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I am working on Quantile Regression (QR) and want to assess models using goodness of fit (GOF) measures. I have come across the post here, here that says, AIC/BIC can be calculated for QR model besides R squared as GOF. My questions are;

  1. Does estimation of QR coefficient involve Maximization of a Likelihood function? If not, how is AIC calculated there? (As per my knowledge, coefficients are estimated using Linear Programming algorithms. I know nothing about the algorithm used there though.)
  2. Can two competing models be tested using AIC values, same as in case of other methods?
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    $\begingroup$ You can certainly write quantile regression as likelihood based estimation. Roughly speaking you take the QR loss function $\ell(\beta)$, take $g=\exp(-\ell(\beta))$, scale to a density $f=cg$ so that $f$ integrates to 1. Then the likelihood will be maximized by minimizing the QR loss. It corresponds to maximizing likelihood for an asymmetric Laplace density for the error term. $\endgroup$ – Glen_b Sep 15 '17 at 6:18
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    $\begingroup$ Note that likelihood maximization is a goal that can be achieved using different algorithms on the given data. Note also that linear programming is an algorithm that can be applied to solve different problems aiming at different goals. Likelihood maximization and linear programming are thus not to be contrasted since they operate on different levels. (I probably cannot express the thought clearly, but I tried.) Also, AIC is not a test, so you do not test models with AIC but rather you compare them. $\endgroup$ – Richard Hardy Sep 15 '17 at 7:23
  • $\begingroup$ @Glen_b I was going to ask, are all the algorithms (Simplex, interior point and finite smoothing) for estimation in R software based on this maximization principle. But, @richard, made it more clear. And, I agree on the comment about comparing rather than testing. My mistake. $\endgroup$ – Enigma Sep 15 '17 at 9:21

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