(**Asymptotic Normality**) $J(\cdot)\sqrt{n}(\hat{{{\mbox{\boldmath $\theta$}}}}(\cdot)-{{\mbox{\boldmath $\theta$}}}(\cdot))$ converge in distribution to a zero mean Gaussian process with covariance function $\Sigma(\tau,\tau')$ defined as $$\Sigma(\tau,\tau')={\mbox{E}}\left[\left(\tau-{\mathrm{1}}\{Y<{{\bm W}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\theta$}}}(\tau)\}\right)
            \left(\tau'-{\mathrm{1}}\{Y<{{\bm W}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\theta$}}}(\tau)\}\right){{\bm W}}{{\bm W}}{^{\mbox{\tiny {\sf T}}}}\right].$$

The proof is given in [@angrist2006quantile], and the asymptotic covariance matrix of $\hat{{{\mbox{\boldmath $\theta$}}}}(\tau)$ can be estimated by either a bootstrap procedure [@hahn1997bayesian] or a nonparametric kernel method [@angrist2006quantile]. We adopt the parametric bootstrap approach to estimate the asymptotic covariance matrix in Section 5. Under model the result of Theorem \[thm:asymptotic\_normality\_pinball\] can be further simplified, which is given in Theorem 5.

Under the condition of Theorem 4, if further we assume $Y=\varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}_0+\epsilon$, and $\epsilon\perp A|{{\bm X}}$, then

1.  $\sup_{\tau}\|\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)-{{\mbox{\boldmath $\beta$}}}_0\|=o_p(1)$;

2.  $\sqrt{n}(\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)-{{\mbox{\boldmath $\beta$}}}_0)\indist N({{\mathbf 0}}, J_{11}^{-1}(\tau)\Sigma_{11}(\tau,\tau)J_{11}^{-1}(\tau))$, where $$\begin{aligned}
            J_{11}(\tau)=& {\mbox{E}}\left[f_{\epsilon}\left(\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}(\tau)-\varphi_0({{\bm X}})|{{\bm X}}\right)
            \pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right],\\
            \Sigma_{11}(\tau,\tau)=& {\mbox{E}}\left\{\left[\tau-{\mathrm{1}}\{\epsilon<\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}(\tau)-\varphi_0({{\bm X}})\}\right]^2
            \pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right\}.
            \end{aligned}$$ Furthermore, we have $\Sigma_{11}(\tau,\tau)\leq\left(\tau^2+|1-2\tau|\right){\mbox{E}}\left[\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right]$.

Comparing the asymptotic normality of $\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)$ with $\hat{{{\mbox{\boldmath $\beta$}}}}^A_{LS}$ yields interesting insights. Assuming that ${\mbox{E}}(Y|{{\bm X}},A)=\varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}_0$ and $({{\mbox{\boldmath $\beta$}}}_0,{{\mbox{\boldmath $\gamma$}}}^*)={\mathrm{argmin}}_{({{\mbox{\boldmath $\beta$}}},{{\mbox{\boldmath $\gamma$}}})}{\mbox{E}}[Y-\varphi({{\bm X}};{{\mbox{\boldmath 