# Maximum likelihood and residuals for quantile regression

If you assume a Gaussian distribution for the "errors" of your regression model, you can express the maximum likelihood, so the coefficients giving the highest likelihood for the observed data, with the sum of squared "errors" of your regression model.

I am trying to calculate the AIC for my quantile regression models, and in order to do that I need to get the likelihood assuming some distribution. The distribution "associated" with the quantile loss function is the asymmetric Laplace distribution.

Of course it is possible to code the pdf and use optimization algorithms to get the maximum likelihood, but I would like to know if something similar to the Gaussian case is available. So a formula linking the maximum likelihood of the asymmetric Laplace and the "errors", or residuals rather than errors since I will be working with predicted and observed values.

I would like you to point out any nonsense terminology I might be using.

• Quantile regression and maximum likelihood estimation (MLE) are not entirely compatible from a theoretical point of view. Or, in other words when performing quantile regression you are most likely not doing MLE, but rather empirical risk minimization (ERM). This is a nuance, but MLE requires the assumption of a conditional distribution, and quantile regression does not do that. In the same sense, when performing OLS you are not necessarily doing MLE: it can also be framed as a ERM, since minimizing the MSE leads to an estimate of the expectation, but both are equivalent mathematically. Commented Jun 27, 2023 at 7:59
• As an example, if you were given a set of data, assumed the conditional distribution was described by the asymmetric Laplace distribution and performed maximum likelihood estimation, you would also estimate the asymmetry parameter (and not vary it to estimate separate conditional quantiles) Commented Jun 27, 2023 at 8:03
• Very interesting, thanks for pointing that out, assuming a likelihood is more restrictive since QR could indeed be framed as a pure loss minimization problem without assuming any conditional distribution. I do not know how restrictive that is practically speaking, although the example of the separate conditional quantiles being constrained by the same asymmetry parameter is telling. Do you have any posts or papers I might read on those considerations, it would be interesting. Commented Jun 27, 2023 at 8:10
• Also that makes my issue of using the AIC or any likelihood based method to compare quantile regression models even more...delicate Commented Jun 27, 2023 at 8:22
• This post stats.stackexchange.com/q/606949/60613 has some discussion on this Commented Jun 27, 2023 at 8:58