Estimation in a Generalization of Bivariate Probit Models with Dummy Endogenous Regressors

abstractThe purpose of this paper is to provide guidelines for empirical researchers who use a class of bivariate threshold crossing models with dummy endogenous variables. A common practice employed by the researchers is the specification of the joint distribution of the unobservables as a bivariate normal distribution, which results in a bivariate probit model. To address the problem of misspecification in this practice, we propose an easy-to-implement semiparametric estimation framework with parametric copula and nonparametric marginal distributions. We establish asymptotic theory, including root-$n$ normality, for the sieve maximum likelihood estimators that can be used to conduct inference on the individual structural parameters and the average treatment effect (ATE). In order to show the practical relevance of the proposed framework, we conduct a sensitivity analysis via extensive Monte Carlo simulation exercises. The results suggest that the estimates of the parameters, especially the ATE, are sensitive to parametric specification, while semiparametric estimation exhibits robustness to underlying data generating processes. We then provide an empirical illustration where we estimate the effect of health insurance on doctor visits. In this paper, we also show that the absence of excluded instruments may result in identification failure, in contrast to what some practitioners believe. \\ Keywords: Triangular threshold crossing model, bivariate probit model, dummy endogenous regressors, binary response, copula, exclusion restriction, sensitivity analysis. JEL Classification Numbers: C14, C35, C36.

assumption$X$ and $Z$ satisfy that $(X,Z)\perp(\varepsilon,\nu)$, where “$\perp$” denotes statistical independence.

assumption$(X^{\prime},Z^{\prime})$ does not lie in a proper linear subspace of $\mathbb{R}^{k+l}$ a.s.\footnote{A proper linear subspace of $\mathbb{R}^{k+l}$ is a linear subspace with a dimension strictly less than $k+l$. The assumption is that if $M$ is a proper linear subspace of $\mathbb{R}^{k+l}$, then $\Pr[(X^{\prime},Z^{\prime})\in M]<1$.}

assumptionThere exists a copula function $C:(0,1)^{2}\rightarrow(0,1)$ such that the joint distribution $F_{\varepsilon\nu}$ of $(\varepsilon,\nu)$ satisfies $F_{\varepsilon\nu}(\varepsilon,\nu)=C(F_{\varepsilon}(\varepsilon),F_{\nu}(\nu))$, where $F_{\varepsilon}$ and $F_{\nu}$ are the marginal distributions of $\varepsilon$ and $\nu$, respectively, that are strictly increasing and absolutely continuous with respect to Lebesgue measure.\footnote{ Sklar's theorem (e.g., nelsen1999introduction) guarantees the existence of such a copula, which is, in fact, unique because $F_{\varepsilon}$ and $F_{\nu}$ are continuous.}

assumptionAs scale and location normalizations, $\alpha_{1}=\beta_{1}=1$ and $\alpha_{0}=\beta_{0}=0$.

definition[Strictly More SI or Less SD]Let $C(u_{2}|u_{1})$ and $\tilde{C}(u_{2}|u_{1})$ be conditional copulas, for which $1-C(u_{2}|u_{1})$ and $1-\tilde{C}(u_{2}|u_{1})$ are either increasing or decreasing in $u_{1}$ for all $u_{2}$. Such copulas are referred to as stochastically increasing (SI) or stochastically decreasing (SD), respectively. Then, $\tilde{C}$ is strictly more SI (or less SD) than $C$ if $\psi(u_{1},u_{2})\equiv\tilde{C}^{-1}(C(u_{2}|u_{1})|u_{1})$ is strictly increasing in $u_{1}$,\footnote{Note that $\psi(u_{1},u_{2})$ is increasing in $u_{2}$ by definition.} which is denoted as $C\prec_{S}\tilde{C}$.

assumptionThe copula in Assumption (ref) satisfies $C(\cdot,\cdot)=C(\cdot,\cdot;\rho)$ with a scalar dependence parameter $\rho\in\Omega$, is twice differentiable in $u_{1}$, $u_{2}$ and $\rho$, and satisfies \begin{equation} C(u_{1}|u_{2};\rho_{1})\prec_{S}C(u_{1}|u_{2};\rho_{2}) for any \rho_{1}<\rho_{2}. \end{equation}

assumption$F_{\varepsilon}$ and $F_{\nu}$ are known up to means $\mu\equiv(\mu_{\varepsilon},\mu_{\nu})$ and variances $\sigma^{2}\equiv(\sigma_{\varepsilon}^{2},\sigma_{\nu}^{2})$.

theoremIn model (ref), suppose Assumptions (ref)\textendash (ref) hold. Then, $(\alpha{}^{\prime},\beta{}^{\prime},\delta_{1},\gamma,\rho,\mu,\sigma)$ are point identified in an open and convex parameter space if (i) $\gamma$ is a nonzero vector, and (ii) $\mathcal{X}$ does not lie in a proper linear subspace of $\mathbb{R}^{k}$ a.s.

assumption(i) The distributions of $X_{j}$ (for $1\leq j\leq k$) and $Z_{j}$ (for $1\leq j\leq l$) are absolutely continuous with respect to Lebesgue measure; (ii) There exists at least one element $X_{j}$ in $X$ such that its support conditional on $(X_{1},...,X_{j-1},X_{j+1},...,X_{k})$ is $\mathbb{R}$ and $\alpha_{j}\neq0$ and $\beta_{j}\neq0$, where, without loss of generality, we let $j=1$.

theoremIn model (ref), suppose Assumptions (ref)\textendash (ref), and (ref)(i) hold. Then $(\alpha{}^{\prime},\beta{}^{\prime},\delta_{1},\gamma,\rho)$ are point identified in an open and convex parameter space if (i) $\gamma$ is a nonzero vector; and (ii) $\mathcal{X}$ does not lie in a proper linear subspace of $\mathbb{R}^{k}$ a.s. In addition, if Assumption (ref)(ii) holds, $F_{\varepsilon}(\cdot)$ and $F_{\nu}(\cdot)$ are identified up to additive constants.

theoremIn model (ref), suppose Assumptions (ref)$^{\prime}$ and (ref) hold. Also, suppose Assumption (ref) holds conditional on $W$, and Assumptions (ref), (ref)\textendash (ref) hold with $C_{w}(u_{1},u_{2})$, $F_{\varepsilon\nu|w}(\varepsilon,\nu)$, $F_{\varepsilon|w}(\varepsilon)$ and $F_{\nu|w}(\nu)$ instead, for all $w\in\text{supp}(W)$. Then, $(\alpha{}^{\prime},\beta{}^{\prime},\delta_{1},\gamma,\rho,\mu,\sigma)$ are point identified in an open and convex parameter space if (i) $\gamma$ is a nonzero vector, and (ii) $\mathcal{X}$ does not lie in a proper linear subspace of $\mathbb{R}^{k}$ a.s. conditional on $W$.

theoremIn model (ref), suppose Assumptions (ref)$^{\prime}$ and (ref) hold. Also, suppose Assumptions (ref) and (ref)(i) hold conditional on $W$, and Assumptions (ref) and (ref) hold with $C_{w}(u_{1},u_{2})$, $F_{\varepsilon\nu|w}(\varepsilon,\nu)$, $F_{\varepsilon|w}(\varepsilon)$ and $F_{\nu|w}(\nu)$ instead, for all $w\in\text{supp}(W)$. Then $(\alpha{}^{\prime},\beta{}^{\prime},\delta_{1},\gamma,\rho)$ are point identified in an open and convex parameter space if (i) $\gamma$ is a nonzero vector; and (ii) $\mathcal{X}$ does not lie in a proper linear subspace of $\mathbb{R}^{k}$ a.s. In addition, if Assumption (ref)(ii) holds conditional on $W$, $F_{\varepsilon|w}(\cdot)$ and $F_{\nu|w}(\cdot)$ are identified up to additive constants for all $w\in\text{supp}(W)$.

theoremIn model (ref) with $X=(1,X_{1})$ where $X_{1}\in\mbox{supp}(X_{1})=\{0,1\}$, suppose that the assumptions in Theorem (ref) hold, except that $\gamma=0$. Then, there exist two element-wise distinct sets of $(\delta_{1},\rho,\mu_{\varepsilon},\sigma_{\varepsilon})$ that generate the same observed data.

itemize• In the bivariate probit model with continuous common exogenous covariates and no excluded instruments, the parameters will be, at best, weakly identified; • This also implies that, in the semiparametric model considered in Theorem (ref), the structural parameters and the marginal distributions are not identified without an exclusion restriction, even if $X_{1}$ has large support.

assumptionThere exists a compact and convex subset $\Psi\subseteq\tilde{\Psi}$ such that $\psi_{0}\in int(\Psi)$, where $int(A)$ is the interior of the set $A$.

assumptionThere exists a measurable function $\underline{p}(X,Z)$ such that for all $\theta\in\Theta$ and for all $y,d=0,1$, $p_{yd,XZ}(\theta)\geq\underline{p}(X,Z)$, with $E|\log(\underline{p}(X,Z))|<\infty$ and $E\left[\frac{1}{\underline{p}(X,Z)^{2}}\right]<\infty$.

assumption$\{W_{i}:i=1,2,...,n\}$ is a random sample, with $E\left[||(X_{i}^{'},Z_{i}^{'})^{'}||_{E}^{2}\right]<\infty$.

assumption(i) $\sqrt{h_{\epsilon0}},\sqrt{h_{\nu0}}\in\Lambda_{R}^{p}([0,1])$, with $p>\frac{1}{2}$ and some $R>0$; (ii) $\mathcal{H}_{\epsilon}=\mathcal{H}_{\nu}=\mathcal{H}$ where $\mathcal{H}\equiv\left\{ h=q^{2}:q\in\Lambda_{R}^{p}([0,1]),\int_{0}^{1}q=1\right\} $, with $R$ being defined as in (i) and $\Lambda_{R}^{p}([0,1])$ being a H\"older ball with radius $R$; (iii) the density functions $h_{\epsilon0}$ and $h_{\nu0}$ are bounded away from zero on $[0,1]$.

assumption(i) $\mathcal{H}_{\epsilon n}=\mathcal{H}_{\nu n}\equiv\{h\in\mathcal{H}:h(x)=p^{k_{n}}(x)^{'}a_{k_{n}},a_{k_{n}}\in\mathbb{R}^{k_{n}},||h||_{\infty}<2R^{2}\}$, where $k_{n}\rightarrow\infty$ and $k_{n}/n\rightarrow0$ as $n\rightarrow\infty$; (ii) for all $j\geq1$, we have $\Theta_{j}\subseteq\Theta_{j+1}$, and there exists a sequence $\{\pi_{j}\theta_{0}\}_{j}$ such that $d_{c}(\pi_{j}\theta_{0},\theta_{0})\rightarrow0$ as $j\rightarrow\infty$.

assumptionFor $j=1,2$, let $C_{j}(u_{1},u_{2};\rho)\equiv\frac{\partial C(u_{1},u_{2};\rho)}{\partial u_{j}}$ and $C_{\rho}(u_{1},u_{2};\rho)\equiv\frac{\partial C(u_{1},u_{2};\rho)}{\partial\rho}$. The derivatives $C_{j}(\cdot,\cdot;\cdot)$ and $C_{\rho}(\cdot,\cdot;\cdot)$ are uniformly bounded for all $j=1,2$.

theoremSuppose that Assumptions (ref)\textendash (ref) and (ref) hold. If Assumptions (ref)\textendash (ref) are satisfied, then $d_{c}(\hat{\theta}_{n},\theta_{0})\overset{p}{\rightarrow}0$.

theoremSuppose that Assumptions (ref)\textendash (ref) and (ref)\textendash (ref) hold. If Assumption (ref) in the online appendix additionally holds, then we have $||\hat{\theta}_{n}-\theta_{0}||_{2}=O_{p}\left(\max\left\{ \sqrt{k_{n}/n},k_{n}^{-p}\right\} \right)$. Furthermore, if we choose $k_{n}\propto n^{\frac{1}{2p+1}}$, then we have $||\hat{\theta}_{n}-\theta_{0}||_{2}=O_{p}\left(n^{-\frac{p}{2p+1}}\right)$.

assumptionThe copula function $C(u_{1},u_{2};\rho)$ is twice continuously differentiable with respect to $u_{1},u_{2},$ and $\rho$, and its first- and second- order partial derivatives are well defined in a neighborhood of $\theta_{0}$.

assumptionThe following conditions hold: (i) there exist constants $w>1+\frac{1}{2p}$ and a small $\epsilon_{0}>0$ such that for any $v\in\mathbb{V}$ with $||v||\leq\epsilon_{0}$, \[ \left|T(\theta_{0}+v)-T(\theta_{0})-\frac{\partial T(\theta_{0})}{\partial\theta^{'}}[v]\right|=O(||v||^{w}); \] (ii) For any $v\in\mathbb{V}$, $T(\theta_{0}+tv)$ is continuously differentiable in $t\in[0,1]$ around $t=0$, and \[ \left\Vert \frac{\partial T(\theta_{0})}{\partial\theta^{'}}\right\Vert \equiv\sup_{v\in\mathbb{V},||v||>0}\frac{\left|\frac{\partial T(\theta_{0})}{\partial\theta^{'}}[v]\right|}{||v||}<\infty. \]

assumptionThere exists $\pi_{n}v^{*}\in\Theta_{n}-\{\theta_{0}\}$ such that $||\pi_{n}v^{*}-v^{*}||=o(n^{-1/4})$.

propositionSuppose that Assumptions (ref)\textendash (ref), (ref)\textendash (ref), (ref)\textendash (ref) are satisfied. If $k_{n}\propto n^{\frac{1}{2p+1}}$, then we have \[ \sqrt{n}(T(\hat{\theta}_{n})-T(\theta_{0}))\overset{d}{\rightarrow}\mathcal{N}\left(0,\left\Vert \frac{\partial T(\theta_{0})}{\partial\theta^{'}}\right\Vert ^{2}\right). \]

theoremSuppose that Assumptions (ref)\textendash (ref), (ref)\textendash (ref), (ref), (ref)\textendash (ref) hold. Then, we have \begin{equation} \sqrt{n}(\hat{\psi}_{n}-\psi_{0})\overset{d}{\rightarrow}\mathcal{N}\left(0,\mathcal{I}_{*}(\psi_{0})^{-1}\right), \end{equation} and the form of $\mathcal{I}_{*}(\psi)$ is given in the online appendix.

theoremSuppose that assumptions in Theorem (ref) hold. Then, $\hat{\mathcal{I}}_{*}(\hat{\psi}_{n})=\mathcal{I}_{*}(\psi_{0})+o_{p}(1)$.

theoremLet $x\in\text{supp}(X)$ be given. Suppose that the conditions in Proposition (ref) hold with $T(\theta_{0})=ATE(\theta_{0};x)$. Then, we have \begin{equation} \sqrt{n}(ATE(\hat{\theta}_{n};x)-ATE(\theta_{0};x))\overset{d}{\rightarrow}\mathcal{N}\left(0,\left\Vert \frac{\partial ATE(\theta_{0};x)}{\partial\theta^{'}}[v]\right\Vert ^{2}\right), \end{equation} where $\left\Vert \frac{\partial ATE(\theta_{0};x)}{\partial\theta^{'}}[v]\right\Vert ^{2}=\sup_{v\in\mathbb{V},||v||>0}\frac{\left|\frac{\partial ATE(\theta_{0};x)}{\partial\theta^{'}}[v]\right|}{||v||}$, and the form of $\frac{\partial ATE(\theta_{0};x)}{\partial\theta^{'}}[v]$ is given by (ref) in the online appendix. Furthermore, the asymptotic variance in (ref) can be estimated as follows: \[ \hat{\sigma}_{ATE(\theta;x)}^{2}\equiv\max_{v\in\Theta_{n}}\left\Vert \frac{\partial ATE(\hat{\theta}_{n};x)}{\partial\theta^{'}}[v]\right\Vert ^{2}. \]

enumerate• For each $b=1,2,...,B$, let $\{B_{i}^{(b)}:i=1,2,...,n\}$ be a random sample generated from a positive random variable $B_{i}$ such that $EB_{i}=1$, $Var(B_{i})=1$, and is independent of $\{W_{i}:i=1,2,...n\}$.\footnote{Note that the condition on the variance of $B_{i}$ can be relaxed. In our empirical example, we use $B_{i}\sim\exp(1)$. } • For each bootstrap iteration $b=1,2,...,B$, define $\hat{\theta}_{n}^{*(b)}$ be a bootstrap estimate of $\theta_{0}$: \[ \hat{\theta}_{n}^{*(b)}\equiv\arg\max_{\theta\in\tilde{\Theta}_{n}}{\textstyle Q_{n}^{*(b)}(\theta)}, \] where $Q_{n}^{*(b)}(\theta)\equiv\frac{1}{n}\sum\limits _{i=1}^{n}B_{i}^{(b)}\cdot l(\theta,W_{i})$. Obtain the bootstrap estimate of the functional of interest by using $\hat{\theta}_{n}^{*(b)}$ and denote it by $T(\hat{\theta}_{n}^{*(b)})$. • The bootstrap standard error of $T(\hat{\theta}_{n})$ is given by $\sqrt{\frac{1}{B}\sum_{b=1}^{B}\left(T(\hat{\theta}_{n}^{*(b)})-\bar{T}_{B}^{*}\right)}$, where $\bar{T}_{B}^{*}\equiv\frac{1}{B}\sum_{b=1}^{B}T(\hat{\theta}_{n}^{*(b)})$.

itemize• When the model is correctly specified, the performance of the sieve ML estimators is comparable to that of the parametric ML estimators. • When the marginal distributions are misspecified, the sieve ML estimation is recommended in order to improve the performance. • The semiparametric ML estimators performs better than the parametric ML estimators under both copula and marginal misspecification. Therefore, the semiparametric models are preferred to the parametric models in such cases. • Especially for the ATE, whenever the marginal distributions are misspecified, the parametric ML estimates can be significantly distorted.

