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Two Sample Unconditional Quantile Effect

This paper proposes a new framework to evaluate unconditional quantile effects (UQE) in a data combination model. The UQE measures the effect of a marginal counterfactual change in the unconditional distribution of a covariate on quantiles of the unconditional distribution of a target outcome. Under rank similarity and conditional independence assumptions, we provide a set of identification results for UQEs when the target covariate is continuously distributed and when it is discrete, respectively. Based on these identification results, we propose semiparametric estimators and establish their large sample properties under primitive conditions. Applying our method to a variant of Mincer's earnings function, we study the counterfactual quantile effect of actual work experience on income.


Introduction
Missing data is a ubiquitous problem in empirical studies. Consider the scenario where a researcher is interested in conducting counterfactual analysis on a target variable, but it is entirely missing from the dataset of interest. In such circumstances, counterfactual policy effects cannot be identified from the primary dataset alone, and therefore, external information and/or stronger identifying assumptions are necessary. In this paper, we utilize both to achieve identification. Specifically, we focus on the situation where the missing variable can be found in another dataset and the information from which can be used to recover target policy parameters in the population of interest, under a set of commonly assumed restrictions on both the data structure and the model primitives.
To fix ideas, consider the following example. Suppose we are interested in studying the effect of a counterfactual change in the distribution of actual labor market experience on some distributional feature of yearly earnings. Our main dataset does not record respondents' work history, and therefore, we cannot recover their actual labor market experience. Suppose the variable is available from a second dataset, but it may not be a reliable source of information on income or it may not be representative of the target population we aim to analyze. In this case, we would benefit from combining information from both samples to identify and estimate our parameter of interest.
Research on counterfactual policy effects under data combination is scarce. Our paper fills this gap by proposing a new framework that accommodates such a data structure.
In this paper, we focus on one particular type of counterfactual policy effects, the unconditional quantile effect (UQE). It measures the effect of a marginal change in the unconditional distribution of a single covariate on the quantiles of a target outcome.
We provide identification results for UQE under various types of marginal distributional change. The key insight of our identification strategy is that some covariates present in both datasets can be excluded from the outcome equation, which would provide a source of exogenous variations that allows us to recover the joint distribution of missing variables, otherwise not identified using the two samples separately.
The second contribution of the paper is to propose novel semiparametric estimators based on these identification results. Departing from the literature on the estimation of counterfactual quantile effects-see, e.g. Firpo et al. (2009b), Sasaki et al. (2020) This paper belongs to the growing literature on the unconditional policy effect. Since Firpo et al. (2009b) introduced the method of unconditional quantile regressions (UQR), the study of unconditional policy effect has gained much attention. In general, this parameter differs from the one identified by the conditional quantile regression (Koenker and Bassett Jr, 1978), where marginal effects on the conditional quantile are the locus of attention. Applied researcher are often interested in the shifts in the quantiles of unconditional distribution of a target outcome. For instance, one may take an interest in how wage distribution changes in response to marginal increases in some characteristics of the labor force, such as education level and experience. Conditional quantile regression cannot be applied to address this type of questions, whereas UQR suits the goal. Rothe (2012) generalizes the method of Firpo et al. (2009b), and analyzes a variety of counterfactual policy effects. He formalizes the idea of ceteris paribus distributional change and provides extensive results for both fixed and marginal policy shifts. Our identification framework is closely related to his treatment of the latter type. Focusing on the special case of quantile effects, we extend his identification results to a data combination setting and provide novel inference theories specifically tailored to the distinct features of combined samples. For recent development in this literature, see Martínez-Iriarte and Sun (2020), Martínez-Iriarte (2020), and Sasaki et al. (2020). For a comprehensive survey on counterfactual distributions and decomposition methods, see Fortin et al. (2011).
Our paper also builds on the econometric methods of data combination. In economics, this strand of literature stems from the two-sample instrumental variables (TSIV) model that was first introduced by Klevmarken (1982), Angrist and Krueger (1992), Arellano and Meghir (1992), and is later extended by Ridder and Moffitt (2007), Inoue and Solon (2010), among others. Conceptually, the semiparametric data combination model we consider here is different from the traditional missing data problem (Robins et al., 1994).
The paper is organized as follows. In the next section, we describe the model and assumptions on the data structure. In Section 3 we introduce the parameter of interest, and then present identification results for continuously distributed and discrete target covariates, respectively. Section 4 discusses the estimation strategy and large sample results. We apply the method to study the income effect of real labor market experience in Section 5. Section 6 concludes.

