Two Sample Unconditional Quantile Effect

Our definition of the unconditional policy effect depends on the notion of a counterfactual experiment, which is formally defined as follows,

The definition of counterfactual experiments does not specify the counterfactural target covariate $\widetilde{X}_{s}$ directly. It is implicitly defined through the first two elements of $\phi$. The first element, $\widetilde{\mathcal{U}}_{s}$, is a set of rank variables associated with the counterfactual target covariate, $\widetilde{X}_{s}$. When $\widetilde{X}_{s}$ is absolutely continuous, $\widetilde{\mathcal{U}}_{s}$ becomes a singleton set, but the set is generally not degenerate when the distribution of $\widetilde{X}_{s}$ contains a mass point. The second component, $\widetilde{G}_{s}$, is the counterfactual distribution of $\widetilde{X}_{s}$ conditional on the study population. When the target covariate is continuously distributed, $\widetilde{G}_{s}$ is continuous and strictly increasing, and therefore, $\widetilde{X}_{s}$ is uniquely determined by $\widetilde{X}_{s}=\widetilde{G}_{s}^{-1}(\widetilde{U}_{s})$, where $\widetilde{U}_{s}$ is the only element in $\widetilde{\mathcal{U}}_{s}$. However, when the target covariate contains mass points, there is a set of counterfactual rank variables that correspond to the same target covariate in the study population. This equivalent class is defined by Condition (i).

Following Rothe (2012), we restrict our attention to counterfactual changes where only the marginal distribution of $X_{s}$ is changed, while the marginal distribution of $Z$ and the dependence structure between $X_{s}$ and $Z$ remain unaffected. This notion of a ceteris paribus change is formally characterized by Conditions (ii)–(iv). Condition (ii) implies that the joint distribution of the observed variables $(Z,R)$ and the latent variable $\epsilon_{s}$ remain unchanged across counterfactual experiments. Under Condition (iii), the structural production function, $g$ is also not affected by the counterfactual change. Condition (iv) imposes a rank similarity condition. It says the conditional rank of the counterfactural target covariate follows the same distribution as the status quo. Due to the possibility of multiplicity of rank variables, the condition is also framed in terms of a set equivalence condition. When we restrict attention to absolutely continuous target covariates, both $\mathcal{U}_{s}$ and $\widetilde{\mathcal{U}}_{s}$ are singleton sets. Hence, this condition reduces to $(\widetilde{X}_{s}|\widetilde{Z}_{1},\widetilde{R}=1)\stackrel{{\scriptstyle d}}{{=}}(X_{s}|Z_{1},R=1)$.

Each counterfactual experiment $\phi$ represents a modification of the underlying economic system. It completely determines the counterfactual outcome in the study population. Yet we remain largely agnostic as to the counterfactual change in the auxiliary population. The definition also leaves the mechanism causing the change in the marginal distribution of the target covariate unspecified.

With the counterfactual experiments defined, we now construct the counterfactual covariate vector by $\widetilde{W}_{G}:=(\widetilde{G}_{s}^{-1}(\widetilde{U}_{s}),\widetilde{Z}_{1}^{\prime})^{\prime}.$ The counterfactual outcome of the study population is then defined as $\widetilde{Y}_{s}=\widetilde{g}_{s}(\widetilde{W}_{G},\widetilde{\epsilon}_{s})$, which follows a marginal distribution, $F_{\widetilde{Y}_{s}},$ and a conditional distribution restricted to the principal population, $F_{\widetilde{Y}_{s}|R=1}$. Note that the unconditional distribution is not well-defined, due to the lack of information on counterfactual changes in the auxiliary population. Therefore, we focus exclusively on the counterfactual distribution conditional on the study population in what follows. When $X$ is discrete, a single counterfactual experiment is mapped to a set of counterfactual outcomes, and we denote the corresponding set of counterfactual distributions by $\operatorname{\mathcal{F}}_{\widetilde{Y}_{s}}$.

In our context, the sequence of counterfactual distributions is defined in terms of the “marginal” distribution of the potential covariate $X_{s}$ in the study population, rather than the true unconditional distribution of the observed $X$. Although $X_{s}$ is missing from the main dataset, and therefore, its marginal distribution cannot be directly identified from the study population, we show in Theorem 3.2 that it can be recovered from the auxiliary data under the rank similarity assumption we impose in Assumption 2.1.

