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!

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    $\begingroup$ 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). $\endgroup$
    – jmbejara
    Jan 24, 2018 at 21:06
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    $\begingroup$ 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. $\endgroup$
    – chan1142
    Jan 25, 2018 at 8:22

1 Answer 1


I can give you literature. An authoritative book on the subject is Koenker, R. (2005). Quantile regression. Cambridge university press.. In there we read (p.126)

4.6 QUANTILE REGRESSION ASYMPTOTICS UNDER DEPENDENT CONDITIONS Relaxation of the independence condition on the observations is obviously possible and offers considerable scope for continuing research. Bloomfield and Steiger (1983) showed the asymptotic normality of a median autoregression estimator for a model in which the observations were assumed to be stationary and ergodic martingale differences. Weiss (1991) considers models under α-mixing conditions. Portnoy (1991) considers a considerably more general class of models with “m-decomposible” errors, a condition that subsumes m-dependent cases with *m* tending to infinity sufficiently slowly... More recently, Koul and Mukherjee (1994) and Mukherjee (2000) have considered the asymptotic behavior of both the primal and dual quantile regression process under conditions of long-range dependence of the error process. This work relies heavily on the theory of weighted empirical processes developed by Koul (1992).

The section continues with a presentation of the properties of Quantile regression for AR, ARMA, and ARCH models.

  • $\begingroup$ Thank you Mr. Papadopoulos. Really helpful answer. I will try to take a thorough look at Bloomfield and Steiger (1983). Is there also information on the issues of autocorrelation and heteroscedasticity of the residuals and their effect in the reference you gave in your answer? $\endgroup$
    – shenflow
    Jan 25, 2018 at 7:38
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    $\begingroup$ @shenflow Yes, as the last sentence of my answer indicates. $\endgroup$ Jan 25, 2018 at 9:24

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