appendix\part*{Online Appendix} \section{Proofs of Results in Section (ref)} \subsection{Proof of Theorem (ref)} Recall \[ \begin{array}{cc} q_{0}\equiv F_{\tilde{\nu}}(-\mu_{\nu}/\sigma_{\nu}), & \quad q_{1}\equiv F_{\tilde{\nu}}((1-\mu_{\nu})/\sigma_{\nu}),\\ t_{0}\equiv F_{\tilde{\varepsilon}}(-\mu_{\varepsilon}/\sigma_{\varepsilon}), & \quad t_{1}\equiv F_{\tilde{\varepsilon}}((1-\mu_{\varepsilon})/\sigma_{\varepsilon}), \end{array} \] and \begin{align*} \tilde{p}_{11,0} & =C(F_{\tilde{\varepsilon}}(F_{\tilde{\varepsilon}}^{-1}(t_{0})+\delta_{1}),q_{0};\rho),\\ \tilde{p}_{11,1} & =C(F_{\tilde{\varepsilon}}(F_{\tilde{\varepsilon}}^{-1}(t_{1})+\delta_{1}),q_{1};\rho),\\ \tilde{p}_{10,0} & =t_{0}-C(t_{0},q_{0};\rho),\\ \tilde{p}_{10,1} & =t_{1}-C(t_{1},q_{1};\rho),\\ \tilde{p}_{00,0} & =1-t_{0}-q_{0}+C(t_{0},q_{0};\rho),\\ \tilde{p}_{00,1} & =1-t_{1}-q_{1}+C(t_{1},q_{1};\rho), \end{align*} where $\tilde{p}_{yd,x}\equiv\Pr[Y=y,D=d|X_{1}=x]$. Again, we want to show that, given $(q_{0},q_{1})$ which are identified from the reduced-form equation, there are two distinct sets of parameter values $(t_{0},t_{1},\delta_{1},\rho)$ and $(t_{0}^{*},t_{1}^{*},\delta_{1}^{*},\rho^{*})$ (with $(t_{0},t_{1},\delta_{1},\rho)$ $\ne$ $(t_{0}^{*},t_{1}^{*},\delta_{1}^{*},\rho^{*})$) that generate the same observed fitted probabilities $\tilde{p}_{yd,0}$ and $\tilde{p}_{yd,1}$ for all $(y,d)\in\{0,1\}^{2}$. In showing this, the following lemma is useful: \begin{lemma}Assumption (ref) implies that, for any $(u_{1},u_{2})\in(0,1)^{2}$ and $\rho\in\Omega$, \begin{equation} C_{\rho}(u_{1},u_{2};\rho)>0. \end{equation} \end{lemma}The proof of this lemma can be found below. Now fix $(q_{0},q_{1})\in(0,1)^{2}$. First, consider the fitted probability $\tilde{p}_{10,0}$. Given $t_{0}\in(0,1)$ and $\rho\in\Omega$, note that, for $\rho^{*}>\rho$,\footnote{The inequality here and other inequalities implied from this (e.g., $t_{0}^{*}>t_{0}$, and etc.) are assumed only for concreteness.} there exists a solution $t_{0}^{*}=t_{0}^{*}(t_{0},q_{0},\rho,\rho^{*})$ such that \begin{align} t_{0}-C(t_{0},q_{0};\rho) & =\Pr[u_{1}\leq t_{0},u_{2}\ge q_{0};\rho]\\ & =\Pr[u_{1}\leq t_{0}^{*},u_{2}\ge q_{0};\rho^{*}]\\ & =t_{0}^{*}-C(t_{0}^{*},q_{0};\rho^{*}),\nonumber \end{align} and note that by Assumption (ref) and a variant of Lemma (ref), we have that $t_{0}^{*}>t_{0}$. Here, $(t_{0},q_{0},\rho)$ and $(t_{0}^{*},q_{0},\rho^{*})$ result in the same observed probability $\tilde{p}_{10,0}=t_{0}-C(t_{0},q_{0};\rho)=t_{0}^{*}-C(t_{0}^{*},q_{0};\rho^{*})$. Now consider the fitted probability $\tilde{p}_{11,0}$. Choose $\delta_{1}=0$. Also let $F_{\tilde{\varepsilon}}\sim Unif(0,1)$ only for simplicity, which is relaxed later. Then there exists a solution $t_{0}^{\dagger}=t_{0}^{\dagger}(t_{0},q_{0},\rho,\rho^{*})$ such that \begin{align} C(t_{0},q_{0};\rho) & =\Pr[u_{1}\leq t_{0},u_{2}\leq q_{0};\rho]\\ & =\Pr[u_{1}\leq t_{0}^{\dagger},u_{2}\leq q_{0};\rho^{*}]\\ & =C(t_{0}^{\dagger},q_{0};\rho^{*}),\nonumber \end{align} and note that $t_{0}^{\dagger}<t_{0}$ by Assumption (ref) and Lemma (ref). Then, by letting $\delta_{1}^{*}=t_{0}^{\dagger}-t_{0}^{*}$, $(t_{0},q_{0},\delta_{1},\rho)$ and $(t_{0}^{*},q_{0},\delta_{1}^{*},\rho^{*})$ satisfy $\tilde{p}_{11,0}=C(t_{0}+0,q_{0};\rho)=C(t_{0}^{*}+\delta_{1}^{*},q_{0};\rho^{*})$. Lastly, note that $\tilde{p}_{00,0}=1-q_{0}-\tilde{p}_{10,0}$ and $\tilde{p}_{01,0}=q_{0}-\tilde{p}_{11,0}$, and so $(t_{0},\delta_{1},\rho)$ and $(t_{0}^{*},\delta_{1}^{*},\rho^{*})$ above will also result in the same values of $\tilde{p}_{00,0}$ and $\tilde{p}_{01,0}$. It is tempting to have a parallel argument for $\tilde{p}_{10,1}$, $\tilde{p}_{11,1}$, $\tilde{p}_{00,1}$, and $\tilde{p}_{01,1}$, but there is a complication. Although other parameters are not, $\delta_{1}$ and $\rho$ are common in both sets of probabilities. Therefore, we proceed as follows. First, consider $\tilde{p}_{10,1}$. Given $t_{1}\in(0,1)$ and the above choice of $\rho^{*}\in\Omega$, note that there exists a solution $t_{1}^{*}=t_{1}^{*}(t_{1},q_{1},\rho,\rho^{*})$ such that \begin{align} t_{1}-C(t_{1},q_{1};\rho) & =\Pr[u_{1}\leq t_{1},u_{2}\ge q_{1};\rho]\\ & =\Pr[u_{1}\leq t_{1}^{*},u_{2}\ge q_{1};\rho^{*}]\\ & =t_{1}^{*}-C(t_{1}^{*},q_{1};\rho^{*}),\nonumber \end{align} and similarly as before, we have $t_{1}^{*}>t_{1}$. Here, $(t_{1},q_{1},\rho)$ and $(t_{1}^{*},q_{1},\rho^{*})$ result in the same observed probability $\tilde{p}_{10,1}=t_{1}-C(t_{1},q_{1};\rho)=t_{1}^{*}-C(t_{1}^{*},q_{1};\rho^{*})$. Now consider $\tilde{p}_{11,1}$. Recall $\delta_{1}=0$ and $F_{\varepsilon}\sim Unif(0,1)$. Then there exists a solution $t_{1}^{\dagger}=t_{1}^{\dagger}(t_{1},q_{1},\rho,\rho^{*})$ such that \begin{align} C(t_{1},q_{1};\rho) & =\Pr[u_{1}\leq t_{1},u_{2}\leq q_{1};\rho]\\ & =\Pr[u_{1}\leq t_{1}^{\dagger},u_{2}\leq q_{1};\rho^{*}]\\ & =C(t_{1}^{\dagger},q_{1};\rho^{*}),\nonumber \end{align} and thus $t_{1}^{\dagger}<t_{1}$. Then, if we can show that \begin{equation} t_{1}^{\dagger}=t_{1}^{*}+\delta_{1}^{*}, \end{equation} where $t_{1}^{*}$ and $\delta_{1}^{*}$ are the values already determined above, then $(t_{1},q_{1},\delta_{1},\rho)$ and $(t_{1}^{*},q_{1},\delta_{1}^{*},\rho^{*})$ result in $\tilde{p}_{11,1}=C(t_{1}+0,q_{1};\rho)=C(t_{1}^{*}+\delta_{1}^{*},q_{1};\rho^{*})$. Then similar as before, the two sets of parameters will generate the same values of $\tilde{p}_{00,1}=1-q_{1}-\tilde{p}_{10,1}$ and $\tilde{p}_{01,1}=q_{1}-\tilde{p}_{11,1}$. Consequently, $(t_{0},t_{1},q_{0},q_{1},\delta_{1},\rho)$ and $(t_{0}^{*},t_{1}^{*},q_{0},q_{1},\delta_{1}^{*},\rho^{*})$ generate the same entire observed fitted probabilities. The remaining question is whether we can find $(t_{0},t_{1},\delta_{1},\rho)$ and $(t_{0}^{*},t_{1}^{*},\delta_{1}^{*},\rho^{*})$ such that (ref) holds. To show this, we choose further specifications. We assume a normal copula.\footnote{This choice is not critical except that we can have $\rho$ reach to 1.} We choose $\rho=0$, $\rho^{*}=1$, $q_{0}=t_{0}=1/3$, and $q_{1}=t_{1}=2/3$. Since $(U_{1},U_{2})$ are jointly uniform, note that when $\rho=0$, the probability of the quadrant in $[0,1]^{2}$ specified by each of (ref), (ref), (ref), and (ref) equals the volume of the quadrant. When $\rho^{*}=1$, all the probability mass lies on the 45 degree line in $[0,1]^{2}$ and no where else, so the probability of a quadrant specified by each of (ref), (ref), (ref), and (ref) equals the length of the 45 line which intersects with that quadrant. Suppose that the following observational equivalence holds: \begin{align*} \Pr[u_{1}\leq t_{0},u_{2}\ge q_{0};\rho] & =\Pr[u_{1}\leq t_{0}^{*},u_{2}\ge q_{0};\rho^{*}]=2/9,\\ \Pr[u_{1}\leq t_{0},u_{2}\leq q_{0};\rho] & =\Pr[u_{1}\leq t_{0}^{\dagger},u_{2}\leq q_{0};\rho^{*}]=1/9,\\ \Pr[u_{1}\leq t_{1},u_{2}\ge q_{1};\rho] & =\Pr[u_{1}\leq t_{1}^{*},u_{2}\ge q_{1};\rho^{*}]=2/9,\\ \Pr[u_{1}\leq t_{1},u_{2}\leq q_{1};\rho] & =\Pr[u_{1}\leq t_{1}^{\dagger},u_{2}\leq q_{1};\rho^{*}]=4/9. \end{align*} One can easily show that these equations yield that $t_{0}^{*}=5/9$, $t_{0}^{\dagger}=1/9$, $t_{1}^{*}=8/9$, and $t_{1}^{\dagger}=4/9$. Consider the equation (ref), which can be rewritten as $t_{1}^{\dagger}=t_{1}^{*}+t_{0}^{\dagger}-t_{0}^{*}$ or $t_{1}^{\dagger}-t_{1}^{*}=t_{0}^{\dagger}-t_{0}^{*}$. Then, note that we have $t_{1}^{\dagger}-t_{1}^{*}=t_{0}^{\dagger}-t_{0}^{*}=-4/9$, which is, in fact, the value of $\delta_{1}^{*}$. In sum, the values of parameters that give the observationally equivalent fitted probabilities are \begin{align} (t_{0},t_{1},q_{0},q_{1},\delta_{1},\rho) & =\left(\frac{1}{3},\frac{2}{3},\frac{1}{3},\frac{2}{3},0,0\right),\\ (t_{0}^{*},t_{1}^{*},q_{0},q_{1},\delta_{1}^{*},\rho^{*}) & =\left(\frac{5}{9},\frac{8}{9},\frac{1}{3},\frac{2}{3},-\frac{4}{9},1\right). \end{align} This argument can be made slightly more general, and thus the counterexample more realistic, by relaxing $F_{\tilde{\varepsilon}}\sim Unif(0,1)$ and $\rho^{*}=1$. We show that a similar argument goes through with $F_{\tilde{\varepsilon}}$ being a general distribution function with a symmetric density function, and $-1\leq\rho^{*}\leq1$ as long as the copula density is symmetric around $u_{2}=u_{1}$ (i.e., the 45 degree line) and $u_{2}=1-u_{1}$. Let $F\equiv F_{\tilde{\varepsilon}}$ be a general distribution whose density function is symmetric. Then there exists a solution $s_{0}^{\dagger}=s_{0}^{\dagger}(t_{0},q_{0},\rho,\rho^{*})$ such that \begin{align*} C(F(F^{-1}(t_{0})+0),q_{0};\rho) & =\Pr[u_{1}\leq t_{0},u_{2}\leq q_{0};\rho]\\ & =\Pr[u_{1}\leq s_{0}^{\dagger},u_{2}\leq q_{0};\rho^{*}]\\ & =C(s_{0}^{\dagger},q_{0};\rho^{*}). \end{align*} Then, by letting $\delta_{1}^{*}=F^{-1}(s_{0}^{\dagger})-F^{-1}(t_{0}^{*})$, we have $s_{0}^{\dagger}=F(F^{-1}(t_{0}^{*})+\delta_{1}^{*})$ and therefore $(t_{0},q_{0},\delta_{1},\rho)$ and $(t_{0}^{*},q_{0},\delta_{1}^{*},\rho^{*})$ result in $p_{11,x}=C(F(F^{-1}(t_{0})+0),q_{0};\rho)=C(F(F^{-1}(t_{0}^{*})+\delta_{1}^{*}),q_{0};\rho^{*})$. Suppose that $\delta_{1}=0$. Then there exists a solution $s_{1}^{\dagger}=s_{1}^{\dagger}(t_{1},q_{1},\rho,\rho^{*})$ such that \begin{align*} C(F(F^{-1}(t_{1})+0),q_{1};\rho) & =\Pr[u_{1}\leq t_{1},u_{2}\leq q_{1};\rho]\\ & =\Pr[u_{1}\leq s_{1}^{\dagger},u_{2}\leq q_{1};\rho^{*}]\\ & =C(s_{1}^{\dagger},q_{1};\rho^{*}). \end{align*} Then, if we can show that \[ F^{-1}(s_{1}^{\dagger})=F^{-1}(t_{1}^{*})+\delta_{1}^{*}, \] then $s_{1}^{\dagger}=F(F^{-1}(t_{1}^{*})+\delta_{1}^{*})$ and therefore $(t_{1},q_{1},\delta_{1},\rho)$ and $(t_{1}^{*},q_{1},\delta_{1}^{*},\rho^{*})$ result in $\tilde{p}_{11,1}=C(F(F^{-1}(t_{1})+0),q_{1};\rho)=C(F(F^{-1}(t_{1}^{*})+\delta_{1}^{*}),q_{1};\rho)$. Note $F^{-1}(s_{1}^{\dagger})=F^{-1}(t_{1}^{*})+\delta_{1}^{*}$ can be rewritten as $F^{-1}(s_{1}^{\dagger})=F^{-1}(t_{1}^{*})+F^{-1}(s_{0}^{\dagger})-F^{-1}(t_{0}^{*})$ or \begin{equation} F^{-1}(s_{1}^{\dagger})-F^{-1}(t_{1}^{*})=F^{-1}(s_{0}^{\dagger})-F^{-1}(t_{0}^{*}). \end{equation} But note that since the density of $F$ is symmetric, any two values $s$ and $\tilde{s}$ in $(0,1)$ that are symmetric around $u_{1}=1/2$ will satisfy $F^{-1}(s)=-F^{-1}(\tilde{s})$. Therefore, since in our example $s_{0}^{\dagger}$ and $t_{1}^{*}$ are symmetric around $u_{1}=1/2$, and so are $s_{1}^{\dagger}$ and $t_{0}^{*}$, we have the desired result (ref), and the counterexample (ref)\textendash (ref) remains valid. Note that the symmetry of the density function of $F$ plays a key role here; the uniform distribution trivially satisfies the condition as does the normal distribution. The above counter-example to identification involves a parameter on the boundary of the parameter space ($\rho^{*}=1$), while the identification results in the paper assume that the parameter space is open and thus that $\rho\in(-1,1)$. We now show that the key idea of the argument remains the same with $-1<\rho^{*}<1$. Suppose that the copula density is symmetric around $u_{2}=u_{1}$ and $u_{2}=1-u_{1}$. The normal copula satisfies this condition for any $\rho\in(-1,1)$. Because of this condition, the symmetry of $s_{0}^{\dagger}$ and $t_{1}^{*}$ (and of $s_{1}^{\dagger}$ and $t_{0}^{*}$) around $u_{1}=1/2$ does not break at a different value of $\rho^{*}$, even though the values of $s_{0}^{\dagger}$, $t_{1}^{*}$, $s_{1}^{\dagger}$, and $t_{0}^{*}$ themselves change. Therefore, (ref) continues to hold with $\rho^{*}\neq1$. \subsection{Proof of Lemma (ref)} The proof of Lemma (ref) is a slight modification of the proof of Theorem 2.14 of Joe1997. Suppose $C_{2|1}\prec_{S}\tilde{C}_{2|1}$. Let $(U_{1},U_{2})\sim C$, $(\tilde{U}_{1},\tilde{U}_{2})\sim\tilde{C}$, with $U_{j}\overset{d}{=}\tilde{U}_{j}$, $j=1,2$. By Theorem 2.9 of Joe1997, $(U_{1},U_{2})\overset{d}{=}(\tilde{U}_{1},\psi(U_{1},U_{2}))$ with $\psi(u_{1},u_{2})=\tilde{C}_{2|1}^{-1}(C_{2|1}(u_{2}|u_{1})|u_{1})$. Since $C_{2|1}\prec_{S}\tilde{C}_{2|1}$, $\psi$ is increasing in $u_{1}$ and $u_{2}$. We consider two cases: \begin{itemize} • Case 1: Suppose that $u_{1}$ and $u_{2}$ are such that $\psi(u_{1},u_{2})\leq u_{2}$. Then \begin{align*} \tilde{C}(u_{1},u_{2}) & =\Pr[\tilde{U}_{1}\leq u_{1},\tilde{U}_{2}\leq u_{2})]=\Pr[\tilde{U}_{1}<u_{1},\tilde{U}_{2}<u_{2})]\\ & =\Pr[U_{1}<u_{1},\psi(U_{1},U_{2})<u_{2})]\geq\Pr[U_{1}<u_{1},\psi(u_{1},U_{2})<u_{2}]\\ & >\Pr[U_{1}<u_{1},U_{2}<u_{2})]=C(u_{1},u_{2}) \end{align*} where the strict inequality holds since $U_{2}<u_{2}$ implies $\psi(u_{1},U_{2})\leq\psi(u_{1},u_{2})\leq u_{2}$ (but not vice versa since $\psi(u_{1},U_{2})\leq u_{2}$ and $\psi(u_{1},u_{2})\leq u_{2}$ does not necessarily imply $U_{2}<u_{2}$ and $\Pr[\psi(u_{1},u_{2})<\psi(u_{1},U_{2})]=\Pr[u_{2}<U_{2}]\neq0$), and the second last inequality holds since, given $U_{1}<u_{1}$, $\psi(U_{1},U_{2})\leq\psi(u_{1},U_{2})<u_{2}$. • Case 2: Suppose that $u_{1}$ and $u_{2}$ are such that $\psi(u_{1},u_{2})>u_{2}$. Then \begin{align*} u_{2}-C(u_{1},u_{2}) & =\Pr[U_{1}>u_{1},U_{2}<u_{2})]>\Pr[U_{1}>u_{1},\psi(u_{1},U_{2})\leq u_{2})]\\ & \geq\Pr[U_{1}>u_{1},\psi(U_{1},U_{2})\leq u_{2})]=\Pr[\tilde{U}_{1}>u_{1},\tilde{U}_{2}<u_{2}]=u_{2}-\tilde{C}(u_{1},u_{2}) \end{align*} where the strict inequality holds since $U_{2}>u_{2}$ implies $\psi(u_{1},U_{2})\geq\psi(u_{1},u_{2})>u_{2}$ or $\psi(u_{1},U_{2})\leq u_{2}$ implies $U_{2}\leq u_{2}$ (but not vice versa). \end{itemize} Therefore in both cases, $C(u_{1},u_{2})<\tilde{C}(u_{1},u_{2})$ for any $u_{1}$ and $u_{2}$. \section{Proofs of Results in Section (ref)} \subsection{Identification under Transformation of Marginal Distribution Functions} Recall that we consider the following specification of the marginal distribution functions to derive the asymptotic theory for the sieve ML estimator: \begin{equation} F_{\epsilon0}(x)=H_{\epsilon0}(G(x)),\quad F_{\nu0}(x)=H_{\nu0}(G_{\nu}(x)), \end{equation} where $G:\mathbb{R}\rightarrow[0,1]$ is a strictly increasing function with its derivative $g(x)\equiv\frac{dG(x)}{dx}$ and $g(x)$ is bounded away from zero on $\mathbb{R}$. We first verify that there exist $H_{\epsilon0}$ and $H_{\nu0}$ that satisfy (ref) for given $F_{\epsilon0}$, $F_{\nu0}$, and $G$. Since $G$ is assumed to be strictly increasing, there exists an inverse function $G^{-1}$. Letting $H_{\epsilon0}(\cdot)=F_{\epsilon0}(G^{-1}(\cdot))$ and $H_{\nu0}(\cdot)=F_{\nu0}(G^{-1}(\cdot))$, it is straightforward to show that $H_{\epsilon0}$ and $H_{\nu0}$ are mappings from $[0,1]$ to $[0,1]$ and that satisfy the relations in (ref). Note too that this transformation does not change the identification results. That is, $F_{0}$ is identified on $\mathbb{R}$ if and only if $H_{0}$ is identified on $[0,1]$. Assuming that $g$ is bounded away from zero on $\mathbb{R}$ and bounded above, the unknown density function $h_{j0}$ can be written as $h_{j0}(x)=\frac{f_{j0}(G^{-1}(x))}{g(G^{-1}(x))}$ for each $j\in\{\epsilon,\nu\}$, which is well-defined on $[0,1]$. In addition, we can see that $h_{\epsilon0}$ and $h_{\nu0}$ are identified if and only if the unknown marginal density functions $f_{\epsilon0}$ and $f_{\nu0}$ are identified. We note that the choice of $G$ depends on the tail behavior of $f_{\epsilon0}$ and $f_{\nu0}$. If researchers believe that the unknown marginal density functions have fat tails, then they should choose a distribution function with fat tails for $G$. On the other hand, one can choose the logistic or the standard normal distribution function for $G$ when $f_{\epsilon0}$ and $f_{\nu0}$ are likely to have thin tails. This is because Assumption (ref) implicitly requires that the unknown marginal density functions and $g$ decay at the same rate at the tails. Specifically, we observe that \[ h_{\epsilon0}(0)=\lim_{x\rightarrow0^{+}}h_{\epsilon0}(x)=\lim_{t\rightarrow-\infty}\frac{f_{\epsilon0}(t)}{g(t)}, \] and the limit exists if the decaying rates are of the same order. We also provide simulation results to examine how the performance of our semiparametric estimator varies across the choice of $G$ when the marginal density functions have fat tails (see Section (ref)). \subsection{Technical Expressions} \subsubsection{H\"older Norm and H\"older Class} Let $\mathcal{C}^{m}(\mathcal{X})$ be the space of $m$-times continuously differentiable real-valued functions on $\mathcal{X}$. Let $\zeta\in(0,1]$ and, given a $d$-tuple $\omega$, let $[\omega]=\omega_{1}+...