The policy parameter we seek to identify in this paper is the pathwise derivative of counterfactual distributional effect conditional on the study population. It is adapted from the definition of the marginal partial distributional policy effect (MPPE) by Rothe (2012).

We consider two specific types of counterfactual distributional changes: MDS and MQS. The defintion of the former is due to Firpo et al. (2009b). It denotes a small perturbation in the distribution of $X_{s}$, in the direction of $G$. MQS, on the other hand, considers a minuscule change in the quantiles of $X_{s}$. This type of policy change includes the MLS, $G^{-1}_{t,ls}(u):=F_{X_{s}|R}^{-1}(u|1)+t$, as a special case.

Turning to the case of quantiles, the quantile operator for a particular $\tau$ is defined by, $\nu_{\tau}(F_{Y_{s}|R=1}):=F_{Y_{s}|R=1}^{-1}(\tau)$. With the understanding that MPPE associated with a counterfactual experiment is generally a set when the X is discretely valued, we suppress the index with respect to $\{\widetilde{Y}_{s,t}\}_{t\geq 0}$ for notational convenience, and denote the MPPE with MDS, $MPPE(\nu_{\tau},\{\widetilde{Y}_{s,t}\}_{t\geq 0})$, and MPPE with MQS, $MPPE(\nu_{\tau},\{\widetilde{Y}_{s,t}\}_{t\geq 0})$, by $UQE_{p}(\tau,G)$ and $UQE_{q}(\tau,G)$, respectively. Here, and in what follows, the qualifier “unconditional” in UQE should be understood as conditional on (or relative to) the study population.

If there is no missing variable, the joint distribution of $(Y,X,Z)$ is directly identifiable from a random sample. Under data combination, however, only the “marginal” conditional distributions: $F_{Y_{s}|ZR=1}$ and $F_{X_{a}|ZR=0}$, can still be separately identified from the two samples, respectively. The conditional distribution, $F_{Y_{s}|X_{s}Z_{1}R=1}$, is generally not identifiable without further cross-population assumptions.

Instead of seeking identification of the entire conditional distribution,³³3Information of the entire distribution is crucial for the conditional quantile regression approach, even if the researcher may be interested in a single fixed quantile. $F_{Y_{s}|X_{s}Z_{1}R=1}(\cdot|\cdot,$ $\cdot,1)$, we demonstrate in Section 3.3 and 3.4 that UQE, and MPPE in general, can be identified using information at a single point of the distribution. This allows us to obtain identification under much milder restrictions on the pseudo-merged population. Identification is achieved through the excluded instrument variables, $Z_{2}$.

To ease notational burden, define $\Lambda(x,z_{1}):=F_{Y_{s}|X_{s}Z_{1}R}(q_{\tau}|x,z_{1},1)$ for all $x\in\operatorname{\mathcal{X}}$ and $z_{1}\in\operatorname{\mathcal{Z}}_{1}$.

Assumption 3.1 implies that $Z_{2}$ can be excluded from the outcome equation, and therefore, can be used as a source of exogenous variation to proxy for the missing covariate in the study population. Note that $\Lambda$ is generally not identified without an exogenous instrument $Z_{2}$. We illustrate this point with the example below.

Under Assumptions 2.1 and 3.1, the following moment-matching equation holds, for all $z\in\mathcal{Z}$,

$$\mathbb{E}\left[1(Y\leq q_{\tau})|Z,R=1\right]-\mathbb{E}\left[\Lambda(W)|Z,R=0\right]=0.$$

(3)

Equivalently, $\Lambda$ can be identified based on a likelihood-ratio-weighting equation,

$$\mathbb{E}\left[\left.R1(Y\leq q_{\tau})-(1-R)\Lambda(W)\dfrac{r(Z)}{1-r(Z)}\right|Z\right]=0.$$

(4)

The next assumption is about the global identification of $\Lambda$.

