# Proving consistency of quantile regression estimators

I have asked the question on the statistic section on stack exchange, but no one was able to give me an answer. I think this is actually is a question that touches econometrics so I am going to ask it here (I would like to use the answer to this in an econometric context):

I am a bit confused. The assumptions that have to be fulfilled so that OLS estimators are 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: https://eml.berkeley.edu/~powell/e241a_sp10/qrnotes.pdf

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 in that case 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!

• On an unrelated note, it seems like there's an error in these notes. On p.1, the definition $\eta_Y = \arg \max_c E[|Y - c| - |Y|]$ seems wrong. If this were the case, $c =0$ would produce zero. Moreover, we can take $c\rightarrow\infty$ and make the objective arbitrarily small. The definition should probably be $\eta_Y = \arg \max_c E[|Y - c|]$. Seems like this error is propagated elsewhere (but doesn't make too much of a difference overall). – jmbejara Jan 24 '18 at 21:06
• iid is sufficient, not necessary. They assume it for convenience. You can allow for heteroskedasticity and autocorrelation as long as some kind of “uniform law of large numbers” holds. Proof for OLS is relatively simple because you have the explicit form. It’s harder in your case. Reading proofs for M estimators would help. – chan1142 Jan 25 '18 at 8:22