+\omega_{d}$. Denote the differential operator by $\mathcal{D}$ and let $\mathcal{D}^{\omega}=\frac{\partial^{[\omega]}}{\partial x_{1}^{\omega_{1}}...\partial x_{d}^{\omega_{d}}}$. Letting $p=m+\zeta$, the H\"older norm of $h\in\mathcal{C}^{m}(\mathcal{X})$ is defined as follows: \[ ||h||_{\Lambda^{p}}\equiv\sup_{[\omega]\leq m,x}|\mathcal{D}^{\omega}h(x)|+\sup_{[\omega]=m}\sup_{x,y\in\mathcal{X},||x-y||_{E}\neq0}\frac{|\mathcal{D}^{\omega}h(x)-\mathcal{D}^{\omega}h(y)|}{||x-y||_{E}^{\zeta}}, \] where $\zeta$ is the H\"older exponent. A H\"older class with smoothness $p>0$, denoted by $\Lambda^{p}(\mathcal{X})$, is defined as $\Lambda^{p}(\mathcal{X})\equiv\{h\in\mathcal{C}^{m}(\mathcal{X}):||h||_{\Lambda^{p}}<\infty\}$. A H\"older ball with radius $R$, $\Lambda_{R}^{p}(\mathcal{X})$, is defined as $\Lambda_{R}^{p}(\mathcal{X})\equiv\{h\in\Lambda^{p}(\mathcal{X}):||h||_{\Lambda^{p}}\leq R<\infty\}$. \subsubsection{Sup-norm and Pseudo-metric $d_{c}$ } For any $h\in\mathcal{H}_{\epsilon}$ (or $\mathcal{H}_{\nu}$), define the sup-norm on $\mathcal{H}_{\epsilon}$ (or $\mathcal{H}_{\nu}$) as follows: \[ ||h||_{\infty}\equiv\sup_{t\in[0,1]}|h(t)|. \] Let $\theta=(\psi^{'},h_{\epsilon},h_{\nu})^{'}\in\Theta$ be given. We define the consistency norm $||\cdot||_{c}$ as follows: \[ ||\theta||_{c}\equiv||\psi||_{E}+||h_{\epsilon}||_{\infty}+||h_{\nu}||_{\mathcal{\infty}}, \] where $||\cdot||_{E}$ is the Euclidean norm. The pseudo-metric $d_{c}(\cdot,\cdot):\Theta\times\Theta\rightarrow[0,\infty)$, which induced by the consistency norm $||\cdot||_{c}$, is defined as \[ d_{c}(\theta_{1},\theta_{2})=||\theta_{1}-\theta_{2}||_{c}. \] \subsubsection{$L^{2}$-norm } \begin{equation} ||\theta-\theta_{0}||_{2}\equiv||\psi-\psi_{0}||_{E}+||h_{\epsilon}-h_{\epsilon0}||_{2}+||h_{\nu}-h_{\nu0}||_{2}, \end{equation} where $||h-\tilde{h}||_{2}^{2}\equiv\int_{0}^{1}(h(t)-\tilde{h}(t))^{2}dt$ for any $h,\tilde{h}\in\mathcal{H}$. It is straightforward to show that $||\theta-\theta_{0}||_{2}\leq d_{c}(\theta,\theta_{0})$, where $d_{c}(\theta,\theta_{0})=||\psi-\psi_{0}||_{E}+||h_{\epsilon}-h_{\epsilon0}||_{\infty}+||h_{\nu}-h_{\nu0}||_{\infty}$. \subsubsection{Fisher inner product and Fisher norm } Recall that $\mathbb{V}$ is the linear span of $\Theta-\{\theta_{0}\}$. Define the Fisher inner product on the space $\mathbb{V}$ as \[ <v,\tilde{v}>\equiv E\left[(\frac{\partial l(\theta_{0},W)}{\partial\theta}[v])(\frac{\partial l(\theta_{0},W)}{\partial\theta}[\tilde{v}])\right] \] for given $v,\tilde{v}\in\mathbb{V}$. Then, the Fisher norm for $v\in\mathbb{V}$ is defined as \[ ||v||^{2}\equiv<v,v>. \] \subsubsection{Relationship between the Fisher norm and $L^{2}$-norm } Note that for any $\theta_{1},\theta_{2}\in\Theta$, we have \begin{align} ||\theta_{1}-\theta_{2}||^{2} & =E\left[(\frac{\partial l(\theta_{0},W_{i})}{\partial\theta}[\theta_{1}-\theta_{2}])^{2}\right]\nonumber \\ & \leq B\left\{ E\left[\{\frac{\partial l(\theta_{0},W_{i})}{\partial\psi^{'}}(\psi_{1}-\psi_{2})\}^{2}\right]+E\left[\{\frac{\partial l(\theta_{0},W_{i})}{\partial h_{\epsilon}}[h_{\epsilon1}-h_{\epsilon2}]\}^{2}\right]+E\left[\{\frac{\partial l(\theta_{0},W_{i})}{\partial h_{\nu}}[h_{\nu1}-h_{\nu2}]\}^{2}\right]\right\} \nonumber \\ & \leq\tilde{B}||\theta_{1}-\theta_{2}||_{2}^{2} \end{align} for some $B,\tilde{B}>0$ under Assumptions (ref), (ref), and (ref). From equation (ref), it is straightforward to see that the convergence rate of the sieve ML estimator with respect to the Fisher norm $||\cdot||$ is at least as fast as the convergence rate with respect to the $L^{2}$-norm. \subsubsection{Directional derivatives of the log-likelihood function } Let $r_{10}=F_{\epsilon0}(x^{'}\beta_{0}+\delta_{10})$, $r_{00}=F_{\epsilon0}(x^{'}\beta_{0})$, and $s_{0}=F_{\nu0}(x^{'}\alpha_{0}+z^{'}\gamma_{0})$. For given $v=(v_{\psi}^{'},v_{\epsilon},v_{\nu})^{'}\in\mathbb{V}$, we have \begin{align} \frac{\partial l(\theta_{0},w)}{\partial\psi^{'}}v_{\psi} & =\sum_{\tilde{y},\tilde{d}\in\{0,1\}}(\mathbf{1}_{\tilde{y},\tilde{d}}\cdot\frac{1}{p_{\tilde{y}\tilde{d},xz}(\theta_{0})}\cdot\frac{\partial p_{\tilde{y}\tilde{d},xz}(\theta_{0})}{\partial\psi^{'}})v_{\psi}, \end{align} \begin{align} \frac{\partial l(\theta_{0},w)}{\partial h_{\epsilon}}[v_{\epsilon}] & =\mathbf{1}_{11}(y,d)\times\left[\frac{1}{p_{11,xz}(\theta_{0})}C_{1}(r_{10},s_{0};\rho_{0})\int_{0}^{G(x^{'}\beta_{0}+\delta_{10})}v_{\epsilon}(t)dt\right]\nonumber \\ & +\mathbf{1}_{10}(y,d)\times\left[\frac{1}{p_{10,xz}(\theta_{0})}\left\{ \left(1-C_{1}(r_{00},s_{0};\rho_{0})\right)\int_{0}^{G(x^{'}\beta_{0})}v_{\epsilon}(t)dt\right\} \right]\nonumber \\ & +\mathbf{1}_{01}(y,d)\times\left[\frac{1}{p_{01,xz}(\theta_{0})}\left\{ -C_{1}(r_{10},s_{0};\rho_{0})\int_{0}^{G(x^{'}\beta_{0}+\delta_{10})}v_{\epsilon}(t)dt\right\} \right]\nonumber \\ & +\mathbf{1}_{00}(y,d)\times\left[\frac{1}{p_{00,xz}(\theta_{0})}\left\{ \left(1-C_{1}(r_{00},s_{0};\rho_{0})\right)\int_{0}^{G(x^{'}\beta_{0})}v_{\epsilon}(t)dt\right\} \right], \end{align} and \begin{align} \frac{\partial l(\theta_{0},w)}{\partial h_{\nu}}[v_{\nu}] & =\left\{ \mathbf{1}_{11}(y,d)\times\frac{1}{p_{11,xz}(\theta_{0})}C_{2}(r_{10},s_{0};\rho_{0})+\mathbf{1}_{10}(y,d)\times\frac{1}{p_{10,xz}(\theta_{0})}(-C_{2}(r_{00},s_{0};\rho_{0}))\right.\nonumber \\ & \left.+\mathbf{1}_{01}(y,d)\times\frac{1}{p_{01,xz}(\theta_{0})}(1-C_{2}(r_{10},s_{0};\rho_{0}))+\mathbf{1}_{00}(y,d)\times\frac{1}{p_{00,xz}(\theta_{0})}(1-C_{2}(r_{00},s_{0};\rho_{0}))\right\} \nonumber \\ & \times\int_{0}^{G(x^{'}\alpha_{0}+z^{'}\gamma_{0})}v_{\nu}(t)dt. \end{align} \subsubsection{Directional derivative of the ATE} Let $v=(v_{\psi}^{'},v_{\epsilon},v_{\nu})^{'}\in\mathbb{V}$. Then, \begin{equation} \frac{\partial ATE(\theta_{0};x)}{\partial\theta^{'}}[v]=\left\{ f_{\epsilon0}(x^{'}\beta_{0}+\delta_{10})(x^{'}v_{\beta}+v_{\delta})-f_{\epsilon0}(x^{'}\beta_{0})x^{'}v_{\beta}\right\} +\int_{G(x^{'}\beta_{0})}^{G(x^{'}\beta_{0}+\delta_{10})}v_{\epsilon}(t)dt, \end{equation} where $f_{\epsilon0}(x)=h_{\epsilon0}(G(x))g(x)$. \subsection{Proof of Theorem (ref)} Define $Q_{0}(\theta)\equiv E[l(\theta,W_{i})]$. The following proposition is a modification of Theorem 3.1 in chen2007large and establishes the consistency of sieve M-estimator.\footnote{See also Remark 3.3 in chen2007large.} \begin{proposition} Let $\hat{\theta}_{n}$ be the sieve extremum estimator defined in (ref). Suppose that the following conditions hold : (i) $Q_{0}(\theta)$ is uniquely maximized at $\theta_{0}$ in $\Theta$ and $Q_{0}(\theta_{0})>-\infty$; (ii) $\Theta$ is compact under $d_{c}(\cdot,\cdot)$, and $Q_{0}(\theta)$ is upper semicontinuous on $\Theta$ under $d_{c}(\cdot,\cdot)$; (iii) The sieve spaces, $\Theta_{n}$, are compact under $d_{c}(\cdot,\cdot)$ ; (iv) $\Theta_{k}\subseteq\Theta_{k+1}\subseteq\Theta$ for all $k\geq1$, and there exists a sequence $\pi_{k}\theta_{0}\in\Theta_{k}$ such that $d_{c}(\theta_{0},\pi_{k}\theta_{0})\rightarrow0$ as $k\rightarrow\infty$ ; (v) For all $k\geq1$, $p\lim_{n\rightarrow\infty}\sup_{\theta\in\Theta_{k}}|Q_{n}(\theta)-Q_{0}(\theta)|=0$. Then, $d_{c}(\hat{\theta}_{n},\theta_{0})=o_{p}(1)$. \end{proposition} We show that the conditions in Theorem (ref) imply those in this proposition to prove consistency of the sieve estimator. We first need to verify that (i) the true parameter $\theta_{0}$ is the unique maximizer of $Q_{0}(\cdot)$ over $\Theta$ and that (ii) the sample log-likelihood function $Q_{n}(\cdot)$ uniformly converges to $Q_{0}(\cdot)$ over the sieve space in probability to establish the consistency of the sieve ML estimator. The following lemma shows that if the model with unknown marginal distributions are identified and some additional conditions are satisfied, then the true parameter $\theta_{0}$ is the unique maximizer of $Q_{0}(\cdot)$ over $\Theta$. \begin{lemma}Suppose that Assumptions (ref)\textendash (ref), (ref), (ref) and (ref) are satisfied. Then the condition (i) in Proposition (ref) is satisfied. \end{lemma} \begin{proof} By Theorem (ref), the model parameter is identified. Under Assumption (ref), we can see that for any $\theta\in\Theta$, $|Q_{0}(\theta)|\leq E|l(\theta,W_{i})|\leq\sum_{y,d\in\{0,1\}}E|\log(p_{yd,XZ}(\theta))|<\infty$, and thus the function $Q_{0}(\theta)$ is well-defined on $\Theta$ and $Q_{0}(\theta)>-\infty$ for all $\theta\in\Theta$; hence $Q_{0}(\theta_{0})>-\infty$. Since the model is identified, it implies that for $\theta\neq\theta_{0}$, there exists a set $E\subset\text{supp}(X,Z)$ such that $\int_{E}dP_{XZ}>0$ and for some $y,d\in\{0,1\}$, $\frac{p_{yd,xz}(\theta)}{p_{yd,xz}(\theta_{0})}\neq1$ on $E$, where $P_{XZ}$ is the distribution function of $(X,Z)$. Thus, we have \begin{align*} Q_{0}(\theta)-Q_{0}(\theta_{0}) & =\int\sum_{y,d\in\{0,1\}}p_{yd,xz}(\theta_{0})\log\left(\frac{p_{yd,xz}(\theta)}{p_{yd,xz}(\theta_{0})}\right)dP_{XZ}<\log\left(\int_{E}\sum_{y,d\in\{0,1\}}p_{yd,xz}(\theta)dP_{XZ}\right)\leq0, \end{align*} where the strict inequality holds by the fact that $p_{yd,xz}(\theta)\neq p_{yd,xz}(\theta_{0})$ on $E$ and Jensen's inequality. Hence, $\theta_{0}$ is the unique maximizer of $Q_{0}(\cdot)$. \end{proof} For any $\omega>0$, let $N(\omega,\Theta_{n},d_{c})$ be the covering numbers without bracketing of $\Theta_{n}$ with respect to the pseudo-metric $d_{c}$. We now establish the uniform convergence of $Q_{n}(\cdot)$ to $Q_{0}$ over the sieve space. \begin{lemma} Suppose that Assumptions (ref)\textendash (ref), (ref) are satisfied. If Assumptions (ref) through (ref) hold, then $\sup_{\theta\in\Theta_{n}}|Q_{n}(\theta)-Q_{0}(\theta)|\overset{p}{\rightarrow}0$ for all $n\geq1$. \end{lemma} \begin{proof} We verify Condition 3.5M in chen2007large. Let $B$ stand for a generic constant and it can be different in each place. By Assumptions (ref) and (ref), the first condition in Condition 3.5M is satisfied. Let $n\geq1$ be a natural number and $\theta,\tilde{\theta}\in\Theta_{n}$. Define $R_{1}(\theta)=F_{\epsilon}(X^{'}\beta+\delta_{1})$, $R_{0}(\theta)=F_{\epsilon}(X^{'}\beta)$, and $S(\theta)=F_{\nu}(X^{'}\alpha+Z^{'}\gamma)$. Similarly, we define $R_{1}(\tilde{\theta})=\tilde{F}_{\epsilon}(X^{'}\tilde{\beta}+\tilde{\delta}_{1})$, $R_{0}(\tilde{\theta})=\tilde{F}_{\epsilon}(X^{'}\tilde{\beta})$, and $S(\tilde{\theta})=\tilde{F}_{\nu}(X^{'}\tilde{\alpha}+Z^{'}\tilde{\gamma})$. For the simplicity of the notations, we write $R_{j}=R_{j}(\theta)$, $\tilde{R}_{j}=R_{j}(\tilde{\theta})$, $S=S(\theta)$, and $\tilde{S}=S(\tilde{\theta})$ for all $j=0,1$. Observe that \begin{align*} |p_{11,XZ}(\theta)-p_{11,XZ}(\tilde{\theta})| & =|C(R_{1},S;\rho)-C(\tilde{R}_{1},\tilde{S};\tilde{\rho})|\\ & \leq|C(R_{1},S;\rho)-C(\tilde{R}_{1},\tilde{S};\rho)|+|C(\tilde{R}_{1},\tilde{S};\rho)-C(\tilde{R}_{1},\tilde{S};\tilde{\rho})|\\ & \leq|R_{1}-\tilde{R}_{1}|+|S-\tilde{S}|+|C_{\rho}(\tilde{R}_{1},\tilde{S};\hat{\rho})||\rho-\tilde{\rho}|\\ & \leq|R_{1}-\tilde{R}_{1}|+|S-\tilde{S}|+B|\rho-\tilde{\rho}|, \end{align*} where $C_{\rho}(\cdot,\cdot;\cdot)$ is the partial derivative of $C(\cdot,\cdot;\cdot)$ with respect to $\rho$ and $\hat{\rho}$ is between $\rho$ and $\tilde{\rho}$ and $B<\infty$. Note that the last inequality holds due to a generic property of copulas (see, e.g. Theorem 2.2.4 in nelsen1999introduction) and the mean value theorem. We also have \begin{align} |R_{1}-\tilde{R}_{1}| & =\left|F_{\epsilon}(X^{'}\beta+\delta_{1})-\tilde{F}_{\epsilon}(X^{'}\tilde{\beta}+\tilde{\delta}_{1})\right|\nonumber \\ & \leq\left|F_{\epsilon}(X^{'}\beta+\delta_{1})-F_{\epsilon}(X^{'}\tilde{\beta}+\tilde{\delta}_{1})\right|+\left|F_{\epsilon}(X^{'}\tilde{\beta}+\tilde{\delta}_{1})-\tilde{F}_{\epsilon}(X^{'}\tilde{\beta}+\tilde{\delta}_{1})\right|\nonumber \\ & \leq\left|f_{\epsilon}(X^{'}\hat{\beta}+\hat{\delta}_{1})\right|\cdot\left|X^{'}(\beta-\tilde{\beta})+(\delta_{1}-\tilde{\delta}_{1})\right|+\int_{0}^{G(X^{'}\tilde{\beta}+\tilde{\delta}_{1})}\left|h_{\epsilon}(t)-\tilde{h}_{\epsilon}(t)\right|dt\nonumber \\ & \leq\sup_{x\in\mathbb{R}}|h_{\epsilon}(G(x))g(x)|\times||(X^{'},1)^{'}||_{E}\cdot||\psi-\tilde{\psi}||_{E}+||h_{\epsilon}-\tilde{h}_{\epsilon}||_{\infty}\nonumber \\ & \leq B\times||(X^{'},1)^{'}||_{E}\times||(\beta^{'},\delta_{1})^{'}-(\tilde{\beta}^{'},\tilde{\delta}_{1})^{'}||_{E}+||h_{\epsilon}-\tilde{h}_{\epsilon}||_{\infty}, \end{align} for