Assumption 3.2 is a high level condition. It is implied by a bounded completeness condition.⁴⁴4For two random element $U$ and $V$, we say $U$ is bounded complete for $V$, relative to a subpopulation $S=s$, if for all bounded measurable functions $\delta(\cdot)$, $\operatorname{\mathbb{E}}[\delta(U)|V,S=s]=0$ implies $\delta(U)\equiv 0$ almost surely. Note that $\Lambda$ is globally identified as long as $\operatorname{\mathbb{E}}[\Lambda(W)-\widetilde{\Lambda}(W)|Z,R=0]=0$ implies $\Lambda=\widetilde{\Lambda}$, which follows immediately if $\Lambda$ is measurable with respect to $W$, and $W$ is bounded complete for $Z$, relative to the auxiliary population. Bounded completeness is weaker than the commonly adopted completeness condition appearing in Newey and Powell (2003) and Fan et al. (2014). We refer readers to Hoeffding et al. (1977), Blundell et al. (2007), and Lehmann (1986) for detailed discussions.

In Section 4, we base our estimation and inference on parametric identification of $\Lambda$. In this case, we assume that $F_{Y_{s}|X_{s}Z_{1}R}(q_{\tau}|x,z_{1},1)=\Lambda(x,z_{1};\beta_{0})$, for $\beta_{0}\in\theta_{\beta}\subset\mathbb{R}^{d_{\beta}}$. In Lemma A.1, we provide a set of sufficient conditions which allow us to establish a global parametric identification condition analogous to Assumption 3.2.

Lemma 3.1 establishes the nonparametric identification of $\Lambda$. The proof for the parametric case follows along exactly the same line so we omit it here. In the next example, we verify the identification assumptions in a conditional normal model.

In this section, we establish the identification of $UQE_{q}$ and $UQE_{p}$ when the distribution of $X$ is absolutely continuous. Before stating the main result, we need some additional identifying assumptions.

Assumption 3.3(a)(i) says that conditional on $Z_{1}$, structural error $\epsilon$ is independent of the rank variable $U_{s}$ in the study population. This is much weaker than the commonly assumed strict independence condition that $X_{s}$ is independent of $\epsilon_{s}$ unconditionally. Conditional exogeneity has also been imposed by Firpo et al. (2009b), Rothe (2012), and Chernozhukov, Fernández-Val and Melly (2013), among others. Assumption 3.3(a)(ii) requires the conditional independence condition of part (a) to hold when counterfactual experiments get sufficiently close to the status quo. Under the rank invariance condition imposed by Rothe (2012), it is automatically implied by Assumption 3.3(a)(i). Assumption 3.3(b) ensures that the conditional distribution of $Y_{s}$ given $W$ is identified over the support of $W$. Assumption 3.4 imposes a smoothness condition on the distribution of target outcome, which implies that $F_{Y|R=1}^{-1}$ is Hadamard differentiable at $F_{Y|R=1}$, tangentially to the set of functions that are continuous at $q_{\tau}$.

The main theoretical result of this section is given as follows.

Let the support of $X$ be $\{x^{1},\dots,x^{l}\}$. When $X$ is discrete, MQS is not well-defined and we consider MDS only, with counterfactual experiments defined through a fixed discrete distribution, $G$.

Assumption 3.5(a) is the counterpart of Assumption 3.3(a) for discrete covariates. Since the rank variables are no longer uniquely pinned down by strictly increasing quantile functions, we strengthen Assumption 3.3(a) so that conditional independence holds for all the rank variables in the equivalent class. With this identifying assumption in hand, we are ready to present the following identification result. For $j=1,\dots,l$, let the period bound generating function be defined by

$$h_{q_{\tau}}(x^{j},x^{j-1},z_{1}):=-\dfrac{(\Lambda(x^{j-1},z_{1})-\Lambda(x^{j},z_{1}))\cdot(G(x^{j-1})-F_{X_{s}|R}(x^{j-1}|1))}{f_{Y_{s}|R}(q_{\tau}|1)}.$$

Theorem 3.3 indicates that $UQE_{p}$ is generally partially identified with bounds generated by $h_{q_{\tau}}.$ In the special case when $\Lambda(x,z_{1})$ is constant in $z_{1}$, the identified set of $UQE_{p}$ reduces to a singleton.

