How to prove consistency of quantile regression estimators?

by shenflow   Last Updated January 23, 2018 15:19 PM

I am a bit confused. The assumptions that have to be fulfilled in order for OLS estimators to be consistent (and efficient) are fairly straightforward.

I am currently trying to prove consistency of quantile regression (QR) estimators.

I have found the following lecture notes:

There are 4 assumptions listed (page 3) for the proof of consistency of the QR estimators. The first one being that the data $x_{t},y_{t}$ given $t=1,2,..,n$ has to be i.i.d. (independent and indentically distributed).

In the case of OLS the data only has to be covariance-stationary. To my understanding the above mentioned assumption of i.i.d. data rules out the possibility of for example autoregressive processes, since the data points are not independent from one another. This in turn puts quite restrictive boundaries on the possible applications of QR.

Am I missing something here? Could someone clarify the assumptions for consistency of the QR estimators for me?

Thank you!

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