Marginal treatment effects in the absence of instrumental variables

abstractWe propose a method for defining, identifying, and estimating the marginal treatment effect (MTE) without imposing the instrumental variable (IV) assumptions of independence, exclusion, and separability (or monotonicity). Under a new definition of the MTE based on reduced-form treatment error that is statistically independent of the covariates, we find that the relationship between the MTE and standard treatment parameters holds in the absence of IVs. We provide a set of sufficient conditions ensuring the identification of the defined MTE in an environment of essential heterogeneity. The key conditions include a linear restriction on potential outcome regression functions, a nonlinear restriction on the propensity score, and a conditional mean independence restriction which will lead to additive separability. We prove this identification using the notion of semiparametric identification based on functional forms. And we provide an empirical application for the Head Start program to illustrate the usefulness of the proposed method in analyzing heterogenous causal effects when IVs are elusive.

exampleWe consider a latent index rule for treatment participation: \begin{equation} D=1\left\{ m\left( X,\varepsilon \right) \geq 0\right\} , \end{equation} where the observed $X$ can be statistically correlated to the unobserved $ \varepsilon $, and no restriction is imposed on the cross-partials of the index function $m$. Without independence and additive separability, model ( (ref)) is completely vacuous, imposing no restrictions on the observed or counterfactual outcomes heckman2001local. This general latent index rule fits into the treatment equation ((ref)) by taking $\mu \left( X\right) =E\left[ m\left( X,\varepsilon \right) \left\vert X\right. \right] $ and $U=\mu \left( X\right) -m\left( X,\varepsilon \right) $.

exampleWe suppose that $X$ is a scalar, and \begin{equation*} \left( \begin{array}{l} U \\ X \end{array} \right) \sim N\left( \left( \begin{array}{l} 0 \\ 0 \end{array} \right) ,\left( \begin{array}{cc} 1 & \sigma _{UX} \\ \sigma _{UX} & \sigma _{X}^{2} \end{array} \right) \right) . \end{equation*} By the property of bivariate normal distribution, we obtain $U\left\vert \left( X=x\right) \right. \sim N\left( \mu _{\left. U\right\vert X}\left( x\right) ,\sigma _{U\left\vert X\right. }^{2}\right) $ and $F_{U\left\vert X\right. }\left( u\left\vert x\right. \right) =\Phi \left( \left. \left[ u-\mu _{U\left\vert X\right. }\left( x\right) \right] \right/ \sigma _{U\left\vert X\right. }\right) $, where $\mu _{U\left\vert X\right. }\left( x\right) =\left( \sigma _{UX}\left/ \sigma _{X}^{2}\right. \right) x$, $ \sigma _{U\left\vert X\right. }^{2}=1-\left( \sigma _{UX}^{2}\left/ \sigma _{X}^{2}\right. \right) $, and $\Phi \left( \cdot \right) $ denotes the standard normal CDF. Hence, \begin{equation*} V=F_{U\left\vert X\right. }\left( U\left\vert X\right. \right) =\Phi \left( \frac{U-\mu _{U\left\vert X\right. }\left( X\right) }{\sigma _{U\left\vert X\right. }}\right) , and V_{x}=\Phi \left( \frac{U-\mu _{U\left\vert X\right. }\left( x\right) }{\sigma _{U\left\vert X\right. }}\right) . \end{equation*} We observe that $V\perp \!\!\!\!\perp X$ because $F_{V\left\vert X\right. }\left( v\left\vert x\right. \right) =v$, but $V_{x}$ is not independent of $ X$ because \begin{equation*} F_{V_{x}\left\vert X\right. }\left( v\left\vert \tilde{x}\right. \right) =\Phi \left( \frac{\left( \sigma _{UX}\left/ \sigma _{X}^{2}\right. \right) \left( x-\tilde{x}\right) }{\sigma _{U\left\vert X\right. }}+\Phi ^{-1}\left( v\right) \right), \end{equation*} and $V_{x}$ is not uniformly distributed because $F_{V_{x}}\left( v\right) =\Phi \left( \mu _{U\left\vert X\right. }\left( x\right) +\sigma _{U\left\vert X\right. }\Phi ^{-1}\left( v\right) \right) $.

exampleSuppose that $X$ is a scalar and \begin{equation*} \left( \begin{array}{c} U_{d} \\ U \\ X \end{array} \right) \sim N\left( \left( \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right) ,\left( \begin{array}{ccc} \sigma _{d}^{2} & \sigma _{dU} & 0 \\ \sigma _{dU} & 1 & \sigma _{UX} \\ 0 & \sigma _{UX} & \sigma _{X}^{2} \end{array} \right) \right) . \end{equation*} In this setting, $U$ is correlated to $X$ with an unconstrained correlation coefficient. Through a property of the multivariate normal distribution, we obtain \begin{equation} \left. \left( \begin{array}{c} U_{d} \\ U \end{array} \right) \right\vert \left( X=x\right) \sim N\left( \left( \begin{array}{c} 0 \\ \mu _{\left. U\right\vert X}\left( x\right) \end{array} \right) ,\left( \begin{array}{cc} \sigma _{d}^{2} & \sigma _{dU} \\ \sigma _{dU} & \sigma _{U\left\vert X\right. }^{2} \end{array} \right) \right) , \end{equation} where $\mu _{U\left\vert X\right. }\left( x\right) =\left( \sigma _{UX}\left/ \sigma _{X}^{2}\right. \right) x$ and $\sigma _{U\left\vert X\right. }^{2}=1-\left( \sigma _{UX}^{2}\left/ \sigma _{X}^{2}\right. \right) $. Hence, \begin{equation*} E\left[ \left. U_{d}\right\vert U=u,X=x\right] =\frac{\sigma _{dU}}{\sigma _{U\left\vert X\right. }^{2}}\left( u-\mu _{\left. U\right\vert X}\left( x\right) \right) . \end{equation*} By Example (ref), we have $V=\Phi \left( \left. \left[ U-\mu _{U\left\vert X\right. }\left( X\right) \right] \right/ \sigma _{U\left\vert X\right. }\right) $, so that $U=\sigma _{U\left\vert X\right. }\Phi ^{-1}\left( V\right) +\mu _{U\left\vert X\right. }\left( X\right) $. Consequently, \begin{equation*} E\left[ \left. U_{d}\right\vert V=v,X=x\right] =E\left[ \left. U_{d}\right\vert U=\sigma _{U\left\vert X\right. }\Phi ^{-1}\left( v\right) +\mu _{U\left\vert X\right. }\left( x\right) ,X=x\right] =\frac{\sigma _{dU} }{\sigma _{U\left\vert X\right. }}\Phi ^{-1}\left( v\right) , \end{equation*} and Assumption CMI holds. More generally, to allow the dependence of $U_{d}$ on $X$ as well, we can set \begin{equation*} \left. \left( \begin{array}{c} U_{d} \\ U \end{array} \right) \right\vert \left( X=x\right) \sim N\left( \left( \begin{array}{c} 0 \\ \mu _{\left. U\right\vert X}\left( x\right) \end{array} \right) ,\left( \begin{array}{cc} \sigma _{d}^{2}\left( x\right) & \sigma _{dU} \\ \sigma _{dU} & \sigma _{U\left\vert X\right. }^{2} \end{array} \right) \right) \end{equation*} in place of ((ref)), where $\sigma _{d}^{2}\left( x\right) $ is the conditional variance of $U_{d}$ given $X=x$. Since $E\left[ \left. U_{d}\right\vert V=v,X=x\right] $ is irrelevant to the variance of $U_{d}$ according to the above analysis, Assumption CMI still holds in the presence of such heteroscedastic $U_{d}$.

exampleConsider the generalized Roy model where individuals choose treatment if the perceived benefit from treatment is greater than the subjective cost of treatment eisenhauer2015generalized. Specifically, \begin{equation} D=1\left\{ Y_{1}-Y_{0}\geq c\left( X\right) +\epsilon \right\} , \end{equation} where $c\left( X\right) $ and $\epsilon $ are the observed and unobserved parts of the cost, respectively. Under Assumption L, the treatment equation ( (ref)) becomes $D=1\left\{ X^{\prime }\left( \beta _{1}-\beta _{0}\right) -c\left( X\right) \geq U\right\} $, where $U=\epsilon -U_{1}+U_{0}$, therefore $\pi \left( x\right) =F_{U\left\vert X\right. }\left( \left. x^{\prime }\left( \beta _{1}-\beta _{0}\right) -c\left( x\right) \right\vert x\right) $, which will be a nonlinear function of $x$ if $c\left( x\right) $ is nonlinear or the conditional distribution function of $U$ given $X=x$ is nonlinear. In economic applications, the cost function $c\left( x\right) $ is typically nonlinear in individual and family characteristics. For instance, the subjective cost of higher education would be U-shaped with respect to the family income, because of the relatively high tuition fee for students with disadvantaged backgrounds and high opportunity cost for students with superior backgrounds. On the other hand, the conditional distribution function $F_{U\left\vert X\right. }\left( \left. u\right\vert x\right) $ will be nonlinear in $x$ in general if $U$ is not independent of $X$, as shown by Example (ref). More generally, the treatment equation ((ref)) is a special case of a utility maximization model of treatment defined as $D=1\left\{ \mathfrak{u} _{1}\left( Y_{1},X,\epsilon _{1}\right) \geq \mathfrak{u}_{0}\left( Y_{0},X,\epsilon _{0}\right) \right\} $, where the utility functions $ \mathfrak{u}_{1}$ and $\mathfrak{u}_{0}$ are quasi-linear with respect to the first argument. An early version of lee2023nonparametric gave further examples illustrating why the nonlinearity of the utility functions makes sense in economic studies.

exampleConsider the case of two continuous covariates. Suppose that for a smooth function $f$, $\pi _{0}\left( x^{C}\right) =f\left( \gamma _{1}x_{1}^{C}+\gamma _{2}x_{2}^{C}+\gamma _{3}x_{1}^{C}x_{2}^{C}\right) $ or $\pi _{0}\left( x^{C}\right) =f\left( \gamma _{1}x_{1}^{C}+\gamma _{2}x_{2}^{C}+\gamma _{3}\left( x_{1}^{C}\right) ^{2}\right) $. Then, we obtain $\left. \partial _{1}\pi _{0}\left( x^{C}\right) \right/ \partial _{2}\pi _{0}\left( x^{C}\right) =\left. \left( \gamma _{1}+\gamma _{3}x_{2}^{C}\right) \right/ \left( \gamma _{2}+\gamma _{3}x_{1}^{C}\right) $ for the interaction case, or $\left. \partial _{1}\pi _{0}\left( x^{C}\right) \right/ \partial _{2}\pi _{0}\left( x^{C}\right) =\left. \left( \gamma _{1}+2\gamma _{3}x_{1}^{C}\right) \right/ \gamma _{2}$ for the quadratic case. In both cases, Assumption NL2.(v) will generally hold for $x^{C}$ and $\tilde{x}^{C}$ satisfying $x_{1}^{C}\neq \tilde{x} _{1}^{C}$.

exampleSuppose the treatment status is determined by a linear-index threshold crossing rule: \begin{equation*} D=1\left\{ X^{\prime }\gamma \geq U\right\} , \end{equation*} where $U$ is not independent of $X$. Namely, suppose $\mu \left( X\right) =X^{\prime }\gamma $ in the structural treatment equation ((ref)). Consider a multiplicatively heteroscedastic $U$ such that $U=\sigma \left( X\right) \tilde{U}$, where $\sigma \left( X\right) $ is a positive function and $\tilde{U}$ is an idiosyncratic error independent of $X$. The normalization derives \begin{equation*} D=1\left\{ \frac{X^{\prime }\gamma }{\sigma \left( X\right) }\geq \tilde{U} \right\} =1\left\{ F_{\tilde{U}}\left(\frac{X^{\prime }\gamma }{\sigma \left( X\right) }\right)\geq V\right\}, \end{equation*} that is, the reduced-form error is defined as $V=F_{\tilde{U}}\left(\tilde{U} \right)$ and the propensity score function is equal to $\pi \left(x\right) = F_{\tilde{U}}\left(\left. \left( x^{\prime }\gamma \right) \right/ \sigma \left( x\right) \right)$. In general, $\pi \left( x\right) $ is a nonlinear function. Another specification we consider is a linear-index model with endogeneity in a certain component $X^{e}$ of $X^{C}$, in which $\pi \left( x\right) =F_{U\left\vert X^{e}\right. }\left( \left. x^{\prime }\gamma \right\vert x^{e}\right) $. Given that $\pi \left( x\right) $ is generally highly nonlinear in $x^{e}$, Assumption NL1 holds straightforwardly, and NL2 holds with $x_{k}^{C}=x^{e}$.

theoremIf Assumptions CMI, L, NL, and S hold, then $\beta _{d}$ and $g_{d}\left( p\right) $ at all $p$ in the support of the propensity score $P$ are identified for $d=0,1$.

theoremSuppose the assumptions of Theorem (ref) hold. Further suppose Assumptions E.1-E.4 given in Appendix (ref) hold. Then for any interior point $v$ of the support of $P$ and any $x$ satisfying $0<\pi \left( x\right) <1$, we have \begin{eqnarray*} \sqrt{nh_{3}^{3}}\left[ \hat{\Delta}^{MTE}\left( x,v\right) -\Delta ^{ MTE}\left( x,v\right) \right] &\rightarrow &N\left( 0,\sigma _{ MTE}^{2}\left( v\right) \right) , \\ \sqrt{nh_{3}}\left[ \hat{\Delta}^{ATE}\left( x\right) -\Delta ^{ ATE}\left( x\right) -b_{ATE}h_{3}^{2}\right] &\rightarrow &N\left( 0,\sigma _{\text{ATE}}^{2}\right) , \\ \sqrt{nh_{3}}\left[ \hat{\Delta}^{\text{TT}}\left( x\right) -\Delta ^{\text{ TT}}\left( x\right) -b_{\text{TT}}\left( \pi \left( x\right) \right) h_{3}^{2}\right] &\rightarrow &N\left( 0,\sigma _{\text{TT}}^{2}\left( \pi \left( x\right) \right) \right) , \\ \sqrt{nh_{3}}\left[ \hat{\Delta}^{\text{TUT}}\left( x\right) -\Delta ^{\text{ TUT}}\left( x\right) -b_{\text{TUT}}\left( \pi \left( x\right) \right) h_{3}^{2}\right] &\rightarrow &N\left( 0,\sigma _{\text{TUT}}^{2}\left( \pi \left( x\right) \right) \right) , \\ \sqrt{nh_{3}}\left[ \hat{\Delta}^{\text{LATE}}\left( x,v_{1},v_{2}\right) -\Delta ^{\text{LATE}}\left( x,v_{1},v_{2}\right) -b_{\text{LATE}}\left( v_{1},v_{2}\right) h_{3}^{2}\right] &\rightarrow &N\left( 0,\sigma _{\text{ LATE}}^{2}\left( v_{1},v_{2}\right) \right) , \end{eqnarray*} where the asymptotic variances $\sigma _{\text{MTE}}^{2},\cdots ,\sigma _{ \text{LATE}}^{2}$ are defined in ((ref))-((ref)), respectively, and the asymptotic bias terms are defined as \begin{eqnarray*} b_{\text{ATE}} &=&\frac{\kappa _{2}}{2}\left( g_{1}^{\left( 2\right) }\left( 1\right) -g_{0}^{\left( 2\right) }\left( 0\right) \right) , \\ b_{\text{TT}}\left( p\right) &=&\frac{\kappa _{2}}{2}\left( g_{1}^{\left( 2\right) }\left( p\right) +\frac{\left( 1-p\right) g_{0}^{\left( 2\right) }\left( p\right) -g_{0}^{\left( 2\right) }\left( 0\right) }{p}\right) , \\ b_{\text{TUT}}\left( p\right) &=&\frac{\kappa _{2}}{2}\left( \frac{ g_{1}^{\left( 2\right) }\left( 1\right) -pg_{1}^{\left( 2\right) }\left( p\right) }{1-p}-g_{0}^{\left( 2\right) }\left( p\right) \right) , \\ b_{\text{LATE}}\left( v_{1},v_{2}\right) &=&\frac{\kappa _{2}}{2}\left( \frac{v_{2}g_{1}^{\left( 2\right) }\left( v_{2}\right) -v_{1}g_{1}^{\left( 2\right) }\left( v_{1}\right) +\left( 1-v_{2}\right) g_{0}^{\left( 2\right) }\left( v_{2}\right) -\left( 1-v_{1}\right) g_{0}^{\left( 2\right) }\left( v_{1}\right) }{v_{2}-v_{1}}\right) , \end{eqnarray*} with $\kappa _{2}=\int k_{3}\left( u\right) u^{2}du$.

appendix\begin{center} { Online Appendices to “Marginal treatment effects in the absence of instrumental variables” } \end{center} \setcounter{table}{0} \setcounter{figure}{0} \section{Additional results of the empirical application} Based on $\hat{\beta}_{1}$ and $\hat{\beta}_{0}$ reported in Table (ref), and the separate estimates of $E\left[ \left. U_{d}\right\vert V=v \right] $ for $d=0,1$ under the semiparametric specification, we can estimate the marginal structural functions \begin{equation*} E\left[ \left. Y_{d}\right\vert V=v\right] =E\left[ X\right] ^{\prime }\beta _{d}+E\left[ \left. U_{d}\right\vert V=v\right] \end{equation*} for potential outcomes $Y_{1}$ and $Y_{0}$, which we plot in Figure (ref). Panel A sheds light on the significantly positive effect of Head Start on education for the respondents with medium-level resistance to treatment, revealing that their gains from the program are driven mainly by the remarkably low educational attainment when untreated. Panel B leads to similar results for wage income, where significant gain emerges for low values of $V$. Moreover, the relatively flat curve of potential wage income in the treated state implies that early childcare attendance serves as an equalizer that diminishes the intergroup difference in the labor market outcome. \begin{figure}[htb] \caption{Semiparametric estimates of marginal structural functions} {-0.2cm} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} { Notes: The marginal structural function refers to the expected potential outcome conditional on the unobserved resistance to treatment. Panels A and B depict the estimated marginal structural function curves for education outcome and labor market outcome, respectively. The 90% confidence interval is based on bootstrapping with 1,000 replications.} \end{figure} In addition, we can semiparametrically estimate conditional expectations of the observed responses given the unobserved determinants of treatment choice, such as the marginal probability of participation \begin{equation*} E\left[ \left. D\right\vert V=v\right] =\Pr \left( \left. P\geq v\right\vert V=v\right) =\Pr \left( P\geq v\right) , \end{equation*} which mirrors the distribution of the propensity score, and the marginal observed outcome \begin{eqnarray*} E\left[ \left. Y\right\vert V=v\right] &=&E\left[ \left. Y_{0}\right\vert V=v \right] +E\left[ \left. D\left( Y_{1}-Y_{0}\right) \right\vert V=v\right] \\ &=&E\left[ \left. Y_{0}\right\vert V=v\right] +E\left[ \left. DX\right\vert V=v\right] ^{\prime }\left( \beta _{1}-\beta _{0}\right) +E\left[ \left. D\left( U_{1}-U_{0}\right) \right\vert V=v\right] \\ &=&E\left[ \left. Y_{0}\right\vert V=v\right] +E\left[ 1\left\{ P\geq v\right\} X\right] ^{\prime }\left( \beta _{1}-\beta _{0}\right) +\Pr \left( P\geq v\right) \cdot E\left[ \left. U_{1}-U_{0}\right\vert V=v\right] , \end{eqnarray*} where the last equality follows from the independence of $X$ and $V$ and from the conditional mean independence of $X$ and $U_{d}$ given $V$. Plugging in proper estimates of each component in the above equations, we obtain the estimated marginal response functions and plot them in Figure (ref). In particular, the marginal observed outcome curves relate the individuals with significant positive returns (with medium $V$ in Panel A and small $V$ in Panel B) to those with low education and low wage income, thereby bridging our results on the MTE and those of de2020head on the distributional treatment effect. \begin{figure}[!htb] \caption{Semiparametric estimates of marginal response functions} {-0.2cm} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} { Notes: This figure plots the (conditional) expectation of the observed outcome (solid line on the left y-axis), and the conditional probability of participation (dashed line on the right y-axis). Panels A and B illustrate the estimated marginal response function curves for education outcome and labor market outcome, respectively. The 90% confidence interval is based on bootstrapping with 1,000 replications.} \end{figure} \setcounter{equation}{0} \section{Proof of Theorem (ref)} Denote $m_{d}\left( x\right) =E\left[ Y\left\vert X=x,D=d\right. \right] $ and $m_{d0}\left( x^{C}\right) =m_{d}\left( x^{C},0\right) $, $d=0,1$. Note that $\pi \left( x\right) $ and $m_{d}\left( x\right) $, and thus $\pi _{0}\left( x^{C}\right) $ and $m_{d0}\left( x^{C}\right) $, are identified functions because they are conditional expectations of observed variables. We first consider identifying $\beta _{d}^{C}$, the coefficients of continuous covariates, from $\pi _{0}\left( x^{C}\right) $ and $m_{d0}\left( x^{C}\right) $. By equation ((ref)), we have \begin{equation} m_{d0}\left( x^{C}\right) =x^{C\prime }\beta _{d}^{C}+g_{d}\left( \pi _{0}\left( x^{C}\right) \right) . \end{equation} When $\dim \left( X^{C}\right) =1$, by differentiating both sides of ((ref)) at $\tilde{x}^{C}$ and invoking Assumption NL1, we obtain \begin{equation*} m_{d0}^{\left( 1\right) }\left( \tilde{x}^{C}\right) =\beta _{d}^{C}+g_{d}^{\left( 1\right) }\left( \pi _{0}\left( \tilde{x}^{C}\right) \right) \pi _{0}^{\left( 1\right) }\left( \tilde{x}^{C}\right) =\beta _{d}^{C}, \end{equation*} which identifies $\beta _{d}^{C}$ for $d=0,1$. When $\dim \left( X^{C}\right) \geq 2$, for $k,j\in \left\{ 1,2,\cdots ,\dim \left( X^{C}\right) \right\} $ satisfying Assumption NL2, taking the partial derivatives of $m_{d0}\left( x^{C}\right) $ with respect to $x_{k}^{C}$ and $ x_{j}^{C}$ yields that \begin{eqnarray*} \partial _{k}m_{d0}\left( x^{C}\right) &=&\beta _{d,k}^{C}+g_{d}^{\left( 1\right) }\left( \pi _{0}\left( x^{C}\right) \right) \partial _{k}\pi _{0}\left( x^{C}\right) , \\ \partial _{j}m_{d0}\left( x^{C}\right) &=&\beta _{d,j}^{C}+g_{d}^{\left( 1\right) }\left( \pi _{0}\left( x^{C}\right) \right) \partial _{j}\pi _{0}\left( x^{C}\right) . \end{eqnarray*} It follows from Assumption NL2.(i)-(ii) that \begin{equation*} \frac{\partial _{k}m_{d0}\left( x^{C}\right) -\beta _{d,k}^{C}}{\partial _{k}\pi _{0}\left( x^{C}\right) }=g_{d}^{\left( 1\right) }\left( \pi _{0}\left( x^{C}\right) \right) =\frac{\partial _{j}m_{d0}\left( x^{C}\right) -\beta _{d,j}^{C}}{\partial _{j}\pi _{0}\left( x^{C}\right) }, \end{equation*} so that \begin{equation} \partial _{k}m_{d0}\left( x^{C}\right) \partial _{j}\pi _{0}\left( x^{C}\right) -\partial _{j}m_{d0}\left( x^{C}\right) \partial _{k}\pi _{0}\left( x^{C}\right) =\partial _{j}\pi _{0}\left( x^{C}\right) \beta _{d,k}^{C}-\partial _{k}\pi _{0}\left( x^{C}\right) \beta _{d,j}^{C}, \end{equation} which is linear in $\beta _{d,k}^{C}$ and $\beta _{d,j}^{C}$. The same equation is obtained if we evaluate the expression at another point $\tilde{x }^{C}$ that satisfies Assumption NL2.(iii)-(iv), which gives \begin{equation} \left( \begin{array}{c} \partial _{k}m_{d0}\left( x^{C}\right) \partial _{j}\pi _{0}\left( x^{C}\right) -\partial _{j}m_{d0}\left( x^{C}\right) \partial _{k}\pi _{0}\left( x^{C}\right) \\ \partial _{k}m_{d0}\left( \tilde{x}^{C}\right) \partial _{j}\pi _{0}\left( \tilde{x}^{C}\right) -\partial _{j}m_{d0}\left( \tilde{x}^{C}\right) \partial _{k}\pi _{0}\left( \tilde{x}^{C}\right) \end{array} \right) =\Psi \left( \begin{array}{c} \beta _{d,k}^{C} \\ \beta _{d,j}^{C} \end{array} \right) , \end{equation} where \begin{equation*} \Psi =\left( \begin{array}{cc} \partial _{j}\pi _{0}\left( x^{C}\right) , & -\partial _{k}\pi _{0}\left( x^{C}\right) \\ \partial _{j}\pi _{0}\left( \tilde{x}^{C}\right) , & -\partial _{k}\pi _{0}\left( \tilde{x}^{C}\right) \end{array} \right) . \end{equation*} Assumption NL2.