If $X$ is binary and $G_{t,p}(x)=1\{0\leq x<1\}(F_{X_{s}|R}(0|1)-t)+1\{x\geq 1\}$, $h_{q_{\tau}}$ reduces to $-\left(\Lambda(1,z_{1})-\Lambda(0,z_{1})\right)/f_{Y|R}(q_{\tau}|1)$. In such circumstance, Theorem 3.3 corresponds to the two-sample generalization of Theorem 5 in Rothe (2012), when $\nu$ in that paper takes on the quantile functional.

Following the discussion in Section 3, we first propose an estimator of the conditional probability, $\Lambda$. Here, we restrict our attention to the parametric setting where $\Lambda$ is indexed by a vector of parameter, $\beta$. We use a moment-matching method based on (4) to estimate $\beta$. The estimation of $\beta$ consists of four-steps. In the first step, we estimate $q_{\tau}$ by solving

$$\widehat{q}_{\tau}:=\arg\min_{q\in\operatorname{\mathcal{Y}}}\mathbb{E}_{n}[R(\tau-1(Y\leq q))\cdot(Y-q)].$$

(5)

The next three steps follow closely the Auxilliary-to-Study Tilting (AST) method proposed by Graham et al. (2016). Using the AST estimator, $\beta$ and the propensity score can be jointly estimated from moment restrictions in (4). To implement the estimator, we first estimate the propensity score, $r(z)$. Towards this end, we assume that the propensity score takes a parametric form, ie. $r(z)=L(k(z)^{\prime}\gamma)$, where $L(\cdot)$ is any link function that satisfies Assumption 4.1(e). Using $L(\cdot)$, $\widehat{\gamma}$ can be obtained by solving the following problem,

$$\widehat{\gamma}:=\arg\max_{\gamma\in\Theta_{\gamma}}\operatorname{\mathbb{E}}_{n}\left[R\log(L(k(Z)^{\prime}\gamma))+(1-R)\log(1-L(k(Z)^{\prime}\gamma))\right].$$

(6)

The AST estimator augments the conditional maximum likelihood estimator $\widehat{\gamma}$ with tilting parameters. The resulting estimator of $\beta$ is more efficient than the one based on $\widehat{\gamma}$ alone. Let $t(z)$ be a vector of known functions of $z$ with a constant term as the first element. Denote the tilting parameters associated with the auxiliary data and the study sample, by $\lambda_{a}$ and $\lambda_{s}$, respectively. They are estimated by solving,

	$\displaystyle\operatorname{\mathbb{E}}_{n}$	$\displaystyle\left[\left(\dfrac{1-R}{1-L(k(Z)^{\prime}\widehat{\gamma}+t(Z)^{\prime}\widehat{\lambda}_{a})}-1\right)L(k(Z)^{\prime}\widehat{\gamma})t(Z)\right]=0,$		(7)
	$\displaystyle\operatorname{\mathbb{E}}_{n}$	$\displaystyle\left[\left(\dfrac{R}{L(k(Z)^{\prime}\widehat{\gamma}+t(Z)^{\prime}\widehat{\lambda}_{s})}-1\right)L(k(Z)^{\prime}\widehat{\gamma})t(Z)\right]=0.$		(8)

Using $\widehat{\lambda}_{s}$ and $\widehat{\lambda}_{a}$, we compute study and auxiliary sample tilts, which are defined as follows

$\displaystyle\widehat{\pi}_{i}^{s}$

$\displaystyle:=\dfrac{L(k(Z_{i})^{\prime}\widehat{\gamma})}{L(k(Z_{i})^{\prime}\widehat{\gamma}+t(Z_{i})^{\prime}\widehat{\lambda}_{s})},\qquad\qquad\widehat{\pi}_{i}^{a}:=\dfrac{L(k(Z_{i})^{\prime}\widehat{\gamma})}{1-L(k(Z_{i})^{\prime}\widehat{\gamma}+t(Z_{i})^{\prime}\widehat{\lambda}_{a})}.$

(9)

