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

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 $t \in T$?

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Nice answer!$\phantom{.}$ –  user14281 Oct 10 '12 at 13:38