(v) ensures that the determinant of $\Psi $ is nonzero, which shows that $\Psi $ is nonsingular. Therefore, equation ((ref)) can be solved for $\beta _{d,k}^{C}$ and $\beta _{d,j}^{C}$ by inverting $\Psi $, thereby identifying $\beta _{d,k}^{C}$ and $\beta _{d,j}^{C}$. Given identification of $\beta _{d,k}^{C}$, we can then identify all other coefficient $\beta _{d,l}^{C}$ in $\beta _{d}^{C}$ by solving ((ref)) with the subscript $j$ replaced by $l$, which gives \begin{equation*} \beta _{d,l}^{C}=\frac{\partial _{l}m_{d0}\left( x^{C}\right) \partial _{k}\pi _{0}\left( x^{C}\right) -\partial _{k}m_{d0}\left( x^{C}\right) \partial _{l}\pi _{0}\left( x^{C}\right) +\partial _{l}\pi _{0}\left( x^{C}\right) \beta _{d,k}^{C}}{\partial _{k}\pi _{0}\left( x^{C}\right) }. \end{equation*} Given the identification of $\beta _{d}^{C}$, the function $g_{d}$ is identified on the support of $\pi _{0}\left( X^{C}\right) $ by \begin{equation*} g_{d}\left( p\right) =E\left[ \left. m_{d0}\left( X^{C}\right) -X^{C\prime }\beta _{d}^{C}\right\vert \pi _{0}\left( X^{C}\right) =p\right] . \end{equation*} Next, we consider identifying $\beta _{d}^{D}$, the coefficients of discrete covariates. For each $k\in \left\{ 1,2,\cdots ,\dim \left( X^{D}\right) \right\} $, we have \begin{equation*} m_{d}\left( x^{C},x^{Dk}\right) =x^{C\prime }\beta _{d}^{C}+x_{k}^{D}\beta _{d,k}^{D}+g_{d}\left( \pi \left( x^{C},x^{Dk}\right) \right) \end{equation*} for any $x^{C}$ in the support of $X^{C}$. By Assumption S, there exists $ x^{C}\left( k\right) $ in the support of $X^{C}$ such that $\pi \left( x^{C}\left( k\right) ,x^{Dk}\right) $ is in the support of $\pi _{0}\left( X^{C}\right) $. It follows from the above identification result that $ g_{d}\left( \pi \left( x^{C}\left( k\right) ,x^{Dk}\right) \right) $ is identified. Consequently, $\beta _{d,k}^{D}$ is identified by \begin{equation*} \beta _{d,k}^{D}=\frac{m_{d}\left( x^{C}\left( k\right) ,x^{Dk}\right) -x^{C}\left( k\right) ^{\prime }\beta _{d}^{C}-g_{d}\left( \pi \left( x^{C}\left( k\right) ,x^{Dk}\right) \right) }{x_{k}^{D}}. \end{equation*} This argument holds for each $k\in \left\{ 1,2,\cdots ,\dim \left( X^{D}\right) \right\} $, thereby identifying $\beta _{d}^{D}$. Finally, given the identification of $\beta _{d}=\left( \beta _{d}^{C},\beta _{d}^{D}\right) $, it follows from ((ref)) that the function $g_{d}$ is identified on the support of $P=\pi \left( X\right) $ by \begin{equation*} g_{d}\left( p\right) =E\left[ \left. m_{d}\left( X\right) -X^{\prime }\beta _{d}\right\vert \pi \left( X\right) =p\right] , \end{equation*} for $d=0,1$. \setcounter{theorem}{0} \setcounter{equation}{0} \section{Further discussion on the nonlinearity assumption} In light of the proof of Theorem (ref), our identification strategy may work even when no continuous covariate is available in the data. In that case, we put a discretely distributed covariate in $X^{C}$. \begin{theorem} Theorem (ref) holds if Assumption NL1 is replaced by that there exist two different constants $x^{C}$, $\tilde{x}^{C}$ \ in the support of $X^{C}$\ such that $\pi _{0}\left( x^{C}\right) =\pi _{0}\left( \tilde{x}^{C}\right) $. \end{theorem} \begin{proof} When $\dim \left( X^{C}\right) =1$, it follows from ((ref)) and the assumption of Theorem (ref) that \begin{equation*} m_{d0}\left( x^{C}\right) -x^{C}\beta _{d}^{C}=g_{d}\left( \pi _{0}\left( x^{C}\right) \right) =g_{d}\left( \pi _{0}\left( \tilde{x}^{C}\right) \right) =m_{d0}\left( \tilde{x}^{C}\right) -\tilde{x}^{C}\beta _{d}^{C}. \end{equation*} Hence, $\beta _{d}^{C}$ is identified as \begin{equation*} \beta _{d}^{C}=\frac{m_{d0}\left( x^{C}\right) -m_{d0}\left( \tilde{x} ^{C}\right) }{x^{C}-\tilde{x}^{C}}. \end{equation*} The remaining part of the proof is the same as the proof of Theorem (ref). \end{proof} The assumption of Theorem (ref) in place of NL1 requires the univariate function $\pi _{0}$ to be not one-to-one, but imposes no other smoothness or continuity restriction on $\pi _{0}$. Therefore, it doesn't require $X^{C}$ to be continuously valued. In the extreme case of $X^{C}$ containing only one binary covariate, this assumption will be equivalent to the full irrelevance of $X^{C}$ to the treatment probability, which is a condition suggested by chamberlain1986asymptotic for identification of the sample selection model. The following two theorems show that Assumption NL1 and the assumption of Theorem (ref) can be extended to the case of multiple continuous covariates. \begin{theorem} Theorem (ref) holds if Assumption NL is replaced by that (i) there are two points $x^{C}$ and $\tilde{x}^{C}$ in the support of $X^{C}$ and an index $k$ such that $x_{k}^{C}\neq \tilde{x} _{k}^{C}$, $x_{-k}^{C}=\tilde{x}_{-k}^{C}$, and $\pi _{0}\left( x^{C}\right) =\pi _{0}\left( \tilde{x}^{C}\right) $, and (ii) there exists $\breve{x}^{C}$ in the support of $X^{C} $ such that functions $\pi _{0}$ and $m_{d0}$ are differentiable at $\breve{x}^{C}$, with $\partial _{k}\pi _{0}\left( \breve{x }^{C}\right) \neq 0 $. \end{theorem} \begin{proof} It follows from equation ((ref)) and $\pi _{0}\left( x^{C}\right) =\pi _{0}\left( \tilde{x}^{C}\right) $ that \begin{equation*} m_{d0}\left( x^{C}\right) -x_{k}^{C}\beta _{d,k}^{C}-x_{-k}^{C\prime }\beta _{d,-k}^{C}=m_{d0}\left( \tilde{x}^{C}\right) -\tilde{x}_{k}^{C}\beta _{d,k}^{C}-\tilde{x}_{-k}^{C^{\prime }}\beta _{d,-k}^{C}. \end{equation*} Hence, $\beta _{d,k}^{C}$ is identified by condition (i) as \begin{equation*} \beta _{d,k}^{C}=\frac{m_{d0}\left( x^{C}\right) -m_{d0}\left( \tilde{x} ^{C}\right) }{x_{k}^{C}-\tilde{x}_{k}^{C}}. \end{equation*} For any index $j\neq k$, taking the partial derivatives of $m_{d0}\left( x^{C}\right) $ with respect to $x_{k}^{C}$ and $x_{j}^{C}$ at $\breve{x}^{C}$ yields that \begin{eqnarray} \partial _{k}m_{d0}\left( \breve{x}^{C}\right) &=&\beta _{d,k}^{C}+g_{d}^{\left( 1\right) }\left( \pi _{0}\left( \breve{x} ^{C}\right) \right) \partial _{k}\pi _{0}\left( \breve{x}^{C}\right) , \\ \partial _{j}m_{d0}\left( \breve{x}^{C}\right) &=&\beta _{d,j}^{C}+g_{d}^{\left( 1\right) }\left( \pi _{0}\left( \breve{x} ^{C}\right) \right) \partial _{j}\pi _{0}\left( \breve{x}^{C}\right) . \end{eqnarray} Since $\partial _{k}\pi _{0}\left( \breve{x}^{C}\right) \neq 0$ by condition (ii), we can find $g_{d}^{\left( 1\right) }\left( \pi _{0}\left( \breve{x} ^{C}\right) \right) $ from ((ref)) to be \begin{equation*} g_{d}^{\left( 1\right) }\left( \pi _{0}\left( \breve{x}^{C}\right) \right) = \frac{\partial _{k}m_{d0}\left( \breve{x}^{C}\right) -\beta _{d,k}^{C}}{ \partial _{k}\pi _{0}\left( \breve{x}^{C}\right) }, \end{equation*} and substitute it into ((ref)) to identify $\beta _{d,j}^{C}$ as \begin{equation*} \beta _{d,j}^{C}=\partial _{j}m_{d0}\left( \breve{x}^{C}\right) -\frac{ \partial _{j}\pi _{0}\left( \breve{x}^{C}\right) }{\partial _{k}\pi _{0}\left( \breve{x}^{C}\right) }\left[ \partial _{k}m_{d0}\left( \breve{x} ^{C}\right) -\beta _{d,k}^{C}\right] . \end{equation*} Therefore, $\beta _{d}^{C}$ is identified. The remaining part of the proof is the same as the proof of Theorem (ref). \end{proof} \begin{theorem} Theorem (ref) holds if Assumption NL is replaced by that there exist two points $x^{C}$ and $\tilde{x}^{C}$ in the support of $X^{C}$ and an index $k$ such that functions $\pi _{0}$ and $ m_{d0}$ are differentiable at $x^{C}$ and $\tilde{x}^{C}$, with $\partial _{k}\pi _{0}\left( x^{C}\right) =0 $ and $\partial _{k}\pi _{0}\left( \tilde{ x}^{C}\right) \neq 0$. \end{theorem} \begin{proof} The identification of $\beta _{d,k}^{C}$ is analogous to Theorem (ref). Then, the identification of all the other coefficients $ \beta _{d,j}^{C}$ in $\beta _{d}^{C}$ and the remaining components of the model is the same as in Theorem (ref) and Theorem (ref), respectively. \end{proof} Alternatively, the identification of coefficients of continuous covariates can be attained by exploiting the local irrelevance of the function $ g_{d}\left( p\right) $. \begin{theorem} Theorem (ref) holds if Assumption NL is replaced by the existence of a vector $x^{C}$ in the support of $X^{C}$ such that $g_{d}^{\left( 1\right) }\left( \pi _{0}\left( x^{C}\right) \right) =0$ for $d=0,1$, and that functions $\pi _{0}$ and $m_{d0}$ are differentiable at $x^{C}$. \end{theorem} \begin{proof} For each $k\in \left\{ 1,2,\cdots ,\dim \left( X^{C}\right) \right\} $, taking the partial derivative of both sides of ((ref)) with respect to $x_{k}^{C}$ gives \begin{equation*} \partial _{k}m_{d0}\left( x^{C}\right) =\beta _{d,k}^{C}+g_{d}^{\left( 1\right) }\left( \pi _{0}\left( x^{C}\right) \right) \partial _{k}\pi _{0}\left( x^{C}\right) =\beta _{d,k}^{C}, \end{equation*} which immediately identifies all $\beta _{d,k}^{C}$ in $\beta _{d}^{C}$. The remaining part of the proof is the same as that of Theorem (ref). \end{proof} The above discussion further illustrates the over-identification feature of our strategy. In fact, the strength of identification of our method is determined by the number of nonlinearity points in the data satisfying Assumption NL or Theorems (ref)-(ref), given different benchmark values of $X^{D}$. \setcounter{theorem}{0} \setcounter{equation}{0} \section{Limited valued outcomes} To accommodate limited valued outcomes, we first generalize the linear additive model to a linear latent index model. Assumption D.L (Linear index). Assume that $Y_{d}=l\left( X^{\prime }\gamma _{d},V_{d}\right) $, $d=0,1$, holds for some known link function $l$\ and unknown coefficients $\gamma _{d} $\textit{, where }$V_{d}$\textit{\ is unobserved error term of }$Y_{d} $ \textit{.} The generalized linear model applies to a variety of frequently encountered limited dependent variables, such as $l\left( t,v\right) =1\left\{ t\geq v\right\} $ for binary $Y\in \left\{ 0,1\right\} $, $l\left( t,v\right) =\max \left( t-v,0\right) $ for censored (at zero) $Y\geq 0$, $l\left( t,v\right) =\exp \left( t+v\right) \left/ \left[ 1+\exp \left( t+v\right) \right] \right. $ for proportion-valued $Y\in \left( 0,1\right) $, $l\left( t,v\right) =\sum_{j=1}^{s}1\left\{ t+v\geq c_{j}\right\} $ for ordered $Y\in \left\{ 0,1,\cdots ,s\right\} $ with known thresholds $c_{1}<c_{2}<\cdots <c_{s}$, and so forth. Although setting $l\left( t,v\right) =t+v$ reduces to the linear additive model, the following discussion doesn't include Theorem (ref) as a special case because $\gamma _{d}$ here can be only identified up to scale for a general link function $l$. Any changes in the scaling of $\gamma _{d}$ can be freely absorbed into $l$, and a scale normalization is needed for identification of $\gamma _{d}$ and $l$. \textbf{Assumption D.N} (Normalization). \textit{Decompose }$X^{\prime }\gamma _{d}=X^{C\prime }\gamma _{d}^{C}+X^{D\prime }\gamma _{d}^{D}$\textit{ , where }$X^{C}$\textit{\ and }$X^{D}$\textit{\ consist of covariates that are continuously and discretely distributed, respectively. Assume that }$ \gamma _{d,1}^{C}$\textit{, the first element of }$\gamma _{d}^{C}$\textit{, equal to 1.} Assumption D.N imposes the convenient normalization that the first continuous covariate has a unit coefficient. This scaling of $\gamma _{d}$ is arbitrary and is innocuous because our focus is on the identification of MTE, which is a difference in the conditional expectations of $l\left( X^{\prime }\gamma _{d},V_{d}\right) $, rather than on separate identification of $\gamma _{d}$ and $l$. Since in the generalized linear model the potential outcomes are not necessarily additive in the error term, the mean independence assumption needs to be strengthened to a stricter distributional independence assumption. \textbf{Assumption D.CI} (Conditional Independence). \textit{Assume that }$ \left. V_{d}\perp \!\!\!\!\perp X\right\vert V$\textit{\ for }$d=0,1$\textit{ , namely, }$V_{d}$\textit{\ is independent of }$X$\textit{\ conditional on }$ V$\textit{, where }$V$\textit{\ is the reduced-form treatment error in equation ((ref)).} Recalling that $V\perp \!\!\!\!\perp X$ by definition, Assumption D.CI is equivalent to the full independence $\left( V_{d},V\right) \perp \!\!\!\!\perp X$ for $d=0,1$, which implies that both the marginal distribution of $V_{d}$ and the copula of $V_{d}$ and $V$ are independent of $X$. Under Assumptions D.L and D.CI, we have \begin{equation*} \Delta ^{\text{MTE}}\left( x,v\right) =E\left[ \left. l\left( x^{\prime }\gamma _{1},V_{1}\right) -l\left( x^{\prime }\gamma _{0},V_{0}\right) \right\vert V=v\right] . \end{equation*} The additive nonseparability of the observables and unobservables constitutes the primary difficulty in identifying MTE in the limited outcome case. Meanwhile, Assumptions D.L and D.CI lead to a double index form of the observable outcome regression functions for each treatment status, that is, \begin{equation} E\left[ \left. Y\cdot 1\left\{ D=d\right\} \right\vert X=x\right] =r_{d}\left( x^{\prime }\gamma _{d},\pi \left( x\right) \right) \end{equation} for $d=0,1$, where \begin{eqnarray} r_{0}\left( t,p\right) &=&\int_{p}^{1}E\left[ \left. l\left( t,V_{0}\right) \right\vert V=v\right] dv, \\ r_{1}\left( t,p\right) &=&\int_{0}^{p}E\left[ \left. l\left( t,V_{1}\right) \right\vert V=v\right] dv. \end{eqnarray} Provided that $\pi \left( x\right) $ is a nonlinear index, $\gamma _{d}$ and thus $r_{d}$ can be identified based on functional forms, which ensures identification of MTE since \begin{equation} \Delta ^{\text{MTE}}\left( x,v\right) =\partial _{2}r_{1}\left( x^{\prime }\gamma _{1},v\right) +\partial _{2}r_{0}\left( x^{\prime }\gamma _{0},v\right) , \end{equation} where $\partial _{2}r_{d}$ is the partial derivative of $r_{d}$ with respect to its second argument. Like in the case of unlimited outcomes, the key powers of this identification strategy are supplied by the nonlinearity of $ \pi \left( x\right) $, as specified by the following assumption. \textbf{Assumption D.NL} (Non-Linearity). \textit{Assume that the functions } $\pi _{0}$\textit{\ and }$r_{d}$\textit{\ are differentiable and denote their partial derivatives with respect to the }$k$\textit{-th argument as }$ \partial _{k}\pi _{0}$\textit{\ and }$\partial _{k}r_{d}$\textit{, where }$ \pi _{0}\left( x^{C}\right) =\pi \left( x^{C},0\right) $\textit{\ and }$ r_{d} $\textit{, }$d=0,1$\textit{, are defined in ((ref)) and ((ref)). Assume that there exist two vectors }$x^{C}$\textit{\ and }$\tilde{x}^{C}$ \textit{\ on the support of }$X^{C}$\textit{\ and two elements }$k$\textit{\ and }$j$\textit{\ of the set }$\left\{ 1,2,\cdots ,\dim \left( X^{C}\right) \right\} $\textit{\ such that }(i)\textit{\ }$\partial _{1}r_{d}\left( x^{C\prime }\gamma _{d}^{C},\pi _{0}\left( x^{C}\right) \right) \neq 0$ \textit{, }(ii)\textit{\ }$\partial _{1}r_{d}\left( \tilde{x}^{C\prime }\gamma _{d}^{C},\pi _{0}\left( \tilde{x}^{C}\right) \right) \neq 0$\textit{ , }(iii)\textit{\ }$\partial _{k}\pi _{0}\left( x^{C}\right) \neq \partial _{1}\pi _{0}\left( x^{C}\right) \gamma _{d,k}^{C}$\textit{, }(iv)\textit{\ }$ \partial _{j}\pi _{0}\left( x^{C}\right) \neq \partial _{1}\pi _{0}\left( x^{C}\right) \gamma _{d,j}^{C}$\textit{, }(v)\textit{\ }$\partial _{l}\pi _{0}\left( \tilde{x}^{C}\right) \neq \partial _{1}\pi _{0}\left( \tilde{x} ^{C}\right) \gamma _{d,l}^{C}$\textit{\ for }$l=2,\cdots ,\dim \left( X^{C}\right) $\textit{, and} \begin{eqnarray*} &\text{(vi) }&\partial _{k}\pi _{0}\left( x^{C}\right) \partial _{j}\pi _{0}\left( \tilde{x}^{C}\right) -\partial _{k}\pi _{0}\left( \tilde{x} ^{C}\right) \partial _{j}\pi _{0}\left( x^{C}\right) \\ &&\neq \left[ \partial _{1}\pi _{0}\left( x^{C}\right) \partial _{j}\pi _{0}\left( \tilde{x}^{C}\right) -\partial _{1}\pi _{0}\left( \tilde{x} ^{C}\right) \partial _{j}\pi _{0}\left( x^{C}\right) \right] \gamma _{d,k}^{C} \\ &&-\left[ \partial _{1}\pi _{0}\left( x^{C}\right) \partial _{k}\pi _{0}\left( \tilde{x}^{C}\right) -\partial _{1}\pi _{0}\left( \tilde{x} ^{C}\right) \partial _{k}\pi _{0}\left( x^{C}\right) \right] \gamma _{d,j}^{C}. \end{eqnarray*} Assumption D.NL.(i)-(ii) require $r_{d}$ to depend on the linear index, and (iii)-(vi) essentially require some nonlinear variation in $\pi _{0}$ under the scale normalization. As in Assumption NL, it is difficult to construct examples other than $\pi _{0}\left( x^{C}\right) =f\left( x^{C\prime }\gamma \right) $ that violates Assumption D.NL. Finally, a support assumption and an invertibility assumption are imposed for technical reasons. \textbf{Assumption D.S} (Support). \textit{For each }$k\in \left\{ 1,2,\cdots ,\dim \left( X^{D}\right) \right\} $\textit{, assume for some }$ x_{k}^{D}\neq 0$\textit{\ in the support of }$X_{k}^{D}$\textit{\ that there exists }$x^{C}\left( k\right) $\textit{\ in the support of }$X^{C}$\textit{\ such that }$\pi \left( x^{C}\left( k\right) ,x^{Dk}\right) $\textit{\ is in the support of }$\pi _{0}\left( X^{C}\right) $\textit{\ and that }$ x^{C}\left( k\right) ^{\prime }\gamma _{d}^{C}+x_{k}^{D}\gamma _{d,k}^{D}$ \textit{\ is in the support of }$X^{C\prime }\gamma _{d}^{C}$\textit{\ for }$ d=0,1$\textit{.} \textbf{Assumption D.I} (Invertibility). \textit{Assume that the function }$ r_{d}$\textit{\ is invertible on its first argument for }$d=0,1$\textit{.} For the case of a binary outcome, we have $r_{0}\left( t,p\right) =\int_{p}^{1}F_{\left. V_{0}\right\vert V}\left( \left. t\right\vert v\right) dv$ and $r_{1}\left( t,p\right) =\int_{0}^{p}F_{\left. V_{1}\right\vert V}\left( \left. t\right\vert v\right) dv$, hence a sufficient condition for Assumption D.I to hold is that $V_{d}$ is continuously distributed with support $ \mathbb{R} $, conditional on $V$, for $d=0,1$. Interestingly, the same continuous distribution condition also suffices for Assumption D.I in the censored case with $l\left( t,v\right) =\max \left( t-v,0\right) $, where $\partial _{1}r_{0}\left( t,p\right) =\int_{p}^{1}F_{\left. V_{0}\right\vert V}\left( \left. t\right\vert v\right) dv$ and $\partial _{1}r_{1}\left( t,p\right) =\int_{0}^{p}F_{\left. V_{1}\right\vert V}\left( \left. t\right\vert v\right) dv$. Under the imposed assumptions, the following identification theorem for the generalized linear model follows immediately from escanciano2016identification. \begin{theorem} If Assumptions D.L, D.NL, D.CI, D.S, D.N, and D.I hold, then $\gamma _{d}$ and $r_{d}\left( t,p\right) $ at all points in the support of $X^{\prime }\gamma _{d}$\ and $P$ are identified for $d=0,1$. \end{theorem} Having established identification of $\gamma _{d}$ and $r_{d}$, the MTE for limited valued outcomes can be identified by ((ref)) and estimated by the semiparametric least squares method of escanciano2016identification. Alternatively, the LIV estimation procedure may be adapted according to \begin{equation*} E\left[ \left. Y\right\vert X=x\right] =s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},\pi \left( x\right) \right) \end{equation*} and \begin{eqnarray*} \Delta ^{\text{MTE}}\left( x,v\right) &=&\frac{\partial s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},v\right) }{\partial v}, \\ \Delta ^{\text{ATE}}\left( x\right) &=&s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},1\right) -s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},0\right) , \\ \Delta ^{\text{TT}}\left( x\right) &=&\frac{s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},\pi \left( x\right) \right) -s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},0\right) }{\pi \left( x\right) }, \\ \Delta ^{\text{TUT}}\left( x\right) &=&\frac{s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},1\right) -s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},\pi \left( x\right) \right) }{1-\pi \left( x\right) }, \\ \Delta ^{\text{LATE}}\left( x,v_{1},v_{2}\right) &=&\frac{s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},v_{2}\right) -s\left( x^{\prime }\gamma _{0},x^{\prime }\gamma _{1},v_{1}\right) }{v_{2}-v_{1}}, \end{eqnarray*} where \begin{equation*} s\left( t_{0},t_{1},p\right) =r_{0}\left( t_{0},p\right) +r_{1}\left( t_{1},p\right) . \end{equation*} \setcounter{equation}{0} \setcounter{theorem}{0} \section{Sufficient conditions and proofs for Theorem (ref)} \subsection{Sufficient conditions} We first give the assumptions that have been mentioned when presenting Theorem (ref). We denote $f_{X}\left( \cdot \right) $ and $ f_{P}\left( \cdot \right) $ as the probability density function (PDF) of $ X=\left( X^{C},X^{D}\right) $ and $P=\pi \left( X\right) $, respectively, and $f_{P}^{\left( 1\right) }\left( p\right) =\left. \partial f_{P}\left( p\right) \right/ \partial p$ as the derivative function of $f_{P}\left( \cdot \right) $ if it is differentiable. For $d=0,1$, we define $ V_{d}=Y-X^{\prime }\beta _{d}-g_{d}\left( P\right) $, which satisfies $E \left[ \left. V_{d}\right\vert X,D=d\right] =0$ because by definition $ g_{d}\left( p\right) =E\left[ \left. Y-X^{\prime }\beta _{d}\right\vert \pi \left( X\right) =p,D=d\right] $. And we denote $\sigma _{V_{d}}^{2}\left( p\right) =Var\left( \left. V_{d}\right\vert P=p,D=d\right) =E\left[ \left. V_{d}^{2}\right\vert P=p,D=d\right] $. Note that \begin{eqnarray*} \sigma _{V_{d}}^{2}\left( p\right) &=&E\left[ \left. \left( U_{d}-g_{d}\left( P\right) \right) ^{2}\right\vert P=p,D=d\right] \\ &=&E\left[ \left. U_{d}^{2}\right\vert P=p,D=d\right] -g_{d}\left( p\right) \\ &=&\sigma _{U_{d}}^{2}\left( p\right) -g_{d}\left( p\right) . \end{eqnarray*} In addition, we denote \begin{equation*} \delta _{d}\left( p\right) =E\left[ \left. 1\left\{ D=d\right\} \right\vert P=p\right] =\left\{ \begin{array}{cc} p, & \text{if }d=1, \\ 1-p, & \text{if }d=0, \end{array} \right. \end{equation*} and \begin{equation*} \delta _{d}^{\left( 1\right) }\left( p\right) =\frac{\partial \delta _{d}\left( p\right) }{\partial p}=\left\{ \begin{array}{cc} 1, & \text{if }d=1, \\ -1, & \text{if }d=0. \end{array} \right. \end{equation*} \textbf{Assumption E.1} (First step). For an integer $s\geq 2$, we assume that: (i) $\pi \left( x^{C},x^{D}\right) $ and $f_{X}\left( x^{C},x^{D}\right) $ are $s$-times differentiable with respect to $x^{C}$ for all $x^{D}$, and the derivative functions are all Lipschitz continuous and bounded; (ii) $\inf_{x^{C}}f_{X}\left( x^{C},x^{D}\right) \geq c$ for a positive constant $c$ for all $x^{D}$; (iii) $k_{1}\left( \cdot \right) $ is a bounded $s$-th order kernel function satisfying that $\int k_{1}\left( u\right) du=1$, $\int u^{t}k_{1}\left( u\right) du=0$ for $t=1,\cdots ,s-1$, and $\int u^{s}k_{1}\left( u\right) du\neq 0$, and the functions $K_{1l}\left( u\right) =\left\vert u\right\vert ^{l}k_{1}\left( u\right) $ are Lipschitz continuous for all $0\leq l\leq s+1$ . (iv) $h_{1l}\rightarrow 0$ for $l=1,2,\cdots ,\dim \left( X^{C}\right) $, and $\left. nh_{11}h_{12}\cdots h_{1,\dim \left( X^{C}\right) }\right/ \ln n\rightarrow \infty $, as $n\rightarrow \infty $. \textbf{Assumption E.2} (Second step). For $d=0,1$, we assume that: (i) $\left\Vert \hat{\beta}_{d}-\beta _{d}\right\Vert =O_{p}\left( n^{-1/2}\right) $; (ii) $g_{d}\left( \cdot \right) $, $f_{P}\left( \cdot \right) $, and $\sigma _{V_{d}}^{2}\left( \cdot \right) $ are twice continuously differentiable over the support of $P$, and $\inf_{p\in \left[ 0,1\right] }f_{P}\left( p\right) \geq c$ for a positive constant $c$. (iii) $k_{3}\left( \cdot \right) $ is a bounded, symmetric, compactly supported, and three times continuously differentiable kernel function; (iv) $h_{3}\rightarrow 0$, $nh_{3}^{3}\rightarrow \infty $, and $ nh_{3}^{7}\rightarrow 0$, as $n\rightarrow \infty $. \textbf{Assumption E.3} (Rate). We assume that as $n\rightarrow \infty $, (i) $nh_{3}\eta _{1}^{2}\rightarrow 0$, (ii) $\left. n\eta _{1}^{6}\right/ h_{3}^{7}\rightarrow 0$, (iii) $nh_{3}^{3}\left\vert h_{1}\right\vert \rightarrow \infty $, and (iv) $n^{3/2}h_{3}^{5}\left\vert h_{1}\right\vert ^{2}\rightarrow \infty $, where \begin{eqnarray*} \eta _{1} &=&tr\left( h_{1}^{s}\right) +\sqrt{\frac{\ln n}{n\left\vert h_{1}\right\vert }}, \\ tr\left( h_{1}^{s}\right) &=&\sum_{l=1}^{\dim \left( X^{C}\right) }h_{1l}^{s}, \\ \left\vert h_{1}\right\vert &=&h_{11}h_{12}\cdots h_{1,\dim \left( X^{C}\right) }. \end{eqnarray*} \textbf{Assumption E.4} (Moment). We assume that $E\left[ \left\Vert X\right\Vert ^{2}\right] <\infty $ and $E\left[ V_{d}^{2}\right] <\infty $ for $d=0,1$. Assumption E.1.