Setup
The objective of our paper is to analyze the effect of a counterfactual change in the marginal distribution of the covariate of interest, X, on the quantiles of the target outcome, Y , under data combination. The precise definition of the counterfactual policy effect is provided in Section 3. When X is exogenous, and all the variables relevant for analysis are observed from a single data source, counterfactual policy effects can be analyzed either directly by applying tools from Firpo et al. (2009b) and Rothe (2012), or indirectly by recovering the structural function using standard identification results such as Matzkin (2003) and Matzkin (2007). However, when the variables of interest are scattered among several different data sources, we face a fundamental identification problem: The conditional distribution of Y given X is not identified from any single sample. In this case, existing methods do not provide an immediate solution.
Throughout this paper, we consider the scenario where our Y and X are sourced from two different data sets. The outcome is contained in the principal or main sample, S s = , from the study population, P s . The target covariate is missing completely from S s . However, it is observed in the auxiliary sample, , from the auxiliary population, P a , which does not contain observations of Y .
We now formally describe our structural model. We allow variables from two populations to be determined by different mechanisms. For the study population, where Y s ∈ Y ⊂ R is the potential outcome in the study population, s ∈ E ⊂ R d is a vector of unobserved heterogeneity term. Equation (1), links the target outcome, a scalar variable, X s ∈ X ⊂ R, and a vector of exogenous variables, Z 1 . Here, X s is the potential covariate of interest in the study population, which is in turn determined by (2). We can think of (2) as the reduced form relationship between X s and Z, where Z := (Z 1 , Z 2 ) ∈ Z := Z 1 × Z 2 ⊂ R dz includes both the exogenous variables in the outcome equation and a vector of excluded instrument, Z 2 . The vector of instrument, Z, is available in both samples, and therefore, it serves to establish a link between two samples. 2 The model in (1) accommodates general nonseparability between covariates and the unobserved heterogeneity. We do not impose any parametric or shape restriction on g s .
Variables in the auxiliary population are determined by Y a = g a (X a , Z 1 , a ), The support of Z is not restricted. We allow the distribution of Z to be continuous, discrete, or mixed. Nevertheless, to ease notational burden, we focus on the continuous case exclusively in what follows.
where g a and h a are generally different from g s and h s .
Let R denote the sample membership indicator. That is, R i = 1, if i-th draw comes from the study population, i = 1, ..., n := n s + n a . Let Y := RY s + (1 − R)Y a and X := RX s + (1 − R)X a . If no variable is missing, we are able to observe (Y s , Y a , X s , X a ).
However, in our context, only RY and (1 − R)X are observed. We then construct a pseudo-merged sample S using the two data sources as Let A := (R, RY, (1 − R)X, Z) and W := (X, Z 1 ) collect the observed variables and the covariates in the outcome equation, respectively. Throughout the paper, we arrange the data in a way such that R i = 1 for i = 1, ..., n s and R i = 0 for i = n s + 1, ..., n.
The merged sample may not correspond to any real-world population. We impose the following set of assumptions on the merged sample so it can mimic a random sample from a pseudo population. These assumptions are largely based on Assumption 1 in Graham et al. (2016).
Assumption 1(a) is a support condition on the commonly observed variables. It ensures that we will be able to find, for all the observations in the study sample, comparable units in the auxiliary sample, Assumption 1(b) imposes a pseudo randomization scheme on R, and therefore, allows us to view the merged data as a random sample from the pseudo-merged population. Let (·) denote the conditional likelihood ratio of Z across two population, i.e. (z) := f Z|R (z|1)/f Z|R (z|0). Assumption 1(c) expresses this likelihood ratio as a function of r(·), which plays the role of the "propensity score" function of R given Z. In our context, this is the probability that one observation belongs to the study population conditional on the value that instrumental variables take. The first part of Assumption 1(d) indicates that n s grows at the same order of magnitude as n a .
The second part of Assumption 1(d) ensures that the pseudo-true merged population is not a degenerate one conditional on all possible values of Z. By Assumption 1(b)-(d) and Bayes' Law, we have r(z) = P(D = 1|Z = z), and thus, r(·) can be viewed as the propensity score function.
Assumption 1(e) is a rank similarity condition. It requires the conditional distribution of X s given Z in the principal population coincide with that of X a in the auxiliary population. Assumption 1(e) is the only cross-population restriction we impose on our data structure, which means the conditional distribution of Y given (X, Z), and therefore, the conditional distribution of Y given Z and the marginal distribution of Z are all allowed to differ across P s and P a . This assumption is weaker than Assumption 1(ii) of Graham et al. (2016), as we do not impose a rank similarity condition on the outcome, which would imply F Ys|ZR=1 (·|·) = F Ya|ZR=0 (·|·).

