimizer interval [1,1].
2. Set Quantile tau (%) to 80. Observe how the positive residuals are penalized more heavily, shifting the optimal prediction interval upwards.
3. Set Constant prediction to 2. The mean pinball loss becomes 14/25, and this prediction falls within the new minimizer interval [2, 4] (since 5(0.8) = 4, an integer, leading to an interval from y_(4) to y_(5)).
4. Move Constant prediction to 4 while tau remains 80. It is still optimal with mean loss 0.560000: the complete minimizer set is an interval, not a single preferred endpoint.
5. Select Dataset = Tied, tau 90 and prediction 1. Every individual loss and the mean loss become zero.