(iii) employs the higher-order kernel device to reduce the bias and improve the convergence rate of the first-step estimation ((ref)). Since the first step is a multivariate kernel regression, the usual symmetric (second-order) kernel will necessarily lead to a rate of convergence that dominates the subsequent local linear and MTE estimation, which is apparently undesirable. By using the higher-order kernel, Lemma (ref) below implies that \begin{equation*} \sup_{x}\left\vert \hat{\pi}\left( x\right) -\pi \left( x\right) \right\vert =O_{p}\left( \eta _{1}\right) . \end{equation*} Therefore, Assumption E.3.(i) ensures the asymptotic negligibility of the first-step estimation error relative to the subsequent local linear estimation. If $h_{1l}$ follows a rate $O\left( n^{-1\left/ \left( 2s+\dim \left( X^{C}\right) \right) \right. }\right) $ that optimally balances the squared bias and the variance, the first-step rate of convergence will become $O_{p}\left( \left( \ln n\right) ^{1/2}n^{-s\left/ \left( 2s+\dim \left( X^{C}\right) \right) \right. }\right) $. Further suppose that $h_{3}$ follows a rule-of-thumb rate $O\left( n^{-1/5}\right) $, then Assumption E.3.(i) will reduce to $s>2\dim \left( X^{C}\right) $. Note that under the supposed rates of $h_{1l}$ and $h_{3}$, Assumption E.3.(ii), (iii), and (iv) will reduce to $s>2\dim \left( X^{C}\right) $, $s>\left( 3/4\right) \dim \left( X^{C}\right) $, and $s>\left( 3/2\right) \dim \left( X^{C}\right) $, respectively, which are all implied by Assumption E.3.(i). Assumption E.2.(i) imposes a weak rate condition on the slope estimation in the second step. The $\sqrt{n}$-consistency of the weighted pairwise difference estimation ((ref)) has been established by ahn1993semiparametric. Actually, most semiparametric estimators for sample selection models satisfy Assumption E.2.(i) as well, such as the Robinson-type partialling-out estimator pagan1999nonparametric , the symmetry-based pairwise difference estimator chen2010semiparametric, and the integrated least squares estimator pan2022semiparametric, among others. \subsection{Proof of Theorem (ref)} We first develop the asymptotic normality of the local linear estimator $ \left( \hat{g}_{d}\left( p\right) ,\hat{g}_{d}^{\left( 1\right) }\left( p\right) \right) $ for $d=0,1$. To this end, we define the infeasible estimator as \begin{equation*} \left( \begin{array}{c} \tilde{g}_{d}\left( p\right) \\ \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) =\left[ \sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ P_{i}-p \end{array} \right) \left( \begin{array}{c} 1 \\ P_{i}-p \end{array} \right) ^{\prime }\right] ^{-1}\left[ \sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ P_{i}-p \end{array} \right) \left( Y_{i}-X_{i}^{\prime }\beta _{d}\right) \right] , \end{equation*} where \begin{equation*} w_{di}\left( p\right) =1\left\{ D_{i}=d\right\} \frac{1}{h_{3}}k_{3}\left( \frac{P_{i}-p}{h_{3}}\right) . \end{equation*} We will analyze $\left( \tilde{g}_{d}\left( p\right) ,\tilde{g}_{d}^{\left( 1\right) }\left( p\right) \right) $ and $\left( \hat{g}_{d}\left( p\right) , \hat{g}_{d}^{\left( 1\right) }\left( p\right) \right) -\left( \tilde{g} _{d}\left( p\right) ,\tilde{g}_{d}^{\left( 1\right) }\left( p\right) \right) $ in order. The asymptotic normality of $\left( \tilde{g}_{d}\left( p\right) ,\tilde{g} _{d}^{\left( 1\right) }\left( p\right) \right) $ can be established by standard arguments for kernel regression li2007nonparametric, except for the presence of the treatment status indicator $1\left\{ D_{i}=d\right\} $ in our case. For completeness, we provide a proof for the asymptotic normality of $\left( \tilde{g}_{d}\left( p\right) ,\tilde{g}_{d}^{\left( 1\right) }\left( p\right) \right) $ in Lemma (ref). In the proof, we obtain an asymptotically linear representation of $\left( \tilde{g}_{d}\left( p\right) ,\tilde{g} _{d}^{\left( 1\right) }\left( p\right) \right) $ in ((ref)): \begin{equation} J_{n}\left[ \left( \begin{array}{c} \tilde{g}_{d}\left( p\right) \\ \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} g_{d}\left( p\right) \\ g_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} \frac{\kappa _{2}}{2}g_{d}^{\left( 2\right) }\left( p\right) h_{3}^{2} \\ 0 \end{array} \right) \right] =\Gamma _{d}J_{n}\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) V_{di}+o_{p}\left( 1\right) , \end{equation} where \begin{eqnarray*} \Gamma _{d} &=&\left( \begin{array}{cc} 1\left/ \left[ \delta _{d}\left( p\right) f_{P}\left( p\right) \right] \right. , & 0 \\ 0, & 1\left/ \left[ \kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) \right] \right. \end{array} \right) , \\ J_{n} &=&\left( \begin{array}{cc} \sqrt{nh_{3}} & 0 \\ 0 & \sqrt{nh_{3}^{3}} \end{array} \right) . \end{eqnarray*} We next consider $J_{n}\left[ \left( \hat{g}_{d}\left( p\right) ,\hat{g} _{d}^{\left( 1\right) }\left( p\right) \right) -\left( \tilde{g}_{d}\left( p\right) ,\tilde{g}_{d}^{\left( 1\right) }\left( p\right) \right) \right] $. For $r=0,1,2$, we denote \begin{eqnarray*} \hat{B}_{r} &=&\frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P}_{i}-p\right) ^{r}, \\ B_{r} &=&\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) ^{r}, \end{eqnarray*} and \begin{equation*} \hat{Q}=\left( \begin{array}{cc} \hat{B}_{0}, & \hat{B}_{1} \\ \hat{B}_{1}, & \hat{B}_{2} \end{array} \right) ,\textQ=\left( \begin{array}{cc} B_{0}, & B_{1} \\ B_{1}, & B_{2} \end{array} \right) , \end{equation*} then \begin{eqnarray*} \left( \begin{array}{c} \hat{g}_{d}\left( p\right) \\ \hat{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) &=&\hat{Q}^{-1}\frac{1}{n}\sum_{i}\hat{w}_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \hat{P}_{i}-p \end{array} \right) \left( Y_{i}-X_{i}^{\prime }\hat{\beta}_{d}\right) \\ &=&\frac{1}{\left\vert \hat{Q}\right\vert }\frac{1}{n}\sum_{i}\hat{w} _{di}\left( p\right) \left( \begin{array}{c} \hat{B}_{2}-\left( \hat{P}_{i}-p\right) \hat{B}_{1} \\ -\hat{B}_{1}+\left( \hat{P}_{i}-p\right) \hat{B}_{0} \end{array} \right) \left( Y_{i}-X_{i}^{\prime }\hat{\beta}_{d}\right) , \\ \left( \begin{array}{c} \tilde{g}_{d}\left( p\right) \\ \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) &=&\frac{1}{\left\vert Q\right\vert }\frac{1}{n}\sum_{i}w_{di}\left( p\right) \left( \begin{array}{c} B_{2}-\left( P_{i}-p\right) B_{1} \\ -B_{1}+\left( P_{i}-p\right) B_{0} \end{array} \right) \left( Y_{i}-X_{i}^{\prime }\beta _{d}\right) , \end{eqnarray*} where \begin{equation*} \left\vert \hat{Q}\right\vert =\hat{B}_{0}\hat{B}_{2}-\hat{B}_{1}^{2},\text\left\vert Q\right\vert =B_{0}B_{2}-B_{1}^{2}. \end{equation*} For $r=0,1$, Further denote \begin{eqnarray*} \hat{L}_{r} &=&\frac{1}{n}\sum_{i}\hat{w}_{di}\left( p\right) \left[ \hat{B} _{r+1}-\left( \hat{P}_{i}-p\right) \hat{B}_{r}\right] \left( Y_{i}-X_{i}^{\prime }\hat{\beta}_{d}\right) , \\ L_{r} &=&\frac{1}{n}\sum_{i}w_{di}\left( p\right) \left[ B_{r+1}-\left( P_{i}-p\right) B_{r}\right] \left( Y_{i}-X_{i}^{\prime }\beta _{d}\right) , \end{eqnarray*} then we have \begin{equation*} \left( \begin{array}{c} \hat{g}_{d}\left( p\right) \\ \hat{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) =\frac{1}{\left\vert \hat{Q}\right\vert }\left( \begin{array}{c} \hat{L}_{1} \\ -\hat{L}_{0} \end{array} \right) ,\text\left( \begin{array}{c} \tilde{g}_{d}\left( p\right) \\ \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) =\frac{1}{\left\vert Q\right\vert }\left( \begin{array}{c} L_{1} \\ -L_{0} \end{array} \right) . \end{equation*} It follows from Lemma (ref) that \begin{eqnarray*} \frac{L_{1}}{\left\vert Q\right\vert } &=&g_{d}\left( p\right) +O_{p}\left( h_{3}^{2}+\frac{1}{\sqrt{nh_{3}}}\right) =g_{d}\left( p\right) +o_{p}\left( h_{3}\right) , \\ \frac{L_{0}}{\left\vert Q\right\vert } &=&-g_{d}^{\left( 1\right) }\left( p\right) +O_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) =-g_{d}^{\left( 1\right) }\left( p\right) +o_{p}\left( 1\right) . \end{eqnarray*} Therefore, \begin{eqnarray} \hat{g}_{d}\left( p\right) -\tilde{g}_{d}\left( p\right) &=&\frac{\hat{L}_{1} }{\left\vert \hat{Q}\right\vert }-\frac{L_{1}}{\left\vert Q\right\vert }= \left[ \left( \hat{L}_{1}-L_{1}\right) -\left( \left\vert \hat{Q}\right\vert -\left\vert Q\right\vert \right) \frac{L_{1}}{\left\vert Q\right\vert } \right] \frac{1}{\left\vert \hat{Q}\right\vert } \notag \\ &=&\frac{\left( \hat{L}_{1}-L_{1}\right) -\left( \left\vert \hat{Q} \right\vert -\left\vert Q\right\vert \right) g_{d}\left( p\right) +o_{p}\left( h_{3}\left( \left\vert \hat{Q}\right\vert -\left\vert Q\right\vert \right) \right) }{\left\vert \hat{Q}\right\vert }, \\ \hat{g}_{d}^{\left( 1\right) }\left( p\right) -\tilde{g}_{d}^{\left( 1\right) }\left( p\right) &=&-\frac{\left( \hat{L}_{0}-L_{0}\right) +\left( \left\vert \hat{Q}\right\vert -\left\vert Q\right\vert \right) g_{d}^{\left( 1\right) }\left( p\right) +o_{p}\left( \left\vert \hat{Q}\right\vert -\left\vert Q\right\vert \right) }{\left\vert \hat{Q}\right\vert }. \end{eqnarray} Denote \begin{equation} V_{di}\left( p\right) =Y_{i}-X_{i}^{\prime }\beta _{d}-g_{d}\left( p\right) =V_{di}+g_{d}\left( P_{i}\right) -g_{d}\left( p\right) , \end{equation} which satisfies $E\left[ \left. V_{di}\left( p\right) \right\vert X_{i},D_{i}=d\right] =g_{d}\left( P_{i}\right) -g_{d}\left( p\right) $ by definition, and denote \begin{eqnarray*} \hat{C}_{r} &=&\frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P}_{i}-p\right) ^{r}V_{di}\left( p\right) , \\ C_{r} &=&\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) ^{r}V_{di}\left( p\right) , \\ \hat{G}_{r} &=&\frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P}_{i}-p\right) ^{r}X_{i}, \end{eqnarray*} for $r=0,1$. By several calculations, \begin{eqnarray*} \left( \hat{L}_{1}-L_{1}\right) -\left( \left\vert \hat{Q}\right\vert -\left\vert Q\right\vert \right) g_{d}\left( p\right) &=&\left( \hat{C}_{0} \hat{B}_{2}-C_{0}B_{2}\right) -\left( \hat{C}_{1}\hat{B}_{1}-C_{1}B_{1} \right) \\ &&-\left( \hat{\beta}_{d}-\beta _{d}\right) ^{\prime }\left( \hat{B}_{2}\hat{ G}_{0}-\hat{B}_{1}\hat{G}_{1}\right) , \\ \hat{L}_{0}-L_{0} &=&\left( \hat{C}_{0}\hat{B}_{1}-C_{0}B_{1}\right) -\left( \hat{C}_{1}\hat{B}_{0}-C_{1}B_{0}\right) \\ &&-\left( \hat{\beta}_{d}-\beta _{d}\right) ^{\prime }\left( \hat{B}_{1}\hat{ G}_{0}-\hat{B}_{0}\hat{G}_{1}\right) , \\ \left\vert \hat{Q}\right\vert -\left\vert Q\right\vert &=&\left( \hat{B}_{0} \hat{B}_{2}-B_{0}B_{2}\right) -\left( \hat{B}_{1}^{2}-B_{1}^{2}\right) . \end{eqnarray*} It follows from Lemmas (ref)-(ref) that \begin{eqnarray*} \hat{C}_{0}\hat{B}_{2}-C_{0}B_{2} &=&\left( \hat{C}_{0}-C_{0}\right) \left( \hat{B}_{2}-B_{2}\right) +C_{0}\left( \hat{B}_{2}-B_{2}\right) +\left( \hat{C }_{0}-C_{0}\right) B_{2} \\ &=&O_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) o_{p}\left( \sqrt{\frac{h_{3}}{ n}}\right) +o_{p}\left( h_{3}\right) o_{p}\left( \sqrt{\frac{h_{3}}{n}} \right) \\ &&+\left( \hat{C}_{0}-C_{0}\right) \left[ \kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) h_{3}^{2}+o_{p}\left( h_{3}^{3}\right) \right] \\ &=&\frac{\kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \left( D_{i}-P_{i}\right) \\ &&+o_{p}\left( \sqrt{\frac{h_{3}^{3}}{n}}\right) , \end{eqnarray*} \begin{eqnarray*} \hat{C}_{1}\hat{B}_{1}-C_{1}B_{1} &=&\left( \hat{C}_{1}-C_{1}\right) \left( \hat{B}_{1}-B_{1}\right) +\left( \hat{C}_{1}-C_{1}\right) B_{1}+C_{1}\left( \hat{B}_{1}-B_{1}\right) \\ &=&O_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) o_{p}\left( \frac{1}{\sqrt{ nh_{3}}}\right) +O_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) O_{p}\left( h_{3}^{2}\right) +O_{p}\left( h_{3}^{2}\right) o_{p}\left( \frac{1}{\sqrt{ nh_{3}}}\right) \\ &=&o_{p}\left( \sqrt{\frac{h_{3}^{3}}{n}}\right) , \end{eqnarray*} \begin{eqnarray*} \left( \hat{\beta}_{d}-\beta _{d}\right) ^{\prime }\left( \hat{B}_{2}\hat{G} _{0}-\hat{B}_{1}\hat{G}_{1}\right) &\leq &\left\Vert \hat{\beta}_{d}-\beta _{d}\right\Vert \left[ \begin{array}{c} \left( \left\vert \hat{B}_{2}-B_{2}\right\vert +\left\vert B_{2}\right\vert \right) \left\Vert \hat{G}_{0}\right\Vert \\ +\left( \left\vert \hat{B}_{1}-B_{1}\right\vert +\left\vert B_{1}\right\vert \right) \left\Vert \hat{G}_{1}\right\Vert \end{array} \right] \\ &=&O_{p}\left( \frac{1}{\sqrt{n}}\right) \left[ \begin{array}{c} \left( o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) +O_{p}\left( h_{3}^{2}\right) \right) O_{p}\left( 1\right) \\ +\left( o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) +O_{p}\left( h_{3}^{2}\right) \right) O_{p}\left( h_{3}\right) \end{array} \right] \\ &=&o_{p}\left( \sqrt{\frac{h_{3}^{3}}{n}}\right) , \end{eqnarray*} \begin{eqnarray*} \hat{C}_{0}\hat{B}_{1}-C_{0}B_{1} &=&\left( \hat{C}_{0}-C_{0}\right) \left( \hat{B}_{1}-B_{1}\right) +C_{0}\left( \hat{B}_{1}-B_{1}\right) +\left( \hat{C }_{0}-C_{0}\right) B_{1} \\ &=&O_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) o_{p}\left( \frac{1}{\sqrt{ nh_{3}}}\right) +o_{p}\left( h_{3}\right) o_{p}\left( \frac{1}{\sqrt{nh_{3}}} \right) +O_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) O_{p}\left( h_{3}^{2}\right) \\ &=&o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) , \end{eqnarray*} \begin{eqnarray*} \hat{C}_{1}\hat{B}_{0}-C_{1}B_{0} &=&\left( \hat{C}_{1}-C_{1}\right) \left( \hat{B}_{0}-B_{0}\right) +C_{1}\left( \hat{B}_{0}-B_{0}\right) +\left( \hat{C }_{1}-C_{1}\right) B_{0} \\ &=&O_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) o_{p}\left( \frac{1}{\sqrt{ nh_{3}^{3}}}\right) +O_{p}\left( h_{3}^{2}\right) o_{p}\left( \frac{1}{\sqrt{ nh_{3}^{3}}}\right) \\ &&+\left( \hat{C}_{1}-C_{1}\right) \left[ \delta _{d}\left( p\right) f_{P}\left( p\right) +o_{p}\left( h_{3}\right) \right] \\ &=&\frac{\delta _{d}\left( p\right) f_{P}\left( p\right) g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) \left[ k_{3}\left( \frac{P_{i}-p}{h_{3}}\right) +k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \frac{P_{i}-p}{h_{3}}\right] \\ &&\cdot \frac{P_{i}-p}{h_{3}}\left( D_{i}-P_{i}\right) +o_{p}\left( \sqrt{ \frac{h_{3}}{n}}\right) , \end{eqnarray*} \begin{eqnarray*} \left( \hat{\beta}_{d}-\beta _{d}\right) ^{\prime }\left( \hat{B}_{1}\hat{G} _{0}-\hat{B}_{0}\hat{G}_{1}\right) &\leq &\left\Vert \hat{\beta}_{d}-\beta _{d}\right\Vert \left[ \begin{array}{c} \left( \left\vert \hat{B}_{1}-B_{1}\right\vert +\left\vert B_{1}\right\vert \right) \left\Vert \hat{G}_{0}\right\Vert \\ +\left( \left\vert \hat{B}_{0}-B_{0}\right\vert +\left\vert B_{0}\right\vert \right) \left\Vert \hat{G}_{1}\right\Vert \end{array} \right] \\ &=&O_{p}\left( \frac{1}{\sqrt{n}}\right) \left[ \begin{array}{c} \left( o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) +O_{p}\left( h_{3}^{2}\right) \right) O_{p}\left( 1\right) \\ +\left( o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) +O_{p}\left( 1\right) \right) O_{p}\left( h_{3}\right) \end{array} \right] \\ &=&o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) , \end{eqnarray*} \begin{eqnarray*} \hat{B}_{0}\hat{B}_{2}-B_{0}B_{2} &=&\left( \hat{B}_{0}-B_{0}\right) \left( \hat{B}_{2}-B_{2}\right) +B_{0}\left( \hat{B}_{2}-B_{2}\right) +\left( \hat{B }_{0}-B_{0}\right) B_{2} \\ &=&o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) o_{p}\left( \sqrt{\frac{ h_{3}}{n}}\right) +O_{p}\left( 1\right) o_{p}\left( \sqrt{\frac{h_{3}}{n}} \right) +o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) O_{p}\left( h_{3}^{2}\right) \\ &=&o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) , \end{eqnarray*} and \begin{eqnarray*} \hat{B}_{1}^{2}-B_{1}^{2} &=&\left( \hat{B}_{1}-B_{1}\right) \left[ \left( \hat{B}_{1}-B_{1}\right) +2B_{1}\right] \\ &=&o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) \left[ o_{p}\left( \frac{1}{ \sqrt{nh_{3}}}\right) +O_{p}\left( h_{3}^{2}\right) \right] \\ &=&o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) . \end{eqnarray*} Therefore, \begin{eqnarray} \left( \hat{L}_{1}-L_{1}\right) -\left( \left\vert \hat{Q}\right\vert -\left\vert Q\right\vert \right) g_{d}\left( p\right) &=&\frac{\kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \notag \\ &&\cdot \left( P_{i}-p\right) \left( D_{i}-P_{i}\right) +o_{p}\left( \sqrt{ \frac{h_{3}^{3}}{n}}\right) , \end{eqnarray} \begin{eqnarray} \hat{L}_{0}-L_{0} &=&-\frac{\delta _{d}\left( p\right) f_{P}\left( p\right) g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) \left[ k_{3}\left( \frac{P_{i}-p}{h_{3}}\right) +k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \frac{P_{i}-p}{h_{3}}\right] \notag \\ &&\cdot \frac{P_{i}-p}{h_{3}}\left( D_{i}-P_{i}\right) +o_{p}\left( \sqrt{ \frac{h_{3}}{n}}\right) , \end{eqnarray} and \begin{equation} \left\vert \hat{Q}\right\vert -\left\vert Q\right\vert =o_{p}\left( \sqrt{ \frac{h_{3}}{n}}\right) . \end{equation} In addition, it follows from Lemma (ref) that \begin{equation*} \left\vert Q\right\vert =\kappa _{2}\delta _{d}^{2}\left( p\right) f_{P}^{2}\left( p\right) h_{3}^{2}+o_{p}\left( h_{3}^{2}\right) . \end{equation*} Hence, \begin{equation*} \frac{\left\vert \hat{Q}\right\vert }{\left\vert Q\right\vert }-1=\frac{ \left\vert \hat{Q}\right\vert -\left\vert Q\right\vert }{\left\vert Q\right\vert }=\frac{o_{p}\left( 1\left/ \sqrt{nh_{3}^{3}}\right. \right) }{ \kappa _{2}\delta _{d}^{2}\left( p\right) f_{P}^{2}\left( p\right) +o_{p}\left( 1\right) }=o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) =o_{p}\left( 1\right) , \end{equation*} and \begin{equation} \frac{1}{\left\vert \hat{Q}\right\vert }=\frac{1}{\left\vert Q\right\vert } \left( 1+o_{p}\left( 1\right) \right) =\frac{1}{\kappa _{2}\delta _{d}^{2}\left( p\right) f_{P}^{2}\left( p\right) h_{3}^{2}}\left( 1+o_{p}\left( 1\right) \right) . \end{equation} Substituting ((ref))-((ref)) into ((ref))-((ref)) yields that \begin{eqnarray*} \hat{g}_{d}\left( p\right) -\tilde{g}_{d}\left( p\right) &=&\left( \frac{ g_{d}^{\left( 1\right) }\left( p\right) }{\delta _{d}\left( p\right) f_{P}\left( p\right) }\right) \frac{1}{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) \frac{P_{i}-p}{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{ P_{i}-p}{h_{3}}\right) \left( D_{i}-P_{i}\right) \\ &&+o_{p}\left( \sqrt{\frac{1}{nh_{3}}}\right) , \\ \hat{g}_{d}^{\left( 1\right) }\left( p\right) -\tilde{g}_{d}^{\left( 1\right) }\left( p\right) &=&\left( \frac{g_{d}^{\left( 1\right) }\left( p\right) }{\kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) } \right) \frac{1}{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) \left[ \frac{ 1}{h_{3}}k_{3}\left( \frac{P_{i}-p}{h_{3}}\right) \right. \\ &&\left. +\frac{P_{i}-p}{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{ P_{i}-p}{h_{3}}\right) \right] \frac{P_{i}-p}{h_{3}^{2}}\left( D_{i}-P_{i}\right) +o_{p}\left( \sqrt{\frac{1}{nh_{3}^{3}}}\right) . \end{eqnarray*} Further combining with ((ref)), we obtain \begin{equation} J_{n}\left[ \left( \begin{array}{c} \hat{g}_{d}\left( p\right) \\ \hat{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} g_{d}\left( p\right) \\ g_{d}^{\left( 1\right) }\left( p\right) \end{array} \right)-\left( \begin{array}{c} \frac{\kappa _{2}}{2}g_{d}^{\left( 2\right) }\left( p\right) h_{3}^{2} \\ 0 \end{array} \right) \right] =\Gamma _{d}A_{dn}\left( p\right) +o_{p}\left( 1\right) , \end{equation} where \begin{eqnarray*} A_{dn}\left( p\right) &=&J_{n}\frac{1}{n}\sum_{i=1}^{n}\left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) \left[ w_{di}\left( p\right) V_{di}+\left( \begin{array}{c} \Lambda _{di}\left( p\right) \\ \Lambda _{di}\left( p\right) +\Upsilon _{di}\left( p\right) \end{array} \right) \left( D_{i}-P_{i}\right) \right] , \\ \Lambda _{di}\left( p\right) &=&g_{d}^{\left( 1\right) }\left( p\right) \delta _{d}\left( P_{i}\right) \frac{P_{i}-p}{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) , \\ \Upsilon _{di}\left( p\right) &=&g_{d}^{\left( 1\right) }\left( p\right) \delta _{d}\left( P_{i}\right) \frac{1}{h_{3}}k_{3}\left( \frac{P_{i}-p}{ h_{3}}\right) . \end{eqnarray*} Since \begin{eqnarray*} E\left[ \left. w_{di}\left( p\right) V_{di}\right\vert P_{i}\right] &=&\frac{ 1}{h_{3}}k_{3}\left( \frac{P_{i}-p}{h_{3}}\right) \delta _{d}\left( P_{i}\right) E\left[ \left. V_{di}\right\vert P_{i},D_{i}=d\right] =0, \\ E\left[ \left. D_{i}-P_{i}\right\vert P_{i}\right] &=&0, \\ E\left[ \left. w_{di}\left( p\right) V_{di}\left( D_{i}-P_{i}\right) \right\vert P_{i}\right] &=&\left( d-P_{i}\right) E\left[ \left. w_{di}\left( p\right) V_{di}\right\vert P_{i}\right] =0, \end{eqnarray*} we have{ \begin{eqnarray*} E\left[ A_{dn}\left( p\right) \right] &=&0, \\ Var\left( A_{dn\left( 1\right) }\left( p\right) \right) &=&h_{3}\left( E \left[ w_{di}^{2}\left( p\right) V_{di}^{2}\right] +E\left[ \Lambda _{di}^{2}\left( p\right) \left( D_{i}-P_{i}\right) ^{2}\right] \right) \\ &=&\kappa \delta _{d}\left( p\right) f_{P}\left( p\right) \sigma _{V_{d}}^{2}\left( p\right) +\kappa _{22}^{\left( 1\right) }p\left( 1-p\right) \delta _{d}^{2}\left( p\right) f_{P}\left( p\right) \left( g_{d}^{\left( 