Identification
In this section, we first introduce the definition of UQE. Then, we develop a set of identification results, for the cases when X is continuously distributed, and when it is discrete, respectively.

Parameter of Interest
Our definition of the unconditional policy effect depends on the notion of a counterfactual experiment, which is formally defined as follows, is the collection of all non-empty closed subsets of the unit interval, and D(X ) denotes the space of distribution functions on X . We say Φ is the set of counterfactual experiments, if for all φ ∈ Φ, we (iii) g s = g s , (iv) for all U s ∈ U s and U s ∈ U s , there exists U s ∈ U s and U s ∈ U s , The definition of counterfactual experiments does not specify the counterfactural target covariate X s directly. It is implicitly defined through the first two elements of φ.
The first element, U s , is a set of rank variables associated with the counterfactual target covariate, X s . When X s is absolutely continuous, U s becomes a singleton set, but the set is generally not degenerate when the distribution of X s contains a mass point. The second component, G s , is the counterfactual distribution of X s conditional on the study population. When the target covariate is continuously distributed, G s is continuous and strictly increasing, and therefore, X s is uniquely determined by X s = G −1 s ( U s ), where U s is the only element in 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 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 U s and U s are singleton sets. Hence, this condition reduces to ( X s | Z 1 , R = 1) d = (X s |Z 1 , R = 1).
Each counterfactual experiment φ represents a modification of the underlying economic system. It completely determines the counterfactual outcome in the study popu- lation. 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.
Remark 1 Our definition of counterfactual experiments relaxes the rank invariance conditions imposed by Rothe (2012). Instead, counterfactual changes in our context only need to satisfy a rank similarity or copula invariance condition.
With the counterfactual experiments defined, we now construct the counterfactual covariate vector by and a conditional distribution restricted to the principal population, F Ys|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 F Ys .
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 2 that it can be recovered from the auxiliary data under the rank similarity assumption we impose in Assumption 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 x 1 x 2 (x, 1)

Figure 1: Marginal Distributional Shift and Marginal Quantile Shift
The MPPE for a given functional ν : D(Y) → R and a sequence of F s,t ∈ F Ys,t is defined by, 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 as a special case.

Definition 3 (Counterfactual Policy Distributions)
Remark 2 Figure 1 illustrates how the rates of change between the two types of counterfactuals are related. Under the condition that F Xs|R=1 is compactly supported with strictly positive density on X , MQS in the direction of q(x) can be approximated in the limit by Turning to the case of quantiles, the quantile operator for a particular τ is defined by, 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 { Y s,t } t≥0 for notational convenience, and denote the MPPE with MDS, U QE q (τ, G), respectively. Here, and in what follows, the qualifier "unconditional" in UQE should be understood as conditional on (or relative to) the study population.

Identification of
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 Ys|ZR=1 and F Xa|ZR=0 , can still be separately identified from the two samples, respectively. The conditional distribution, F Ys|XsZ 1 R=1 , is generally not identifiable without further cross-population assumptions.
Instead of seeking identification of the entire conditional distribution, 3 F Ys|XsZ 1 R=1 (·|·, ·, 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 Λ(x, z 1 ) := F Ys|XsZ 1 R (q τ |x, z 1 , 1) for all x ∈ X and Assumption 2 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 Λ is generally not identified without an exogenous instrument Z 2 . We illustrate this point with the example below.
Under Assumptions 1 and 2, the following moment-matching equation holds, for all Equivalently, Λ can be identified based on a likelihood-ratio-weighting equation, The next assumption is about the global identification of Λ.