some constant $B<\infty$. Similarly, we can show that \begin{equation} |R_{0}-\tilde{R}_{0}|\leq B\times||X||_{E}\times||\beta-\tilde{\beta}||_{E}+||h_{\epsilon}-\tilde{h}_{\epsilon}||_{\infty} \end{equation} and \begin{equation} |S-\tilde{S}|\leq B\times||(X^{'},Z^{'})^{'}||_{E}\times||(\alpha^{'},\gamma^{'})^{'}-(\tilde{\alpha}^{'},\tilde{\gamma}^{'})^{'}||_{E}+||h_{\nu}-\tilde{h}_{\nu}||_{\infty}. \end{equation} Note that, for any comparable subvectors $\psi_{s}$ and $\tilde{\psi}_{s}$ of $\psi$ and $\tilde{\psi}$, respectively, we have $||\psi_{s}-\tilde{\psi}_{s}||_{E}\leq||\psi-\tilde{\psi}||_{E}$ and that, for any subvector $W_{s}$ of $W$, we have $||W_{S}||_{E}\leq||W||_{E}$ a.s. Thus we have \begin{align*} |p_{11,XZ}(\theta)-p_{11,XZ}(\tilde{\theta})| & \leq B||(X^{'},1)^{'}||_{E}\cdot||\psi-\tilde{\psi}||_{E}+||h_{\epsilon}-\tilde{h}_{\epsilon}||_{\infty}\\ & \leq B||(X^{'},1)^{'}||_{E}d_{c}(\theta,\tilde{\theta}). \end{align*} Consequently, it follows that \begin{align*} |p_{10,XZ}(\theta)-p_{10,XZ}(\tilde{\theta})| & \leq|R_{0}-\tilde{R}_{0}|+|C(R_{0},S;\rho)-C(\tilde{R}_{0},\tilde{S};\tilde{\rho})|\\ & \leq2|R_{0}-\tilde{R}_{0}|+|S-\tilde{S}|+B|\rho-\tilde{\rho}|\\ & \leq B\{||X||_{E}||\beta-\tilde{\beta}||_{E}+||(X^{'},Z^{'})^{'}||_{E}||(\alpha^{'},\gamma^{'})^{'}-(\tilde{\alpha}^{'},\tilde{\gamma}^{'})^{'}||_{E}\\ & +||h_{\epsilon}-\tilde{h}_{\epsilon}||_{\infty}+||h_{\nu}-\tilde{h}_{\nu}||_{\infty}+|\rho-\tilde{\rho}|\}\\ & \leq B\cdot||(X^{'},Z^{'},1)^{'}||_{E}d_{c}(\theta,\tilde{\theta}),\\ |p_{01,XZ}(\theta)-p_{01,XZ}(\tilde{\theta})| & \leq2|S-\tilde{S}|+|R_{1}-\tilde{R}_{1}|+B|\rho-\tilde{\rho}|\\ & \leq B||(X^{'},Z^{'},1)^{'}||_{E}d_{c}(\theta,\tilde{\theta}),\\ |p_{00,XZ}(\theta)-p_{00,XZ}(\tilde{\theta})| & \leq|p_{11,XZ}(\theta)-p_{11,XZ}(\tilde{\theta})|+|p_{10,XZ}(\theta)-p_{10,XZ}(\tilde{\theta})|+|p_{01,XZ}(\theta)-p_{01,XZ}(\tilde{\theta})|\\ & \leq B||(X^{'},Z^{'},1)^{'}||_{E}d_{c}(\theta,\tilde{\theta}). \end{align*} In all, we have \begin{align} |l(\theta,W_{i})-l(\tilde{\theta},W_{i})| & \leq{\textstyle \sum\limits _{y,d=0,1}}\mathbf{1}_{yd}(Y_{i},D_{i})\cdot\left|\log p_{yd}(X_{i},Z_{i};\theta)-\log p_{yd}(X_{i},Z_{i};\tilde{\theta})\right|\nonumber \\ & \leq\frac{1}{p(X_{i},Z_{i})}\sum_{y,d=0,1}\mathbf{1}_{yd}(Y_{i},D_{i})\left|p_{yd}(X_{i},Z_{i};\theta)-p_{yd}(X_{i},Z_{i};\tilde{\theta})\right|\nonumber \\ & \leq\frac{B}{p(X_{i},Z_{i})}||(X_{i}^{'},Z_{i}^{'},1)^{'}||_{E}d_{c}(\theta,\tilde{\theta})\nonumber \\ & \equiv U(W_{i})d_{c}(\theta,\tilde{\theta}), \end{align} where $E[U(W_{i})^{2}]<\infty$ by Assumptions (ref) and (ref). This results in \begin{equation} \sup_{\theta,\tilde{\theta}\in\Theta_{n},d_{c}(\theta,\tilde{\theta})\leq\epsilon_{0}}\left|l(\theta,W_{i})-l(\tilde{\theta},W_{i})\right|\leq U(W_{i})\epsilon_{0} \end{equation} and thus the second condition in Condition 3.5M is satisfied with $s=1$. For the last condition in Condition 3.5M, note that for any $\omega>0$, we have \begin{align*} N(\omega,\Theta_{n},d_{c}) & \leq N(\frac{\omega}{2},\Psi,||\cdot||_{E})\cdot N(\frac{\omega}{4},\mathcal{H}_{\epsilon n},||\cdot||_{\infty})\cdot N(\frac{\omega}{4},\mathcal{H}_{\nu n},||\cdot||_{\infty}). \end{align*} By Lemma 2.5 in geer2000empirical, we have $\log N\left(\frac{\omega}{4},\mathcal{H}_{\epsilon n},||\cdot||_{\infty}\right)\leq k_{n}\log\left(1+\frac{32R}{\omega}\right)$ under Assumption (ref)-(i); and hence \begin{align*} \log N\left(\omega,\Theta_{n},d_{c}\right) & \leq const.\times k_{n}\times\log\left(1+\frac{32R}{\omega}\right)=o(n) \end{align*} if $k_{n}/n\rightarrow0$. Since the condition $k_{n}/n=o(1)$ is imposed by Assumption (ref)-(i), the last condition in Condition 3.5M is also satisfied. In all, we have the uniform convergence of $Q_{n}$ to $Q_{0}$ over $\Theta_{k}$. \end{proof} To finish proving Theorem (ref), we verify the conditions in Proposition (ref). By Lemmas (ref) and (ref), the conditions (i) and (v) in Proposition (ref) are satisfied. Using (ref) and Jensen's inequality, we can see that, for any $\theta,\tilde{\theta}\in\Theta$, \begin{align*} |Q_{0}(\theta)-Q_{0}(\tilde{\theta})| & \leq E|l(\theta,W_{i})-l(\tilde{\theta},W_{i})|\leq E[U(W_{i})]d_{c}(\theta,\tilde{\theta})=B\cdot d_{c}(\theta,\tilde{\theta}) \end{align*} for some $B<\infty$. Thus, $Q_{0}(\cdot)$ is continuous with respect to $d_{c}$. Note that since the parameter space of the finite-dimensional parameter $\psi$, $\Psi$, is assumed to be compact in Assumption (ref), the original parameter space $\Theta$ is compact under the $d_{c}$, by Theorems 1 and 2 in freyberger2015compactness, and thus the conditions (ii) and (iii) are satisfied with the specified parameter space and the norm. Since the condition (iv) is directly imposed, we have $d(\hat{\theta}_{n},\theta_{0})=o_{p}(1)$ by Proposition (ref). \subsection{Proof of Theorem (ref)} To establish the convergence rate with respect to the norm $||\cdot||_{2}$, we consider the following assumption: \begin{assumption} Let $K(\theta_{0},\theta)\equiv E[l(\theta_{0},W_{i})-l(\theta,W_{i})]$. Then, there exist $B_{1},B_{2}>0$ such that \[ B_{1}K(\theta_{0},\theta)\leq||\theta-\theta_{0}||_{2}^{2}\leq B_{2}K(\theta_{0},\theta) \] for all $\theta\in\Theta_{n}$ with $d_{c}(\theta,\theta_{0})=o(1)$. \end{assumption} Assumption (ref) implies that the $L^{2}$-norm $||\cdot||_{2}$ and the square-root of the KL divergence are equivalent. We derive the convergence rate of the sieve M-estimator with respect to the norm $||\cdot||_{2}$ by checking the conditions in Theorem 3.2 in chen2007large. Since $\{W_{i}\}_{i=1}^{n}$ is assumed to be i.i.d by Assumption (ref), Condition 3.6 in chen2007large is satisfied. For Condition 3.7 in chen2007large, we note that for a small $\epsilon_{1}>0$ and for any $\theta\in\Theta_{n}$ such that $||\theta-\theta_{0}||_{2}\leq\epsilon_{1}$, we have \begin{align*} Var\left(l(\theta,W_{i})-l(\theta_{0},W_{i})\right) & \leq E\left[l(\theta,W_{i})-l(\theta_{0},W_{i})\right]^{2}\\ & \leq E\left[\frac{1}{p(X_{i},Z_{i})^{2}}\sum_{y,d=0,1}\mathbf{1}_{yd}(Y_{i},D_{i})|p_{yd}(X_{i},Z_{i};\theta)-p_{yd}(X_{i},Z_{i};\theta_{0})|^{2}\right]\\ & \leq E\left[\frac{1}{p(X_{i},Z_{i})^{2}}\sum_{y,d\in\{0,1\}}|p_{yd}(X_{i},Z_{i};\theta)-p_{yd}(X_{i},Z_{i};\theta_{0})|^{2}\right]. \end{align*} By the same logic in (ref), we have \[ Var\left(l(\theta,W_{i})-l(\theta_{0},W_{i})\right)\leq E\left[U(W_{i})^{2}\right]d_{c}(\theta,\theta_{0})^{2}. \] Note that \begin{align*} d_{c}(\theta,\theta_{0})^{2} & =(||\psi-\psi_{0}||_{E}+||h_{\epsilon}-h_{\epsilon0}||_{\infty}+||h_{\nu}-h_{\nu0}||_{\infty})^{2}\\ & \leq4(||\psi-\psi_{0}||_{E}^{2}+||h_{\epsilon}-h_{\epsilon0}||_{\infty}^{2}+||h_{\nu}-h_{\nu0}||_{\infty}^{2}). \end{align*} By Lemma 2 in chen1998sieve, we have \begin{equation} ||h_{j}-h_{j0}||_{\infty}^{2}\leq||h_{j}-h_{j0}||_{2}^{\frac{4p}{2p+1}} \end{equation} for all $j\in\{\epsilon,\nu\}$. Since $\frac{4p}{2p+1}>1$ under Assumption (ref), we can show that \[ \sup_{\{\theta\in\Theta_{n}:||\theta-\theta_{0}||_{2}\leq\epsilon_{1}\}}Var\left(l(\theta,W_{i})-l(\theta_{0},W_{i})\right)\leq B_{1}\epsilon_{1}^{2} \] with $\epsilon_{1}\leq1$ and some constant $B_{1}$, and thus Condition 3.7 in chen2007large is satisfied. We recall equation (ref) to verify Condition 3.8 in chen2007large. Let $\epsilon_{2}>0$ be given and consider \begin{align} |l(\theta,W_{i})-l(\theta_{0},W_{i})| & \leq U(W_{i})d_{c}(\theta,\theta_{0})\nonumber \\ & =U(W_{i})\left\{ ||\psi-\psi_{0}||_{E}+||h_{\epsilon}-h_{\epsilon0}||_{\infty}+||h_{\nu}-h_{\nu0}||_{\infty}\right\} \nonumber \\ & \leq U(W_{i})\left\{ ||\psi-\psi_{0}||_{E}+||h_{\epsilon}-h_{\epsilon0}||_{2}^{\frac{2p}{2p+1}}+||h_{\nu}-h_{\nu0}||_{2}^{\frac{2p}{2p+1}}\right\} \nonumber \\ & \leq U(W_{i})\left\{ ||\psi-\psi_{0}||_{E}^{\frac{2p+1}{2p}}+||h_{\epsilon}-h_{\epsilon0}||_{2}+||h_{\nu}-h_{\nu0}||_{2}\right\} ^{\frac{2p}{2p+1}}\nonumber \\ & \leq U(W_{i})\left\{ ||\psi-\psi_{0}||_{E}\times(\sup_{\psi\in\Psi}||\psi||_{E}+||\psi_{0}||_{E})^{\frac{1}{2p}}+||h_{\epsilon}-h_{\epsilon0}||_{2}+||h_{\nu}-h_{\nu0}||_{2}\right\} ^{\frac{2p}{2p+1}}\nonumber \\ & \leq\tilde{U}(W_{i})\left\{ ||\psi-\psi_{0}||_{E}+||h_{\epsilon}-h_{\epsilon0}||_{2}+||h_{\nu}-h_{\nu0}||_{2}\right\} ^{\frac{2p}{2p+1}}, \end{align} where $\tilde{U}(W_{i})=\max\{1,(\sup_{\psi\in\Psi}||\psi||_{E}+||\psi_{0}||_{E})^{\frac{1}{2p}}\}\times U(W_{i})$. Since the parameter space for $\psi$, $\Psi$, is compact under Assumption (ref), $E[\tilde{U}(W_{i})^{2}]<\infty$. Thus, we have \[ \sup_{\{\theta\in\Theta_{n}:||\theta-\theta_{0}||_{2}\leq\epsilon_{2}\}}\left|l(\theta,W_{i})-l(\theta_{0},W_{i})\right|\leq\epsilon_{2}^{\frac{2p}{2p+1}}\tilde{U}(W_{i}) \] with $E[\tilde{U}_{i}(W_{i})^{2}]<\infty$ and this implies that, under Assumption (ref), Condition 3.8 in chen2007large is satisfied with $s=\frac{2p}{2p+1}\in(0,2)$ and $\gamma=2$. Let $\mathcal{L}_{n}\equiv\{l(\theta_{0},W_{i})-l(\theta,W_{i}):\theta\in\Theta_{n},||\theta-\theta_{0}||_{2}\leq\epsilon_{2}\}$. For given $\omega>0$, let $N_{[]}(\omega,\mathcal{L}_{n},||\cdot||_{L^{2}})$ be the covering number with bracketing of $\mathcal{L}_{n}$ with respect to the norm $||\cdot||_{L^{2}}$. We now need to calculate $\kappa_{n}$ which is defined as \[ \kappa_{n}\equiv\inf\left\{ \kappa\in(0,1):\frac{1}{\sqrt{n}\kappa^{2}}\int_{b\kappa^{2}}^{\kappa}\sqrt{H_{[]}(\omega,\mathcal{L}_{n},||\cdot||_{L^{2}})}d\omega\leq const.\right\} , \] where, for $f\in\mathcal{L}_{n}$, $||f(\theta,W_{i})||_{L^{2}}^{2}\equiv E[f(\theta,W_{i})^{2}]$ is the $L^{2}$-norm on $\mathcal{L}_{n}$ and $H_{[]}(\omega,\mathcal{L}_{n},||\cdot||_{L^{2}})$ is the $L_{2}$-metric entropy with bracketing of the class $\mathcal{L}_{n}$ (see vvw96 or geer2000empirical for the definition of $L_{2}$-metric entropy with bracketing). Let $B_{0}=E[U(W_{i})^{2}]$, where $U(W_{i})$ is the same to the one in (ref). By Theorem 2.7.11 in vvw96 and equation (ref), we can show that \begin{align*} N_{[]}\left(\omega,\mathcal{L}_{n},||\cdot||_{L^{2}}\right) & \leq N\left(\frac{\omega}{2B_{0}},\Theta_{n},d_{c}\right)\\ & \leq N\left(\frac{\omega}{4B_{0}},\Psi,||\cdot||_{E}\right)\cdot N\left(\frac{\omega}{8B_{0}},\mathcal{H}_{\epsilon n},||\cdot||_{\infty}\right)\cdot N\left(\frac{\omega}{8B_{0}},\mathcal{H}_{\nu n},||\cdot||_{\infty}\right), \end{align*} and this leads to \begin{align*} H_{[]}\left(\omega,\mathcal{L}_{n},||\cdot||_{L^{2}}\right) & =\log\left(N_{[]}\left(\omega,\mathcal{L}_{n},||\cdot||_{L^{2}}\right)\right)\leq const.\times k_{n}\times\log(1+\frac{64B_{0}R}{\omega}). \end{align*} In all, $\kappa_{n}$ solves \begin{align*} \frac{1}{\sqrt{n}\kappa_{n}^{2}}\int_{b\kappa_{n}^{2}}^{\kappa_{n}}\sqrt{H_{[]}(\omega,\mathcal{L}_{n},||\cdot||_{L^{2}})}d\omega & \leq\frac{const.}{\sqrt{n}\kappa_{n}^{2}}\int_{b\kappa_{n}^{2}}^{\kappa_{n}}\sqrt{k_{n}\cdot\log(1+\frac{64B_{0}R}{\omega})}d\omega\\ & \leq\frac{const.}{\sqrt{n}\kappa_{n}^{2}}\sqrt{k_{n}}\int_{b\kappa_{n}^{2}}^{\kappa_{n}}\sqrt{\frac{1}{\omega}}d\omega\leq const.\times\frac{1}{\sqrt{n}\kappa_{n}^{2}}\sqrt{k_{n}}\kappa_{n}\leq const., \end{align*} and thus $\kappa_{n}\propto\sqrt{\frac{k_{n}}{n}}$. Lastly, since $||\theta_{0}-\pi_{n}\theta_{0}||_{2}\leq||\theta_{0}-\pi_{n}\theta_{0}||_{c}=O(k_{n}^{-p})$ by lorentz1966approximation, we have \[ ||\hat{\theta}_{n}-\theta_{0}||_{2}=O_{p}\left(\max\left\{ \sqrt{\frac{k_{n}}{n}},k_{n}^{-p}\right\} \right) \] by Theorem 3.2 in chen2007large. By choosing $k_{n}\propto n^{\frac{1}{2p+1}}$, we have \[ ||\hat{\theta}_{n}-\theta_{0}||_{2}=O_{p}\left(n^{-\frac{p}{2p+1}}\right). \] \subsection{Proof of Proposition (ref)} We first provide some technical assumptions for the asymptotic normality. Let $\mu_{n}(g)=\frac{1}{n}\sum_{i=1}^{n}\{g(W_{i})-E[g(W_{i})]\}$ be the empirical process indexed by $g$. Let the convergence rate of the sieve estimator be $\delta_{n}$ (i.e., $||\hat{\theta}_{n}-\theta_{0}||=O_{p}(\delta_{n})$). \begin{assumption} There exist $\xi_{1}>0$ and $\xi_{2}>0$ with $2\xi_{1}+\xi_{2}<1$ and a constant $K$, such that $(\delta_{n})^{3-(2\xi_{1}+\xi_{2})}=o(n^{-1})$. In addition, the following hold for all $\tilde{\theta}\in\Theta_{n}$ with $||\tilde{\theta}-\theta_{0}||\leq\delta_{n}$, and all $v\in\mathbb{V}$ with $||v||\leq\delta_{n}$: (i) $\left|E\left[\frac{\partial^{2}l(\tilde{\theta},W)}{\partial\psi\partial\psi^{'}}-\frac{\partial^{2}l(\theta_{0},W)}{\partial\psi\partial\psi^{'}}\right]\right|<K\left\Vert \tilde{\theta}-\theta_{0}\right\Vert ^{1-\xi_{2}}$; (ii) $\left|E\left[\sum_{j\in\{\epsilon,\nu\}}\left\{ \frac{\partial^{2}l(\tilde{\theta},W)}{\partial\psi\partial h_{j}}[v_{j}]-\frac{\partial^{2}l(\theta_{0},W)}{\partial\psi\partial h_{j}}[v_{j}]\right\} \right]\right|\leq K\left\Vert v\right\Vert ^{1-\xi_{1}}\left\Vert \tilde{\theta}-\theta_{0}\right\Vert ^{1-\xi_{2}}$; (iii) $\left|E\left[\sum_{i,j\in\{\epsilon,\nu\}}\left\{ \frac{\partial^{2}l(\tilde{\theta},W)}{\partial h_{i}\partial h_{j}}[v,v]-\frac{\partial^{2}l(\theta_{0},W)}{\partial h_{i}\partial h_{j}}[v,v]\right\} \right]\right|\leq K||v||^{2(1-\xi_{1})}||\tilde{\theta}-\theta_{0}||^{1-\xi_{2}}$. \end{assumption} \begin{assumption} The following hold: (i) $\sup_{\theta\in\Theta_{n}:||\theta-\theta_{0}||=O(\delta_{n})}\mu_{n}\left(\frac{\partial l(\theta,W)}{\partial\psi^{'}}-\frac{\partial l(\theta_{0},W)}{\partial\psi^{'}}\right)=o_{p}\left(n^{-\frac{1}{2}}\right)$ ; (ii) For all $j\in\{\epsilon,\nu\}$, $\sup_{\theta\in\Theta_{n}:||\theta-\theta_{0}||=O(\delta_{n})}\mu_{n}\left(\frac{\partial l(\theta,W)}{\partial h_{j}}[\pi_{n}v_{j}^{*}]-\frac{\partial l(\theta_{0},W)}{\partial h_{j}}[\pi_{n}v_{j}^{*}]\right)=o_{p}\left(n^{-\frac{1}{2}}\right)$. \end{assumption} Assumptions (ref) and (ref) are modifications of Assumptions 5 and 6 in CFT06, which are needed to control for the second-order expansion of the log-likelihood function $l(\theta,W)$. Under Assumption (ref), these conditions require that the unknown marginal density functions be sufficiently smooth. For example, the sieve estimator needs to converge at a faster rate than $1/(3-(2\xi_{1}+\xi_{2}))$ to satisfy $(\delta_{n})^{3-(2\xi_{1}+\xi_{2})}=o(n^{-1})$. Usually, the convergence rate depends positively on the smoothness parameter $p$ in Assumption (ref) and thus the class of models should be restricted