1\right) }\left( p\right) \right) ^{2}+o\left( 1\right) , \\ Var\left( A_{dn\left( 2\right) }\left( p\right) \right) &=&h_{3}^{3}\left( E \left[ \frac{\left( P_{i}-p\right) ^{2}}{h_{3}^{4}}w_{di}^{2}\left( p\right) V_{di}^{2}\right] +E\left[ \frac{\left( P_{i}-p\right) ^{2}}{h_{3}^{4}} \left( \Lambda _{di}\left( p\right) +\Upsilon _{di}\left( p\right) \right) ^{2}\left( D_{i}-P_{i}\right) ^{2}\right] \right) \\ &=&\kappa _{22}\delta _{d}\left( p\right) f_{P}\left( p\right) \sigma _{V_{d}}^{2}\left( p\right) +\left( \kappa _{24}^{\left( 1\right) }-2\kappa _{22}\right) p\left( 1-p\right) \delta _{d}^{2}\left( p\right) f_{P}\left( p\right) \left( g_{d}^{\left( 1\right) }\left( p\right) \right) ^{2}+o\left( 1\right) , \\ E\left[ A_{dn\left( 1\right) }\left( p\right) A_{dn\left( 2\right) }\left( p\right) \right] &=&h_{3}^{2}\left( E\left[ \frac{P_{i}-p}{h_{3}^{2}} w_{di}^{2}\left( p\right) V_{di}^{2}\right] +E\left[ \frac{P_{i}-p}{h_{3}^{2} }\left( \Lambda _{di}^{2}\left( p\right) +\Lambda _{di}\left( p\right) \Upsilon _{di}\left( p\right) \right) \left( D_{i}-P_{i}\right) ^{2}\right] \right) \\ &=&O\left( h_{3}\right) =o\left( 1\right) , \end{eqnarray*} }where \begin{eqnarray*} \kappa _{22}^{\left( 1\right) } &=&\int \left( k_{3}^{\left( 1\right) }\left( u\right) \right) ^{2}u^{2}du, \\ \kappa _{24}^{\left( 1\right) } &=&\int \left( k_{3}^{\left( 1\right) }\left( u\right) \right) ^{2}u^{4}du. \end{eqnarray*} It then follows from Liapunov's central limit theorem that \begin{equation*} A_{dn}\left( p\right) \rightarrow N\left( 0,\text{diag}\left( \begin{array}{c} \kappa \delta _{d}\left( p\right) f_{P}\left( p\right) \sigma _{V_{d}}^{2}\left( p\right) +\kappa _{22}^{\left( 1\right) }p\left( 1-p\right) \delta _{d}^{2}\left( p\right) f_{P}\left( p\right) \left( g_{d}^{\left( 1\right) }\left( p\right) \right) ^{2} \\ \kappa _{22}\delta _{d}\left( p\right) f_{P}\left( p\right) \sigma _{V_{d}}^{2}\left( p\right) +\left( \kappa _{24}^{\left( 1\right) }-2\kappa _{22}\right) p\left( 1-p\right) \delta _{d}^{2}\left( p\right) f_{P}\left( p\right) \left( g_{d}^{\left( 1\right) }\left( p\right) \right) ^{2} \end{array} \right) \right) , \end{equation*} and thus from ((ref)) that \begin{equation*} J_{n}\left[ \left( \begin{array}{c} \hat{g}_{d}\left( p\right) \\ \hat{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} g_{d}\left( p\right) \\ g_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} \frac{\kappa _{2}}{2}g_{d}^{\left( 2\right) }\left( p\right) h_{3}^{2} \\ 0 \end{array} \right) \right] \rightarrow N\left( 0,\Sigma _{d}\left( p\right) \right) , \end{equation*} where \begin{equation*} \Sigma _{d}\left( p\right) =\left( \begin{array}{cc} \frac{\kappa \sigma _{V_{d}}^{2}\left( p\right) +\kappa _{22}^{\left( 1\right) }p\left( 1-p\right) \delta _{d}\left( p\right) \left( g_{d}^{\left( 1\right) }\left( p\right) \right) ^{2}}{\delta _{d}\left( p\right) f_{P}\left( p\right) } & 0 \\ 0 & \frac{\kappa _{22}\sigma _{V_{d}}^{2}\left( p\right) +\left( \kappa _{24}^{\left( 1\right) }-2\kappa _{22}\right) p\left( 1-p\right) \delta _{d}\left( p\right) \left( g_{d}^{\left( 1\right) }\left( p\right) \right) ^{2}}{\kappa _{2}^{2}\delta _{d}\left( p\right) f_{P}\left( p\right) } \end{array} \right) . \end{equation*} Moreover, we can deduce the joint asymptotic normality of $\left( \hat{g} _{1}\left( p\right) ,\hat{g}_{1}^{\left( 1\right) }\left( p\right) \right) $ and $\left( \hat{g}_{0}\left( p\right) ,\hat{g}_{0}^{\left( 1\right) }\left( p\right) \right) $ from the asymptotically linear representation ((ref)). Due to \begin{eqnarray*} E\left[ A_{1n\left( 1\right) }\left( p\right) A_{0n\left( 1\right) }\left( p\right) \right] &=&h_{3}E\left[ \Lambda _{1i}\left( p\right) \Lambda _{0i}\left( p\right) \left( D_{i}-P_{i}\right) ^{2}\right] =\kappa _{22}^{\left( 1\right) }g_{1}^{\left( 1\right) }\left( p\right) g_{0}^{\left( 1\right) }\left( p\right) p^{2}\left( 1-p\right) ^{2}f_{P}\left( p\right) +o\left( 1\right) , \\ E\left[ A_{1n\left( 1\right) }\left( p\right) A_{0n\left( 2\right) }\left( p\right) \right] &=&h_{3}^{2}E\left[ \frac{P_{i}-p}{h_{3}^{2}}\Lambda _{1i}\left( p\right) \left( \Lambda _{0i}\left( p\right) +\Upsilon _{0i}\left( p\right) \right) \left( D_{i}-P_{i}\right) ^{2}\right] =O\left( h_{3}\right) =o\left( 1\right) , \\ E\left[ A_{1n\left( 2\right) }\left( p\right) A_{0n\left( 1\right) }\left( p\right) \right] &=&h_{3}^{2}E\left[ \frac{P_{i}-p}{h_{3}^{2}}\left( \Lambda _{1i}\left( p\right) +\Upsilon _{1i}\left( p\right) \right) \Lambda _{0i}\left( p\right) \left( D_{i}-P_{i}\right) ^{2}\right] =O\left( h_{3}\right) =o\left( 1\right) , \\ E\left[ A_{1n\left( 2\right) }\left( p\right) A_{0n\left( 2\right) }\left( p\right) \right] &=&h_{3}^{3}E\left[ \frac{\left( P_{i}-p\right) ^{2}}{ h_{3}^{4}}\left( \Lambda _{1i}\left( p\right) +\Upsilon _{1i}\left( p\right) \right) \left( \Lambda _{0i}\left( p\right) +\Upsilon _{0i}\left( p\right) \right) \left( D_{i}-P_{i}\right) ^{2}\right] \\ &=&\left( \kappa _{24}^{\left( 1\right) }-2\kappa _{22}\right) g_{1}^{\left( 1\right) }\left( p\right) g_{0}^{\left( 1\right) }\left( p\right) p^{2}\left( 1-p\right) ^{2}f_{P}\left( p\right) +o\left( 1\right) , \end{eqnarray*} it follows from Liapunov's central limit theorem that \begin{equation} \left( \begin{array}{cc} J_{n} & 0 \\ 0 & J_{n} \end{array} \right) \left[ \left( \begin{array}{c} \hat{g}_{1}\left( p\right) \\ \hat{g}_{1}^{\left( 1\right) }\left( p\right) \\ \hat{g}_{0}\left( p\right) \\ \hat{g}_{0}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} g_{1}\left( p\right) \\ g_{1}^{\left( 1\right) }\left( p\right) \\ g_{0}\left( p\right) \\ g_{0}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} \frac{\kappa _{2}}{2}g_{1}^{\left( 2\right) }\left( p\right) h_{3}^{2} \\ 0 \\ \frac{\kappa _{2}}{2}g_{0}^{\left( 2\right) }\left( p\right) h_{3}^{2} \\ 0 \end{array} \right) \right] \rightarrow N\left( 0,\Sigma \left( p\right) \right) , \end{equation} where \begin{equation*} \Sigma \left( p\right) =\left( \begin{array}{cc} \Sigma _{1}\left( p\right) & \Sigma _{10}\left( p\right) \\ \Sigma _{10}\left( p\right) & \Sigma _{0}\left( p\right) \end{array} \right) , \end{equation*} and \begin{equation*} \Sigma _{10}\left( p\right) =\left( \begin{array}{cc} \frac{\kappa _{22}^{\left( 1\right) }p\left( 1-p\right) g_{1}^{\left( 1\right) }\left( p\right) g_{0}^{\left( 1\right) }\left( p\right) }{f_{P}\left( p\right) } & 0 \\ 0 & \frac{\left( \kappa _{24}^{\left( 1\right) }-2\kappa _{22}\right) p\left( 1-p\right) g_{1}^{\left( 1\right) }\left( p\right) g_{0}^{\left( 1\right) }\left( p\right) }{\kappa _{2}^{2}f_{P}\left( p\right) } \end{array} \right) . \end{equation*} Now the asymptotic normality of the estimators for MTE and relevant causal parameters can be readily deduced from ((ref)). For the MTE estimator, we have \begin{eqnarray*} &&\sqrt{nh_{3}^{3}}\left[ \hat{\Delta}^{\text{MTE}}\left( x,v\right) -\Delta ^{\text{MTE}}\left( x,v\right) \right] \\ &=&x^{\prime }\left[ \sqrt{nh_{3}^{3}}\left( \hat{\beta}_{1}-\beta _{1}\right) -\sqrt{nh_{3}^{3}}\left( \hat{\beta}_{0}-\beta _{0}\right) \right] \\ &&+\left[ \sqrt{nh_{3}^{3}}\left( \hat{g}_{1}\left( v\right) -g_{1}\left( v\right) \right) -\sqrt{nh_{3}^{3}}\left( \hat{g}_{0}\left( v\right) -g_{0}\left( v\right) \right) \right] \\ &&+v\sqrt{nh_{3}^{3}}\left( \hat{g}_{1}^{\left( 1\right) }\left( v\right) -g_{1}^{\left( 1\right) }\left( v\right) \right) +\left( 1-v\right) \sqrt{ nh_{3}^{3}}\left( \hat{g}_{0}^{\left( 1\right) }\left( v\right) -g_{0}^{\left( 1\right) }\left( v\right) \right) \\ &=&O_{p}\left( \sqrt{h_{3}^{3}}\right) +O\left( \sqrt{nh_{3}^{7}}\right) +O_{p}\left( h_{3}\right) \\ &&+v\sqrt{nh_{3}^{3}}\left( \hat{g}_{1}^{\left( 1\right) }\left( v\right) -g_{1}^{\left( 1\right) }\left( v\right) \right) +\left( 1-v\right) \sqrt{ nh_{3}^{3}}\left( \hat{g}_{0}^{\left( 1\right) }\left( v\right) -g_{0}^{\left( 1\right) }\left( v\right) \right) \\ &\rightarrow &N\left( 0,\sigma _{\text{MTE}}^{2}\left( v\right) \right) , \end{eqnarray*} where \begin{equation} \sigma _{\text{MTE}}^{2}\left( v\right) =\frac{\kappa _{22}\left[ v\sigma _{V_{1}}^{2}\left( v\right) +\left( 1-v\right) \sigma _{V_{0}}^{2}\left( v\right) \right] +\left( \kappa _{24}^{\left( 1\right) }-2\kappa _{22}\right) v\left( 1-v\right) \left[ vg_{1}^{\left( 1\right) }\left( v\right) +\left( 1-v\right) g_{0}^{\left( 1\right) }\left( v\right) \right] ^{2}}{\kappa _{2}^{2}f_{P}\left( v\right) }. \end{equation} For the relevant causal parameter estimators, we have \begin{eqnarray*} &&\sqrt{nh_{3}}\left[ \hat{\Delta}^{\text{ATE}}\left( x\right) -\Delta ^{ \text{ATE}}\left( x\right) -\frac{\kappa _{2}}{2}\left( g_{1}^{\left( 2\right) }\left( 1\right) -g_{0}^{\left( 2\right) }\left( 0\right) \right) h_{3}^{2}\right] \\ &=&x^{\prime }\left[ \sqrt{nh_{3}}\left( \hat{\beta}_{1}-\beta _{1}\right) - \sqrt{nh_{3}}\left( \hat{\beta}_{0}-\beta _{0}\right) \right] \\ &&+\sqrt{nh_{3}}\left[ \hat{g}_{1}\left( 1\right) -g_{1}\left( 1\right) - \frac{\kappa _{2}}{2}g_{1}^{\left( 2\right) }\left( 1\right) h_{3}^{2}\right] -\sqrt{nh_{3}}\left[ \hat{g}_{0}\left( 0\right) -g_{0}\left( 0\right) -\frac{ \kappa _{2}}{2}g_{0}^{\left( 2\right) }\left( 0\right) h_{3}^{2}\right] \\ &=&O_{p}\left( \sqrt{h_{3}}\right) +\sqrt{nh_{3}}\left[ \hat{g}_{1}\left( 1\right) -g_{1}\left( 1\right) -\frac{\kappa _{2}}{2}g_{1}^{\left( 2\right) }\left( 1\right) h_{3}^{2}\right] -\sqrt{nh_{3}}\left[ \hat{g}_{0}\left( 0\right) -g_{0}\left( 0\right) -\frac{\kappa _{2}}{2}g_{0}^{\left( 2\right) }\left( 0\right) h_{3}^{2}\right] \\ &\rightarrow &N\left( 0,\sigma _{\text{ATE}}^{2}\right) , \end{eqnarray*} \begin{eqnarray*} &&\sqrt{nh_{3}}\left[ \hat{\Delta}^{\text{TT}}\left( x\right) -\Delta ^{ \text{TT}}\left( x\right) -\frac{\kappa _{2}}{2}\left( g_{1}^{\left( 2\right) }\left( \pi \left( x\right) \right) +\frac{\left( 1-\pi \left( x\right) \right) g_{0}^{\left( 2\right) }\left( \pi \left( x\right) \right) -g_{0}^{\left( 2\right) }\left( 0\right) }{\pi \left( x\right) }\right) h_{3}^{2}\right] \\ &=&x^{\prime }\left[ \sqrt{nh_{3}}\left( \hat{\beta}_{1}-\beta _{1}\right) - \sqrt{nh_{3}}\left( \hat{\beta}_{0}-\beta _{0}\right) \right] +\sqrt{nh_{3}} \left[ \hat{g}_{1}\left( \hat{\pi}\left( x\right) \right) -g_{1}\left( \pi \left( x\right) \right) \right] \\ &&+\sqrt{nh_{3}}\left[ \frac{\left( 1-\hat{\pi}\left( x\right) \right) \hat{g }_{0}\left( \hat{\pi}\left( x\right) \right) -\hat{g}_{0}\left( 0\right) }{ \hat{\pi}\left( x\right) }-\frac{\left( 1-\pi \left( x\right) \right) g_{0}\left( \pi \left( x\right) \right) -g_{0}\left( 0\right) }{\pi \left( x\right) }\right] \\ &&-\sqrt{nh_{3}}\frac{\kappa _{2}}{2}\left[ g_{1}^{\left( 2\right) }\left( \pi \left( x\right) \right) +\frac{1-\pi \left( x\right) }{\pi \left( x\right) }g_{0}^{\left( 2\right) }\left( \pi \left( x\right) \right) -\frac{1 }{\pi \left( x\right) }g_{0}^{\left( 2\right) }\left( 0\right) \right] h_{3}^{2} \\ &=&O_{p}\left( \sqrt{h_{3}}\right) +O_{p}\left( \sqrt{nh_{3}}\eta _{1}\right) +\sqrt{nh_{3}}\left[ \hat{g}_{1}\left( \pi \left( x\right) \right) -g_{1}\left( \pi \left( x\right) \right) -\frac{\kappa _{2}}{2} g_{1}^{\left( 2\right) }\left( \pi \left( x\right) \right) h_{3}^{2}\right] \\ &&+O_{p}\left( \sqrt{nh_{3}}\eta _{1}\right) +\frac{1-\pi \left( x\right) }{ \pi \left( x\right) }\sqrt{nh_{3}}\left[ \hat{g}_{0}\left( \pi \left( x\right) \right) -g_{0}\left( \pi \left( x\right) \right) -\frac{\kappa _{2} }{2}g_{0}^{\left( 2\right) }\left( \pi \left( x\right) \right) h_{3}^{2} \right] \\ &&+O_{p}\left( \sqrt{nh_{3}}\eta _{1}\right) -\frac{1}{\pi \left( x\right) } \sqrt{nh_{3}}\left[ \hat{g}_{0}\left( 0\right) -g_{0}\left( 0\right) -\frac{ \kappa _{2}}{2}g_{0}^{\left( 2\right) }\left( 0\right) h_{3}^{2}\right] \\ &\rightarrow &N\left( 0,\sigma _{\text{TT}}^{2}\left( \pi \left( x\right) \right) \right) , \end{eqnarray*} \begin{eqnarray*} &&\sqrt{nh_{3}}\left[ \hat{\Delta}^{\text{TUT}}\left( x\right) -\Delta ^{ \text{TUT}}\left( x\right) -\frac{\kappa _{2}}{2}\left( \frac{g_{1}^{\left( 2\right) }\left( 1\right) -\pi \left( x\right) g_{1}^{\left( 2\right) }\left( \pi \left( x\right) \right) }{1-\pi \left( x\right) }-g_{0}^{\left( 2\right) }\left( \pi \left( x\right) \right) \right) h_{3}^{2}\right] \\ &=&-\sqrt{nh_{3}}\left[ \hat{g}_{0}\left( \pi \left( x\right) \right) -g_{0}\left( \pi \left( x\right) \right) -\frac{\kappa _{2}}{2}g_{0}^{\left( 2\right) }\left( \pi \left( x\right) \right) h_{3}^{2}\right] \\ &&-\frac{\pi \left( x\right) }{1-\pi \left( x\right) }\sqrt{nh_{3}}\left[ \hat{g}_{1}\left( \pi \left( x\right) \right) -g_{1}\left( \pi \left( x\right) \right) -\frac{\kappa _{2}}{2}g_{1}^{\left( 2\right) }\left( \pi \left( x\right) \right) h_{3}^{2}\right] \\ &&+\frac{1}{1-\pi \left( x\right) }\sqrt{nh_{3}}\left[ \hat{g}_{1}\left( 1\right) -g_{1}\left( 1\right) -\frac{\kappa _{2}}{2}g_{1}^{\left( 2\right) }\left( 1\right) h_{3}^{2}\right] +o_{p}\left( 1\right) \\ &\rightarrow &N\left( 0,\sigma _{\text{TUT}}^{2}\left( \pi \left( x\right) \right) \right) , \end{eqnarray*} \begin{equation*} \sqrt{nh_{3}}\left[ \begin{array}{c} \hat{\Delta}^{\text{LATE}}\left( x,v_{1},v_{2}\right) -\Delta ^{\text{LATE} }\left( x,v_{1},v_{2}\right) \\ -\frac{\kappa _{2}}{2}\left( \frac{v_{2}g_{1}^{\left( 2\right) }\left( v_{2}\right) -v_{1}g_{1}^{\left( 2\right) }\left( v_{1}\right) +\left( 1-v_{2}\right) g_{0}^{\left( 2\right) }\left( v_{2}\right) -\left( 1-v_{1}\right) g_{0}^{\left( 2\right) }\left( v_{1}\right) }{v_{2}-v_{1}} \right) h_{3}^{2} \end{array} \right] \rightarrow N\left( 0,\sigma _{\text{LATE}}^{2}\left( v_{1},v_{2}\right) \right) , \end{equation*} where \begin{equation} \sigma _{\text{ATE}}^{2}=\frac{\kappa \sigma _{V_{1}}^{2}\left( 1\right) }{ f_{P}\left( 1\right) }+\frac{\kappa \sigma _{V_{0}}^{2}\left( 0\right) }{ f_{P}\left( 0\right) }, \end{equation} \begin{eqnarray} \sigma _{\text{TT}}^{2}\left( p\right) &=&\frac{\kappa \sigma _{V_{1}}^{2}\left( p\right) +\kappa _{22}^{\left( 1\right) }p^{2}\left( 1-p\right) \left( g_{1}^{\left( 1\right) }\left( p\right) \right) ^{2}}{ pf_{P}\left( p\right) } \notag \\ &&+\left( \frac{1-p}{p}\right) ^{2}\frac{\kappa \sigma _{V_{0}}^{2}\left( p\right) +\kappa _{22}^{\left( 1\right) }p\left( 1-p\right) ^{2}\left( g_{0}^{\left( 1\right) }\left( p\right) \right) ^{2}}{\left( 1-p\right) f_{P}\left( p\right) } \notag \\ &&+\frac{2\kappa _{22}^{\left( 1\right) }\left( 1-p\right) ^{2}g_{1}^{\left( 1\right) }\left( p\right) g_{0}^{\left( 1\right) }\left( p\right) }{ f_{P}\left( p\right) }+\frac{\kappa \sigma _{V_{0}}^{2}\left( 0\right) }{ p^{2}f_{P}\left( 0\right) }, \end{eqnarray} \begin{eqnarray} \sigma _{\text{TUT}}^{2}\left( p\right) &=&\frac{\kappa \sigma _{V_{0}}^{2}\left( p\right) +\kappa _{22}^{\left( 1\right) }p\left( 1-p\right) ^{2}\left( g_{0}^{\left( 1\right) }\left( p\right) \right) ^{2}}{ \left( 1-p\right) f_{P}\left( p\right) } \notag \\ &&+\left( \frac{p}{1-p}\right) ^{2}\frac{\kappa \sigma _{V_{1}}^{2}\left( p\right) +\kappa _{22}^{\left( 1\right) }p^{2}\left( 1-p\right) \left( g_{1}^{\left( 1\right) }\left( p\right) \right) ^{2}}{pf_{P}\left( p\right) } \notag \\ &&+\frac{2\kappa _{22}^{\left( 1\right) }p^{2}g_{1}^{\left( 1\right) }\left( p\right) g_{0}^{\left( 1\right) }\left( p\right) }{f_{P}\left( p\right) }+ \frac{\kappa \sigma _{V_{1}}^{2}\left( 1\right) }{\left( 1-p\right) ^{2}f_{P}\left( 1\right) }, \end{eqnarray} \begin{eqnarray} \sigma _{\text{LATE}}^{2}\left( v_{1},v_{2}\right) &=&\frac{\kappa }{\left( v_{2}-v_{1}\right) ^{2}f_{P}\left( v_{1}\right) }\left[ v_{1}\sigma _{V_{1}}^{2}\left( v_{1}\right) +\left( 1-v_{1}\right) \sigma _{V_{0}}^{2}\left( v_{1}\right) \right] \notag \\ &&+\frac{\kappa }{\left( v_{2}-v_{1}\right) ^{2}f_{P}\left( v_{2}\right) } \left[ v_{2}\sigma _{V_{1}}^{2}\left( v_{2}\right) +\left( 1-v_{2}\right) \sigma _{V_{0}}^{2}\left( v_{2}\right) \right] \notag \\ &&+\frac{\kappa _{22}^{\left( 1\right) }v_{1}\left( 1-v_{1}\right) }{\left( v_{2}-v_{1}\right) ^{2}f_{P}\left( v_{1}\right) }\left[ v_{1}g_{1}^{\left( 1\right) }\left( v_{1}\right) +\left( 1-v_{1}\right) g_{0}^{\left( 1\right) }\left( v_{1}\right) \right] ^{2} \notag \\ &&+\frac{\kappa _{22}^{\left( 1\right) }v_{2}\left( 1-v_{2}\right) }{\left( v_{2}-v_{1}\right) ^{2}f_{P}\left( v_{2}\right) }\left[ v_{2}g_{1}^{\left( 1\right) }\left( v_{2}\right) +\left( 1-v_{2}\right) g_{0}^{\left( 1\right) }\left( v_{2}\right) \right] ^{2}, \end{eqnarray} which completes the proof. \subsection{Auxiliary lemmas} \begin{lemma} Under Assumption E.1, we have \begin{eqnarray*} \sup_{x}\left\vert \hat{\pi}\left( x\right) -\pi \left( x\right) \right\vert &=&O\left( \sum_{l=1}^{\dim \left( X^{C}\right) }h_{1l}^{s}+\sqrt{\frac{\ln n }{nh_{11}h_{12}\cdots h_{1,\dim \left( X^{C}\right) }}}\right) \text{ almost surely,} \\ \sup_{x}\left\vert \hat{f}_{X}\left( x\right) -f_{X}\left( x\right) \right\vert &=&O\left( \sum_{l=1}^{\dim \left( X^{C}\right) }h_{1l}^{s}+ \sqrt{\frac{\ln n}{nh_{11}h_{12}\cdots h_{1,\dim \left( X^{C}\right) }}} \right) \text{ almost surely,} \end{eqnarray*} where $\hat{\pi}\left( x\right) $ is defined in ((ref)) and $\hat{ f}_{X}\left( x\right) $ is the normalized denominator of $\hat{\pi}\left( x\right) $, namely, \begin{equation} \hat{f}_{X}\left( x\right) =\frac{1}{nh_{11}h_{12}\cdots h_{1,\dim \left( X^{C}\right) }}\sum_{i=1}^{n}\left[ \prod_{l=1}^{\dim \left( X^{C}\right) }k_{1}\left( \frac{X_{il}^{C}-x_{l}^{C}}{h_{1l}}\right) \right] 1\left\{ X_{i}^{D}=x^{D}\right\} . \end{equation} \end{lemma} \begin{proof} This proof follows straightforwardly from combining Subsection 1.11 and Theorems 1.4 and 2.6 of li2007nonparametric, thus is omitted here. \end{proof} \begin{lemma} Under Assumption E.2.(ii)-(iv), the infeasible local linear estimators $\left( \tilde{g}_{d}\left( p\right) ,\tilde{g} _{d}^{\left( 1\right) }\left( p\right) \right) $, $d=0,1$, are asymptotically normally distributed for any interior point $p$ of the support of $P$: \begin{equation*} \left( \begin{array}{cc} \sqrt{nh_{3}} & 0 \\ 0 & \sqrt{nh_{3}^{3}} \end{array} \right) \left[ \left( \begin{array}{c} \tilde{g}_{d}\left( p\right) \\ \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} g_{d}\left( p\right) \\ g_{d}^{\left( 1\right) }\left( p\right) \end{array} \right)-\left( \begin{array}{c} \frac{\kappa _{2}}{2}g_{d}^{\left( 2\right) }\left( p\right) h_{3}^{2} \\ 0 \end{array} \right) \right] \rightarrow N\left( 0,\Xi _{d}\left( p\right) \right) , \end{equation*} where \begin{equation*} \Xi _{d}\left( p\right) =\left( \begin{array}{cc} \left. \kappa \sigma _{V_{d}}^{2}\left( p\right) \right/ \left[ \delta _{d}\left( p\right) f_{P}\left( p\right) \right] & 0 \\ 0 & \left. \kappa _{22}\sigma _{V_{d}}^{2}\left( p\right) \right/ \left[ \kappa _{2}^{2}\delta _{d}\left( p\right) f_{P}\left( p\right) \right] \end{array} \right) , \end{equation*} and \begin{eqnarray*} \kappa &=&\int k_{3}^{2}\left( u\right) du, \\ \kappa _{2} &=&\int k_{3}\left( u\right) u^{2}du, \\ \kappa _{22} &=&\int k_{3}^{2}\left( u\right) u^{2}du. \end{eqnarray*} \end{lemma} \begin{proof} Denote \begin{equation*} Q_{d}=\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ P_{i}-p \end{array} \right) \left( \begin{array}{c} 1 \\ P_{i}-p \end{array} \right) ^{\prime } \end{equation*} as the normalized denominator of $\left( \tilde{g}_{d}\left( p\right) , \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \right) $. By $Y-X^{\prime }\beta _{d}=g_{d}\left( P\right) +V_{d}$, we have \begin{eqnarray*} \left( \begin{array}{c} \tilde{g}_{d}\left( p\right) \\ \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) &=&Q_{d}^{-1}\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ P_{i}-p \end{array} \right) \left[ g_{d}\left( P_{i}\right) +V_{di}\right] \\ &=&\left( \begin{array}{c} g_{d}\left( p\right) \\ g_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) +Q_{d}^{-1}\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ P_{i}-p \end{array} \right) \left[ \frac{1}{2}g_{d}^{\left( 2\right) }\left( p\right) \left( P_{i}-p\right) ^{2}+R_{di}+V_{di}\right] , \end{eqnarray*} where the second equality follows from a Taylor expansion of $g_{d}\left( P_{i}\right) $ around $P_{i}=p$, and $R_{di}=\frac{1}{6}g_{d}^{\left( 3\right) }\left( P_{i}^{\ast }\right) \left( P_{i}-p\right) ^{3}$ with $ P_{i}^{\ast }$ an intermediate value. Note that the three times differentiability of $g_{d}\left( \cdot \right) $ required in the expansion is stronger than Assumption E.2.(ii) and is imposed only for notational simplicity. Actually, the twice continuous differentiability of $g_{d}\left( \cdot \right) $ imposed by Assumption E.2.