Assumption 3 is a high level condition. It is implied by a bounded completeness
condition. 4 Note that Λ is globally identified as long as implies Λ = Λ, which follows immediately if Λ 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 Λ.
In this case, we assume that F Ys|XsZ 1 R (q τ |x, z 1 , 1) = Λ(x, z 1 ; β 0 ), for β 0 ∈ θ β ⊂ R d β . 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.
Lemma 1 establishes the nonparametric identification of Λ. 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.
Example 1 (Conditional Normal Model) Let the structural equations of the study population be given by where s and η s are jointly normally distributed. Specifically, for positive-valued functions ψ y (·) and ψ x (·), we have contains an open set and that h(z 1 , ·) maps open sets of z 2 into open sets.
Assumption 3 then follows by Theorem 2.2 in Newey and Powell (2003).
Turning to the linear case, let g s As a consequence, (γ s 1 , γ s 2 ) are uniquely determined by (3) or (4), if and only if δ 2 = 0.

Identification with Continuously Distributed X
In this section, we establish the identification of U QE q and U QE p when the distribution of X is absolutely continuous. Before stating the main result, we need some additional identifying assumptions.
Assumption 4 (a) (i) s ⊥ ⊥ U s |Z 1 , R = 1; (ii) there exists a t 0 sufficiently close to 0, such that for all Assumption 4(a)(i) says that conditional on Z 1 , structural error 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 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 4(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 4(a)(i). Assumption 4(b) ensures that the conditional distribution of Y s given W is identified over the support of W . Assumption 5 imposes a smoothness condition on the distribution of target outcome, which implies that F −1 Y |R=1 is Hadamard differentiable at F Y |R=1 , tangentially to the set of functions that are continuous at q τ .
The main theoretical result of this section is given as follows.
Theorem 2 Suppose that Assumptions 1-5 hold, and that the distribution of X is absolutely continuous with respect to the Lebesgue measure, both U QE p (τ, G) and U QE q (τ, G) Suppose in addition that X is compact, and F Xs|R=1 is continuously differentiable on X with strictly positive density function f Xs|R=1 . Then we have,

Remark 3
The compactness condition on X is assumed to ensure the existence of pathwise derivative of the inverse map. It can be relaxed by imposing a boundary condition on Λ x . Specifically, we may assume that Λ x vanishes when x ∈ [F Xs|R=1 (q 1 )+ , F Xs|R=1 (q 2 )− ], for 0 < q 1 < q 2 < 1 and some > 0.

Identification with Discrete Covariate
Let the support of X be {x 1 , . . . , 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 6 (a) (i) s ⊥ ⊥ U s |Z 1 , R = 1, for all U s ∈ U s ; (ii) there exists a t 0 sufficiently close to 0, Assumption 6(a) is the counterpart of Assumption 4(a) for discrete covariates. Since the rank variables are no longer uniquely pinned down by strictly increasing quantile functions, we strengthen Assumption 4(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, . . . , l, let the period bound generating function be defined by Theorem 3 Suppose that Assumptions 1-3, 5, and 6 hold, U QE p (τ, G) is partially identified, with Theorem 3 indicates that U QE p is generally partially identified with bounds generated by h qτ . In the special case when Λ(x, z 1 ) is constant in z 1 , the identified set of U QE p reduces to a singleton.
If X is binary and G t, In such circumstance, Theorem 3 corresponds to the two-sample generalization of Theorem 5 in Rothe (2012), when ν in that paper takes on the quantile functional.

Estimation and Inference
In this section, we discuss estimation and inference for our two-sample UQE. First, we describe an estimation procedure for U QE q and U QE p as identified in Theorem 2. We then show that our estimator is consistent and asymptotically normal in Theorem 5. 5

