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I hear it said [1] that QR makes no distribution assumptions about its error term.

Question 1: Does this mean that heteroscedastic and serially correlated disturbances do not effect the unbiasedness, consistency and minimum variance property of the coefficient estimates? What about the QR standard error validity?

Question 2: If QR is robust to all of this, then what's the point of the multitude of bootstraps that have been invented specifically for QR?

Feel free to only answer Question 1 if you're unable to answer Question 2.

[1] The answer to this: What diagnostic plots exists for quantile regression?

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Quantile regression does not assume that the error terms are normally distributed (nor does it assume some other shape for them). There are tons of references for this, e.g. Koenker 2005. However, while I haven't found anywhere a statement that quantile regression assumes independence, several papers e.g. Ceraci and Bottai make that assumption implicitly. It also aligns with my intuition. I think this perhaps comes from our usual lumping of distributional assumptions being iid - but the independence is a separate thing.

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The asymptotic results come from the general theory of extremum estimators which allows for the complications you bring up. However, the criterion function in this case is not smooth, so there might be issues with estimation of the derivatives, while the bootstrap is still reliable. Some modifications allow to speed it up while preserving validity which is relevant for huge data-sets or when you are interested in the band for the whole process.

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