(ii) suffices for the same conclusion to hold. By Lemma (ref), \begin{equation*} Q_{d}=\left( \begin{array}{cc} \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) , & \frac{1}{n} \sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) \\ \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) , & \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) ^{2} \end{array} \right) =\left( \begin{array}{cc} O_{p}\left( 1\right) & O_{p}\left( h_{3}^{2}\right) \\ O_{p}\left( h_{3}^{2}\right) & O_{p}\left( h_{3}^{2}\right) \end{array} \right) , \end{equation*} which is an asymptotically singular matrix. Hence, we further normalize $ Q_{d}$ by a matrix, \begin{equation*} G=\left( \begin{array}{cc} 1, & 0 \\ 0, & h_{3}^{-2} \end{array} \right) , \end{equation*} such that \begin{eqnarray*} GQ_{d} &=&\left( \begin{array}{cc} \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) , & \frac{1}{n} \sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) \\ \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) , & \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) ^{2} \end{array} \right) \\ &\overset{p}{\longrightarrow }&T_{d}\equiv \left( \begin{array}{cc} \delta _{d}\left( p\right) f_{P}\left( p\right) , & 0 \\ \kappa _{2}\left[ \delta _{d}^{\left( 1\right) }\left( p\right) f_{P}\left( p\right) +\delta _{d}\left( p\right) f_{P}^{\left( 1\right) }\left( p\right) \right] , & \kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) \end{array} \right) , \end{eqnarray*} Since $T_{d}$ is invertible, we have \begin{equation*} \left( GQ_{d}\right) ^{-1}\overset{p}{\longrightarrow }T_{d}^{-1}=\left( \begin{array}{cc} 1, & 0 \\ -\left[ \left. \delta _{d}^{\left( 1\right) }\left( p\right) \right/ \delta _{d}\left( p\right) +\left. f_{P}^{\left( 1\right) }\left( p\right) \right/ f_{P}\left( p\right) \right] , & 1\left/ \kappa _{2}\right. \end{array} \right) \frac{1}{\delta _{d}\left( p\right) f_{P}\left( p\right) }. \end{equation*} Therefore, \begin{eqnarray*} &&J_{n}\left[ \left( \begin{array}{c} \tilde{g}_{d}\left( p\right) \\ \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} g_{d}\left( p\right) \\ g_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) \right] \\ &=&J_{n}\left( GQ_{d}\right) ^{-1}G\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ P_{i}-p \end{array} \right) \left[ \frac{1}{2}g_{d}^{\left( 2\right) }\left( p\right) \left( P_{i}-p\right) ^{2}+R_{di}+V_{di}\right] \\ &=&J_{n}T_{d}^{-1}\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) \left[ \frac{1}{2}g_{d}^{\left( 2\right) }\left( p\right) \left( P_{i}-p\right) ^{2}+R_{di}+V_{di}\right] +o_{p}\left( 1\right) , \end{eqnarray*} where \begin{equation*} J_{n}=\left( \begin{array}{cc} \sqrt{nh_{3}} & 0 \\ 0 & \sqrt{nh_{3}^{3}} \end{array} \right) . \end{equation*} Note that for \begin{equation*} \Gamma _{d}=diag\left( T_{d}^{-1}\right) =\left( \begin{array}{cc} 1\left/ \left[ \delta _{d}\left( p\right) f_{P}\left( p\right) \right] \right. , & 0 \\ 0, & 1\left/ \left[ \kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) \right] \right. \end{array} \right) , \end{equation*} we have \begin{eqnarray*} &&J_{n}T_{d}^{-1}\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) \left[ \frac{1}{2}g_{d}^{\left( 2\right) }\left( p\right) \left( P_{i}-p\right) ^{2}+R_{di}+V_{di}\right] \\ &=&J_{n}\Gamma _{d}\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) \left[ \frac{1}{2}g_{d}^{\left( 2\right) }\left( p\right) \left( P_{i}-p\right) ^{2}+R_{di}+V_{di}\right] +o_{p}\left( 1\right) , \end{eqnarray*} because \begin{eqnarray*} \sqrt{nh_{3}^{3}}T_{d\left( 2,1\right) }^{-1}\frac{1}{n}\sum_{i=1}^{n}w_{di} \left( p\right) \frac{1}{2}g_{d}^{\left( 2\right) }\left( p\right) \left( P_{i}-p\right) ^{2} &=&O\left( \sqrt{nh_{3}^{7}}+h_{3}^{3}\right) =o\left( 1\right) , \\ \sqrt{nh_{3}^{3}}T_{d\left( 2,1\right) }^{-1}\frac{1}{n}\sum_{i=1}^{n}w_{di} \left( p\right) R_{di} &=&O\left( \sqrt{nh_{3}^{9}}+h_{3}^{4}\right) =o\left( 1\right) , \\ \sqrt{nh_{3}^{3}}T_{d\left( 2,1\right) }^{-1}\frac{1}{n}\sum_{i=1}^{n}w_{di} \left( p\right) V_{di} &=&O\left( h_{3}\right) =o\left( 1\right) , \end{eqnarray*} where \begin{equation*} T_{d\left( 2,1\right) }^{-1}=-\frac{\left. \delta _{d}^{\left( 1\right) }\left( p\right) \right/ \delta _{d}\left( p\right) +\left. f_{P}^{\left( 1\right) }\left( p\right) \right/ f_{P}\left( p\right) }{\delta _{d}\left( p\right) f_{P}\left( p\right) }. \end{equation*} Therefore, \begin{eqnarray} &&J_{n}\left[ \left( \begin{array}{c} \tilde{g}_{d}\left( p\right) \\ \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} g_{d}\left( p\right) \\ g_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) \right] \notag \\ &=&J_{n}\Gamma _{d}\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) \left[ \frac{1}{2}g_{d}^{\left( 2\right) }\left( p\right) \left( P_{i}-p\right) ^{2}+R_{di}+V_{di}\right] +o_{p}\left( 1\right) \notag \\ &=&\Gamma _{d}\left[ A_{1d}\left( p\right) +A_{2d}\left( p\right) +A_{3d}\left( p\right) \right] +o_{p}\left( 1\right) , \end{eqnarray} where \begin{eqnarray*} A_{1d}\left( p\right) &=&\left( \begin{array}{c} A_{1d\left( 1\right) }\left(p\right) \\ A_{1d\left( 2\right) }\left( p\right) \end{array} \right) \equiv J_{n}\frac{1 }{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) \frac{1}{2}g_{d}^{\left( 2\right) }\left( p\right) \left( P_{i}-p\right) ^{2}, \\ A_{2d}\left( p\right) &=&\left( \begin{array}{c} A_{2d\left( 1\right) }\left(p\right) \\ A_{2d\left( 2\right) }\left( p\right) \end{array} \right) \equiv J_{n}\frac{1 }{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) R_{di} \\ &=&J_{n}\frac{1}{6n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) g_{d}^{\left( 3\right) }\left( P_{i}^{\ast }\right) \left( P_{i}-p\right) ^{3}, \\ A_{3d}\left( p\right) &=&\left( \begin{array}{c} A_{3d\left( 1\right) }\left(p\right) \\ A_{3d\left( 2\right) }\left( p\right) \end{array} \right) \equiv J_{n}\frac{1 }{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \begin{array}{c} 1 \\ \frac{P_{i}-p}{h_{3}^{2}} \end{array} \right) V_{di}. \end{eqnarray*} We first consider $A_{1d}\left( p\right) $. Since \begin{eqnarray*} E\left[ A_{1d\left( 1\right) }\left( p\right) \right] &=&\frac{\sqrt{nh_{3}} }{2}g_{d}^{\left( 2\right) }\left( p\right) E\left[ w_{di}\left( p\right) \left( P_{i}-p\right) ^{2}\right] =\frac{\kappa _{2}}{2}g_{d}^{\left( 2\right) }\left( p\right) \delta _{d}\left( p\right) f_{P}\left( p\right) \sqrt{nh_{3}^{5}}+o\left( 1\right) , \\ E\left[ A_{1d\left( 2\right) }\left( p\right) \right] &=&\frac{\sqrt{ nh_{3}^{3}}}{2}g_{d}^{\left( 2\right) }\left( p\right) E\left[ w_{di}\left( p\right) \frac{P_{i}-p}{h_{3}^{2}}\left( P_{i}-p\right) ^{2}\right] =O\left( \sqrt{nh_{3}^{7}}\right) =o\left( 1\right) , \end{eqnarray*} and \begin{eqnarray*} Var\left( A_{1d\left( 1\right) }\left( p\right) \right) &\leq &\frac{h_{3}}{4 }\left[ g_{d}^{\left( 2\right) }\left( p\right) \right] ^{2}E\left[ w_{di}^{2}\left( p\right) \left( P_{i}-p\right) ^{4}\right] =O\left( h_{3}^{2}\right) =o\left( 1\right) , \\ Var\left( A_{1d\left( 2\right) }\left( p\right) \right) &\leq &\frac{ h_{3}^{3}}{4}\left[ g_{d}^{\left( 2\right) }\left( p\right) \right] ^{2}E \left[ w_{di}^{2}\left( p\right) \frac{\left( P_{i}-p\right) ^{6}}{h_{3}^{4}} \right] =O\left( h_{3}^{4}\right) =o\left( 1\right) , \end{eqnarray*} it follows from Markov's inequality that \begin{eqnarray*} A_{1d\left( 1\right) }\left( p\right) &=&\frac{\kappa _{2}}{2}g_{d}^{\left( 2\right) }\left( p\right) \delta _{d}\left( p\right) f_{P}\left( p\right) \sqrt{nh_{3}^{5}}+o_{p}\left( 1\right) , \\ A_{1d\left( 2\right) }\left( p\right) &=&o_{p}\left( 1\right) , \end{eqnarray*} namely, \begin{equation} A_{1d}\left( p\right) =J_{n}\left( \begin{array}{c} \frac{\kappa _{2}}{2}g_{d}^{\left( 2\right) }\left( p\right) \delta _{d}\left( p\right) f_{P}\left( p\right) h_{3}^{2} \\ 0 \end{array} \right) +o_{p}\left( 1\right) . \end{equation} Similarly, we can show that \begin{equation} A_{2d}\left( p\right) =o_{p}\left( 1\right) . \end{equation} We next consider $A_{3d}\left( p\right) $. By the definition of $V_{di}$, we have $E\left[ A_{3d}\left( p\right) \right] =0$. For the variance-covariance term, we have \begin{gather*} Var\left( A_{3d\left( 1\right) }\left( p\right) \right) =h_{3}E\left[ w_{di}^{2}\left( p\right) V_{di}^{2}\right] =\kappa \sigma _{V_{d}}^{2}\left( p\right) \delta _{d}\left( p\right) f_{P}\left( p\right) +O\left( h_{3}\right) , \\ Var\left( A_{3d\left( 2\right) }\left( p\right) \right) =h_{3}^{3}E\left[ w_{di}^{2}\left( p\right) \frac{\left( P_{i}-p\right) ^{2}}{h_{3}^{4}} V_{di}^{2}\right] =\kappa _{22}\sigma _{V_{d}}^{2}\left( p\right) \delta _{d}\left( p\right) f_{P}\left( p\right) +O\left( h_{3}\right) , \\ Cov\left( A_{3d\left( 1\right) }\left( p\right) ,A_{3d\left( 2\right) }\left( p\right) \right) =h_{3}^{2}E\left[ w_{di}^{2}\left( p\right) \frac{ P_{i}-p}{h_{3}^{2}}V_{di}^{2}\right] =O\left( h_{3}\right) . \end{gather*} It then follows from Liapunov's central limit theorem that \begin{equation} A_{3d}\left( p\right) \rightarrow N\left( 0,\Omega _{d}\right) , \end{equation} where \begin{equation*} \Omega _{d}=\left( \begin{array}{cc} \kappa \sigma _{V_{d}}^{2}\left( p\right) \delta _{d}\left( p\right) f_{P}\left( p\right) & 0 \\ 0 & \kappa _{22}\sigma _{V_{d}}^{2}\left( p\right) \delta _{d}\left( p\right) f_{P}\left( p\right) \end{array} \right) . \end{equation*} Substituting ((ref)) and ((ref)) into ((ref)) yields that \begin{equation} J_{n}\left[ \left( \begin{array}{c} \tilde{g}_{d}\left( p\right) \\ \tilde{g}_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) -\left( \begin{array}{c} g_{d}\left( p\right) \\ g_{d}^{\left( 1\right) }\left( p\right) \end{array} \right) \right] =J_{n}\left( \begin{array}{c} \frac{\kappa _{2}}{2}g_{d}^{\left( 2\right) }\left( p\right) h_{3}^{2} \\ 0 \end{array} \right) +\Gamma _{d}A_{3d}\left( p\right) +o_{p}\left( 1\right) , \end{equation} and the conclusion follows from ((ref)). \end{proof} \begin{lemma} Under Assumption E.2, we have \begin{eqnarray*} \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) &=&\delta _{d}\left( p\right) f_{P}\left( p\right) +o_{p}\left( h_{3}\right) , \\ \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) &=&\kappa _{2}\left[ \delta _{d}^{\left( 1\right) }\left( p\right) f_{P}\left( p\right) +\delta _{d}\left( p\right) f_{P}^{\left( 1\right) }\left( p\right) \right] +O_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}} \right) , \\ \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) ^{2} &=&\kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) +o_{p}\left( h_{3}\right) . \end{eqnarray*} \end{lemma} \begin{proof} Since the data are i.i.d., we have \begin{eqnarray*} E\left[ \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \right] &=&E\left[ 1\left\{ D_{i}=d\right\} \frac{1}{h_{3}}k_{3}\left( \frac{P_{i}-p}{h_{3}} \right) \right] \\ &=&E\left[ \delta _{d}\left( P_{i}\right) \frac{1}{h_{3}}k_{3}\left( \frac{ P_{i}-p}{h_{3}}\right) \right] \\ &=&\int \delta _{d}\left( v\right) \frac{1}{h_{3}}k_{3}\left( \frac{v-p}{ h_{3}}\right) f_{P}\left( v\right) dv \\ &=&\int \delta _{d}\left( p+uh_{3}\right) k_{3}\left( u\right) f_{P}\left( p+uh_{3}\right) du \\ &=&\delta _{d}\left( p\right) f_{P}\left( p\right) +O\left( h_{3}^{2}\right) , \end{eqnarray*} where the second equality follows from the law of iterated expectation. On the other hand, \begin{eqnarray*} Var\left( \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \right) &=&\frac{1}{ nh_{3}^{2}}Var\left( 1\left\{ D_{i}=d\right\} k_{3}\left( \frac{P_{i}-p}{ h_{3}}\right) \right) \\ &\leq &\frac{1}{nh_{3}^{2}}E\left[ 1\left\{ D_{i}=d\right\} k_{3}^{2}\left( \frac{P_{i}-p}{h_{3}}\right) \right] \\ &=&O\left( \frac{1}{nh_{3}}\right) . \end{eqnarray*} Therefore, \begin{equation*} \left( E\left[ \left( \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) -\delta _{d}\left( p\right) f_{P}\left( p\right) \right) ^{2}\right] \right) ^{1/2}=O\left( h_{3}^{2}+\frac{1}{\sqrt{nh_{3}}}\right) =o\left( h_{3}\right) , \end{equation*} where the last equality is by Assumption E.2.(iv). The first assertion of the lemma then follows from Markov's inequality. For the second assertion of the lemma, we have \begin{eqnarray*} E\left[ \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) \right] &=&E\left[ \delta _{d}\left( P_{i}\right) \frac{1}{ h_{3}^{3}}k_{3}\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \right] \\ &=&\int \delta _{d}\left( v\right) \frac{1}{h_{3}^{3}}k_{3}\left( \frac{v-p}{ h_{3}}\right) \left( v-p\right) f_{P}\left( v\right) dv \\ &=&\frac{1}{h_{3}}\int \delta _{d}\left( p+uh_{3}\right) f_{P}\left( p+uh_{3}\right) k_{3}\left( u\right) udu \\ &=&\kappa _{2}\left[ \delta _{d}^{\left( 1\right) }\left( p\right) f_{P}\left( p\right) +\delta _{d}\left( p\right) f_{P}^{\left( 1\right) }\left( p\right) \right] +O\left( h_{3}^{2}\right) , \end{eqnarray*} and \begin{eqnarray*} Var\left( \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) \right) &\leq &\frac{1}{nh_{3}^{6}}E\left[ 1\left\{ D_{i}=d\right\} k_{3}^{2}\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) ^{2}\right] \\ &=&O\left( \frac{1}{nh_{3}^{3}}\right) . \end{eqnarray*} Therefore, \begin{eqnarray*} \left( E\left[ \left( \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) -\kappa _{2}\left[ \delta _{d}^{\left( 1\right) }\left( p\right) f_{P}\left( p\right) +\delta _{d}\left( p\right) f_{P}^{\left( 1\right) }\left( p\right) \right] \right) ^{2}\right] \right) ^{1/2} &=&O\left( h_{3}^{2}+\frac{1}{\sqrt{nh_{3}^{3}}}\right) \\ &=&O\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) , \end{eqnarray*} and the second assertion of the lemma follows. Analogously, the third assertion of the lemma can be readily shown. \end{proof} \begin{lemma} Under Assumption E.2, we have \begin{eqnarray*} \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) V_{di}\left( p\right) &=&o_{p}\left( h_{3}\right) , \\ \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) V_{di}\left( p\right) &=&\kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) g_{d}^{\left( 1\right) }\left( p\right) +O_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) , \end{eqnarray*} where $V_{di}\left( p\right) $ is defined in ((ref)). \end{lemma} \begin{proof} Note that $E\left[ \left. V_{di}\left( p\right) \right\vert P_{i},D_{i}=d\right] =g_{d}\left( P_{i}\right) -g_{d}\left( p\right) $ because $E\left[ \left. V_{di}\right\vert P_{i},D_{i}=d\right] =0$ by definition. Hence, \begin{eqnarray*} E\left[ \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) V_{di}\left( p\right) \right] &=&E\left[ \delta _{d}\left( P_{i}\right) \left( g_{d}\left( P_{i}\right) -g_{d}\left( p\right) \right) \frac{1}{h_{3}}k_{3}\left( \frac{ P_{i}-p}{h_{3}}\right) \right] \\ &=&O\left( h_{3}^{2}\right) , \\ E\left[ \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) V_{di}\left( p\right) \right] &=&E\left[ \delta _{d}\left( P_{i}\right) \left( g_{d}\left( P_{i}\right) -g_{d}\left( p\right) \right) \frac{1}{h_{3}^{3}}k_{3}\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \right] \\ &=&\kappa _{2}\delta _{d}\left( p\right) f_{P}\left( p\right) g_{d}^{\left( 1\right) }\left( p\right) +O\left( h_{3}^{2}\right) . \end{eqnarray*} For the variances, the same arguments as in Lemma (ref) show that \begin{eqnarray*} Var\left( \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) V_{di}\left( p\right) \right) &=&O\left( \frac{1}{nh_{3}}\right) , \\ Var\left( \frac{1}{nh_{3}^{2}}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) V_{di}\left( p\right) \right) &=&O\left( \frac{1}{nh_{3}^{3}} \right) . \end{eqnarray*} The conclusion then follows from Markov's inequality. \end{proof} \begin{lemma} Under Assumptions E.1, E.2, and E.3, we have \begin{eqnarray*} \frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) -\frac{1}{n} \sum_{i=1}^{n}w_{di}\left( p\right) &=&o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}} }\right) , \\ \frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) -\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) &=&o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) , \\ \frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) ^{2}-\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) ^{2} &=&o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) . \end{eqnarray*} \end{lemma} \begin{proof} By a Taylor expansion, \begin{eqnarray*} &&\frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) -\frac{1}{n} \sum_{i=1}^{n}w_{di}\left( p\right) \\ &=&\frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{2}} k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( \hat{P} _{i}-P_{i}\right) \\ &&+\frac{1}{2n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{3}} k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( \hat{P} _{i}-P_{i}\right) ^{2} \\ &&+\frac{1}{6n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{4}} k_{3}^{\left( 3\right) }\left( \frac{\hat{P}_{i}^{\ast }-p}{h_{3}}\right) \left( \hat{P}_{i}-P_{i}\right) ^{3}, \end{eqnarray*} where $\hat{P}_{i}^{\ast }$ is an intermediate value between $\hat{P}_{i}$ and $P_{i}$. For any i.i.d. nonnegative random variables $\left\{ Z_{ni}:1\leq i\leq n\right\} $ with $0<EZ_{ni}<\infty $, we have $\left. \left( 1/n\right) \sum_{i=1}^{n}Z_{ni}\right/ EZ_{ni}=O_{p}\left( 1\right) $ by Markov's inequality. Thus, it follows from Lemma (ref) and Assumption E.3.(i)-(ii) that \begin{eqnarray*} \left\vert \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( \hat{P}_{i}-P_{i}\right) \right\vert &\leq &\sup_{i}\left\vert \hat{P} _{i}-P_{i}\right\vert \frac{1}{n}\sum_{i=1}^{n}\frac{1}{h_{3}^{2}}\left\vert k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \right\vert \\ &=&O_{p}\left( \eta _{1}\right) O_{p}\left( \frac{1}{h_{3}^{2}}E\left\vert k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \right\vert \right) \\ &=&O_{p}\left( \frac{\eta _{1}}{h_{3}}\right) =o_{p}\left( \frac{1}{\sqrt{ nh_{3}^{3}}}\right) , \end{eqnarray*} \begin{eqnarray*} \left\vert \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{3}}k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( \hat{P}_{i}-P_{i}\right) ^{2}\right\vert &\leq &O_{p}\left( \eta _{1}^{2}\right) O_{p}\left( \frac{1}{h_{3}^{3}}E\left\vert k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \right\vert \right) \\ &=&O_{p}\left( \frac{\eta _{1}^{2}}{h_{3}^{2}}\right) =o_{p}\left( \frac{1}{ nh_{3}^{3}}\right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) , \end{eqnarray*} and \begin{equation*} \left\vert \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{4}}k_{3}^{\left( 3\right) }\left( \frac{\hat{P}_{i}^{\ast }-p}{h_{3}} \right) \left( \hat{P}_{i}-P_{i}\right) ^{3}\right\vert \leq O_{p}\left( \frac{\eta _{1}^{3}}{h_{3}^{4}}\right) =o_{p}\left( \frac{1}{h_{3}^{4}}\sqrt{ \frac{h_{3}^{7}}{n}}\right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) . \end{equation*} Therefore, \begin{equation*} \frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) -\frac{1}{n} \sum_{i=1}^{n}w_{di}\left( p\right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}} \right) . \end{equation*} For the second claim of the lemma, we decompose the left-hand side into three terms: \begin{eqnarray} &&\frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) -\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) \notag \\ &=&\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \hat{P} _{i}-P_{i}\right) +\frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( P_{i}-p\right) \notag \\ &&+\frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) . \end{eqnarray} For the first term, we have \begin{equation} \left\vert \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \hat{P} _{i}-P_{i}\right) \right\vert \leq O_{p}\left( \eta _{1}\right) O_{p}\left( E\left\vert w_{di}\left( p\right) \right\vert \right) =O_{p}\left( \eta _{1}\right) O_{p}\left( 1\right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) . \end{equation} For the second term, a Taylor expansion yields that \begin{eqnarray*} &&\frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( P_{i}-p\right) \\ &=&\frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{2}} k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) \\ &&+\frac{1}{2n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{3}} k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{2} \\ &&+\frac{1}{6n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{4}} k_{3}^{\left( 3\right) }\left( \frac{\hat{P}_{i}^{\ast }-p}{h_{3}}\right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{3}. \end{eqnarray*} It follows that \begin{eqnarray*} &&\left\vert \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) \right\vert \\ &\leq &\sup_{i}\left\vert \hat{P}_{i}-P_{i}\right\vert \frac{1}{n} \sum_{i=1}^{n}\frac{1}{h_{3}^{2}}\left\vert k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \right\vert \left\vert P_{i}-p\right\vert \\ &=&O_{p}\left( \eta _{1}\right) O_{p}\left( \frac{1}{h_{3}}E\left[ \left\vert k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \right\vert \left\vert \frac{P_{i}-p}{h_{3}}\right\vert \right] \right) \\ &=&O_{p}\left( \eta _{1}\right) O_{p}\left( 1\right) =o_{p}\left( \frac{1}{ \sqrt{nh_{3}}}\right) , \end{eqnarray*} \begin{eqnarray*} &&\left\vert \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{3}}k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{2}\right\vert \\ &\leq &O_{p}\left( \eta _{1}^{2}\right) O_{p}\left( \frac{1}{h_{3}^{2}} E\left\vert k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \right\vert \left\vert \frac{P_{i}-p}{h_{3}}\right\vert \right) \\ &=&O_{p}\left( \frac{\eta _{1}^{2}}{h_{3}}\right) =o_{p}\left( \frac{1}{ nh_{3}^{2}}\right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) , \end{eqnarray*} and \begin{equation*} \left\vert \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{4}}k_{3}^{\left( 3\right) }\left( \frac{\hat{P}_{i}^{\ast }-p}{h_{3}} \right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{3}\right\vert \leq O_{p}\left( \frac{\eta _{1}^{3}}{h_{3}^{4}}\right) =o_{p}\left( \frac{1}{h_{3}^{4}}\sqrt{\frac{h_{3}^{7}}{n}}\right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) . \end{equation*} Therefore, \begin{equation} \frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( P_{i}-p\right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}}} \right) . \end{equation} For the third term of ((ref)), we have \begin{equation*} \left\vert \frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) \right\vert \leq \sup_{i}\left\vert \hat{P}_{i}-P_{i}\right\vert \frac{1}{n} \sum_{i=1}^{n}\left\vert \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right\vert . \end{equation*} As for the first claim of the lemma, we can show that \begin{equation} \frac{1}{n}\sum_{i=1}^{n}\left\vert \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right\vert =o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}} \right) =o_{p}\left( 1\right) . \end{equation} It then follows that \begin{equation} \left\vert \frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) \right\vert \leq O_{p}\left( \eta _{1}\right) o_{p}\left( 1\right) =o_{p}\left( \frac{1}{ \sqrt{nh_{3}}}\right) . \end{equation} Substituting ((ref)), ((ref)), and ((ref)) into ((ref)), we obtain \begin{equation*} \frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) -\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) . \end{equation*} For the third claim of the lemma, we decompose the left-hand side into five terms: \begin{eqnarray} &&\frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) ^{2}-\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) ^{2} \notag \\ &=&\frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) ^{2}+\frac{2}{n} \sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) \left( P_{i}-p\right) \notag \\ &&+\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \hat{P} _{i}-P_{i}\right) ^{2}+\frac{2}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \hat{P}_{i}-P_{i}\right) \left( P_{i}-p\right) \notag \\ &&+\frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( P_{i}-p\right) ^{2}. \end{eqnarray} For the first term, it follows from ((ref)) that \begin{eqnarray} \left\vert \frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) ^{2}\right\vert &\leq &\sup_{i}\left\vert \hat{P}_{i}-P_{i}\right\vert ^{2} \frac{1}{n}\sum_{i=1}^{n}\left\vert \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right\vert \notag \\ &=&O_{p}\left( \eta _{1}^{2}\right) o_{p}\left( 1\right) =o_{p}\left( \frac{1 }{nh_{3}}\right) \notag \\ &=&o_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\sqrt{\frac{h_{3}}{n}}\right) =o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) . \end{eqnarray} For the second term, a Taylor expansion yields that \begin{eqnarray*} &&\frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) \left( P_{i}-p\right) \\ &=&\frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{2}} k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{2} \\ &&+\frac{1}{2n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{3}} k_{3}^{\left( 2\right) }\left( \frac{\hat{P}_{i}^{\ast }-p}{h_{3}}\right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{3}. \end{eqnarray*} Analogously, it follows that \begin{eqnarray*} \left\vert \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{2}\right\vert &\leq &O_{p}\left( \eta _{1}^{2}\right) =o_{p}\left( \frac{1}{nh_{3}}\right) =o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) , \\ \left\vert \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{3}}k_{3}^{\left( 2\right) }\left( \frac{\hat{P}_{i}^{\ast }-p}{h_{3}} \right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{3}\right\vert &\leq &O_{p}\left( \frac{\eta _{1}^{3}}{h_{3}^{3}}\right) =o_{p}\left( \frac{\sqrt{h_{3}^{7}}}{h_{3}^{3}\sqrt{n}}\right) =o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) . \end{eqnarray*} Therefore, \begin{equation} \frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) \left( P_{i}-p\right) =o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) . \end{equation} For the third and fourth terms of ((ref)), we have \begin{eqnarray} \left\vert \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \hat{P} _{i}-P_{i}\right) ^{2}\right\vert &\leq &O_{p}\left( \eta _{1}^{2}\right) O_{p}\left( E\left\vert w_{di}\left( p\right) \right\vert \right) \notag \\ &=&O_{p}\left( \eta _{1}^{2}\right) O_{p}\left( 1\right) =o_{p}\left( \sqrt{ \frac{h_{3}}{n}}\right) , \\ \left\vert \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \hat{P} _{i}-P_{i}\right) \left( P_{i}-p\right) \right\vert &\leq &O_{p}\left( \eta _{1}\right) O_{p}\left( E\left\vert w_{di}\left( p\right) \left( P_{i}-p\right) \right\vert \right) \notag \\ &=&O_{p}\left( \eta _{1}h_{3}\right) =o_{p}\left( \sqrt{\frac{h_{3}}{n}} \right) . \end{eqnarray} For the last term of ((ref)), we can show as well by a Taylor expansion that \begin{equation} \frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( P_{i}-p\right) ^{2}=o_{p}\left( \sqrt{\frac{h_{3}}{n} }\right) . \end{equation} The conclusion follows from combining ((ref))-((ref)). \end{proof} \begin{lemma} Under Assumptions E.1, E.2, E.3, and E.4, we have \begin{eqnarray*} &&\frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) V_{di}\left( p\right) - \frac{1}{n}\sum_{i}^{n}w_{di}\left( p\right) V_{di}\left( p\right) \\ &=&\frac{g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) \frac{1}{h_{3}}k_{3}^{\left( 1\right) }\left( \frac{ P_{i}-p}{h_{3}}\right) \frac{P_{i}-p}{h_{3}}\left( D_{i}-P_{i}\right) +o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) \\ &=&O_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) , \end{eqnarray*} and \begin{eqnarray*} &&\frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) V_{di}\left( p\right) -\frac{1}{n}\sum_{i}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) V_{di}\left( p\right) \\ &=&\frac{g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) \left[ k_{3}\left( \frac{P_{i}-p}{h_{3}}\right) +k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \frac{P_{i}-p}{ h_{3}}\right] \frac{P_{i}-p}{h_{3}}\left( D_{i}-P_{i}\right) +o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) \\ &=&O_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) , \end{eqnarray*} where $V_{di}\left( p\right) $ is defined in ((ref)). \end{lemma} \begin{proof} By a Taylor expansion, \begin{eqnarray} &&\frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) V_{di}\left( p\right) -\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) V_{di}\left( p\right) \notag \\ &=&\frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{2}} k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( \hat{P} _{i}-P_{i}\right) V_{di}\left( p\right) \notag \\ &&+\frac{1}{2n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{3}} k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( \hat{P} _{i}-P_{i}\right) ^{2}V_{di}\left( p\right) \notag \\ &&+\frac{1}{6n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{4}} k_{3}^{\left( 3\right) }\left( \frac{\hat{P}_{i}^{\ast }-p}{h_{3}}\right) \left( \hat{P}_{i}-P_{i}\right) ^{3}V_{di}\left( p\right) \notag \\ &\triangleq &\left( \text{I}\right) +\left( \text{II}\right) +\left( \text{ III}\right) , \end{eqnarray} where $\hat{P}_{i}^{\ast }$ is an intermediate value between $\hat{P}_{i}$ and $P_{i}$. Note that $\left( \text{I}\right) $ and $\left( \text{II} \right) $ can be represented as U-statistics of order two and three, respectively, plus higher order terms. Specifically, by denoting $ W_{i}=\left( Y_{i},D_{i},X_{i}\right) $ as the $i$-th observation, and \begin{eqnarray*} K_{1h}\left( X_{j},X_{i}\right) &=&\prod_{l=1}^{\dim \left( X^{C}\right) } \frac{1}{\left\vert h_{1}\right\vert }k_{1}\left( \frac{X_{jl}^{C}-X_{il}^{C} }{h_{1l}}\right) 1\left\{ X_{j}^{D}=X_{i}^{D}\right\} , \\ m_{1}\left( W_{i},W_{j}\right) &=&1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{2} }k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) V_{di}\left( p\right) \left( D_{j}-P_{i}\right) K_{1h}\left( X_{j},X_{i}\right) , \\ m_{2}\left( W_{i},W_{j},W_{l}\right) &=&1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{3}}k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) V_{di}\left( p\right) \left( D_{j}-P_{i}\right) K_{1h}\left( X_{j},X_{i}\right) \left( D_{l}-P_{i}\right) K_{1h}\left( X_{l},X_{i}\right) , \\ M_{1} &=&\frac{1}{n\left( n-1\right) }\sum_{i}\sum_{j\neq i}\frac{ m_{1}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }, \\ M_{2} &=&\frac{1}{n\left( n-1\right) \left( n-2\right) }\sum_{i}\sum_{j\neq i}\sum_{l\neq j,i}\frac{m_{2}\left( W_{i},W_{j},W_{l}\right) }{ f_{X}^{2}\left( X_{i}\right) }, \end{eqnarray*} we have \begin{eqnarray} \left( \text{I}\right) &=&\frac{1}{n\left( n-1\right) }\sum_{i}\sum_{j\neq i} \frac{m_{1}\left( W_{i},W_{j}\right) }{\hat{f}_{X}\left( X_{i}\right) } =M_{1}\cdot \left( 1+O_{p}\left( \eta _{1}\right) \right) , \\ \left( \text{II}\right) &=&\frac{1}{n\left( n-1\right) ^{2}} \sum_{i}\sum_{j\neq i}\sum_{l\neq i}\frac{m_{2}\left( W_{i},W_{j},W_{l}\right) }{\hat{f}_{X}^{2}\left( X_{i}\right) }=M_{2}\cdot \left( 1+O_{p}\left( \eta _{1}\right) \right) , \end{eqnarray} where $\hat{f}_{X}\left( x\right) $ is defined in ((ref)) and the last equalities follow from Lemma (ref). To analyze the U-statistics $M_{1}$ and $M_{2}$, we need to calculate the conditional expectations of their kernels: \begin{eqnarray*} E\left[ \left. \frac{m_{1}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{i}\right] &=&O\left( tr\left( h_{1}^{s}\right) \right) \cdot 1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) V_{di}\left( p\right) , \\ E\left[ \left. \frac{m_{1}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{j}\right] &=&\delta _{d}\left( P_{j}\right) \frac{g_{d}\left( P_{j}\right) -g_{d}\left( p\right) }{h_{3}^{2}} k_{3}^{\left( 1\right) }\left( \frac{P_{j}-p}{h_{3}}\right) \left( D_{j}-P_{j}\right) \left[ 1+O\left( tr\left( h_{1}^{s}\right) \right) \right] , \\ E\left[ \left. \frac{m_{2}\left( W_{i},W_{j},W_{l}\right) }{f_{X}^{2}\left( X_{i}\right) }\right\vert W_{i}\right] &=&O\left( tr\left( h_{1}^{2s}\right) \right) \cdot 1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{3}} k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) V_{di}\left( p\right) , \\ E\left[ \left. \frac{m_{2}\left( W_{i},W_{j},W_{l}\right) }{f_{X}^{2}\left( X_{i}\right) }\right\vert W_{j}\right] &=&O\left( tr\left( h_{1}^{s}\right) \right) \cdot \delta _{d}\left( P_{j}\right) \frac{g_{d}\left( P_{j}\right) -g_{d}\left( p\right) }{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{ P_{j}-p}{h_{3}}\right) \left( D_{j}-P_{j}\right) \\ &&\cdot \left[ 1+O\left( tr\left( h_{1}^{s}\right) \right) \right] , \\ E\left[ \left. \frac{m_{2}\left( W_{i},W_{j},W_{l}\right) }{f_{X}^{2}\left( X_{i}\right) }\right\vert W_{l}\right] &=&O\left( tr\left( h_{1}^{s}\right) \right) \cdot \delta _{d}\left( P_{l}\right) \frac{g_{d}\left( P_{l}\right) -g_{d}\left( p\right) }{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{ P_{l}-p}{h_{3}}\right) \left( D_{l}-P_{l}\right) \\ &&\cdot \left[ 1+O\left( tr\left( h_{1}^{s}\right) \right) \right] , \end{eqnarray*} where we use $E\left[ \left. V_{di}\left( p\right) \right\vert X_{i},D_{i}=d \right] =g_{d}\left( P_{i}\right) -g_{d}\left( p\right) $, and hence, \begin{eqnarray*} E\left[ M_{1}\right] &=&E\left[ E\left[ \left. \frac{m_{1}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{j}\right] \right] =0, \\ E\left[ M_{2}\right] &=&E\left[ E\left[ \left. \frac{m_{2}\left( W_{i},W_{j},W_{l}\right) }{f_{X}^{2}\left( X_{i}\right) }\right\vert W_{j} \right] \right] =0. \end{eqnarray*} Since by Assumptions E.1.(ii) and E.3.(iii)-(iv), \begin{eqnarray*} E\left[ \frac{m_{1}^{2}\left( W_{i},W_{j}\right) }{f_{X}^{2}\left( X_{i}\right) }\right] &=&O\left( \frac{1}{h_{3}^{3}\left\vert h_{1}\right\vert }\right) =o\left( n\right) , \\ E\left[ \frac{m_{2}^{2}\left( W_{i},W_{j},W_{l}\right) }{f_{X}^{4}\left( X_{i}\right) }\right] &=&O\left( \frac{1}{h_{3}^{5}\left\vert h_{1}\right\vert ^{2}}\right) =o\left( n^{3/2}\right) , \end{eqnarray*} it follows from the projection method for U-statistics powell1989semiparametric that \begin{equation*} M_{1}=\frac{1}{n}\sum_{i=1}^{n}E\left[ \left. \frac{m_{1}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{i}\right] + \frac{1}{n}\sum_{j=1}^{n}E\left[ \left. \frac{m_{1}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{j}\right] +o_{p}\left( \frac{1}{ \sqrt{n}}\right) , \end{equation*} and from the variances of U-statistics shao2003 that \begin{eqnarray*} Var\left( M_{2}\right) &\leq &\frac{3}{n}\left[ Var\left( E\left[ \left. \frac{m_{2}\left( W_{i},W_{j},W_{l}\right) }{f_{X}^{2}\left( X_{i}\right) } \right\vert W_{i}\right] \right) +2Var\left( E\left[ \left. \frac{ m_{2}\left( W_{i},W_{j},W_{l}\right) }{f_{X}^{2}\left( X_{i}\right) } \right\vert W_{j}\right] \right) \right] +O\left( \frac{1}{n^{2}}\right) \\ &=&O\left( \frac{tr\left( h_{1}^{4s}\right) }{nh_{3}^{5}}+\frac{tr\left( h_{1}^{2s}\right) }{nh_{3}}+\frac{1}{n^{2}}\right) =O\left( \frac{1}{ n^{3}h_{3}^{7}}+\frac{1}{n^{2}h_{3}^{2}}+\frac{1}{n^{2}}\right) =O\left( \frac{1}{n^{3}h_{3}^{7}}+\frac{1}{n^{2}h_{3}^{2}}\right) , \end{eqnarray*} where we use $tr\left( h_{1}^{s}\right) =O\left( \eta _{1}\right) =o\left( 1\left/ \sqrt{nh_{3}}\right. \right) $ by Assumption E.3.(i). Since \begin{equation*} \frac{1}{n}\sum_{i=1}^{n}E\left[ \left. \frac{m_{1}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{i}\right] =O\left( tr\left( h_{1}^{s}\right) \right) \cdot \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{ P_{i}-p}{h_{3}}\right) V_{di}\left( p\right) , \end{equation*} and \begin{eqnarray*} E\left[ \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{h_{3}^{2}} k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) V_{di}\left( p\right) \right] &=&E\left[ \delta _{d}\left( P_{i}\right) \frac{ g_{d}\left( P_{i}\right) -g_{d}\left( p\right) }{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \right] \\ &=&O\left( 1\right) , \\ Var\left( \frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} \frac{1}{ h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) V_{di}\left( p\right) \right) &\leq &\frac{1}{n}E\left[ \frac{1}{h_{3}^{4}} \left\vert k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \right\vert ^{2}V_{di}^{2}\left( p\right) \right] \\ &=&O\left( \frac{1}{nh_{3}^{3}}\right) =o\left( 1\right) , \end{eqnarray*} it follows from Markov's inequality that \begin{equation*} \frac{1}{n}\sum_{i=1}^{n}E\left[ \left. \frac{m_{1}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{i}\right] =O\left( tr\left( h_{1}^{s}\right) \right) \cdot O_{p}\left( 1\right) =O_{p}\left( tr\left( h_{1}^{s}\right) \right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) , \end{equation*} Hence, \begin{eqnarray} M_{1} &=&\frac{1}{n}\sum_{j=1}^{n}\delta _{d}\left( P_{j}\right) \frac{ g_{d}\left( P_{j}\right) -g_{d}\left( p\right) }{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{j}-p}{h_{3}}\right) \left( D_{j}-P_{j}\right) +o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) \notag \\ &=&\frac{g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{j=1}^{n}\delta _{d}\left( P_{j}\right) \frac{P_{j}-p}{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{j}-p}{h_{3}}\right) \left( D_{j}-P_{j}\right) +o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) . \end{eqnarray} On the other hand, \begin{equation*} E\left[ M_{2}^{2}\right] =Var\left( M_{2}\right) =O\left( \frac{1}{ n^{3}h_{3}^{7}}+\frac{1}{n^{2}h_{3}^{2}}\right) =o\left( \frac{1}{nh_{3}} \right) , \end{equation*} and thus by Markov's inequality, \begin{equation} M_{2}=o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) . \end{equation} Substituting ((ref)) and ((ref)) into ((ref)) and ((ref)), respectively, we obtain \begin{equation} \left( \text{I}\right) +\left( \text{II}\right) =\frac{g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) \frac{1}{h_{3}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \frac{P_{i}-p}{h_{3}}\left( D_{i}-P_{i}\right) +o_{p}\left( \frac{1}{\sqrt{ nh_{3}}}\right) . \end{equation} For the term $\left( \text{III}\right) $, it follows from Lemma (ref) that \begin{equation} \left( \text{III}\right) \leq \frac{1}{6n}\sum_{i=1}^{n}\frac{1}{h_{3}^{4}} \left\vert k_{3}^{\left( 3\right) }\left( \frac{\hat{P}_{i}^{\ast }-p}{h_{3}} \right) \right\vert \left\vert \hat{P}_{i}-P_{i}\right\vert ^{3}=O_{p}\left( \frac{\eta _{1}^{3}}{h_{3}^{4}}\right) =o_{p}\left( \frac{1}{\sqrt{nh_{3}}} \right) , \end{equation} where the last equality follows from Assumption E.3.(ii). The first claim of the lemma follows from substituting ((ref)) and ((ref)) into ( (ref)). For the second claim, we decompose the left-hand side into three terms: \begin{eqnarray} &&\frac{1}{n}\sum_{i=1}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) V_{di}\left( p\right) -\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( P_{i}-p\right) V_{di}\left( p\right) \notag \\ &=&\frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \hat{P} _{i}-P_{i}\right) V_{di}\left( p\right) +\frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( P_{i}-p\right) V_{di}\left( p\right) \notag \\ &&+\frac{1}{n}\sum_{i=1}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) V_{di}\left( p\right) . \end{eqnarray} Denote \begin{eqnarray*} m_{3}\left( W_{i},W_{j}\right) &=&\frac{1}{h_{3}}w_{di}\left( p\right) V_{di}\left( p\right) \left( D_{j}-P_{i}\right) K_{1h}\left( X_{j},X_{i}\right) , \\ M_{3} &=&\frac{1}{n\left( n-1\right) }\sum_{i}\sum_{j\neq i}\frac{ m_{3}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }, \end{eqnarray*} then it follows from Lemma (ref) that \begin{equation*} \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \hat{P} _{i}-P_{i}\right) V_{di}\left( p\right) =\frac{h_{3}}{n\left( n-1\right) } \sum_{i}\sum_{j\neq i}\frac{m_{3}\left( W_{i},W_{j}\right) }{\hat{f} _{X}\left( X_{i}\right) }=h_{3}M_{3}\cdot \left( 1+O_{p}\left( \eta _{1}\right) \right) . \end{equation*} Since \begin{eqnarray*} E\left[ \frac{m_{3}^{2}\left( W_{i},W_{j}\right) }{f_{X}^{2}\left( X_{i}\right) }\right] &=&O\left( \frac{1}{h_{3}^{3}\left\vert h_{1}\right\vert }\right) =o\left( n\right) , \\ E\left[ \left. \frac{m_{3}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{i}\right] &=&O\left( tr\left( h_{1}^{s}\right) \right) \cdot \frac{1}{h_{3}}w_{di}\left( p\right) V_{di}\left( p\right) , \\ E\left[ \left. \frac{m_{3}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{j}\right] &=&\delta _{d}\left( P_{j}\right) \left[ g_{d}\left( P_{j}\right) -g_{d}\left( p\right) \right] \frac{1}{ h_{3}^{2}}k_{3}\left( \frac{P_{j}-p}{h_{3}}\right) \left( D_{j}-P_{j}\right) \left[ 1+O\left( tr\left( h_{1}^{s}\right) \right) \right] , \\ E\left[ \frac{m_{3}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) } \right] &=&0, \end{eqnarray*} it follows from the projection of U-statistics that \begin{eqnarray*} M_{3} &=&\frac{1}{n}\sum_{i=1}^{n}E\left[ \left. \frac{m_{3}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{i}\right] + \frac{1}{n}\sum_{j=1}^{n}E\left[ \left. \frac{m_{3}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }\right\vert W_{j}\right] +o_{p}\left( \frac{1}{ \sqrt{n}}\right) \\ &=&\frac{1}{n}\sum_{j=1}^{n}\delta _{d}\left( P_{j}\right) \frac{g_{d}\left( P_{j}\right) -g_{d}\left( p\right) }{h_{3}^{2}}k_{3}\left( \frac{P_{j}-p}{ h_{3}}\right) \left( D_{j}-P_{j}\right) +o_{p}\left( \frac{1}{\sqrt{nh_{3}}} \right) \\ &=&\frac{g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{j=1}^{n}\delta _{d}\left( P_{j}\right) \frac{P_{j}-p}{h_{3}^{2}}k_{3}\left( \frac{P_{j}-p}{ h_{3}}\right) \left( D_{j}-P_{j}\right) +o_{p}\left( \frac{1}{\sqrt{nh_{3}}} \right) . \end{eqnarray*} Therefore, \begin{eqnarray} \frac{1}{n}\sum_{i=1}^{n}w_{di}\left( p\right) \left( \hat{P} _{i}-P_{i}\right) V_{di}\left( p\right) &=&\frac{g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) k_{3}\left( \frac{P_{i}-p}{h_{3}}\right) \frac{P_{i}-p}{h_{3}}\left( D_{i}-P_{i}\right) \notag \\ &&+o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) . \end{eqnarray} For the second term in the decomposition ((ref)), a Taylor expansion gives \begin{eqnarray*} &&\frac{1}{n}\sum_{i}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( P_{i}-p\right) V_{di}\left( p\right) \\ &=&\frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} V_{di}\left( p\right) \frac{1}{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}} \right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) \\ &&+\frac{1}{2n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} V_{di}\left( p\right) \frac{1}{h_{3}^{3}}k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}} \right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{2} \\ &&+\frac{1}{6n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} V_{di}\left( p\right) \frac{1}{h_{3}^{4}}k_{3}^{\left( 3\right) }\left( \frac{\hat{P}_{i}^{\ast }-p }{h_{3}}\right) \left( P_{i}-p\right) \left( \hat{P}_{i}-P_{i}\right) ^{3} \\ &\triangleq &\left( \text{IV}\right) +\left( \text{V}\right) +\left( \text{VI }\right) . \end{eqnarray*} Denote \begin{eqnarray*} m_{4}\left( W_{i},W_{j}\right) &=&1\left\{ D_{i}=d\right\} V_{di}\left( p\right) \frac{1}{h_{3}^{3}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{ h_{3}}\right) \left( P_{i}-p\right) \left( D_{j}-P_{i}\right) K_{1h}\left( X_{j},X_{i}\right) , \\ M_{4} &=&\frac{1}{n\left( n-1\right) }\sum_{i}\sum_{j\neq i}\frac{ m_{4}\left( W_{i},W_{j}\right) }{f_{X}\left( X_{i}\right) }, \end{eqnarray*} then we have \begin{equation*} \left( \text{IV}\right) =\frac{h_{3}}{n\left( n-1\right) } \sum_{i}\sum_{j\neq i}\frac{m_{4}\left( W_{i},W_{j}\right) }{\hat{f} _{X}\left( X_{i}\right) }=h_{3}M_{4}\cdot \left( 1+O_{p}\left( \eta _{1}\right) \right) . \end{equation*} Analogously, it follows from the projection of U-statistics that \begin{eqnarray*} M_{4} &=&\frac{1}{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) \frac{ g_{d}\left( P_{i}\right) -g_{d}\left( p\right) }{h_{3}^{3}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \left( D_{i}-P_{i}\right) +o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) \\ &=&\frac{g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) \frac{1}{h_{3}}k_{3}^{\left( 1\right) }\left( \frac{ P_{i}-p}{h_{3}}\right) \left( \frac{P_{i}-p}{h_{3}}\right) ^{2}\left( D_{i}-P_{i}\right) +o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) , \end{eqnarray*} and hence \begin{equation*} \left( \text{IV}\right) =\frac{g_{d}^{\left( 1\right) }\left( p\right) }{n} \sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( \frac{P_{i}-p}{h_{3}}\right) ^{2}\left( D_{i}-P_{i}\right) +o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) . \end{equation*} Moreover, it follows from Lemma (ref) that \begin{eqnarray*} \left\vert \left( \text{V}\right) \right\vert &\leq &O_{p}\left( \eta _{1}^{2}\right) \cdot O_{p}\left( E\left\vert V_{di}\left( p\right) \frac{1}{ h_{3}^{3}}k_{3}^{\left( 2\right) }\left( \frac{P_{i}-p}{h_{3}}\right) \left( P_{i}-p\right) \right\vert \right) =O_{p}\left( \eta _{1}^{2}\right) =o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) , \\ \left\vert \left( \text{VI}\right) \right\vert &\leq &O_{p}\left( \eta _{1}^{3}\right) \cdot O_{p}\left( E\left\vert V_{di}\left( p\right) \frac{1}{ h_{3}^{4}}k_{3}^{\left( 3\right) }\left( \frac{\hat{P}_{i}^{\ast }-p}{h_{3}} \right) \left( P_{i}-p\right) \right\vert \right) =O_{p}\left( \frac{\eta _{1}^{3}}{h_{3}^{3}}\right) =o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) . \end{eqnarray*} Therefore, we have \begin{eqnarray} &&\frac{1}{n}\sum_{i}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( P_{i}-p\right) V_{di}\left( p\right) \notag \\ &=&\frac{g_{d}^{\left( 1\right) }\left( p\right) }{n}\sum_{i=1}^{n}\delta _{d}\left( P_{i}\right) k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}} \right) \left( \frac{P_{i}-p}{h_{3}}\right) ^{2}\left( D_{i}-P_{i}\right) +o_{p}\left( \frac{1}{\sqrt{nh_{3}}}\right) . \end{eqnarray} For the third term in the decomposition ((ref)), a Taylor expansion gives \begin{eqnarray} &&\frac{1}{n}\sum_{i}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left( \hat{P}_{i}-P_{i}\right) V_{di}\left( p\right) \notag \\ &=&\frac{1}{n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} V_{di}\left( p\right) \frac{1}{h_{3}^{2}}k_{3}^{\left( 1\right) }\left( \frac{P_{i}-p}{h_{3}} \right) \left( \hat{P}_{i}-P_{i}\right) ^{2} \notag \\ &&+\frac{1}{2n}\sum_{i=1}^{n}1\left\{ D_{i}=d\right\} V_{di}\left( p\right) \frac{1}{h_{3}^{3}}k_{3}^{\left( 2\right) }\left( \frac{\hat{P}_{i}^{\ast }-p }{h_{3}}\right) \left( \hat{P}_{i}-P_{i}\right) ^{3} \notag \\ &=&O_{p}\left( \eta _{1}^{2}\right) +O_{p}\left( \frac{\eta _{1}^{3}}{ h_{3}^{3}}\right) =o_{p}\left( \sqrt{\frac{h_{3}}{n}}\right) . \end{eqnarray} The second claim of the lemma then follows from inserting ((ref) ), ((ref)), and ((ref)) into the decomposition ((ref)). \end{proof} \begin{lemma} Under Assumptions E.1, E.2, E.3, and E.4, we have \begin{eqnarray*} \frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) X_{i} &=&O_{p}\left( 1\right) , \\ \frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) X_{i} &=&O_{p}\left( h_{3}\right) . \end{eqnarray*} \end{lemma} \begin{proof} Note that \begin{equation} \frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) X_{i}=\frac{1}{n} \sum_{i}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] X_{i}+\frac{1}{n}\sum_{i}^{n}w_{di}\left( p\right) X_{i}. \end{equation} Similar to the first claim of Lemma (ref), we can show that \begin{equation} \frac{1}{n}\sum_{i}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] X_{i}=O_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) =o_{p}\left( 1\right) . \end{equation} For the second term, we have \begin{equation*} E\left[ \frac{1}{n}\sum_{i}^{n}w_{di}\left( p\right) X_{i}\right] =O\left( 1\right) \end{equation*} and \begin{equation*} Var\left( \frac{1}{n}\sum_{i}^{n}w_{di}\left( p\right) X_{i}\right) =O\left( \frac{1}{nh_{3}}\right) =o\left( 1\right) , \end{equation*} hence, \begin{equation} \frac{1}{n}\sum_{i}^{n}w_{di}\left( p\right) X_{i}=O_{p}\left( 1\right) . \end{equation} The first claim of the lemma follows from substituting ((ref)) and ( (ref)) into ((ref)). For the second claim, it follows from the compactness of the support of $ k_{3}\left( \cdot \right) $ imposed in Assumption E.2.