Estimation Procedure
Following the discussion in Section 3, we first propose an estimator of the conditional probability, Λ. Here, we restrict our attention to the parametric setting where Λ is indexed by a vector of parameter, β. We use a moment-matching method based on (4) to estimate β. The estimation of β consists of four-steps. In the first step, we estimate q τ by solving The next three steps follow closely the Auxilliary-to-Study Tilting (AST) method proposed by Graham et al. (2016). Using the AST estimator, β and the propensity score can 5 Here we focus on the scenario where the distribution of X is absolutely continuous. When X is discrete, the problem features partially identified parameters defined by the intersection bounds. Chernozhukov, Lee and Rosen (2013) provide an extensive treatment of this topic. We omit discussion here and refer readers to Appendix D in Rothe (2012) for a detailed discussion on how to apply their method.
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) γ), where L(·) is any link function that satisfies Assumption 7(e). Using L(·), γ can be obtained by solving the following problem, The AST estimator augments the conditional maximum likelihood estimator γ with tilting parameters. The resulting estimator of β is more efficient than the one based on γ 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 λ a and λ s , respectively. They are estimated by solving, Using λ s and λ a , we compute study and auxiliary sample tilts, which are defined as follows π s i := Also let e(z) be a d β -dimensional vector of known functions of z, and g(a; q τ , γ, λ s , λ s , β) := ( π s r1(y ≤ q τ ) − π a (1 − r)Λ(w; β))e(z). Now, in the last step, β can be estimated by where L n (β) := E n [g(A; q τ , γ, λ s , λ a , β)] 2 Ωn and x 2 Ωn := x Ω n x, for a sequence of positive definite weighting matrices Ω n .
For g p , we need an estimator for f X|R=1 (·). 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. 6 For a kernel K x with compact support, and some bandwidth b x , we let x,x, and ρ x are the lower and upper bound, of X , and the diameter of Supp(K x ), respectively. The density, f Y |R , can also be estimated using kernel density estimator. Specifically, for any kernel function K y (·) that satisfies Now, plugging in the estimators of nuisance quantities, U QE j (τ, G) can thus be estimated by, We summarize the estimation procedure in the following algorithm.
6 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.

Large Sample Results
In this section, we present inference results for the estimators introduced in the previous section. We first establish large sample properties of β, for which purpose, some additional regularity conditions are in order.
Assumption 7(c) imposes mild smoothness conditions on the parametric function Λ(·, ·; ·), requiring it to be bounded between the unit interval, thus behaving like a distribution function. Assumption 7(d) states that Ω n is consistent for Ω, which is positive definite.
Assumption 7(e) implies that the true "propensity score" is known up to finite dimensional γ 0 . It also specifies smoothness and boundedness conditions on the parametric propensity score. Finally, due to the estimation of q τ , we impose a finite fourth moment condition in Assumption 7(f), which is stronger than the usual square-integrability condition.
Lemma 4 Suppose that Assumptions 1-5, and Assumption 7 hold, then (i) β p → β 0 ; furthermore, (ii) suppose that the Jacobian, M Ω , as defined in the Supplementary Appendix Lemma 4 shows that the parameters of F Ys|XsZR=1 are consistently estimated by β.
Furthermore, it admits an asymptotic linear representation with influence function given by ψ β (A; θ 0 , q τ ), which plays a key role in establishing the large sample properties of UQE. Towards this ends, we need the following set of assumptions.
Assumption 8 is uniformly continuous; (iii) the density f Y |R=1 (y) is strictly bounded away from 0, three times continuously differentiable in y with uniformly bounded derivatives for y in Y 0 , such that q τ ∈ Y 0 .
(d) (i) X is compact; (ii) G is continuously differentiable on X with strictly positive density.

Assumption 8(a) strengthens Assumption 5 and imposes stronger smoothness conditions
on the distribution of Y s . Assumption 8(b) states several regularity conditions on kernel functions, which is standard in the literature. Assumption 8(c) specifies admissible rate for the bandwidth parameter. We can choose b y = O(n −κ s ), for κ ∈ [1/5, 1/2). Assumption 8(d) imposes support and smoothness conditions for the counterfactual target covariate.
Asymptotic properties of U QE are formally characterized in the next theorem.
Theorem 5 Under Assumptions 1-5, 7, and 8, (i) the following linear expansions hold, Suppose in addition that Assumption B.1 holds, (ii) then we have · y 2 K y (y)dy, From the linear expansions in Theorem 5, we conclude that U QE converges at a rate that is slower than root-n. This result is mainly driven by the nonparametric estimation of the density f Y |R=1 , and therefore, the estimator is nonparametric in essence. Moreover, the asymptotic expansion includes an asymptotic bias term, B(τ, G, b y ). If we assume, as in Firpo et al. (2009a), nb 5 y → 0 or κ < 1/5, the bias vanishes asymptotically.
Remark 4 Estimators for the asymptotic variance of U QE p (τ, G) and U QE q (τ, G) can be constructed using their empirical counterparts. Specifically, let . Under a suitable rate condition on b y , consistency of Σ j follows directly from the first two parts of Theorem 5. To achieve better finite-sample performance, we can add the root-n terms of the influence functions to the variance estimator, based on which, we propose the following improved variance estimator, In the above definition, ψ j (a; θ, q τ , b y ) is a plug-in estimator of the influence function, ψ j (A; θ 0 , q τ , b y ), for j = p, q. A detailed description of the construction of ψ can be found in the Supplementary Appendix C. As an alternative, we may conduct inference using a bootstrap procedure, e.g. exchangeable bootstrap. It can be shown that bootstrap approximates not only the leading term but also the higher-order terms in the asymptotic linear representation, which would immediately lead to an improvement over the simple plug-in estimator.
Remark 5 Theorem 5 implies that tests of the unconditional quantile effect converges at a non-parametric rate in general. Nonetheless, for the null of zero, positive, and negative effects, we can still construct tests that have power against departures of the null at the parametric rate. For example, to test the null: H 0 : U QE j (τ, G) = 0, it is equivalent to test H 0 : d j (β 0 , G) = 0, as U QE j (τ, G) = 0 ⇔ d j (β 0 , G) = 0, for j = p, q. From Theorem 5, we know that d j,n ( β, G) converges at the parametric rate. Moreover, we have (13) can be used to test H 0 , applying standard testing procedures.

