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My goal is to estimate the market beta (so exposure of an asset returns to market shocks) in quantiles : $Q_{r_i|r_M} = a_0(\tau) + \beta_i(\tau)r_M+\varepsilon_i(\tau)$ where $r_i$ are asset returns (so about 252 observations per year with daily data). I conduct this regression for $N$ assets separately. Then, I use a Wald test to see if the coefficients in the lower/upper tails are significantly different than the coefficients in the median and I categorize the asset based on this (unstable versus stable).

In order to have an idea of the dynamics of the stability, I use rolling windows.

Question: how many observations should I have in my quantile regression? Is there a strict minimum (such as having ~60 observations per estimated quantile) ?

I can't find any resource about the sufficient number of observations to be included in a quantile regression. I know that for OLS, 60+ observations can be enough, but is it the same for quantile regression ? This article claims that they use windows with 24 observations as they provide the "necessary number of degrees of freedom" but I would like to have a good reference to cite !

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    $\begingroup$ There would be multiple possible explanations of those changes. Naturally small samples mean more variable estimates, just as with any model, but there are many other possibilities that could lead to this sort of circumstance. Those alternative explanations are impossible to rule out with the information here. $\endgroup$ Commented May 6 at 23:05
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    $\begingroup$ Beyond the "tail length" (a 5% quantile is harder to estimate in QR than a 50% quantile, so the coefficients for the 5% quantile should be more variable, though I don't know of any theoretical work on the variance of QR coefficients), of course the other ingredients are (a) the number of predictors, (b) what is a "sufficiently stable" coefficient for your purposes, (c) the (probably heteroskedastic, otherwise why use QR?) data generating process, and (d) how many observations you have in a window (daily data gives more observations than monthly). I don't see how "strict" guidance is possible. $\endgroup$ Commented May 7 at 6:38
  • $\begingroup$ Thank you ! I'll add further details in my question which may help to answer ! $\endgroup$ Commented May 7 at 9:48
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    $\begingroup$ You still haven't given us any information about what would be "enough" for your purposes. Even if you would, about the only reasonable answer for such a complicated analysis would be "run a simulation study to find out." $\endgroup$ Commented May 7 at 15:39
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    $\begingroup$ That's only a generic rule of thumb for flagging one possible shortcoming of a regression analysis. It cannot be any more than that because that would suppose everybody has identical requirements for the precision of their estimates and has the same general pattern of values of their explanatory variables. $\endgroup$ Commented May 9 at 14:55

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