(iii) that \begin{equation*} \left\Vert \frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) X_{i}\right\Vert \leq \frac{1}{n}\sum_{i}^{n}\hat{w} _{di}\left( p\right) \left\vert \hat{P}_{i}-p\right\vert \left\Vert X_{i}\right\Vert =O\left( h_{3}\right) \cdot \frac{1}{n}\sum_{i}^{n}\hat{w} _{di}\left( p\right) \left\Vert X_{i}\right\Vert . \end{equation*} As above, we can show that \begin{eqnarray*} \frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) \left\Vert X_{i}\right\Vert &=&\frac{1}{n}\sum_{i}^{n}\left[ \hat{w}_{di}\left( p\right) -w_{di}\left( p\right) \right] \left\Vert X_{i}\right\Vert +\frac{1 }{n}\sum_{i}^{n}w_{di}\left( p\right) \left\Vert X_{i}\right\Vert \\ &=&O_{p}\left( \frac{1}{\sqrt{nh_{3}^{3}}}\right) +O_{p}\left( 1\right) =O_{p}\left( 1\right) . \end{eqnarray*} Therefore, \begin{equation*} \frac{1}{n}\sum_{i}^{n}\hat{w}_{di}\left( p\right) \left( \hat{P} _{i}-p\right) X_{i}=O_{p}\left( h_{3}\right) . \end{equation*} \end{proof} \section{Implementation guidance} Table (ref) summarizes our estimation procedure and preliminary tests for the assumptions. In general, we recommend that the data cell we choose for testing should have at least 50 observations. For example, there are 40 among all 60 data cells in Table (ref) satisfying this criterion. If we examine the supports of the 40 data cells one by one using the verification method as in our application, we will find that Assumption S holds for all of them. Moreover, we plot the conditional propensity score functions for all the 40 data cells in Figure (ref), which verifies that Assumption NL1 (nonmonotonicity) holds for all of them as well. In other words, we can verify the validity of Assumptions S and NL1 in our application regardless of which data cell we choose. This shows that Assumptions S and NL1 are quite weak and hardly limit the applicability of our identification strategy in practice. On the other hand, if the number of discrete covariates is not small and/or all data cells have less than 50 observations, we should consider to use a semiparametric or parametric approach that imposes a linear index structure on the treatment function. In this case, we should remember to include nonlinear terms of continuous covariates in the linear index to ensure the identification. \setcounter{equation}{0} \section{Simulation} We examine the finite sample property of the MTE estimation based on our non-IV identification strategy by a Monte Carlo simulation. The treatment variable data are generated according to the structural treatment equation ( (ref)): \begin{equation*} D_{i}=1\left\{ \mu \left( X_{i}\right) \geq U_{i}\right\} ,\text i=1,2,\cdots ,n, \end{equation*} where the function $\mu \left( \cdot \right) $ and the distribution of error $U_{i}$ will vary across different designs. The vector $X_{i}$ consists of one continuous covariate and five discrete covariates, namely, $X_{i}=\left( X_{i}^{C},X_{1i}^{D},X_{2i}^{D},X_{3i}^{D},X_{4i}^{D},X_{5i}^{D}\right) ^{\prime }$, where $X_{i}^{C}$ follows a standard normal distribution and $ X_{ji}^{D}$ ($j=1,\cdots ,5$) are independent $\left\{ 0,1\right\} $-valued binary random variables with identical distribution that $\Pr \left( X_{ji}^{D}=1\right) =0.5$. The potential and observed outcomes are generated by \begin{eqnarray*} Y_{1i} &=&\beta _{1}^{C}X_{i}^{C}+\sum_{j=1}^{5}\left( X_{ji}^{D}-0.5\right) +U_{1i}, \\ Y_{0i} &=&\beta _{0}^{C}X_{i}^{C}+\sum_{j=1}^{5}\frac{1}{6-j}\left( X_{ji}^{D}-0.5\right) +U_{0i}, \\ Y_{i} &=&D_{i}Y_{1i}+\left( 1-D_{i}\right) Y_{0i},\texti=1,2,\cdots ,n, \end{eqnarray*} where $U_{1i}$ is generated by $U_{1i}=-0.75U_{i}+\sqrt{1-0.75^{2}}\varrho _{i}$ with a standard normal random variable $\varrho _{i}$ that is independent of $U_{i}$ and $X_{i}$, and $U_{0i}$ follows an independent standard normal distribution as well. The sample size $n$ is set 4000. We consider 16 different designs. In the benchmark design (Design 1), the structural treatment error $U_{i}$ follows a standard normal distribution that is independent of $X_{i}$, and the treatment function $\mu \left( \cdot \right) $ is quadratic in the continuous covairate and linear in the discrete covariates. Specifically, we set \begin{equation} \mu \left( X_{i}\right) =X_{i}^{C}+\frac{1}{2}\left[ \left( X_{i}^{C}\right) ^{2}-1\right] +\sum_{j=1}^{5}\frac{1}{j}\left( X_{ji}^{D}-0.5\right) . \end{equation} Note that $E\left[ \mu \left( X_{i}\right) \right] =0$ and the probability of treatment is about $0.5$. In addition, we impose the exclusion restriction $\beta _{1}^{C}=\beta _{0}^{C}=0$ in Design 1. Other designs include cases of no exclusion restriction, linear or cubic treatment function $\mu \left( \cdot \right) $, nonnormal treatment error $U_{i}$, nonzero correlation between $U_{i}$ and $X_{i}^{C}$ as in Example (ref), heteroskedastic $U_{i}$ or $U_{di}$, nonnormal forms of MTE, and nonlinear outcome equations. Table (ref) collects detailed descriptions of the designs. For each design, the simulation replicates 1000 times. In every replication $b=1,\cdots ,1000$, we compute different estimators for the MTE evaluated at $x=\bar{X}$, the sample mean of $X_{i}$, and $ v=0.01,0.02,\cdots ,0.99$. The first estimator is fully parametric, assuming jointly normally distributed errors and quadratic $\mu \left( \cdot \right) $ : $\mu \left( X_{i}\right) =\gamma _{0}+\gamma _{1}X_{i}^{C}+\gamma _{2}\left( X_{i}^{C}\right) ^{2}+\sum_{j=1}^{5}\gamma _{2+j}X_{ji}^{D}$. The normal assumption leads to the widely-used Heckman's two-step estimation. Nevertheless, the parametric estimator has a high risk of misspecification. More flexible estimators comprise of a nonparametric first step ((ref)) and a parametric second step. In the first step, we use the second-order ($s=2$) Gaussian kernel as $k_{1}\left( \cdot \right) $ because higher-order kernels are likely to result in unstable estimates in practice pan2022semiparametric, and use the rule-of-thumb bandwidth as given in Table (ref). In the second step, we consider three parametric specifications on the unobservable heterogeneity of MTE, namely, a normal specification that $E\left[ \left. U_{di}\right\vert V_{i}=v\right] =\rho _{d0}+\rho _{dV}\Phi ^{-1}\left( v\right) $, a linear specification that $E\left[ \left. U_{di}\right\vert V_{i}=v\right] =\theta _{d0}+\theta _{d1}v$, and a quadratic specification that $E\left[ \left. U_{di}\right\vert V_{i}=v\right] =\theta _{d0}+\theta _{d1}v+\theta _{d2}v^{2}$, where $V_{i}=F_{\left. U\right\vert X}\left( \left. U_{i}\right\vert X_{i}\right) $ is the reduced-form treatment error. The most flexible estimator under Assumptions L and NL is the semiparametric one that comprises of a nonparametric first step and a partially linear second step, leading to the kernel-weighted pairwise difference estimator ((ref))-((ref)). We use the Gaussian kernel and rule-of-thumb bandwidths as well. Theorem (ref) has established the asymptotic normality of this semiparametric MTE estimator. Recall that all the above estimators impose no exclusion restriction and base the identification on different functional forms between the treatment and outcome equations. As a comparison, we also consider the conventional IV-based MTE estimator under the semiparametric specification, whose implementation differs from the semiparametric non-IV MTE estimator only in that it assumes the exclusion restriction $\beta _{1}^{C}=\beta _{0}^{C}=0$. For each MTE estimator, we calculate the average over replications, that is, $\left. \sum_{b=1}^{1000} \hat{\Delta}_{b}^{\text{MTE}}\left( \bar{X},v\right) \right/ 1000$. And we depict the averaged MTE curves, along with the true MTE curve, for each design in Figures (ref)-(ref). In the benchmark design, the parametric and normal MTE estimators are correctly specified. Therefore, we can see from the upper-left panel of Figure (ref) that the parametric MTE performs perfectly and the normal MTE comes in second, and both of them are superior to the semiparametric one which exploits the least information of the model. Since the exclusion restriction holds in this design, the IV-based MTE is also consistent and performs well. However, by comparing the two semiparametric estimators, we find little improvement of the IV-based MTE over the non-IV MTE, implying that the exclusion restriction provides not much extra identifying power when the identification based on functional forms holds. The misspecified linear and quadratic MTE estimators perform satisfactorily as well, mainly because the true MTE curve is close to linear. When the exclusion restriction is violated as in Design 2, the upper-right panel of Figure (ref) shows that the IV-based MTE estimator deteriorates drastically due to misspecification. In Design 3, Assumption NL is violated and the identification based on functional forms fails to hold. The bottom-left panel of Figure (ref) confirms that in this case, the normal, linear, quadratic, and semiparametric non-IV MTE estimators are underidentified, while the IV-based MTE provides much improvement over its non-IV counterpart. Somewhat surprisingly, however, the parametric MTE still performs perfectly, suggesting a potential success of the classical parametric identification based on functional forms, even when the semiparametric identification fails. The bottom-right panel of Figure (ref) shows that estimation based on functional forms is more robust than that based on IV to the violation of identifying conditions. Besides the curves, we also summarize the simulation results of the MTE estimators by the bias $\left. \sum_{b=1}^{1000}\left[ \hat{\Delta}_{b}^{ \text{MTE}}\left( \bar{X},v\right) -\Delta ^{\text{MTE}}\left( \bar{X} ,v\right) \right] \right/ 1000$, the root mean squared error (RMSE) $\sqrt{ \left. \sum_{b=1}^{1000}\left[ \hat{\Delta}_{b}^{\text{MTE}}\left( \bar{X} ,v\right) -\Delta ^{\text{MTE}}\left( \bar{X},v\right) \right] ^{2}\right/ 1000}$, and the coverage probability of 95% confidence interval $\left. \sum_{b=1}^{1000}1\left\{ \left\vert \hat{\Delta}_{b}^{\text{MTE}}\left( \bar{X},v\right) -\Delta ^{\text{MTE}}\left( \bar{X},v\right) \right\vert \leq 1.96\cdot SE\left( \hat{\Delta}^{\text{MTE}}\left( \bar{X},v\right) \right) \right\} \right/ 1000$, where $SE\left( \hat{\Delta}^{\text{MTE} }\left( \bar{X},v\right) \right) $ is approximated by the simulated standard deviation of the estimator. Table (ref) reports the summary statistics of $\hat{\Delta}_{b}^{\text{MTE}}\left( \bar{X},0.5\right) $ for Designs 1-4, along with those of the ATE estimator and of $\hat{\beta} _{1}^{C}-\hat{\beta}_{0}^{C}$ that is the estimated partial effect of the continuous covariate on treatment effects. In Design 1, the coverage probabilities of all the estimators are close to the true size 95%, partly demonstrating the validity of a simple $t$-test when the identification is attained. In Design 2, the IV-based estimator is heavily biased in both estimation and inference, while in Design 3, the non-IV estimators are biased except the fully parametric one, which is consistent with the finding of Figure (ref). The summary statistics in Design 4 show that estimation based on functional forms is much more robust than that based on IV in terms of inference as well. In the remaining designs, we relax the exclusion restriction and no longer report the results for the IV-based estimator. Design 5 considers a case that allows for nonzero correlation between the continuous covariate and structural treatment error but Assumption CMI holds, as in Example (ref). The upper-left panel of Figure (ref) shows that the parametric MTE curve is still almost completely overlapped with the true MTE curve, while the other four MTE curves are more biased than in the uncorrelated case (Designs 1-2). Although biased, the semiparametric MTE estimator is fairly robust in terms of inference as shown by Table (ref). Design 6 is inspired by Example (ref), where nonlinearity comes from the multiplicative heteroscedasticity in the treatment error. In this design, the quadratic treatment functional form of the parametric MTE is wrongly specified. Although the misspecification does not lead to large bias in estimation, it does lead to apparent invalidation of inference revealed by the severely biased coverage probability of the parametric estimator. In comparison, the normal estimator that allows for a nonparametric treatment equation performs satisfactorily in terms of the coverage probability. Designs 7 and 8 specify normal polynomial functional forms on the unobservable heterogeneity of MTE, under which the considered parametric second steps are all misspecified. Therefore, the semiparametric estimator that allows for nonparametric functional form of the unobservable heterogeneity of MTE has the best overall performance in these two designs, as illustrated by the lower panels of Figure (ref) and the right half part of Table (ref). Figures (ref) and (ref) depict the MTE curves in additional designs. Designs 9 and 10 consider nonnormally distributed treatment error $U_{i}$, under which the parametric MTE estimator is misspecified and expected to be inconsistent. However, the upper-left panel of Figure (ref) shows that the parametric MTE still works in Design 9. This is mainly because the MTE estimator depends only on the propensity score (namely, the conditional expectation of treatment) but not on regression coefficients of the treatment equation. The estimation of the propensity score is actually a prediction problem. Some forms of misspecification may have a less serious impact on the prediction than on the regression. In Design 9, the $t\left( 3\right) $ distribution of the treatment error is symmetric about zero and resembles the normal distribution, thus exerting only a small influence on the parametric MTE. In comparison, the $t\left( 2\right) $ distributed treatment error in Design 10 is more fat-tailed and therefore leads to more biased parametric MTE. The nonnormally distributed treatment error also results in nonnormal unobservable heterogeneity of MTE. As in Designs 7 and 8, the semiparametric MTE performs the best overall in such case. Design 11 considers a cubic treament function $\mu \left( \cdot \right) $, under which the parametric MTE is notably biased due to severe misspecification. The last panel of Figure (ref) for Design 12 shows that the heteroskedasticity in outcome errors makes little difference to the performance of the MTE estimators by comparing with the benchmark design. Figure (ref) investigates the impact of nonlinearity in the outcome equation, and the results are also as expected. That is, if we correctly include the nonlinear terms during implementation, the MTE estimator still performs well as shown by the dotted lines that stand for the correctly specified semiparametric estimator. However, if we omit the nonlinear terms in the outcome equation, the MTE estimator may be biased, and the magnitude of bias depends on the specific forms of nonlinearity. When the nonlinear term is an interaction of two discrete covariates as in Design 13, the omission of it has quite limited influence on the MTE estimators. Nevertheless, the omitted variable problem will become obvious if the interaction term pertains to the continuous covariate as in Design 14. The lower panels of Figure (ref) further show that omitting a quadratic term or an exponential term in the outcome equation will induce more substantial bias in the estimation of MTE. In summary, the fully parametric estimator and the estimators that specify nonparametric first step and parametric second step are the first choice when we are confident that the specification is correct or at least nearly correct. Otherwise, the semiparametric estimator is perferred because its performance is the most robust across different designs. \setcounter{table}{0} \setcounter{figure}{0} \begin{sidewaystable}[htb] \caption{An overview of the testing and estimation procedures} \resizebox{\linewidth}{!}{ \begin{tabular}{lrL{27cm}} \hline\hline \multicolumn{3}{l}{\textbf{Step 0}: Test Assumptions S and NL} \\ \cline{2-3} & \textbf{Step 0.1}: & List the data cells for all possible values of discrete covariates and the corresponding conditional supports of the propensity score, as in Table (ref). \\ & \textbf{Step 0.2}: & Find a data cell that satisfies Assumption S and has at least 50 observations. \\ & & \textit{Note}: A method of quick validation of Assumption S is to check whether the present data cell has full support or whether its support is overlapped with all other data cells' supports. \\ & \textbf{Step 0.3}: & Test Assumption NL for the data cell selected by Step 0.2. \\ & & \textit{If there is only one continuous covariate}, then plot the propensity score function for the selected data cell as in Figure (ref), and visually inspect whether the function is nonmonotonic. \\ & & \textit{If there are at least two continuous covariates}, then test the nonlinearity of the propensity score function by methods from the literature on single-index models fan1996consistent, stute2005nonparametric, xia2009model, maistre2019nonparametric. \\ & \textbf{Step 0.4}: & If Assumption NL holds, go to Step 1. Otherwise, return to Step 0.2-0.3 until finding one data cell for which Assumptions S and NL both hold. \\ \hline \multicolumn{3}{l}{\textbf{Step 1}: Estimate the treatment equation $ D=1\left\{ \mu \left( X\right) \geq U\right\} =1\left\{ \pi \left( X\right) \geq V\right\} $} \\ \cline{2-3} & \textbf{Case 1.1}: & Nonparametric (kernel) estimation. Suppose that both the functional form of $\mu \left( \cdot \right) $ and the distribution of the structural treatment error $U$ are nonparametrically specified. Then, estimate the propensity score $\pi \left( X\right) $ by the kernel estimator ((ref)). \\ & & \textit{Note}: We recommend to use the Gaussian kernel (i.e., standard normal density) as $k_{1}\left( \cdot \right) $ and the rule-of-thumb bandwidth $h_{1}=sd\left( X^{C}\right) \cdot n^{-1\left/ \left( 4+\dim \left( X^{C}\right) \right) \right. }$, where $sd\left( X^{C}\right) $ is the sample standard deviation of the continuous covariate. \\ & \textbf{Case 1.2}: & Semiparametric (single-index) estimation. Suppose that $\mu \left( \cdot \right) $ is parametrically specified as a linear index and the distribution of $U$ is nonparametrically specified. Then, estimate $\pi \left( X\right) $ by methods from the literature on single-index models powell1989semiparametric, ichimura1993semiparametric, klein1993efficient, lewbel2000semiparametric. \\ & \textbf{Case 1.3}: & Parametric (Probit/Logit) estimation. Suppose that $ \mu \left( \cdot \right) $ is parametrically specified and $U$ is assumed to follow a normal/logistic distribution. Then, perform a Probit/Logit regression on the treatment equation and predict $\pi \left( X\right) $. \\ \hline \multicolumn{3}{l}{\textbf{Step 2}: Estimate the outcome equation ((ref)) and MTE ((ref))} \\ \cline{2-3} & \textbf{Case 2.1}: & Semiparametric (partially linear) estimation. Suppose that the functional form of $g_{d}\left( \cdot \right) $ is nonparametrically specified.