to that in which the density functions are sufficiently smooth. Note that since the sieve ML estimator $\hat{\theta}_{n}$ is consistent with respect to the pseudo-metric $d_{c}$ by Theorem (ref), it is consistent with respect to the norm $||\cdot||_{2}$ and thus with respect to the Fisher norm by equation (ref). We also point out that $||\hat{\theta}_{n}-\theta_{0}||=O_{p}(n^{-\frac{p}{2p+1}})$ by equation (ref) and Theorem (ref) under the given set of Assumptions. We follow the proof of Theorem 1 in CFT06. Assumptions 1 and 2 in CFT06 are implied by Assumption (ref)-(ref), (ref)-(ref), and (ref). The first two parts in Assumption (ref) correspond to Assumption 3 in CFT06. Since $p>1/2$ by Assumption (ref), $||\hat{\theta}_{n}-\theta_{0}||=o_{p}(n^{-1/4})$ by Theorem (ref) and this implies that $||\hat{\theta}_{n}-\theta_{0}||\times||\pi_{n}v^{*}-v^{*}||=o(n^{-1/2})$ under Assumption (ref). In addition, since $w>1+\frac{1}{2p}$, $\delta_{n}^{w}=o(n^{-1/2})$ by that $||\hat{\theta}_{n}-\theta_{0}||=O_{p}(n^{-\frac{p}{2p+1}})$. Hence, Assumptions 3 and 4 in CFT06 are satisfied. Define $r[\theta,\theta_{0},W_{i}]\equiv l(\theta,W_{i})-l(\theta_{0},Z_{i})-\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[\theta-\theta_{0}]$ and $\xi_{0}=2\xi_{1}+\xi_{2}$. Let $\zeta_{n}$ be a positive sequence with $\zeta_{n}=o(n^{-1/2})$ and $(\delta_{n})^{3-(2\xi_{1}+\xi_{2})}=\zeta_{n}o(n^{-1/2})$. Then we have \begin{align} 0 & \leq\frac{1}{n}\sum_{i=1}^{n}l(\hat{\theta}_{n},W_{i})-l(\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*},W_{i})\leq\mp\zeta_{n}\frac{1}{n}\sum_{i=1}^{n}\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[\pi_{n}v^{*}]\nonumber \\ & +\mu_{n}(r[\hat{\theta}_{n},\theta_{0},W_{i}]-r[\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*},\theta_{0},W_{i}])+E[r[\hat{\theta}_{n},\theta_{0},W_{i}]-r[\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*},\theta_{0},W_{i}]]. \end{align} We first note that, by Assumption (ref), \begin{align} E\left[\frac{1}{n}\sum_{i=1}^{n}\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[\pi_{n}v^{*}-v^{*}]\right]^{2} & \leq\frac{1}{n}E\left[\left\{ \frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[\pi_{n}v^{*}-v^{*}]\right\} ^{2}\right]\nonumber \\ & =\frac{1}{n}||\pi_{n}v^{*}-v^{*}||^{2}=o(n^{-1}), \end{align} and hence $\frac{1}{n}\sum_{i=1}^{n}\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[\pi_{n}v^{*}-v^{*}]=o_{p}(n^{-1/2})$. Observe that, by the mean value theorem, \begin{align} E\left[r[\theta,\theta_{0},W_{i}]\right] & =E\left[l(\theta,W_{i})-l(\theta_{0},W_{i})-\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[\theta-\theta_{0}]\right]\nonumber \\ & =E\left[\frac{1}{2}\frac{\partial^{2}l(\theta_{0},W_{i})}{\partial\theta\partial\theta^{'}}[\theta-\theta_{0},\theta-\theta_{0}]\right]\nonumber \\ & +\frac{1}{2}E\left[\frac{\partial^{2}l(\tilde{\theta},W_{i})}{\partial\theta\partial\theta^{'}}[\theta-\theta_{0},\theta-\theta_{0}]-\frac{\partial^{2}l(\theta_{0},W_{i})}{\partial\theta\partial\theta^{'}}[\theta-\theta_{0},\theta-\theta_{0}]\right], \end{align} where $\theta,\tilde{\theta}\in\Theta_{n}$ and $\tilde{\theta}$ is between $\theta$ and $\theta_{0}$. In addition, for any $v=(v_{\psi}^{'},v_{\epsilon},v_{\nu})^{'}\in\mathbb{V}$ and $\tilde{\theta}\in\Theta_{n}$ with $||\tilde{\theta}-\theta_{0}||=O(\delta_{n})$, we have \begin{align*} E\left[\frac{\partial^{2}l(\tilde{\theta},W_{i})}{\partial\theta\partial\theta^{'}}[v,v]-\frac{\partial^{2}l(\theta_{0},W_{i})}{\partial\theta\partial\theta^{'}}[v,v]\right] & =v_{\psi}^{'}E\left[\frac{\partial^{2}l(\tilde{\theta},W_{i})}{\partial\psi\partial\psi^{'}}-\frac{\partial^{2}l(\theta_{0},W_{i})}{\partial\psi\partial\psi^{'}}\right]v_{\psi}\\ & +\sum_{j\in\{\epsilon,\nu\}}2v_{\theta}^{'}E\left[\frac{\partial^{2}l(\tilde{\theta},W_{i})}{\partial\psi\partial h_{j}}[v_{j}]-\frac{\partial^{2}l(\theta_{0},W_{i})}{\partial\psi\partial h_{j}}[v_{j}]\right]\\ & +\sum_{k\in\{\epsilon,\nu\}}\sum_{j\in\{\epsilon,\nu\}}E\left[\frac{\partial^{2}l(\tilde{\theta},W_{i})}{\partial h_{k}\partial h_{j}}[v_{k},v_{j}]-\frac{\partial^{2}l(\theta_{0},W_{i})}{\partial h_{k}\partial h_{j}}[v_{k},v_{j}]\right], \end{align*} and this term can be controlled under Assumption (ref) in the same way of CFT06. This leads us to that \begin{align} E[r[\hat{\theta}_{n},\theta_{0},W_{i}]-r[\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*},\theta_{0},W_{i}]] & =-\frac{1}{2}(||\hat{\theta}_{n}-\theta_{0}||^{2}-||\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*}-\theta_{0}||)+\zeta_{n}o(n^{-1/2})\nonumber \\ & =\pm\zeta_{n}\times<\hat{\theta}_{n}-\theta_{0},v^{*}>+\zeta_{n}o(n^{-1/2}) \end{align} because we have $<\hat{\theta}_{n}-\theta_{0},\pi_{n}v^{*}-v^{*}>=o_{p}(n^{-1/2})$ and $||\pi_{n}v^{*}||^{2}\rightarrow||v^{*}||^{2}<\infty$. We also have that \begin{align*} & \mu_{n}\left(r[\hat{\theta}_{n},\theta_{0},W_{i}]-r[\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*},\theta_{0},W_{i}]\right)\\ = & \mu_{n}\left(l(\hat{\theta}_{n},W_{i})-l(\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*},W_{i})-\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[\mp\zeta_{n}\pi_{n}v^{*}]\right)\\ = & \mp\zeta_{n}\cdot\mu_{n}\left(\frac{\partial l(\tilde{\theta},W_{i})}{\partial\theta^{'}}[\pi_{n}v^{*}]-\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[\pi_{n}v^{*}]\right), \end{align*} where $\tilde{\theta}\in\Theta_{n}$ is between $\hat{\theta}_{n}$ and $\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*}$. By Assumption (ref), we have \begin{equation} \mu_{n}\left(r[\hat{\theta}_{n},\theta_{0},W_{i}]-r[\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*},\theta_{0},W_{i}]\right)=o_{p}(\zeta_{n}n^{-1/2}). \end{equation} Combining equations (ref) through (ref) with the fact that $E\left[\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[v^{*}]\right]=0$, we have \begin{align*} 0 & \leq\frac{1}{n}\sum_{i=1}^{n}l(\hat{\theta}_{n},W_{i})-l(\hat{\theta}_{n}\pm\zeta_{n}\pi_{n}v^{*},W_{i})\\ & =\mp\zeta_{n}\cdot\mu_{n}\left(\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[v^{*}]\right)\pm\zeta_{n}<\hat{\theta}_{n}-\theta_{0},v^{*}>+\zeta_{n}\cdot o_{p}(n^{-1/2}), \end{align*} and this results in that \begin{align*} \sqrt{n}<\hat{\theta}_{n}-\theta_{0},v^{*}> & =\sqrt{n}\mu_{n}\left(\frac{\partial l(\theta_{0},W_{i})}{\partial\theta^{'}}[v^{*}]\right)+o_{p}(1)\overset{d}{\rightarrow}\mathcal{N}\left(0,||v^{*}||^{2}\right). \end{align*} By Assumption (ref), we have \[ \sqrt{n}\left(T(\hat{\theta}_{n})-T(\theta_{0})\right)=\sqrt{n}<\hat{\theta}_{n}-\theta_{0},v^{*}>\overset{d}{\rightarrow}\mathcal{N}\left(0,||v^{*}||^{2}\right) \] by the same way in CFT06. \subsection{Proof of Theorem (ref)} Define \begin{equation} \mathcal{S}_{\psi_{0}}^{'}\equiv\frac{\partial l(\theta_{0},W)}{\partial\psi^{'}}-\left\{ \frac{\partial l(\theta_{0},W)}{\partial h_{\epsilon}}[b_{\epsilon}^{*}]+\frac{\partial l(\theta_{0},W)}{\partial h_{\nu}}[b_{\nu}^{*}]\right\} , \end{equation} where $b_{\epsilon}^{*}=(b_{\epsilon1}^{*},...,b_{\epsilon d_{\psi}}^{*})\in\Pi_{k=1}^{d_{\psi}}(\mathcal{H}_{\epsilon}-\{h_{\epsilon0}\})$ and $b_{\nu}^{*}=(b_{\nu1}^{*},...,b_{\nu d_{\psi}}^{*})\in\Pi_{k=1}^{d_{\psi}}(\mathcal{H_{\nu}}-\{h_{\nu0}\})$ are the solutions to the following optimization problems for $k=1,2,...,d_{\psi}$: \[ \inf_{(b_{\epsilon k},b_{\nu k})\in\bar{\mathbb{V}}_{\epsilon}\times\bar{\mathbb{V}}_{\nu}}E\left[\left(\frac{\partial l(\theta_{0},W)}{\partial\theta_{k}}-\left\{ \frac{\partial l(\theta_{0},W)}{\partial h_{\epsilon}}[b_{\epsilon k}]+\frac{\partial l(\theta_{0},W)}{\partial h_{\nu}}[b_{\nu k}]\right\} \right)^{2}\right]. \] We consider the following assumption to establish the asymptotic normality for $\psi_{0}$. \begin{assumption} $\mathcal{I}_{*}(\psi_{0})\equiv E[\mathcal{S}_{\psi_{0}}\mathcal{S}_{\psi_{0}}^{'}]$ is non-singular. \end{assumption} To prove Theorem (ref), take any arbitrary $\lambda\in\mathbb{R}^{d_{\psi}}-\{0\}$ with $|\lambda|\in(0,\infty)$ and let $T:\Theta\rightarrow\mathbb{R}$ be a functional of the form $T(\theta)=\lambda^{'}\psi$. Then, for any $v\in\mathbb{V}$, we have $\frac{\partial T(\theta_{0})}{\partial\theta}[v]=\lambda^{'}v_{\psi}$ and there exist a small $\eta>0$ such that $||v||\leq\eta$ and a constant $\tilde{c}>0$ such that \begin{equation} \left|T(\theta_{0}+v)-T(\theta_{0})-\frac{\partial T(\theta_{0})}{\partial\theta}\right|\leq\tilde{c}||v||^{w} \end{equation} with $w=\infty$. Therefore, Assumption (ref)-(i) is satisfied with $w=\infty$ in this case. In addition, we have \begin{align*} \sup_{v\in\mathbb{V}:||v||>0}\frac{|\lambda^{'}v_{\psi}|^{2}}{||v||^{2}} & =\sup_{v\in\mathbb{V}:||v||>0}\frac{|\lambda^{'}v_{\psi}|^{2}}{E\left[(\frac{\partial l(\theta_{0},W)}{\partial\psi^{'}}v_{\psi}+\sum_{j\in\{\epsilon,\nu\}}\frac{\partial l(\theta_{0},W)}{\partial h_{j}}[v_{j}])^{2}\right]}\\ & =\lambda^{'}E[\mathcal{S}_{\psi_{0}}\mathcal{S}_{\psi_{0}}^{'}]^{-1}\lambda=\lambda^{'}\mathcal{I}_{*}(\theta_{0})^{-1}\lambda. \end{align*} Note that the Riesz representer $v^{*}$ exists if and only if $\lambda^{'}E[\mathcal{S}_{\psi_{0}}\mathcal{S}_{\psi_{0}}^{'}]^{-1}\lambda$ is finite. Since Assumption (ref) implies that $\lambda^{'}E[\mathcal{S}_{\psi_{0}}\mathcal{S}_{\psi_{0}}^{'}]^{-1}\lambda$ is finite, Assumption (ref)-(ii) holds. Hence, by Proposition (ref), we have \[ \sqrt{n}\left(\lambda^{'}\hat{\psi}_{n}-\lambda^{'}\psi_{0}\right)\overset{d}{\rightarrow}\mathcal{N}\left(0,\lambda^{'}\mathcal{I}_{*}(\psi_{0})^{-1}\lambda\right). \] Since $\lambda$ was arbitrary, we obtain the result by Cram\'er-Wold device. \subsection{H�lder ball } Suppose that $h\in\Lambda_{R}^{p}([0,1])$, where $p=m+\zeta$, $m\geq0$ is an integer and $\zeta\in(0,1]$ is the H�lder exponent. We want to show that $h^{2}\in\Lambda_{\tilde{R}}^{p}([0,1])$, where $\tilde{R}=R^{2}2^{m+1}$ . Recall that $\mathcal{D}$ is the differential operator. We note that $||h||_{\infty}\leq R$ and thus $\sup_{x}|\mathcal{D}^{\omega}h(x)|\leq R$ for all $\omega\leq m$. By Leibniz's formula, we have \begin{align*} \left|\mathcal{D}^{\omega}h^{2}(x)\right| & =\left|\sum_{\iota\leq\omega}\begin{pmatrix}\omega\\ \iota \end{pmatrix}\mathcal{D}^{\iota}h\mathcal{D}^{\omega-\iota}h\right|\leq R^{2}\sum_{\iota\leq\omega}\begin{pmatrix}\omega\\ \iota \end{pmatrix}=R^{2}2^{\omega}\leq K^{2}2^{m}<\infty \end{align*} for all $\omega\leq m$. Observe that, by Leibniz's formula, for any $x,y\in[0,1]$ with $x\neq y$, \begin{align*} \left|\mathcal{D}^{m}h^{2}(x)-\mathcal{D}^{m}h^{2}(y)\right| & =\left|\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}\mathcal{D}^{\omega}h(x)\mathcal{D}^{m-\omega}h(x)-\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}\mathcal{D}^{\omega}h(y)\mathcal{D}^{m-\omega}h(y)\right|\\ & \leq\left|\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}\mathcal{D}^{\omega}h(x)\mathcal{D}^{m-\omega}h(x)-\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}\mathcal{D}^{\omega}h(y)\mathcal{D}^{m-\omega}h(x)\right|\\ & \ \ +\left|\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}\mathcal{D}^{\omega}h(y)\mathcal{D}^{m-\omega}h(x)-\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}\mathcal{D}^{\omega}h(y)\mathcal{D}^{m-\omega}h(y)\right|\\ & \leq2\times\{\sup_{\omega\leq m}\sup_{x}\left|\mathcal{D}^{\omega}h(x)\right|\}\times\left|\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}\{\mathcal{D}^{\omega}h(x)-\mathcal{D}^{\omega}h(y)\}\right|\\ & \leq2R\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}\left|\mathcal{D}^{\omega}h(x)-\mathcal{D}^{\omega}h(y)\right|. \end{align*} We also have that, for all $\omega<m$, \begin{align*} \frac{|\mathcal{D}^{\omega}h(x)-D^{\omega}h(y)|}{|x-y|^{\zeta}} & =\frac{|\mathcal{D}^{\omega}h(x)-\mathcal{D}^{\omega}h(y)|}{|x-y|}|x-y|^{1-\zeta}=|\mathcal{D}^{\omega+1}h(\tilde{x})||x-y|^{1-\zeta}\leq R, \end{align*} where $\tilde{x}$ is between $x$ and $y$. Note that $\zeta\in(0,1]$ and thus $|x-y|^{1-\zeta}\leq1$ for all $x,y\in[0,1]$. Since $h\in\Lambda_{R}^{p}([0,1])$, we have $\frac{|\mathcal{D}^{m}h(x)-\mathcal{D}^{m}h(y)|}{|x-y|^{\zeta}}\leq R$. Hence, \begin{align*} \frac{|\mathcal{D}^{m}h^{2}(x)-\mathcal{D}^{m}h^{2}(y)|}{|x-y|^{\zeta}} & \leq2R\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}\frac{|\mathcal{D}^{\omega}h(x)-\mathcal{D}^{\omega}h(y)|}{|x-y|^{\zeta}}\leq2R^{2}\sum_{\omega\leq m}\begin{pmatrix}m\\ \omega \end{pmatrix}=R^{2}2^{m+1}<\infty, \end{align*} and this implies that $h^{2}\in\Lambda_{\tilde{R}}^{p}([0,1])$ with $\tilde{R}=R^{2}2^{m+1}$. \section{Additional Simulation Results } \subsection{A Larger Sample Size} Tables (ref) and (ref) show the simulation results with a larger sample size ($n=1000$). We can see that the main findings in the main text remain the same even with this larger sample size. \subsection{Copula and Marginal Misspecification } We consider the simulation results when both the copula and the marginal distributions are misspecified, reported in Tables (ref)\textendash (ref) and (ref)\textendash (ref). If both the copula and the marginal distributions are misspecified, the performance of the parametric ML estimators are comparable to, or slightly worse than that under marginal misspecification. Consider, for example, the case where the true copula function is the Frank copula and the sample size is 500. The estimators of $\psi$ under both the copula and marginal misspecification (Table (ref)) have slightly larger root mean squared errors (RMSEs) than the corresponding estimators under the marginal misspecification (Table (ref)). On the other hand, the performance of the estimators of the ATE varies across copula specifications. In particular, when the true data generating process (DGP) is based on the Gumbel copula, the copula and marginal misspecification has a significant effect on the performance of the parametric estimators of the ATE. The RMSEs of the estimators of the ATE under the copula and marginal misspecification (Table (ref)) are larger than those under the marginal misspecification (Table (ref)). Specifically, the RMSE of the parametric estimator of the ATE under the marginal misspecification is 0.1637 (Table (ref)), whereas the RMSEs of the corresponding estimators under both the copula and marginal misspecification are 0.1835, 0.2178, and 0.2732 when the