$Y$ takes an additive error term $\epsilon$ as in , we assume the conditional quantile function $Q_\tau({{\bm X}},A)=\varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C({{\bm X}};{{\mbox{\boldmath $\beta$}}}_0(\tau))$, where we redundantly represent the baseline function and contrast function as $\varphi_0(\cdot)$ and $C(\cdot)$ respectively. Notice that we use ${{\mbox{\boldmath $\beta$}}}_0(\tau)$ instead of ${{\mbox{\boldmath $\beta$}}}_0$ to emphasize that the true ${{\mbox{\boldmath $\beta$}}}$ may vary with respect to $\tau$. The proposed model is $\hat{Q}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})=\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})+\{A-\pi({{\bm X}})\}C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ with $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ correctly specified. Define $$\label{eq:population_beta_tau}
\left({{\mbox{\boldmath $\beta$}}}(\tau),{{\mbox{\boldmath $\gamma$}}}(\tau)\right)=\underset{{{\mbox{\boldmath $\beta$}}}\in\mathcal{B},{{\mbox{\boldmath $\gamma$}}}\in\Gamma}{\mathrm{argmin}}{\mbox{E}}\left[\rho_{\tau}\{Y-\hat{Q}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\}
-\rho_{\tau}\{Y-\hat{Q}({{\mbox{\boldmath $\beta$}}}',{{\mbox{\boldmath $\gamma$}}}')\}\right]$$ where $({{\mbox{\boldmath $\beta$}}}',{{\mbox{\boldmath $\gamma$}}}')$ is any fixed point in $\mathcal{B}\times\Gamma$. Define the QR specification error as $\Delta_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\triangleq\hat{Q}({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})-Q_\tau({{\bm X}},A)$. Define the quantile-specific residual as $\epsilon_{\tau}\triangleq Y-Q_\tau({{\bm X}},A)$ with conditional density function $f_{\epsilon_{\tau}}(\cdot|{{\bm X}},A)$. Then we have the following approximation theorem. The proof of the theorem follows Theorem 1 of [@angrist2006quantile], and is omitted for brevity.

Suppose that (i) the conditional density $f_{Y}(y|{{\bm X}},A)$ exists a.s.; (ii)${\mbox{E}}[Q_{\tau}({{\bm X}},A)]$ and ${\mbox{E}}[\Delta^2_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})]$ are finite; (iii) $\left({{\mbox{\boldmath $\beta$}}}(\tau),{{\mbox{\boldmath $\gamma$}}}(\tau)\right)$ uniquely solves . Then $$\left({{\mbox{\boldmath $\beta$}}}(\tau),{{\mbox{\boldmath $\gamma$}}}(\tau)\right)=\underset{{{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}}}{\mathrm{argmin}}{\mbox{E}}[w_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})\Delta^2_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})]$$ where $$w_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})=\int_{0}^{1}(1-u)f_{\epsilon_{\tau}}(u\Delta_{\tau}({{\bm X}},A;{{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})|{{\bm X}},A)du.$$ \[thm:approximation\_quantile\]

**Remarks:**

1.  