# What are the assumptions for applying a quantile regression model?

The question has been asked (one time) on CV before, but the answer is really imprecise and does not really answer the question in my opinion.

So: What are the assumptions for estimating a linear regression model via quantile regression?

To my understanding (and as several CV users have mentioned), quantile regression does not assume any specific distribution of the error terms - does that mean that, in a time series model, autocorrelation and heteroscedasticity do not have to be accounted for?

What about the other Gauss-Markov assumptions? I would assume that the assumption of no perfect multicollinearity has to be met when applying quantile regression, but do the parameters have to be linear? The linearity assumption only has to hold for the specific quantile I would assume.

Anyways - I do not find any backup for any of my thoughts in the scientific literature and I would appreciate a comprehensive answer. Thank you!

• You should link to the previous question that you mention. Dec 31, 2017 at 14:17
• shenflow, Could you include the link to the other question? Feb 21, 2018 at 19:08
• stats.stackexchange.com/questions/47929/… The answer is really imprecise: The auther does not say which properties of the estimators he is referring to. This in turn kind of makes the answer "wrong" in the sense that one can not generally state those things. Heteroskedasticity for example has an effect on the efficiency of QR estimators. There are a lot of theoretical constructs that have been proposed to deal with this. Feb 22, 2018 at 9:52
• Actually, one can differentiate quite precisely which assumptions have to be fulfilled so that the QR estimators do inherit certain properties. There are very specific derivations of for instance consistency. To just say "it does not have strong assumptions when it comes to the error term" is not really an answer. At least in my opinion. Feb 22, 2018 at 9:52

Quantile regression assumes

• the normal regression assumptions of linearity and additivity (unless you add more terms to the model)
• independence of observations
• very large sample size, as quantile regression is not very efficient
• $$Y$$ is very continuous; quantile regression doesn't work well when there are many ties at one or more values of $$Y$$

You might also consider semiparametric regression (e.g., proportional odds or hazards models) which are more efficient and also allow you to estimate the mean.

My RMS course notes goes a bit more into quantile and semiparametric regression in the chapter on ordinal models for continuous $$Y$$.

• Thank you for the answer Mr Harrell! Is there any comprehensive literature on this? In most (basic) econometric textbooks I have read, quantile regression is not mentioned.The original paper on quantile regression by Koenker kind of implies what you are saying, but does not explicitly mention the assumptions (at least to my understanding). Dec 31, 2017 at 13:24
• See the link I just added Dec 31, 2017 at 13:54
• Sorry but I have a follow up question @FrankHarrell. I am trying to prove consistency of quantile regression estimators. I have found the following lecture notes:eml.berkeley.edu/~powell/e241a_sp10/qrnotes.pdf . Assumption A1 says that the data has to be iid. In the case of ols, the data only has to be covariance stationary. Does this imply that data that shows some sort of autocorrelation structure (autoregressive processes) can not be used to estimate consistent coefficients by applying quantile regression? Jan 23, 2018 at 14:08
• Possibly but I'm not well versed in that aspect. More interested in small sample properties. But start with simplest case: one predictor $X$ that is categorical with $k$ levels. This is completely equivalent to computing $k$ ordinary sample quantiles. Jan 24, 2018 at 13:06
• Which immediately tells you that quantile regression is inefficient because I've never seen a distribution that looks anything like a double exponential. Jan 7, 2021 at 0:24