Gaussian, Frank, and Clayton copulas are used, respectively (Table (ref)). On the other hand, there is no clear evidence that the performance of the sieve ML estimators under both the copula and marginal misspecification is worse than that under misspecification of the marginal distributions. For example, when the true copula belongs to the Frank family but the copula is specified as the Gaussian or Gumbel copula, we can see that the RMSEs of the sieve ML estimators of the finite-dimensional parameters other than $\gamma$ and the ATE under the copula and marginal misspecification (Table (ref)) are lower than those under the marginal misspecification (Table (ref)). In contrast, we can see from the same tables that the Clayton copula specification draws the opposite conclusion when the true copula is the Frank. In general, no matter whether the copula is misspecified, we find that the sieve ML estimators outperform the parametric estimators in terms of the RMSE when the marginal distributions are misspecified. \subsection{Unknown Marginal Density Functions with Fat Tails} We examine the finite sample performance of the sieve ML estimator of $\theta_{0}$ when the unknown marginal density functions $f_{\epsilon0}$ and $f_{\nu0}$ have fat tails. We consider the $t$ distribution with 3 degree of freedom as the true marginal distributions. While the marginal distributions in the parametric models are specified by normal distributions, we consider two specifications for the semiparametric models. These specifications differ in the choice of $G$: we choose the standard normal distribution and the distribution function of $t(3)$ for $G$ in the first and second specifications, respectively. All simulation results are obtained with 500 observations and 2000 simulation iterations. Table (ref) presents simulation results. While the parametric estimates have larger standard deviations, the biases of the semiparametric estimates are larger than those of the parametric estimates. However, the resulting RMSEs of the semiparametric estimates are slightly larger than those of the parametric estimates. This is because the semiparametric specification does not satisfy the assumptions required for the asymptotic theory. Table (ref) shows simulation results where $G$ is the distribution function of $t(3)$. The performance of semiparametric estimator is comparable to that of parametric estimator in terms of the RMSE. The biases of semiparametric estimates in Table (ref) are much smaller than those in Table (ref), and the standard deviations of semiparametric estimates are very similar to those of parametric estimates. The simulation results in Tables (ref) and (ref) suggest that if a researcher has a prior belief about the tail behavior of the unknown marginal density functions, it should be reflected in the choice of $G$ for semiparametric models. If it is believed that the marginal density functions have fat tails, one may choose a distribution function with fat tails for $G$, such as the distribution function of $t(3)$. \subsection{Different Degrees of Dependence } Tables (ref) through (ref) provide simulation results across various degrees of dependence between $\epsilon$ and $\nu$. The dependence measure is unified into the Spearman's $\rho$, and we consider cases of $\rho_{sp}\in\{-0.5,0.2,0.7\}$.\footnote{Note that we only consider the Gaussian and Frank copulas for $\rho_{sp}=-0.5$ as the Clayton or the Gumbel copula does not allow for negative dependence.} We find that regardless of degrees of dependence, the results in our main paper remains the same: (i) the performance of the semiparametric estimator is comparable to that of the parametric estimator under correct specification, (ii) the semiparametric estimators outperform the parametric estimators under misspecification of the marginals. \subsection{Coverage Probabilities of Bootstrap Confidence Intervals} We conduct simulations to investigate coverage probabilities of bootstrap confidence intervals (CIs). We consider the following design: \begin{align*} Y_{i} & =\mathbf{1}\{-X_{1i}+X_{2i}\beta+D_{i}\delta\geq\epsilon_{i}\},\quad D_{i}=\mathbf{1}\{-X_{1i}+X_{2i}\alpha+Z_{i}\gamma\geq\nu_{i}\}, \end{align*} where $(\alpha,\gamma,\beta,\delta)=(0.5,0.8,0.8,1.1)$ and $(\epsilon,\nu)$ are generated from the Gaussian copula and normal marginals with $\rho_{sp}=0.5$. $(X_{1i},X_{2i},Z_{i})$ is drawn from a multivariate normal distribution. Note that the coefficients on $X_{1i}$ are fixed for scale normalization. The sample size, number of bootstrap iterations, and number of simulations are 500, 200, and 200, respectively. We consider two types of CIs: (i) CIs using the normal approximation, (ii) the percentile bootstrap CIs. Table (ref) presents the coverage probabilities of both CIs. We find that the bootstrap percentile CIs performs better than the CIs based on the normal approximation and that their coverage probabilities are close to the nominal level (95%). \begin{table}[H] \caption{Correct Specification ($n=1,000$) (True marginal: normal)} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8025} & {1.1165} & {0.4996} & {0.3632} & {Estimate} & {0.8026} & {1.1205} & {0.5031} & {0.3596}\tabularnewline \hline {S.D} & {0.0654} & {0.2737} & {0.1081} & {0.0656} & {S.D} & {0.0655} & {0.2939} & {0.1092} & {0.0668}\tabularnewline \hline {Bias} & {0.0025} & {0.0165} & {-0.0004} & {-0.0011} & {Bias} & {0.0026} & {0.0205} & {0.0031} & {-0.0048}\tabularnewline \hline {RMSE} & {0.0655} & {0.2742} & {0.1081} & {0.0656} & {RMSE} & {0.0655} & {0.2946} & {0.1092} & {0.0670}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8017} & {1.1188} & {0.5010} & {0.3635} & {Estimate} & {0.8007} & {1.1164} & {0.5042} & {0.3594}\tabularnewline \hline {S.D} & {0.0658} & {0.2605} & {0.1023} & {0.0620} & {S.D} & {0.0652} & {0.2663} & {0.1066} & {0.0652}\tabularnewline \hline {Bias} & {0.0017} & {0.0188} & {0.0010} & {-0.0009} & {Bias} & {0.0007} & {0.0164} & {0.0042} & {-0.0049}\tabularnewline \hline {RMSE} & {0.0658} & {0.2612} & {0.1023} & {0.0620} & {RMSE} & {0.0652} & {0.2668} & {0.1067} & {0.0653}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8030} & {1.1055} & {0.5007} & {0.3621} & {Estimate} & {0.8029} & {1.1100} & {0.5035} & {0.3572}\tabularnewline \hline {S.D} & {0.0658} & {0.2329} & {0.0958} & {0.0566} & {S.D} & {0.0659} & {0.2524} & {0.0964} & {0.0560}\tabularnewline \hline {Bias} & {0.0030} & {0.0055} & {0.0007} & {-0.0023} & {Bias} & {0.0029} & {0.0100} & {0.0035} & {-0.0071}\tabularnewline \hline {RMSE} & {0.0659} & {0.2330} & {0.0958} & {0.0567} & {RMSE} & {0.0660} & {0.2526} & {0.0965} & {0.0565}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8022} & {1.1192} & {0.4963} & {0.3644} & {Estimate} & {0.8025} & {1.1240} & {0.4986} & {0.3626}\tabularnewline \hline {S.D} & {0.0668} & {0.2655} & {0.1057} & {0.0635} & {S.D} & {0.0665} & {0.2818} & {0.1086} & {0.0684}\tabularnewline \hline {Bias} & {0.0022} & {0.0192} & {-0.0037} & {0.0001} & {Bias} & {0.0025} & {0.0240} & {-0.0014} & {-0.0017}\tabularnewline \hline {RMSE} & {0.0669} & {0.2662} & {0.1057} & {0.0635} & {RMSE} & {0.0665} & {0.2829} & {0.1086} & {0.0684}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Misspecification of Marginals ($n=1,000$) (True marginal: mixture of normals)} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7941} & {1.0549} & {0.4496} & {0.2447} & {Estimate} & {0.8641} & {1.3030} & {0.4778} & {0.1262}\tabularnewline \hline {S.D} & {0.0911} & {0.4256} & {0.1156} & {0.0807} & {S.D} & {0.0778} & {0.2576} & {0.0721} & {0.0463}\tabularnewline \hline {Bias} & {-0.0059} & {-0.0451} & {-0.0504} & {0.1381} & {Bias} & {0.0641} & {0.2030} & {-0.0222} & {0.0195}\tabularnewline \hline {RMSE} & {0.0913} & {0.4279} & {0.1261} & {0.1599} & {RMSE} & {0.1008} & {0.3279} & {0.0755} & {0.0502}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8044} & {1.3066} & {0.3940} & {0.2919} & {Estimate} & {0.8525} & {1.2802} & {0.4777} & {0.1291}\tabularnewline \hline {S.D} & {0.0899} & {0.3876} & {0.0966} & {0.0684} & {S.D} & {0.0837} & {0.2577} & {0.0690} & {0.0500}\tabularnewline \hline {Bias} & {0.0044} & {0.2066} & {-0.1060} & {0.1853} & {Bias} & {0.0525} & {0.1802} & {-0.0223} & {0.0225}\tabularnewline \hline {RMSE} & {0.0901} & {0.4392} & {0.1434} & {0.1975} & {RMSE} & {0.0988} & {0.3145} & {0.0725} & {0.0549}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8065} & {1.1207} & {0.4240} & {0.2553} & {Estimate} & {0.8547} & {1.2669} & {0.4851} & {0.1219}\tabularnewline \hline {S.D} & {0.0906} & {0.3704} & {0.1047} & {0.0677} & {S.D} & {0.0801} & {0.2622} & {0.0706} & {0.0456}\tabularnewline \hline {Bias} & {0.0065} & {0.0207} & {-0.0761} & {0.1487} & {Bias} & {0.0547} & {0.1669} & {-0.0150} & {0.0153}\tabularnewline \hline {RMSE} & {0.0908} & {0.3710} & {0.1294} & {0.1634} & {RMSE} & {0.0969} & {0.3108} & {0.0722} & {0.0481}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7849} & {1.0104} & {0.4606} & {0.2391} & {Estimate} & {0.8618} & {1.2980} & {0.4791} & {0.1268}\tabularnewline \hline {S.D} & {0.0893} & {0.3566} & {0.0950} & {0.0695} & {S.D} & {0.0781} & {0.2516} & {0.0684} & {0.0463}\tabularnewline \hline {Bias} & {-0.0151} & {-0.0896} & {-0.0393} & {0.1325} & {Bias} & {0.0618} & {0.1980} & {-0.0208} & {0.0201}\tabularnewline \hline {RMSE} & {0.0906} & {0.3677} & {0.1028} & {0.1496} & {RMSE} & {0.0996} & {0.3202} & {0.0715} & {0.0504}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Copula and Marginals Misspecification 1 ($n=500$) (True copula: Gaussian, true marginal: mixture of normals)} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8140} & {1.3080} & {0.3775} & {0.2916} & {Estimate} & {0.8463} & {1.3514} & {0.4499} & {0.1351}\tabularnewline \hline {S.D} & {0.1257} & {0.4899} & {0.1202} & {0.0862} & {S.D} & {0.1137} & {0.3502} & {0.0964} & {0.0686}\tabularnewline \hline {Bias} & {0.0140} & {0.2080} & {-0.1225} & {0.1849} & {Bias} & {0.0463} & {0.2514} & {-0.0501} & {0.0285}\tabularnewline \hline {RMSE} & {0.1265} & {0.5322} & {0.1716} & {0.2040} & {RMSE} & {0.1227} & {0.4311} & {0.1087} & {0.0743}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8244} & {1.5699} & {0.3691} & {0.3176} & {Estimate} & {0.8534} & {1.4386} & {0.4945} & {0.1586}\tabularnewline \hline {S.D} & {0.1271} & {0.6609} & {0.1697} & {0.0999} & {S.D} & {0.1128} & {0.3398} & {0.1044} & {0.0734}\tabularnewline \hline {Bias} & {0.0244} & {0.4699} & {-0.1308} & {0.2110} & {Bias} & {0.0534} & {0.3386} & {-0.0054} & {0.0520}\tabularnewline \hline {RMSE} & {0.1294} & {0.8109} & {0.2143} & {0.2335} & {RMSE} & {0.1248} & {0.4797} & {0.1046} & {0.0899}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7981} & {1.0706} & {0.4232} & {0.2448} & {Estimate} & {0.8546} & {1.2025} & {0.4697} & {0.1137}\tabularnewline \hline {S.D} & {0.1281} & {0.5795} & {0.1519} & {0.1077} & {S.D} & {0.1118} & {0.3611} & {0.1027} & {0.0600}\tabularnewline \hline {Bias} & {-0.0019} & {-0.0294} & {-0.0767} & {0.1382} & {Bias} & {0.0546} & {0.1025} & {-0.0302} & {0.0070}\tabularnewline \hline {RMSE} & {0.1281} & {0.5802} & {0.1702} & {0.1752} & {RMSE} & {0.1244} & {0.3754} & {0.1070} & {0.0604}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Copula and Marginals Misspecification 2 ($n=500$) (True copula: Frank, true marginal: mixture of normals)} {} \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{5}{|c|}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7992} & {1.1673} & {0.4517} & {0.2527} & {Estimate} & {0.8500} & {1.1788} & {0.5173} & {0.1192}\tabularnewline \hline {S.D} & {0.1342} & {0.6901} & {0.1680} & {0.1179} & {S.D} & {0.1158} & {0.3602} & {0.1000} & {0.0652}\tabularnewline \hline {Bias} & {-0.0008} & {0.0673} & {-0.0483} & {0.1461} & {Bias} & {0.0500} & {0.0788} & {0.0173} & {0.0126}\tabularnewline \hline {RMSE} & {0.1342} & {0.6934} & {0.1748} & {0.1877} & {RMSE} & {0.1262} & {0.3687} & {0.1015} & {0.0664}\tabularnewline \hline \multicolumn{5}{|c|}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8235} & {1.6132} & {0.3870} & {0.3184} & {Estimate} & {0.8484} & {1.3679} & {0.5212} & {0.1548}\tabularnewline \hline {S.D} & {0.1329} & {0.7039} & {0.1670} & {0.1018} & {S.D} & {0.1188} & {0.3416} & {0.1012} & {0.0755}\tabularnewline \hline {Bias} & {0.0235} & {0.5132} & {-0.1130} & {0.2118} & {Bias} & {0.0484} & {0.2679} & {0.0212} & {0.0482}\tabularnewline \hline {RMSE} & {0.1350} & {0.8711} & {0.2017} & {0.2350} & {RMSE} & {0.1283} & {0.4341} & {0.1034} & {0.0896}\tabularnewline \hline \multicolumn{5}{|c|}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8001} & {1.1697} & {0.4202} & {0.2564} & {Estimate} & {0.8485} & {1.1059} & {0.4997} & {0.1071}\tabularnewline \hline {S.D} & {0.1347} & {0.6697} & {0.1608} & {0.1165} & {S.D} & {0.1161} & {0.3548} & {0.0997} & {0.0601}\tabularnewline \hline {Bias} & {0.0001} & {0.0697} & {-0.0798} & {0.1498} & {Bias} & {0.0485} & {0.0059} & {-0.0003} & {0.0005}\tabularnewline \hline {RMSE} & {0.1347} & {0.6733} & {0.1795} & {0.1897} & {RMSE} & {0.1258} & {0.3548} & {0.0997} & {0.0601}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Copula and Marginals Misspecification 3 ($n=500$) (True copula: Clayton, true marginal: mixture of normals)} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7986} & {1.0471} & {0.4017} & {0.2392} & {Estimate} & {0.8533} & {1.1780} & {0.4493} & {0.1076}\tabularnewline \hline {S.D} & {0.1346} & {0.6366} & {0.1731} & {0.1181} & {S.D} & {0.1164} & {0.3438} & {0.1033} & {0.0569}\tabularnewline \hline {Bias} & {-0.0014} & {-0.0529} & {-0.0983} & {0.1325} & {Bias} & {0.0533} & {0.0780} & {-0.0508} & {0.0009}\tabularnewline \hline {RMSE} & {0.1346} & {0.6388} & {0.1991} & {0.1775} & {RMSE} & {0.1281} & {0.3525} & {0.1151} & {0.0569}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8083} & {1.1559} & {0.3611} & {0.2712} & {Estimate} & {0.8412} & {1.2404} & {0.4199} & {0.1160}\tabularnewline \hline {S.D} & {0.1318} & {0.4453} & {0.1143} & {0.0856} & {S.D} & {0.1166} & {0.3408} & {0.0965} & {0.0611}\tabularnewline \hline {Bias} & {0.0083} & {0.0559} & {-0.1389} & {0.1646} & {Bias} & {0.0412} & {0.1404} & {-0.0802} & {0.0094}\tabularnewline \hline {RMSE} & {0.1321} & {0.4488} & {0.1799} & {0.1855} & {RMSE} & {0.1237} & {0.3686} & {0.1255} & {0.0619}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8046} & {1.1937} & {0.3316} & {0.2680} & {Estimate} & {0.8542} & {1.1610} & {0.4148} & {0.1046}\tabularnewline \hline {S.D} & {0.1355} & {0.6663} & {0.1748} & {0.1220} & {S.D} & {0.1166} & {0.3283} & {0.1032} & {0.0557}\tabularnewline \hline {Bias} & {0.0046} & {0.0937} & {-0.1684} & {0.1613} & {Bias} & {0.0542} & {0.0610} & {-0.0852} & {-0.0020}\tabularnewline \hline {RMSE} & {0.1356} & {0.6728} & {0.2427} & {0.2022} & {RMSE} & {0.1285} & {0.3339} & {0.1339} & {0.0557}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Copula and Marginals Misspecification 4 ($n=500$) (True copula: Gumbel, true marginal: mixture of normals)} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7978} & {1.1488} & {0.4658} & {0.2523} & {Estimate} & {0.8609} & {1.3801} & {0.4957} & {0.1460}\tabularnewline \hline {S.D} & {0.1304} & {0.6489} & {0.1598} & {0.1117} & {S.D} & {0.1132} & {0.3749} & {0.1052} & {0.0730}\tabularnewline \hline {Bias} & {-0.0022} & {0.0488} & {-0.0342} & {0.1456} & {Bias} & {0.0609} & {0.2801} & {-0.0042} & {0.0393}\tabularnewline \hline {RMSE} & {0.1304} & {0.6508} & {0.1634} & {0.1835} & {RMSE} & {0.1286} & {0.4679} & {0.1053} & {0.0829}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8140} & {1.4128} & {0.3834} & {0.3064} & {Estimate} & {0.8532} & {1.4755} & {0.4543} & {0.1611}\tabularnewline \hline {S.D} & {0.1290} & {0.5211} & {0.1184} & {0.0867} & {S.D} & {0.1177} & {0.3466} & {0.0969} & {0.0752}\tabularnewline \hline {Bias} & {0.0140} & {0.3128} & {-0.1166} & {0.1998} & {Bias} & {0.0532} & {0.3755} & {-0.0457} & {0.0545}\tabularnewline \hline {RMSE} & {0.1297} & {0.6078} & {0.1662} & {0.2178} & {RMSE} & {0.1292} & {0.5110} & {0.1072} & {0.0929}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8276} & {1.8999} & {0.3208} & {0.3614} & {Estimate} & {0.8603} & {1.6010} & {0.4823} & {0.1960}\tabularnewline \hline {S.D} & {0.1321} & {0.7365} & {0.1753} & {0.0986} & {S.D} & {0.1172} & {0.3103} & {0.1065} & {0.0799}\tabularnewline \hline {Bias} & {0.0276} & {0.7999} & {-0.1791} & {0.2548} & {Bias} & {0.0603} & {0.5010} & {-0.0177} & {0.0894}\tabularnewline \hline {RMSE} & {0.1350} & {1.0873} & {0.2506} & {0.2732} & {RMSE} & {0.1318} & {0.5893} & {0.1079} & {0.1199}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Copula and Marginals Misspecification 1 ($n=1,000$) (True copula: Gaussian, true marginal: mixture of normals)} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8086} & {1.3159} & {0.3652} & {0.2975} & {Estimate} & {0.8549} & {1.3936} & {0.4376} & {0.1371}\tabularnewline \hline {S.D} & {0.0897} & {0.3636} & {0.0927} & {0.0650} & {S.D} & {0.0830} & {0.2548} & {0.0689} & {0.0506}\tabularnewline \hline {Bias} & {0.0086} & {0.2159} & {-0.1347} & {0.1909} & {Bias} & {0.0549} & {0.2936} & {-0.0623} & {0.0305}\tabularnewline \hline {RMSE} & {0.0901} & {0.4229} & {0.1636} & {0.2017} & {RMSE} & {0.0995} & {0.3887} & {0.0929} & {0.0591}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8193} & {1.5478} & {0.3661} & {0.3205} & {Estimate} & {0.8613} & {1.4684} & {0.4886} & {0.1574}\tabularnewline \hline {S.D} & {0.0906} & {0.4574} & {0.1217} & {0.0705} & {S.D} & {0.0812} & {0.2351} & {0.0710} & {0.0514}\tabularnewline \hline {Bias} & {0.0193} & {0.4478} & {-0.1338} & {0.2139} & {Bias} & {0.0613} & {0.3684} & {-0.0113} & {0.0508}\tabularnewline \hline {RMSE} & {0.0927} & {0.6401} & {0.1809} & {0.2252} & {RMSE} & {0.1018} & {0.4370} & {0.0719} & {0.0722}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7930} & {1.0391} & {0.4210} & {0.2453} & {Estimate} & {0.8620} & {1.2302} & {0.4574} & {0.1157}\tabularnewline \hline {S.D} & {0.0911} & {0.4010} & {0.1070} & {0.0771} & {S.D} & {0.0790} & {0.2554} & {0.0709} & {0.0439}\tabularnewline \hline {Bias} & {-0.0070} & {-0.0609} & {-0.0789} & {0.1386} & {Bias} & {0.0620} & {0.1302} & {-0.0426} & {0.0090}\tabularnewline \hline {RMSE} & {0.0914} & {0.4056} & {0.1330} & {0.1586} & {RMSE} & {0.1004} & {0.2867} & {0.0827} & {0.0449}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Copula and Marginals Misspecification 2 ($n=1,000$) (True copula: Frank, true marginal: mixture of normals)} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7935} & {1.0825} & {0.4653} & {0.2465} & {Estimate} & {0.8601} & {1.1832} & {0.5145} & {0.1196}\tabularnewline \hline {S.D} & {0.0926} & {0.4333} & {0.1152} & {0.0803} & {S.D} & {0.0768} & {0.2641} & {0.0723} & {0.0450}\tabularnewline \hline {Bias} & {-0.0065} & {-0.0175} & {-0.0347} & {0.1399} & {Bias} & {0.0601} & {0.0832} & {0.0145} & {0.0130}\tabularnewline \hline {RMSE} & {0.0929} & {0.4336} & {0.1203} & {0.1613} & {RMSE} & {0.0976} & {0.2769} & {0.0738} & {0.0468}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8188} & {1.5580} & {0.3941} & {0.3173} & {Estimate} & {0.8583} & {1.3743} & {0.5200} & {0.1542}\tabularnewline \hline {S.D} & {0.0919} & {0.4621} & {0.1194} & {0.0708} & {S.D} & {0.0794} & {0.2439} & {0.0718} & {0.0526}\tabularnewline \hline {Bias} & {0.0188} & {0.4580} & {-0.1059} & {0.2106} & {Bias} & {0.0583} & {0.2743} & {0.0200} & {0.0476}\tabularnewline \hline {RMSE} & {0.0938} & {0.6506} & {0.1595} & {0.2222} & {RMSE} & {0.0985} & {0.3671} & {0.0746} & {0.0709}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7954} & {1.0843} & {0.4327} & {0.2496} & {Estimate} & {0.8596} & {1.1105} & {0.4959} & {0.1082}\tabularnewline \hline {S.D} & {0.0927} & {0.4252} & {0.1119} & {0.0796} & {S.D} & {0.0765} & {0.2578} & {0.0708} & {0.0413}\tabularnewline \hline {Bias} & {-0.0046} & {-0.0157} & {-0.0673} & {0.1429} & {Bias} & {0.0596} & {0.0105} & {-0.0041} & {0.0016}\tabularnewline \hline {RMSE} & {0.0928} & {0.4255} & {0.1306} & {0.1636} & {RMSE} & {0.0970} & {0.2580} & {0.0709} & {0.0413}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Copula and Marginals Misspecification 3 ($n=1,000$) (True copula: Clayton, true marginal: mixture of normals)} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7928} & {0.9952} & {0.4102} & {0.2370} & {Estimate} & {0.8618} & {1.2015} & {0.4441} & {0.1097}\tabularnewline \hline {S.D} & {0.0929} & {0.4262} & {0.1233} & {0.0837} & {S.D} & {0.0764} & {0.2527} & {0.0737} & {0.0411}\tabularnewline \hline {Bias} & {-0.0072} & {-0.1048} & {-0.0898} & {0.1303} & {Bias} & {0.0618} & {0.1015} & {-0.0559} & {0.0030}\tabularnewline \hline {RMSE} & {0.0932} & {0.4389} & {0.1525} & {0.1549} & {RMSE} & {0.0983} & {0.2723} & {0.0925} & {0.0412}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8048} & {1.1667} & {0.3480} & {0.2754} & {Estimate} & {0.8510} & {1.2695} & {0.4101} & {0.1152}\tabularnewline \hline {S.D} & {0.0910} & {0.3362} & {0.0918} & {0.0649} & {S.D} & {0.0825} & {0.2578} & {0.0701} & {0.0453}\tabularnewline \hline {Bias} & {0.0048} & {0.0667} & {-0.1520} & {0.1688} & {Bias} & {0.0510} & {0.1695} & {-0.0899} & {0.0086}\tabularnewline \hline {RMSE} & {0.0911} & {0.3428} & {0.1776} & {0.1808} & {RMSE} & {0.0970} & {0.3085} & {0.1140} & {0.0461}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8046} & {1.1937} & {0.3316} & {0.2680} & {Estimate} & {0.8594} & {1.1883} & {0.4090} & {0.1054}\tabularnewline \hline {S.D} & {0.1355} & {0.6663} & {0.1748} & {0.1220} & {S.D} & {0.0784} & {0.2373} & {0.0727} & {0.0412}\tabularnewline \hline {Bias} & {0.0046} & {0.0937} & {-0.1684} & {0.1613} & {Bias} & {0.0594} & {0.0883} & {-0.0911} & {-0.0013}\tabularnewline \hline {RMSE} & {0.1356} & {0.6728} & {0.2427} & {0.2022} & {RMSE} & {0.0984} & {0.2532} & {0.1165} & {0.0412}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Copula and Marginals Misspecification 4 ($n=1,000$) (True copula: Gumbel, true DGP marginal: mixture of normals)} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7905} & {1.1059} & {0.4669} & {0.2520} & {Estimate} & {0.8660} & {1.4046} & {0.4893} & {0.1428}\tabularnewline \hline {S.D} & {0.0896} & {0.4412} & {0.1167} & {0.0815} & {S.D} & {0.0775} & {0.2644} & {0.0723} & {0.0508}\tabularnewline \hline {Bias} & {-0.0095} & {0.0059} & {-0.0330} & {0.1454} & {Bias} & {0.0660} & {0.3046} & {-0.0107} & {0.0362}\tabularnewline \hline {RMSE} & {0.0901} & {0.4412} & {0.1213} & {0.1667} & {RMSE} & {0.1018} & {0.4034} & {0.0730} & {0.0624}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8123} & {1.4374} & {0.3701} & {0.3149} & {Estimate} & {0.8628} & {1.5142} & {0.4473} & {0.1582}\tabularnewline \hline {S.D} & {0.0901} & {0.3917} & {0.0930} & {0.0651} & {S.D} & {0.0817} & {0.2377} & {0.0697} & {0.0545}\tabularnewline \hline {Bias} & {0.0123} & {0.3374} & {-0.1299} & {0.2083} & {Bias} & {0.0628} & {0.4142} & {-0.0526} & {0.0515}\tabularnewline \hline {RMSE} & {0.0910} & {0.5169} & {0.1597} & {0.2182} & {RMSE} & {0.1030} & {0.4776} & {0.0874} & {0.0750}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8228} & {1.8913} & {0.3197} & {0.3656} & {Estimate} & {0.8645} & {1.6249} & {0.4851} & {0.1894}\tabularnewline \hline {S.D} & {0.0927} & {0.5234} & {0.1336} & {0.0714} & {S.D} & {0.0808} & {0.2084} & {0.0742} & {0.0550}\tabularnewline \hline {Bias} & {0.0228} & {0.7913} & {-0.1803} & {0.2589} & {Bias} & {0.0645} & {0.5249} & {-0.0149} & {0.0828}\tabularnewline \hline {RMSE} & {0.0955} & {0.9488} & {0.2244} & {0.2686} & {RMSE} & {0.1034} & {0.5648} & {0.0757} & {0.0994}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \begin{centering} \caption{ Misspecification of Marginals ($n=500$) (True Marginal: $t(3)$)} \end{centering} \begin{centering} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation$\dagger$, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242}\tabularnewline \hline {Estimate} & {0.8060} & {1.1937} & {0.4938} & {0.3098} & {Estimate} & {0.7288} & {0.8480} & {0.5762} & {0.2499}\tabularnewline \hline {S.D} & {0.1119} & {0.5749} & {0.1647} & {0.1068} & {S.D} & {0.1037} & {0.3832} & {0.1339} & {0.1085}\tabularnewline \hline {Bias} & {0.0060} & {0.0937} & {-0.0062} & {-0.0143} & {Bias} & {-0.0712} & {-0.2520} & {0.0763} & {-0.0742}\tabularnewline \hline {RMSE} & {0.0125} & {0.3306} & {0.0271} & {0.0116} & {RMSE} & {0.0108} & {0.1468} & {0.0179} & {0.0173}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation$\dagger$, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242}\tabularnewline \hline {Estimate} & {0.8080} & {1.0964} & {0.5164} & {0.2919} & {Estimate} & {0.7370} & {0.8618} & {0.5763} & {0.2459}\tabularnewline \hline {S.D} & {0.1139} & {0.4602} & {0.1347} & {0.0922} & {S.D} & {0.1030} & {0.3226} & {0.1041} & {0.0884}\tabularnewline \hline {Bias} & {0.0080} & {-0.0036} & {0.0164} & {-0.0323} & {Bias} & {-0.0630} & {-0.2382} & {0.0763} & {-0.0783}\tabularnewline \hline {RMSE} & {0.0130} & {0.2118} & {0.0181} & {0.0096} & {RMSE} & {0.0106} & {0.1041} & {0.0108} & {0.0139}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation$\dagger$, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242}\tabularnewline \hline {Estimate} & {0.8000} & {1.0202} & {0.5386} & {0.2786} & {Estimate} & {0.7330} & {0.8446} & {0.5689} & {0.2547}\tabularnewline \hline {S.D} & {0.1145} & {0.4385} & {0.1357} & {0.0946} & {S.D} & {0.1044} & {0.3479} & {0.1250} & {0.0989}\tabularnewline \hline {Bias} & {0.0000} & {-0.0798} & {0.0385} & {-0.0456} & {Bias} & {-0.0670} & {-0.2554} & {0.0689} & {-0.0695}\tabularnewline \hline {RMSE} & {0.0131} & {0.1923} & {0.0184} & {0.0110} & {RMSE} & {0.0109} & {0.1210} & {0.0156} & {0.0146}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation$\dagger$, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242}\tabularnewline \hline {Estimate} & {0.8098} & {1.2599} & {0.4767} & {0.3205} & {Estimate} & {0.7344} & {0.8905} & {0.5628} & {0.2559}\tabularnewline \hline {S.D} & {0.1153} & {0.6137} & {0.1732} & {0.1140} & {S.D} & {0.1045} & {0.4106} & {0.1461} & {0.1144}\tabularnewline \hline {Bias} & {0.0098} & {0.1599} & {-0.0233} & {-0.0037} & {Bias} & {-0.0656} & {-0.2095} & {0.0628} & {-0.0682}\tabularnewline \hline {RMSE} & {0.0133} & {0.3767} & {0.0300} & {0.0130} & {RMSE} & {0.0109} & {0.1686} & {0.0213} & {0.0177}\tabularnewline \hline \end{tabular} \end{centering} \begin{centering} \end{centering} $\dagger$: The semiparametric models are specified with $G=\Phi$, where $\Phi(\cdot)$ is the standard normal distribution function. \end{table} \begin{table}[H] \begin{centering} \caption{ Misspecification of Marginals ($n=500$) (True marginal: $t(3)$)} \end{centering} \begin{centering} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation$\dagger$, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242}\tabularnewline \hline {Estimate} & {0.8124} & {1.1948} & {0.4913} & {0.3098} & {Estimate} & {0.8098} & {1.1957} & {0.4930} & {0.3252}\tabularnewline \hline {S.D} & {0.1149} & {0.5540} & {0.1626} & {0.1068} & {S.D} & {0.1146} & {0.5905} & {0.1639} & {0.1107}\tabularnewline \hline {Bias} & {0.0124} & {0.0948} & {-0.0086} & {-0.0143} & {Bias} & {0.0098} & {0.0957} & {-0.0069} & {0.0010}\tabularnewline \hline {RMSE} & {0.0132} & {0.3069} & {0.0264} & {0.0116} & {RMSE} & {0.0131} & {0.3487} & {0.0269} & {0.0123}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation$\dagger$, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242}\tabularnewline \hline {Estimate} & {0.8063} & {1.0832} & {0.5246} & {0.2873} & {Estimate} & {0.8087} & {1.1877} & {0.4953} & {0.3257}\tabularnewline \hline {S.D} & {0.1152} & {0.4741} & {0.1354} & {0.0941} & {S.D} & {0.1153} & {0.5132} & {0.1397} & {0.0971}\tabularnewline \hline {Bias} & {0.0063} & {-0.0168} & {0.0246} & {-0.0369} & {Bias} & {0.0087} & {0.0877} & {-0.0047} & {0.0015}\tabularnewline \hline {RMSE} & {0.0133} & {0.2247} & {0.0183} & {0.0102} & {RMSE} & {0.0133} & {0.2633} & {0.0195} & {0.0094}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation$\dagger$, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242}\tabularnewline \hline {Estimate} & {0.8067} & {1.0312} & {0.5354} & {0.2797} & {Estimate} & {0.8117} & {1.1871} & {0.4972} & {0.3172}\tabularnewline \hline {S.D} & {0.1161} & {0.4525} & {0.1365} & {0.0950} & {S.D} & {0.1163} & {0.5845} & {0.1500} & {0.0998}\tabularnewline \hline {Bias} & {0.0067} & {-0.0688} & {0.0354} & {-0.0445} & {Bias} & {0.0117} & {0.0871} & {-0.0028} & {-0.0070}\tabularnewline \hline {RMSE} & {0.0135} & {0.2048} & {0.0186} & {0.0110} & {RMSE} & {0.0135} & {0.3416} & {0.0225} & {0.0100}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation$\dagger$, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242} & {True Values} & {0.8000} & {1.1000} & {0.5000} & {0.3242}\tabularnewline \hline {Estimate} & {0.8101} & {1.2629} & {0.4780} & {0.3213} & {Estimate} & {0.8062} & {1.1713} & {0.5024} & {0.3225}\tabularnewline \hline {S.D} & {0.1153} & {0.5991} & {0.1711} & {0.1113} & {S.D} & {0.1153} & {0.5477} & {0.1561} & {0.1103}\tabularnewline \hline {Bias} & {0.0101} & {0.1629} & {-0.0220} & {-0.0029} & {Bias} & {0.0062} & {0.0713} & {0.0024} & {-0.0017}\tabularnewline \hline {RMSE} & {0.0133} & {0.3589} & {0.0293} & {0.0124} & {RMSE} & {0.0133} & {0.3000} & {0.0244} & {0.0122}\tabularnewline \hline \end{tabular} \end{centering} \begin{centering} \end{centering} $\dagger$: The semiparametric models are specified with $G=F_{t_{3}}$, where $F_{t_{3}}$ is the distribution function of $t(3)$. \end{table} \begin{table}[H] \caption{Correct Specification ($n=500,\ \rho_{sp}=0.2$) (True marginal: normal) } \begin{centering} \end{centering} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8026} & {1.1342} & {0.2093} & {0.3643} & {Estimate} & {0.8026} & {1.1342} & {0.2093} & {0.3526}\tabularnewline \hline {S.D} & {0.0945} & {0.4199} & {0.1840} & {0.0963} & {S.D} & {0.0945} & {0.4199} & {0.1840} & {0.0952}\tabularnewline \hline {Bias} & {0.0026} & {0.0342} & {0.0093} & {0.0000} & {Bias} & {0.0026} & {0.0342} & {0.0093} & {-0.0117}\tabularnewline \hline {RMSE} & {0.0089} & {0.1763} & {0.0339} & {0.0093} & {RMSE} & {0.0089} & {0.1763} & {0.0339} & {0.0092}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8037} & {1.0818} & {0.2216} & {0.3517} & {Estimate} & {0.8051} & {1.0905} & {0.2278} & {0.3448}\tabularnewline \hline {S.D} & {0.0974} & {0.3309} & {0.1468} & {0.0807} & {S.D} & {0.0981} & {0.3591} & {0.1443} & {0.0856}\tabularnewline \hline {Bias} & {0.0037} & {-0.0182} & {0.0215} & {-0.0126} & {Bias} & {0.0051} & {-0.0095} & {0.0277} & {-0.0195}\tabularnewline \hline {RMSE} & {0.0095} & {0.1095} & {0.0215} & {0.0067} & {RMSE} & {0.0096} & {0.1290} & {0.0208} & {0.0077}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.1999} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.1999} & {0.3643}\tabularnewline \hline {Estimate} & {0.8036} & {1.0973} & {0.2138} & {0.3571} & {Estimate} & {0.8046} & {1.1040} & {0.2216} & {0.3492}\tabularnewline \hline {S.D} & {0.0934} & {0.3170} & {0.1498} & {0.0773} & {S.D} & {0.0936} & {0.3593} & {0.1533} & {0.0818}\tabularnewline \hline {Bias} & {0.0036} & {-0.0027} & {0.0139} & {-0.0072} & {Bias} & {0.0046} & {0.0040} & {0.0217} & {-0.0151}\tabularnewline \hline {RMSE} & {0.0087} & {0.1005} & {0.0224} & {0.0060} & {RMSE} & {0.0088} & {0.1291} & {0.0235} & {0.0069}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8017} & {1.0886} & {0.2175} & {0.3524} & {Estimate} & {0.8033} & {1.1245} & {0.2170} & {0.3495}\tabularnewline \hline {S.D} & {0.0940} & {0.3640} & {0.1519} & {0.0867} & {S.D} & {0.0954} & {0.4176} & {0.1578} & {0.0927}\tabularnewline \hline {Bias} & {0.0017} & {-0.0114} & {0.0175} & {-0.0119} & {Bias} & {0.0033} & {0.0245} & {0.0170} & {-0.0149}\tabularnewline \hline {RMSE} & {0.0088} & {0.1325} & {0.0231} & {0.0077} & {RMSE} & {0.0091} & {0.1744} & {0.0249} & {0.0088}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Misspecification of Marginals ($n=500,\ \rho_{sp}=0.2$) (True marginal: mixture of normals)} \begin{centering} \end{centering} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8038} & {0.9013} & {0.2088} & {0.2108} & {Estimate} & {0.8544} & {1.2755} & {0.1821} & {0.1256}\tabularnewline \hline {S.D} & {0.1308} & {0.5666} & {0.1823} & {0.1137} & {S.D} & {0.1166} & {0.3865} & {0.1271} & {0.0638}\tabularnewline \hline {Bias} & {0.0038} & {-0.1987} & {0.0088} & {0.1041} & {Bias} & {0.0544} & {0.1755} & {-0.0179} & {0.0190}\tabularnewline \hline {RMSE} & {0.0171} & {0.3210} & {0.0332} & {0.0238} & {RMSE} & {0.0136} & {0.1494} & {0.0161} & {0.0044}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8056} & {1.0026} & {0.1732} & {0.2366} & {Estimate} & {0.8391} & {1.2759} & {0.1854} & {0.1218}\tabularnewline \hline {S.D} & {0.1306} & {0.3979} & {0.1086} & {0.0781} & {S.D} & {0.1198} & {0.3588} & {0.0936} & {0.0573}\tabularnewline \hline {Bias} & {0.0056} & {-0.0974} & {-0.0268} & {0.1299} & {Bias} & {0.0391} & {0.1759} & {-0.0146} & {0.0152}\tabularnewline \hline {RMSE} & {0.0170} & {0.1583} & {0.0118} & {0.0230} & {RMSE} & {0.0143} & {0.1288} & {0.0088} & {0.0035}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.1999} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.1999} & {0.1066}\tabularnewline \hline {Estimate} & {0.8038} & {0.9008} & {0.1951} & {0.2144} & {Estimate} & {0.8459} & {1.2556} & {0.1878} & {0.1172}\tabularnewline \hline {S.D} & {0.1310} & {0.4511} & {0.1508} & {0.0920} & {S.D} & {0.1185} & {0.3701} & {0.1214} & {0.0573}\tabularnewline \hline {Bias} & {0.0038} & {-0.1992} & {-0.0048} & {0.1077} & {Bias} & {0.0459} & {0.1556} & {-0.0122} & {0.0105}\tabularnewline \hline {RMSE} & {0.0172} & {0.2035} & {0.0228} & {0.0201} & {RMSE} & {0.0140} & {0.1370} & {0.0147} & {0.0034}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.2000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7925} & {0.7687} & {0.2430} & {0.1884} & {Estimate} & {0.8523} & {1.2767} & {0.1840} & {0.1245}\tabularnewline \hline {S.D} & {0.1330} & {0.4382} & {0.1344} & {0.0939} & {S.D} & {0.1202} & {0.3913} & {0.1128} & {0.0629}\tabularnewline \hline {Bias} & {-0.0075} & {-0.3313} & {0.0430} & {0.0817} & {Bias} & {0.0523} & {0.1767} & {-0.0160} & {0.0178}\tabularnewline \hline {RMSE} & {0.0177} & {0.1920} & {0.0181} & {0.0155} & {RMSE} & {0.0144} & {0.1532} & {0.0127} & {0.0043}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Correct Specification ($n=500,\ \rho_{sp}=0.7$)(True marginal: normal) } \begin{centering} \end{centering} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8032} & {1.1403} & {0.6979} & {0.3660} & {Estimate} & {0.8038} & {1.1475} & {0.7059} & {0.3615}\tabularnewline \hline {S.D} & {0.0932} & {0.3663} & {0.1161} & {0.0860} & {S.D} & {0.0942} & {0.3909} & {0.1167} & {0.0928}\tabularnewline \hline {Bias} & {0.0032} & {0.0403} & {-0.0020} & {0.0016} & {Bias} & {0.0038} & {0.0475} & {0.0060} & {-0.0028}\tabularnewline \hline {RMSE} & {0.0087} & {0.1342} & {0.0135} & {0.0074} & {RMSE} & {0.0089} & {0.1528} & {0.0136} & {0.0086}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8094} & {1.2165} & {0.6676} & {0.3858} & {Estimate} & {0.8097} & {1.2244} & {0.6738} & {0.3783}\tabularnewline \hline {S.D} & {0.0928} & {0.3185} & {0.0912} & {0.0675} & {S.D} & {0.0930} & {0.3138} & {0.0856} & {0.0748}\tabularnewline \hline {Bias} & {0.0094} & {0.1165} & {-0.0324} & {0.0214} & {Bias} & {0.0097} & {0.1244} & {-0.0262} & {0.0140}\tabularnewline \hline {RMSE} & {0.0086} & {0.1015} & {0.0083} & {0.0050} & {RMSE} & {0.0086} & {0.0985} & {0.0073} & {0.0058}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8055} & {1.1382} & {0.6952} & {0.3666} & {Estimate} & {0.8065} & {1.1581} & {0.7002} & {0.3598}\tabularnewline \hline {S.D} & {0.0946} & {0.3188} & {0.0939} & {0.0709} & {S.D} & {0.0946} & {0.3441} & {0.0910} & {0.0750}\tabularnewline \hline {Bias} & {0.0055} & {0.0382} & {-0.0049} & {0.0023} & {Bias} & {0.0065} & {0.0581} & {0.0002} & {-0.0045}\tabularnewline \hline {RMSE} & {0.0090} & {0.1017} & {0.0088} & {0.0050} & {RMSE} & {0.0090} & {0.1184} & {0.0083} & {0.0057}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8036} & {1.1517} & {0.6945} & {0.3688} & {Estimate} & {0.8055} & {1.1806} & {0.6979} & {0.3702}\tabularnewline \hline {S.D} & {0.0937} & {0.3644} & {0.1185} & {0.0841} & {S.D} & {0.0942} & {0.3941} & {0.1197} & {0.0942}\tabularnewline \hline {Bias} & {0.0036} & {0.0517} & {-0.0055} & {0.0045} & {Bias} & {0.0055} & {0.0806} & {-0.0021} & {0.0058}\tabularnewline \hline {RMSE} & {0.0088} & {0.1328} & {0.0140} & {0.0071} & {RMSE} & {0.0089} & {0.1553} & {0.0143} & {0.0089}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Misspecification of Marginals ($n=500,\ \rho_{sp}=0.7$) (True marginal: mixture of normals)} \begin{centering} \end{centering} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7942} & {1.1740} & {0.6323} & {0.2582} & {Estimate} & {0.8565} & {1.2619} & {0.6932} & {0.1252}\tabularnewline \hline {S.D} & {0.1276} & {0.6180} & {0.1331} & {0.1080} & {S.D} & {0.1084} & {0.3714} & {0.0835} & {0.0661}\tabularnewline \hline {Bias} & {-0.0058} & {0.0740} & {-0.0676} & {0.1515} & {Bias} & {0.0565} & {0.1619} & {-0.0068} & {0.0186}\tabularnewline \hline {RMSE} & {0.0163} & {0.3820} & {0.0177} & {0.0346} & {RMSE} & {0.0118} & {0.1379} & {0.0070} & {0.0047}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8164} & {1.5022} & {0.5823} & {0.3157} & {Estimate} & {0.8566} & {1.3039} & {0.6787} & {0.1411}\tabularnewline \hline {S.D} & {0.1237} & {0.5841} & {0.1151} & {0.0918} & {S.D} & {0.1094} & {0.3071} & {0.0619} & {0.0685}\tabularnewline \hline {Bias} & {0.0164} & {0.4022} & {-0.1177} & {0.2091} & {Bias} & {0.0566} & {0.2039} & {-0.0212} & {0.0345}\tabularnewline \hline {RMSE} & {0.0153} & {0.3412} & {0.0132} & {0.0521} & {RMSE} & {0.0120} & {0.0943} & {0.0038} & {0.0059}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Clayton Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Clayton Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8219} & {1.3357} & {0.6006} & {0.2820} & {Estimate} & {0.8569} & {1.2553} & {0.6888} & {0.1272}\tabularnewline \hline {S.D} & {0.1297} & {0.5681} & {0.1200} & {0.0902} & {S.D} & {0.1109} & {0.3197} & {0.0714} & {0.0628}\tabularnewline \hline {Bias} & {0.0219} & {0.2357} & {-0.0995} & {0.1754} & {Bias} & {0.0569} & {0.1553} & {-0.0113} & {0.0206}\tabularnewline \hline {RMSE} & {0.0168} & {0.3227} & {0.0144} & {0.0389} & {RMSE} & {0.0123} & {0.1022} & {0.0051} & {0.0044}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Gumbel Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gumbel Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {0.7000} & {0.1066}\tabularnewline \hline {Estimate} & {0.7874} & {1.1463} & {0.6389} & {0.2567} & {Estimate} & {0.8556} & {1.2614} & {0.6953} & {0.1251}\tabularnewline \hline {S.D} & {0.1235} & {0.5168} & {0.1135} & {0.0942} & {S.D} & {0.1081} & {0.3526} & {0.0810} & {0.0661}\tabularnewline \hline {Bias} & {-0.0126} & {0.0463} & {-0.0611} & {0.1501} & {Bias} & {0.0556} & {0.1614} & {-0.0047} & {0.0184}\tabularnewline \hline {RMSE} & {0.0152} & {0.2670} & {0.0129} & {0.0314} & {RMSE} & {0.0117} & {0.1243} & {0.0066} & {0.0047}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Correct Specification ($n=500,\ \rho_{sp}=-0.5$)(True marginal: normal) } \begin{centering} \end{centering} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {-0.5000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {-0.5000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8090} & {1.1134} & {-0.4912} & {0.3560} & {Estimate} & {0.8101} & {1.1164} & {-0.4822} & {0.3448}\tabularnewline \hline {S.D} & {0.0970} & {0.4097} & {0.1727} & {0.0871} & {S.D} & {0.0974} & {0.4248} & {0.1708} & {0.0840}\tabularnewline \hline {Bias} & {0.0090} & {0.0134} & {0.0088} & {-0.0084} & {Bias} & {0.0101} & {0.0164} & {0.0177} & {-0.0196}\tabularnewline \hline {RMSE} & {0.0094} & {0.1678} & {0.0298} & {0.0077} & {RMSE} & {0.0095} & {0.1805} & {0.0292} & {0.0074}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {-0.5000} & {0.3643} & {True Values} & {0.8000} & {1.1000} & {-0.5000} & {0.3643}\tabularnewline \hline {Estimate} & {0.8049} & {1.1135} & {-0.4887} & {0.3582} & {Estimate} & {0.8060} & {1.1335} & {-0.4855} & {0.3524}\tabularnewline \hline {S.D} & {0.0946} & {0.3451} & {0.1389} & {0.0733} & {S.D} & {0.0943} & {0.3835} & {0.1399} & {0.0745}\tabularnewline \hline {Bias} & {0.0049} & {0.0135} & {0.0113} & {-0.0062} & {Bias} & {0.0060} & {0.0335} & {0.0145} & {-0.0119}\tabularnewline \hline {RMSE} & {0.0090} & {0.1191} & {0.0193} & {0.0054} & {RMSE} & {0.0089} & {0.1471} & {0.0196} & {0.0057}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Misspecification of Marginals ($n=500,\ \rho_{sp}=-0.5$) (True marginal: mixture of normals)} \begin{centering} \end{centering} {} \begin{tabular}{|c|c|c|c|c||c|c|c|c|c|} \hline \multicolumn{5}{|c||}{Parametric Estimation, Gaussian Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Gaussian Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {-0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {-0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8385} & {0.3931} & {-0.3171} & {0.1028} & {Estimate} & {0.8389} & {1.2832} & {-0.5237} & {0.1168}\tabularnewline \hline {S.D} & {0.1301} & {0.4122} & {0.1665} & {0.1052} & {S.D} & {0.1123} & {0.4094} & {0.1232} & {0.0627}\tabularnewline \hline {Bias} & {0.0385} & {-0.7069} & {0.1829} & {-0.0038} & {Bias} & {0.0389} & {0.1832} & {-0.0238} & {0.0101}\tabularnewline \hline {RMSE} & {0.0169} & {0.1699} & {0.0277} & {0.0111} & {RMSE} & {0.0126} & {0.1676} & {0.0152} & {0.0040}\tabularnewline \hline \multicolumn{5}{|c||}{Parametric Estimation, Frank Copula} & \multicolumn{5}{c|}{Semiparametric Estimation, Frank Copula}\tabularnewline \hline & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$} & & {$\gamma$} & {$\delta_{1}$} & {$\rho_{sp}$} & {$ATE$}\tabularnewline \hline {True Values} & {0.8000} & {1.1000} & {-0.5000} & {0.1066} & {True Values} & {0.8000} & {1.1000} & {-0.5000} & {0.1066}\tabularnewline \hline {Estimate} & {0.8432} & {0.3734} & {-0.3450} & {0.1030} & {Estimate} & {0.8375} & {1.2838} & {-0.5234} & {0.1153}\tabularnewline \hline {S.D} & {0.1373} & {0.3150} & {0.1032} & {0.0791} & {S.D} & {0.1153} & {0.3758} & {0.0917} & {0.0531}\tabularnewline \hline {Bias} & {0.0432} & {-0.7266} & {0.1550} & {-0.0037} & {Bias} & {0.0375} & {0.1838} & {-0.0234} & {0.0087}\tabularnewline \hline {RMSE} & {0.0189} & {0.0992} & {0.0106} & {0.0063} & {RMSE} & {0.0133} & {0.1412} & {0.0084} & {0.0029}\tabularnewline \hline \end{tabular} \end{table} \begin{table}[H] \caption{Coverage Probabilities of Bootstrap Confidence Intervals (Nominal Level = 0.95)} \begin{centering} \end{centering} \begin{tabular}{ccc} \hline & Normal Approximation & Bootstrap Percentile\tabularnewline \hline $ATE$ & 0.9050 & 0.9300\tabularnewline $\alpha$ & 0.9700 & 0.9500\tabularnewline $\gamma$ & 0.9600 & 0.9250\tabularnewline $\beta$ & 0.9400 & 0.9300\tabularnewline $\delta$ & 0.8750 & 0.9200\tabularnewline $\rho$ & 0.9000 & 0.9500\tabularnewline \hline \end{tabular} \end{table}