Quantile regression with dummy variable that's equal to 0 over most $t$

Question

I have the model $Y_t = a + b*X_t + c*D_t + e_t$, where $t \in T = \{1,...,3000\}$ and $D_t$ is a binary variable equal to $0$ over $T \backslash \{20,21,...,30\}$, and equal to $1$ over $\{20,21,...,30\}$.

$c$ is highly significant in OLS.

However if I run quantile regression $Q_{Y_t}(\tau | X_t,D_t) = a(\tau) + b(\tau)*X_t + c(\tau)*D_t + e_t$ then $c(\tau)$ is insignificant for any $\tau \in [0,0.4]\cup[0.6,1]$.

Now what I want to know is if QR could give false tail results for a dummy variable that's equal to unity over a tiny portion of $T$? As in, could the population relationship be strong, but QR isn't "strong" enough to pick up any tail relationship when $D_t = 0$ over almost all $t \in T$?

Answer 1

First off, to quote Andrew Gelman "The difference between 'significant' and 'not significant' is not itself statistically significant" (a great title, in my opinion). I would, instead, compare effect sizes.

Next, you should expect different answers when you ask different questions. Are the coefficients you get for the different quantile regressions very different from each other? Is the one for the median very different from those for the 40 and 60th percentiles?

To your specific question: I think the next step for you to take is to look at how Y behaves when D = 1 (where there are only 10 cases) vs. D = 0. A parallel box plot may be useful. If all the Y for D = 1 are concentrated in a small range, that could mess up quantile regression.