$\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}\inprob{{\mbox{\boldmath $\beta$}}}_0$ for all $\tau\in(0,1)$, where $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}$ is the solution of when $M(x)=\rho_{\tau}(x)$.

**Remarks:**

1.  Theorem 1 doesn’t assume the finiteness of $E(Y)$. Therefore it can be applied to the cases when $\epsilon_i$ follows a Cauchy distribution.

2.  After fitting the model, the Assumption (A2), $\epsilon\perp A|{{\bm X}}$, can be verified by applying conditional independence test with $\hat{r}(\hat{{{\mbox{\boldmath $\beta$}}}},\hat{{{\mbox{\boldmath $\gamma$}}}})$ and $A$ given ${{\bm X}}$, where $\hat{r}(\hat{{{\mbox{\boldmath $\beta$}}}},\hat{{{\mbox{\boldmath $\gamma$}}}})$ is the estimated residual and $\hat{r}(\hat{{{\mbox{\boldmath $\beta$}}}},\hat{{{\mbox{\boldmath $\gamma$}}}})=Y-\varphi({{\bm X}};\hat{{{\mbox{\boldmath $\gamma$}}}})-\{A-\pi({{\bm X}})\}C({{\bm X}};\hat{{{\mbox{\boldmath $\beta$}}}})$. See [@lawrance1976conditional; @su2007consistent; @song2007testing; @huang2010testing; @zhang2012kernel] for more discussion of conditional independence hypothesis tests. In particular, we demonstrate the usefulness of the test by applying the Kernel-based conditional independence test (KCI-test, [@zhang2012kernel]) in Section 5. KCI-test doesn’t assume functional forms among variables and thus suits our need.

When the conditional independence assumption ($\epsilon\perp A|{{\bm X}}$) does not hold, $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}$ may no longer be a consistent estimator of ${{\mbox{\boldmath $\beta$}}}_0$. This is intuitively reasonable as $\epsilon$ contains extra information with respect to $A$. In fact, a general result which can be derived in this case is that, $(\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)},\hat{{{\mbox{\boldmath $\gamma$}}}}^{R}_{\rho(\tau)})$ minimizes a weighed mean-square error loss function with specification error [@angrist2006quantile; @lee2013interpretation].

Instead of assuming response 