Also let $e(z)$ be a $d_{\beta}$-dimensional vector of known functions of $z$, and $g(a;\widehat{q}_{\tau},\widehat{\gamma},\widehat{\lambda}_{s},\widehat{\lambda}_{s},\beta)$ $:=(\widehat{\pi}^{s}r1(y\leq\widehat{q}_{\tau})-\widehat{\pi}^{a}(1-r)\Lambda(w;\beta))e(z)$. Now, in the last step, $\beta$ can be estimated by

$$\widehat{\beta}:=\arg\inf_{\beta\in\Theta_{\beta}}\widehat{\operatorname{\mathcal{L}}}_{n}(\beta),$$

(10)

where $\widehat{\operatorname{\mathcal{L}}}_{n}(\beta):=\left\lVert\operatorname{\mathbb{E}}_{n}[g(A;\widehat{q}_{\tau},\widehat{\gamma},\widehat{\lambda}_{s},\widehat{\lambda}_{a},\beta)]\right\rVert^{2}_{\Omega_{n}}$ and $\left\lVert x\right\rVert^{2}_{\Omega_{n}}:=x^{\prime}\Omega_{n}x$, for a sequence of positive definite weighting matrices $\Omega_{n}$.

Using these quantities, we can obtain $\Lambda_{x}(W;\widehat{\beta}):=\partial\Lambda(W;\widehat{\beta})/\partial x$, and $\widehat{\ell}(z):=\frac{n_{a}}{n_{s}}\cdot\frac{L(k(z)^{\prime}\widehat{\gamma}+t(z)^{\prime}\widehat{\lambda}_{s})}{1-L(k(z)^{\prime}\widehat{\gamma}+t(z)^{\prime}\widehat{\lambda}_{a})}.$ Throughout this section, we assume that the counterfactual distribution $G$ is known. In practice, if $G$ is not known, it may be estimated from an independent sample; see e.g. Rothe (2010). Using the above estimates and $\widehat{F}_{X_{s}|R=1}(\cdot):=\operatorname{\mathbb{E}}_{n_{a}}[\widehat{\ell}(z)1(X\leq\cdot)]$, where $\operatorname{\mathbb{E}}_{n_{a}}[X]$ denotes $n_{a}^{-1}\sum_{i=n_{s}+1}^{n}X_{i}$, $\widehat{g}_{q}$ can be obtained as the plug-in estimator. For $g_{p}$, we need an estimator for $f_{X|R=1}(\cdot)$. Our identification relies on a compact support condition, and it is well known that the Prazen-Rosenblatt density estimator is not valid near the boundary of support. To overcome this challenge, we introduce trimming.⁶⁶6 Trimming is widely adopted in the literature; see e.g. Härdle and Stoker (1989), Powell et al. (1989) among others. This specific trimming function is inspired by Guerre et al. (2000) and Li et al. (2002). As an alternative, we can use a local polynomial density estimator that adjusts for the boundary bias adaptively; see Cattaneo et al. (2020) for details. For a kernel $K_{x}$ with compact support, and some bandwidth $b_{x}$, we let

$$\widehat{f}_{X|R}(x|1):=\operatorname{\mathbb{E}}_{n_{a}}\left[\widehat{\ell}(Z)I_{b_{x}}K_{b_{x}}\left(X-x\right)\right],$$

where $K_{b_{x}}(\cdot):=b_{x}^{-1}K_{x}(\cdot/b_{x})$. $I_{b_{x}}$ is a trimming indicator, which equals one for $x\in\{[\underline{x}+\rho_{x}b_{x}/2,\bar{x}-\rho_{x}b_{x}/2]\}$, where $\underline{x}$, $\bar{x}$, and $\rho_{x}$ are the lower and upper bound, of $\operatorname{\mathcal{X}}$, and the diameter of $\operatorname{Supp}(K_{x})$, respectively. The density, $f_{Y|R}$, can also be estimated using kernel density estimator. Specifically, for any kernel function $K_{y}(\cdot)$ that satisfies Assumption 4.2(b), let $\widehat{f}_{Y|R}(y|1):=\operatorname{\mathbb{E}}_{n_{s}}[K_{b_{y}}\left(Y_{i}-y\right)],$ where $\operatorname{\mathbb{E}}_{n_{s}}[X]:=n_{s}^{-1}\sum_{i=1}^{n_{s}}X_{i}$ and $K_{b_{y}}(y):=b_{y}^{-1}K_{y}(y/b_{y})$.