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This question already has an answer here:

I'd like to know if the following explanation is a correct way to introduce Quantile regression to someone who just know OLS (the goal is just to give an intuition)

  1. Quantile regression minimise MAE instead of MSE so this is more robus to outlier.
  2. We can give more or less penalty to over estimation than under estimation.

For instance with Q=0.9, under estimation penalty is 9 times higher than over estimation penalty. In this case 90% of the distance between the dots and the regression line is under the line and 10% over the line.

  1. If we don't care about giving more penalty to over or under estimation we can use Q=0.5 and then we do a LAD regression. In this case 50% of the distance between the dots and the regression line is under the line and 50% over the line.

ps: Unfortunately we can obtain an infinite number of solution with this technique so we can ensure we really have 90% or 50% of residual over the regression line but that's the idea

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marked as duplicate by Tim, Sycorax, gung, Xi'an, Sven Hohenstein Oct 23 '15 at 18:45

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I will say that QR gives you the posibility to study the relationship between all portions of the distribution of the response variable with the predictor variables.

You can take a look to the article "A gentle introduction to quantile regression for ecologist" and also visit the R. Koenker's webpage for more examples and material.

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