\ Then, estimate $\beta _{d}$ by the kernel-weighted pairwise difference estimator ((ref)) and $g_{d}\left( \cdot \right) $ by the local linear estimator. \\ & & \textit{Note}: We recommend to use the Gaussian kernel and rule-of-thumb bandwidths $h_{2}=h_{3}=sd\left( \hat{P}\right) \cdot n^{-1\left/ 5\right. }$ when estimating $\beta _{d}$ and $g_{d}\left( \cdot \right) $. \\ & \textbf{Case 2.2}: & Parametric estimation. Suppose that $g_{d}\left( \cdot \right) $ is parametrically specified as $g_{d}\left( \cdot ;\theta _{d}\right) $ for a finite-dimensional parameter $\theta _{d}$. We consider three commonly used specifications. \\ & & \textbf{Case 2.2.1}: Polynomial specification. Suppose that $ g_{d}\left( \cdot \right) $ is a polynomial function. Then, regress $Y$ on $ X $, $\hat{P}$, and the powers of $\hat{P}$ for the treated and untreated groups to estimate $\beta _{d}$ and $g_{d}\left( \cdot \right) $. \\ & & \textbf{Case 2.2.2}: Normal specification. Suppose that $\left( U,U_{d}\right) $ follows a bivariate normal distribution, which implies $ g_{d}\left( p\right) =\theta _{1d}\left. \phi \left( \Phi ^{-1}\left( p\right) \right) \right/ \left( p-1+d\right) $. Therefore, regress $Y$ on $X$ and $\left. \phi \left( \Phi ^{-1}\left( \hat{P}\right) \right) \right/ \left( \hat{P}-1+d\right) $ for each group $d=0,1$ to estimate $\beta _{d}$ and $g_{d}\left( \cdot \right) $. \\ & & \textbf{Case 2.2.3}: Normal polynomial specification, which includes the normal specification as a special case when the order of polynomial is one. If the order is set two, then add a term $\Phi ^{-1}\left( \hat{P} \right) \left. \phi \left( \Phi ^{-1}\left( \hat{P}\right) \right) \right/ \left( \hat{P}-1+d\right) $ to the above regression. If the order is set three, further add $\left( \Phi ^{-2}\left( \hat{P}\right) +2\right) \left. \phi \left( \Phi ^{-1}\left( \hat{P}\right) \right) \right/ \left( \hat{P} -1+d\right) $. If the order is set four, further add $\left( \Phi ^{-3}\left( \hat{P}\right) +3\Phi ^{-1}\left( \hat{P}\right) \right) \left. \phi \left( \Phi ^{-1}\left( \hat{P}\right) \right) \right/ \left( \hat{P} -1+d\right) $, and so on. \\ & \textbf{Step 2.3}: & Estimate the MTE. After obtaining the estimates of $ \beta _{d}$ and $g_{d}\left( \cdot \right) $, plug them into the equation ( (ref)) to construct the estimator for the MTE. \\ \hline\hline \end{tabular}} \end{sidewaystable} \begin{table}[htb] \caption{Support of the nonparametrically estimated propensity score in each data cell} { \begin{tabular}{ccccccc} \hline\hline Data cell & Age & Gender & Race & Parental education & Observations & Support of $\hat{\pi} _{0}\left( X^{C}\right) $ \\ \hline 1 & 30 & Male & Hispanic & High school or lower & 70 & [0, 0.208] \\ 2 & 30 & Male & Hispanic & College or higher & 10 & [0, 0.288] \\ 3 & 30 & Male & Black & High school or lower & 93 & [0, 0.588] \\ 4 & 30 & Male & Black & College or higher & 21 & [0, 0.813] \\ 5 & 30 & Male & White & High school or lower & 147 & [0, 0.177] \\ 6 & 30 & Male & White & College or higher & 78 & [0, 0.208] \\ 7 & 30 & Female & Hispanic & High school or lower & 65 & [0, 0.434] \\ 8 & 30 & Female & Hispanic & College or higher & 11 & 0 \\ 9 & 30 & Female & Black & High school or lower & 100 & [0, 0.671] \\ 10 & 30 & Female & Black & College or higher & 20 & [0, 0.641] \\ 11 & 30 & Female & White & High school or lower & 125 & [0, 0.759] \\ 12 & 30 & Female & White & College or higher & 73 & [0, 0.179] \\ 13 & 31 & Male & Hispanic & High school or lower & 87 & [0, 0.374] \\ 14 & 31 & Male & Hispanic & College or higher & 19 & [0, 0.442] \\ 15 & 31 & Male & Black & High school or lower & 112 & [0.030, 0.548] \\ 16 & 31 & Male & Black & College or higher & 24 & [0, 0.531] \\ 17 & 31 & Male & White & High school or lower & 148 & [0, 0.242] \\ 18 & 31 & Male & White & College or higher & 92 & [0, 0.125] \\ 19 & 31 & Female & Hispanic & High school or lower & 97 & [0, 0.737] \\ 20 & 31 & Female & Hispanic & College or higher & 14 & [0, 0.788] \\ 21 & 31 & Female & Black & High school or lower & 121 & [0.249, 0.817] \\ 22 & 31 & Female & Black & College or higher & 24 & [0, 0.639] \\ 23 & 31 & Female & White & High school or lower & 149 & [0, 0.178] \\ 24 & 31 & Female & White & College or higher & 75 & [0, 0.221] \\ 25 & 32 & Male & Hispanic & High school or lower & 65 & [0, 0.771] \\ 26 & 32 & Male & Hispanic & College or higher & 13 & 0 \\ 27 & 32 & Male & Black & High school or lower & 133 & [0, 0.623] \\ 28 & 32 & Male & Black & College or higher & 23 & [0, 0.949] \\ 29 & 32 & Male & White & High school or lower & 175 & [0, 0.194] \\ 30 & 32 & Male & White & College or higher & 90 & [0, 0.061] \\ \hline\hline \end{tabular} } \end{table} \setcounter{table}{1} \begin{table}[htb] \caption{Support of the nonparametrically estimated propensity score in each data cell} { \begin{tabular}{ccccccc} \hline\hline Data cell & Age & Gender & Race & Parental education & Observations & Support of $\hat{\pi} _{0}\left( X^{C}\right) $ \\ \hline 31 & 32 & Female & Hispanic & High school or lower & 94 & [0, 0.642] \\ 32 & 32 & Female & Hispanic & College or higher & 17 & [0, 0.351] \\ 33 & 32 & Female & Black & High school or lower & 130 & [0.021, 0.999] \\ 34 & 32 & Female & Black & College or higher & 34 & [0, 1] \\ 35 & 32 & Female & White & High school or lower & 155 & [0, 0.526] \\ 36 & 32 & Female & White & College or higher & 81 & [0, 0.118] \\ 37 & 33 & Male & Hispanic & High school or lower & 72 & [0, 0.567] \\ 38 & 33 & Male & Hispanic & College or higher & 15 & [0, 0.743] \\ 39 & 33 & Male & Black & High school or lower & 123 & [0, 0.818] \\ 40 & 33 & Male & Black & College or higher & 25 & [0, 0.596] \\ 41 & 33 & Male & White & High school or lower & 141 & [0, 0.154] \\ 42 & 33 & Male & White & College or higher & 95 & [0, 0.133] \\ 43 & 33 & Female & Hispanic & High school or lower & 83 & [0, 0.481] \\ 44 & 33 & Female & Hispanic & College or higher & 10 & [0, 0.447] \\ 45 & 33 & Female & Black & High school or lower & 125 & [0.258, 0.628] \\ 46 & 33 & Female & Black & College or higher & 27 & [0, 0.925] \\ 47 & 33 & Female & White & High school or lower & 152 & [0, 0.140] \\ 48 & 33 & Female & White & College or higher & 93 & [0, 0.350] \\ 49 & 34 & Male & Hispanic & High school or lower & 85 & [0, 0.864] \\ 50 & 34 & Male & Hispanic & College or higher & 16 & [0, 0.673] \\ 51 & 34 & Male & Black & High school or lower & 101 & [0.176, 1] \\ 52 & 34 & Male & Black & College or higher & 27 & [0, 0.790] \\ 53 & 34 & Male & White & High school or lower & 162 & [0, 0.204] \\ 54 & 34 & Male & White & College or higher & 85 & [0, 0.057] \\ 55 & 34 & Female & Hispanic & High school or lower & 73 & [0, 0.443] \\ 56 & 34 & Female & Hispanic & College or higher & 11 & 0 \\ 57 & 34 & Female & Black & High school or lower & 109 & [0.271, 0.586] \\ 58 & 34 & Female & Black & College or higher & 24 & [0, 0.856] \\ 59 & 34 & Female & White & High school or lower & 143 & [0, 0.5] \\ 60 & 34 & Female & White & College or higher & 76 & [0, 0.055] \\ \hline\hline \end{tabular} } \end{table} \begin{figure}[tbh] \caption{Test of Assumption NL1 for all data cells with size larger than 50} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \end{figure} \setcounter{figure}{0} \begin{figure}[tbh] \caption{Test of Assumption NL1 for all data cells with size larger than 50} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.245\linewidth} \centerline \end{minipage} \end{figure} \setcounter{table}{0} \setcounter{figure}{0} \begin{sidewaystable}[htb] \caption{Simulation designs} \resizebox{\linewidth}{!}{ \begin{tabular}{clC{11.8cm}cc} \hline\hline Design & Exclusion restriction & Other difference from Design 1 & Consistent estimators & Inconsistent estimators \\ \hline 1 & Yes: $\beta _{1}^{C}=\beta _{0}^{C}=0$ & - & P, N, S, SIV & L, Q \\ 2 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & - & P, N, S & L, Q, SIV \\ 3 & Yes: $\beta _{1}^{C}=\beta _{0}^{C}=0$ & Linear treatment function: $\mu \left( X_{i}\right) =X_{i}^{C}+\sum_{j=1}^{5}\frac{1}{j}\left( X_{ji}^{D}-0.5\right) $. & SIV & P, N, L, Q, S \\ 4 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & Linear treatment function: $ \mu \left( X_{i}\right) =X_{i}^{C}+\sum_{j=1}^{5}\frac{1}{j}\left( X_{ji}^{D}-0.5\right) $. & - & P, N, L, Q, S, SIV \\ 5 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & $X_{i}^{C}=\sqrt{1-0.75^{2}} U_{i}+0.75\varrho _{i}$ s.t. $Cov\left( U_{i},X_{i}^{C}\right) \neq 0$. & P, N, S & L, Q \\ 6 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & (i) Linear treatment function: $\mu \left( X_{i}\right) =X_{i}^{C}+\sum_{j=1}^{5}\frac{1}{j}\left( X_{ji}^{D}-0.5\right) $, and (ii) Heteroskedastic $U_{i}$: $U_{i}=\exp \left( X_{i}^{C}+\frac{1}{2}\left[ \left( X_{i}^{C}\right) ^{2}-1\right] \right) \epsilon _{i}$, where $\epsilon _{i}\sim N\left( 0,1\right) $ and is independent of $X_{i}^{C}$. & N, S & P, L, Q \\ 7 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & Normal quadratic MTE: $ U_{1i}=-\left( \frac{3}{8}U_{i}^{2}+\frac{3}{4}U_{i}-\frac{3}{8}\right) + \sqrt{1-\frac{27}{32}}\varrho _{i}$ s.t. $E\left[ \left. U_{1i}\right\vert V_{i}=v\right] =\frac{3}{8}-\frac{3}{4}\Phi ^{-1}\left( v\right) -\frac{3}{8} \Phi ^{-2}\left( v\right) $, where $V_{i}=\Phi \left( U_{i}\right) $. & S & P, N, L, Q \\ 8 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & Normal cubic MTE: $U_{1i}=- \frac{1}{4}U_{i}^{3}+\frac{1}{4}\varrho _{i}$ s.t. $E\left[ \left. U_{1i}\right\vert V_{i}=v\right] =-\frac{1}{4}\Phi ^{-3}\left( v\right) $. & S & P, N, L, Q \\ 9 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & $U_{i}\sim t\left( 3\right) $ and is standardized to have unit variance. & S & P, N, L, Q \\ 10 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & $U_{i}\sim t\left( 2\right) $. & S & P, N, L, Q \\ 11 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & $\mu \left( X_{i}\right) =X_{i}^{C}+\frac{1}{2}\left[ \left( X_{i}^{C}\right) ^{2}-1\right] +\frac{1}{ 3}\left( X_{i}^{C}\right) ^{3}+\sum_{j=1}^{5}\frac{1}{j}\left( X_{ji}^{D}-0.5\right) $. & N, S & P, L, Q \\ 12 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & Heteroskedastic $U_{di}$: $ U_{1i}=-0.75U_{i}+\sqrt{1-0.75^{2}}\varrho _{i}$ with $\varrho _{i}\sim N\left( 0,\exp \left( 2X_{i}^{C}-2\right) \right) $, and $U_{0i}\sim N\left( 0,\left( X_{i}^{C}\right) ^{2}\right) $. & P, N, S & L, Q \\ 13 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & $ Y_{1i}=2X_{i}^{C}+\sum_{j=1}^{5}\left( X_{ji}^{D}-0.5\right) +\left( X_{1i}^{D}-0.5\right) \left( X_{2i}^{D}-0.5\right) +U_{1i}$. & Correctly specified S & P, N, L, Q, S \\ 14 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & $ Y_{1i}=2X_{i}^{C}+\sum_{j=1}^{5}\left( X_{ji}^{D}-0.5\right) +X_{i}^{C}\left( X_{1i}^{D}-0.5\right) +U_{1i}$. & Correctly specified S & P, N, L, Q, S \\ 15 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & $ Y_{1i}=2X_{i}^{C}+\sum_{j=1}^{5}\left( X_{ji}^{D}-0.5\right) +\left[ \left( X_{i}^{C}\right) ^{2}-1\right] +U_{1i}$. & Correctly specified S & P, N, L, Q, S \\ 16 & No: $\beta _{1}^{C}=2,\beta _{0}^{C}=1$ & $ Y_{1i}=2X_{i}^{C}+\sum_{j=1}^{5}\left( X_{ji}^{D}-0.5\right) +0.5\left[ \exp \left( X_{i}^{C}\right) -\exp \left( 0.5\right) \right] +U_{1i}$. & Correctly specified S & P, N, L, Q, S \\ \hline\hline \end{tabular} } \newline { Notes: P, N, L, Q, S, and SIV stand for parametric, normal, linear, quadratic, semiparametric, and semiparametric IV estimators for the MTE, respectively. Please refer to Table (ref) for details of the estimators. In Design 1, $U_{i}$ follows a standard normal distribution that is independent of $X_{i}$, and $\mu \left( \cdot \right) $ is quadratic as in ( (ref)). Design 2 differs from Design 1 only in relaxing the exclusion restriction.} \end{sidewaystable} \begin{sidewaystable}[htb] \caption{MTE estimators considered in the simulation} \resizebox{\linewidth}{!}{ \begin{tabular}{C{4cm}cL{4.8cm}cL{9.6cm}} \hline\hline Estimator & Step 1 & & Step 2 & \\ \hline Parametric MTE & \textit{Case 1.3}: & Probit regression of $D_{i}$ on $ X_{i}^{C}$, $\left( X_{i}^{C}\right) ^{2}$, and $X_{ji}^{D}$. & \textit{Case 2.2.2}: & Linear regression of $Y_{i}$ on $X_{i}$ and $\left. \phi \left( \Phi ^{-1}\left( \hat{P}_{i}\right) \right) \right/ \left( \hat{P} _{i}-1+d\right) $ for each group $d=0,1$. \\ Normal MTE & \textit{Case 1.1}: & Nonparametric regression of $D_{i}$ on $ X_{i}^{C}$ and $X_{ji}^{D}$. & \textit{Case 2.2.2}: & Linear regression of $ Y_{i}$ on $X_{i}$ and $\left. \phi \left( \Phi ^{-1}\left( \hat{P} _{i}\right) \right) \right/ \left( \hat{P}_{i}-1+d\right) $ for each group $ d=0,1$. \\ Linear MTE & \textit{Case 1.1}: & Nonparametric regression of $D_{i}$ on $ X_{i}^{C}$ and $X_{ji}^{D}$. & \textit{Case 2.2.1}: & Linear regression of $ Y_{i}$ on $X_{i}$ and $\hat{P}_{i}$ for each group. \\ Quadratic MTE & \textit{Case 1.1}: & Nonparametric regression of $D_{i}$ on $ X_{i}^{C}$ and $X_{ji}^{D}$. & \textit{Case 2.2.1}: & Linear regression of $ Y_{i}$ on $X_{i}$, $\hat{P}_{i}$, and $\hat{P}_{i}^{2}$ for each group. \\ Semiparametric MTE & \textit{Case 1.1}: & Nonparametric regression of $D_{i}$ on $X_{i}^{C}$ and $X_{ji}^{D}$. & \textit{Case 2.1}: & Partially linear regression of $Y_{i}$ on $X_{i}$ and $g_{d}\left( \hat{P}_{i}\right) $ by ( (ref)). \\ Semiparametric IV-MTE & \textit{Case 1.1}: & Nonparametric regression of $ D_{i}$ on $X_{i}^{C}$ and $X_{ji}^{D}$. & \textit{Case 2.1}: & Partially linear regression of $Y_{i}$ on $X_{ji}^{D}$ and $g_{d}\left( \hat{P} _{i}\right) $ by ((ref)) with $X_{i}^{C}$ excluded. \\ Correctly specified semiparametric MTE & \textit{Case 1.1}: & Nonparametric regression of $D_{i}$ on $X_{i}^{C}$ and $X_{ji}^{D}$. & \textit{Case 2.1}: & Partially linear regression of $Y_{i}$ on $X_{i}$, $g_{d}\left( \hat{P} _{i}\right) $, and the true nonlinear term in Designs 13-16. \\ \hline\hline \end{tabular} } \newline { Notes: The italics refer to the cases presented in Table (ref) . The semiparametric IV-MTE estimator is considered only in Designs 1-4, while the correctly specified semiparametric MTE estimator is considered only in Designs 13-16.} \end{sidewaystable} \begin{figure}[tbh] \caption{Simulated MTE curves for Designs 1-4} {-0.1cm} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} { Notes: The parametric MTE estimator specifies both parametric first step and parametric second step. The normal, linear, quadratic MTE estimators specify nonparametric first step and parametric second step. The semiparametric MTE estimator specifies nonparametric first step and partially linear (semiparametric) second step. The above five MTE estimators impose no exclusion restriction and base the identification on different functional forms between the treatment and outcome equations, while the semiparametric IV-MTE estimator imposes the exclusion restriction $\beta _{1}^{C}=\beta _{0}^{C}=0$ but otherwise is the same as the semiparametric MTE. Please refer to Table (ref) for more details about the estimators} \end{figure} \begin{sidewaystable}[htb] \caption{Simulation results for Designs 1-4} \resizebox{\linewidth}{!}{ \begin{tabular}{ccccccccccccccccc} \hline\hline & & \multicolumn{3}{c}{Design 1} & & \multicolumn{3}{c}{Design 2} & & \multicolumn{3}{c}{Design 3} & & \multicolumn{3}{c}{Design 4} \\ \cline{3-5}\cline{7-9}\cline{11-13}\cline{15-17} & & Bias & RMSE & Coverage & & Bias & RMSE & Coverage & & Bias & RMSE & Coverage & & Bias & RMSE & Coverage \\ \hline \multicolumn{17}{l}{\underline{Parametric estimator}} \\ MTE$\left( v=0.5\right) $ & & 0.014 & 0.101 & 0.962 & & -0.004 & 0.101 & 0.958 & & 0.008 & 0.214 & 0.945 & & 0.008 & 0.214 & 0.946 \\ ATE & & 0.051 & 0.107 & 0.927 & & 0.033 & 0.101 & 0.942 & & 0.045 & 0.206 & 0.948 & & 0.045 & 0.207 & 0.944 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & -0.001 & 0.054 & 0.948 & & -0.001 & 0.054 & 0.948 & & -0.003 & 0.125 & 0.949 & & -0.003 & 0.125 & 0.949 \\ \multicolumn{17}{l}{\underline{Normal estimator}} \\ MTE$\left( v=0.5\right) $ & & 0.006 & 0.118 & 0.954 & & -0.003 & 0.119 & 0.951 & & 0.257 & 0.309 & 0.674 & & 0.256 & 0.309 & 0.677 \\ ATE & & 0.041 & 0.119 & 0.931 & & 0.032 & 0.117 & 0.939 & & 0.278 & 0.323 & 0.601 & & 0.278 & 0.323 & 0.601 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & -0.035 & 0.067 & 0.909 & & -0.035 & 0.067 & 0.909 & & -0.168 & 0.193 & 0.574 & & -0.168 & 0.193 & 0.574 \\ \multicolumn{17}{l}{\underline{Linear estimator}} \\ MTE$\left( v=0.5\right) $ & & -0.027 & 0.123 & 0.951 & & -0.037 & 0.126 & 0.949 & & 0.293 & 0.351 & 0.669 & & 0.293 & 0.351 & 0.665 \\ ATE & & -0.015 & 0.118 & 0.951 & & -0.024 & 0.121 & 0.954 & & 0.300 & 0.355 & 0.652 & & 0.300 & 0.355 & 0.651 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & -0.024 & 0.064 & 0.924 & & -0.024 & 0.064 & 0.924 & & -0.188 & 0.215 & 0.576 & & -0.188 & 0.215 & 0.576 \\ \multicolumn{17}{l}{\underline{Quadratic estimator}} \\ MTE$\left( v=0.5\right) $ & & -0.102 & 0.176 & 0.899 & & -0.111 & 0.182 & 0.891 & & 0.257 & 0.323 & 0.734 & & 0.257 & 0.323 & 0.738 \\ ATE & & 0.043 & 0.142 & 0.934 & & 0.034 & 0.141 & 0.937 & & 0.538 & 0.584 & 0.350 & & 0.537 & 0.584 & 0.349 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & -0.028 & 0.067 & 0.923 & & -0.028 & 0.067 & 0.923 & & -0.261 & 0.284 & 0.363 & & -0.261 & 0.284 & 0.363 \\ \multicolumn{17}{l}{\underline{Semiparametric estimator}} \\ MTE$\left( v=0.5\right) $ & & -0.049 & 0.309 & 0.947 & & -0.056 & 0.312 & 0.947 & & 0.329 & 0.450 & 0.809 & & 0.336 & 0.455 & 0.807 \\ ATE & & 0.041 & 0.168 & 0.940 & & 0.044 & 0.194 & 0.947 & & 0.621 & 0.684 & 0.441 & & 0.629 & 0.693 & 0.440 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & -0.029 & 0.068 & 0.917 & & -0.029 & 0.068 & 0.922 & & -0.282 & 0.307 & 0.358 & & -0.287 & 0.311 & 0.345 \\ \multicolumn{17}{l}{\underline{Semiparametric IV estimator}} \\ MTE$\left( v=0.5\right) $ & & -0.094 & 0.307 & 0.939 & & 4.394 & 4.427 & 0 & & -0.082 & 0.264 & 0.940 & & 4.410 & 4.424 & 0 \\ ATE & & -0.016 & 0.119 & 0.947 & & 6.324 & 6.330 & 0 & & 0.005 & 0.108 & 0.955 & & 6.868 & 6.872 & 0 \\ \hline\hline \end{tabular} } \newline { Notes: RMSE stands for the root mean squared error, and Coverage stands for the coverage probability of asymptotic 95% confidence interval. For the semiparametric IV estimator, $\beta _{1}^{C}$ and $\beta _{0}^{C}$ are assumed to be zero and thus not estimated.} \end{sidewaystable} \begin{figure}[tbh] \caption{Simulated MTE curves for Designs 5-8} {-0.1cm} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} { Notes: The parametric MTE estimator specifies both parametric first step and parametric second step. The normal, linear, quadratic MTE estimators specify nonparametric first step and parametric second step. And the semiparametric MTE estimator specifies nonparametric first step and partially linear (semiparametric) second step.} \end{figure} \begin{sidewaystable}[htb] \caption{Simulation results for Designs 5-8} \resizebox{\linewidth}{!}{ \begin{tabular}{ccccccccccccccccc} \hline\hline & & \multicolumn{3}{c}{Design 5} & & \multicolumn{3}{c}{Design 6} & & \multicolumn{3}{c}{Design 7} & & \multicolumn{3}{c}{Design 8} \\ \cline{3-5}\cline{7-9}\cline{11-13}\cline{15-17} & & Bias & RMSE & Coverage & & Bias & RMSE & Coverage & & Bias & RMSE & Coverage & & Bias & RMSE & Coverage \\ \hline \multicolumn{17}{l}{\underline{Parametric estimator}} \\ MTE$\left( v=0.5\right) $ & & -0.007 & 0.081 & 0.947 & & -0.045 & 0.198 & 0.347 & & -0.122 & 0.152 & 0.725 & & -0.076 & 0.133 & 0.891 \\ ATE & & 0.042 & 0.087 & 0.911 & & -0.011 & 0.183 & 0.347 & & 0.267 & 0.280 & 0.124 & & -0.039 & 0.110 & 0.932 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & -0.002 & 0.038 & 0.949 & & 0.064 & 0.098 & 0.315 & & -0.062 & 0.078 & 0.724 & & 0.038 & 0.070 & 0.899 \\ \multicolumn{17}{l}{\underline{Normal estimator}} \\ MTE$\left( v=0.5\right) $ & & -0.135 & 0.164 & 0.701 & & 0.075 & 0.137 & 0.912 & & -0.102 & 0.146 & 0.833 & & -0.102 & 0.164 & 0.876 \\ ATE & & -0.083 & 0.121 & 0.846 & & 0.106 & 0.152 & 0.844 & & 0.285 & 0.301 & 0.176 & & -0.066 & 0.138 & 0.916 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & 0.005 & 0.038 & 0.954 & & -0.013 & 0.054 & 0.949 & & -0.083 & 0.097 & 0.628 & & 0.018 & 0.065 & 0.943 \\ \multicolumn{17}{l}{\underline{Linear estimator}} \\ MTE$\left( v=0.5\right) $ & & -0.147 & 0.176 & 0.671 & & 0.025 & 0.117 & 0.949 & & -0.226 & 0.249 & 0.421 & & -0.008 & 0.130 & 0.952 \\ ATE & & -0.128 & 0.159 & 0.735 & & 0.037 & 0.118 & 0.943 & & 0.156 & 0.187 & 0.670 & & 0.002 & 0.127 & 0.957 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & 0.007 & 0.039 & 0.950 & & -0.013 & 0.054 & 0.956 & & -0.033 & 0.061 & 0.890 & & -0.024 & 0.066 & 0.932 \\ \multicolumn{17}{l}{\underline{Quadratic estimator}} \\ MTE$\left( v=0.5\right) $ & & -0.288 & 0.311 & 0.321 & & -0.058 & 0.149 & 0.920 & & -0.021 & 0.126 & 0.952 & & -0.245 & 0.289 & 0.632 \\ ATE & & -0.044 & 0.121 & 0.936 & & 0.168 & 0.222 & 0.795 & & 0.000 & 0.124 & 0.959 & & 0.181 & 0.234 & 0.792 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & 0.003 & 0.039 & 0.950 & & -0.005 & 0.055 & 0.956 & & -0.020 & 0.055 & 0.933 & & -0.041 & 0.074 & 0.892 \\ \multicolumn{17}{l}{\underline{Semiparametric estimator}} \\ MTE$\left( v=0.5\right) $ & & -0.197 & 0.334 & 0.890 & & 0.103 & 0.321 & 0.922 & & -0.049 & 0.266 & 0.944 & & -0.014 & 0.313 & 0.958 \\ ATE & & -0.076 & 0.163 & 0.917 & & 0.130 & 0.209 & 0.887 & & 0.035 & 0.155 & 0.943 & & 0.042 & 0.179 & 0.951 \\ $\hat{\beta}_{1}^{C}-\hat{\beta}_{0}^{C}$ & & 0.006 & 0.039 & 0.946 & & -0.013 & 0.059 & 0.941 & & -0.027 & 0.058 & 0.914 & & -0.018 & 0.066 & 0.947 \\ \hline\hline \end{tabular} } \newline { Notes: RMSE stands for the root mean squared error, and Coverage stands for the coverage probability of asymptotic 95% confidence interval.} \end{sidewaystable} \begin{figure}[tbh] \caption{Simulated MTE curves for Designs 9-12} {-0.1cm} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} { Notes: The parametric MTE estimator specifies both parametric first step and parametric second step. The normal, linear, quadratic MTE estimators specify nonparametric first step and parametric second step. And the semiparametric MTE estimator specifies nonparametric first step and partially linear (semiparametric) second step.} \end{figure} \begin{figure}[tbh] \caption{Simulated MTE curves for Designs 13-16} {-0.1cm} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} \begin{minipage}[t]{0.48\linewidth} \centerline \end{minipage} { Notes: The parametric, normal, linear, quadratic, and semiparametric MTE estimators are defined as before without including any nonlinear term in the outcome equation, while the correctly specified semiparametric MTE estimator includes the true nonlinear term in the outcome equation.} \end{figure}