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What goodness of fit tests are usually used for quantile regression? Ideally I need something similar to F-test in linear regression, but something like AIC in logistic regression will suite as well. I use quantreg R package, but found only some Khmaladze test in there. To be fair I hardly understand what is does.

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quantreg includes several AIC functions: "AIC.nlrq", "AIC.rq", "AIC.rqs" and "AIC.rqss" and similar log likelihood functions.

It also has a vignette at vignette("rq",package="quantreg").

Do these do what you want?

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