Empirical Application
We apply our identification and estimation methods to a variant of the Mincer's regression. Our main goal here is to demonstrate the bias from using potential instead of actual labor experience in human capital earnings models.
Identifying the causal relationship between earnings and human capital accumulation has been a focus of labor economic studies for decades. Traditionally, Mincer's regression has been widely used to quantify the link between labor wage, education and labor market experience.
Most datasets do not provide respondents' actual work histories. Therefore, many researchers choose to proxy the variable with potential work experience. The potential experience measure is usually calculated by subtracting years of schooling plus some constant (typically 6 years) from age. Despite the popularity of this practice, many labor economists believe that the return to actual experience tends to be biased when we employ the potential experience as proxy; see e.g. Regan and Oaxaca (2009). One of their main argument is that any lapse in labor force participation would be implicitly assumed away when potential experience instead of the actual one is used. There is little reason to believe that the return to employed experience is the same as that of the unemployed period. Hence, it is still preferable to use the actual labor experience.
We use the 1970 wave of IPUMS as our main sample. The data is a 1-in-10,000 national random sample of the population. The outcome of interest is the natural log of yearly earnings. The target covariate, actual work experience, is missing from IPUMS.
To apply the procedure described in Section 4, we need a dataset where the actual work experience is available. For that purpose, we use the 1972 wave of PSID as cleaned by Hirukawa et al. (2020). Detailed work histories are available in PSID. Therefore, it allows us to recover the actual labor market experience. However, running analysis directly with PSID may not be ideal due to the fact that it is not nationally representative. Our method is able to address this issue by combining information from both samples.
To estimate F Ys|XsZ 1 R=1 , we consider the following specification, P(log(Income) ≤ y) = Λ(β 0 + β 1 educ + β 2 black + β 3 south + β 4 married + β 5 exper r + β 6 exper 2 r ) where exper r stands for individual's actual or realized work experience, educ denotes the highest grade completed by the respondent, black, married, and south are dummy variables which take one if the person is black, married, and lives in the south, respectively. The actual work experience serves as our X s . It enters (14) with linear and quadratic terms. We let (educ, black, south, married) be the set of included instruments Z 1 , and the potential experience, exper p , be the excluded instrument Z 2 . We provide estimation results when Λ(·) takes either the logistic link or the probit link. To implement the AST estimator, we choose j(Z) = (Z 1 , Z 2 ) , for j = k, t, e. The density of Y s is estimated is the usual "rule-of-thumb" bandwidth. Table 1 reports the descriptive statistics for the two samples. We use only the data on men aged between 23 and 50 when the surveys are taken, to ensure that common support condition holds. This leaves us with a sample of n s = 3, 504 respondents for IPUMS and n a = 1, 697 for PSID. There are considerable differences between the two datasets. Individuals who are black, married and/or lives in the south are over-represented in PSID compared to the nationally representative IPUMS. On average, an individual in PSID has 0.36 years more potential experience than actual experience.
For UQE with MDS and MQS, we choose the smoothed empirical distribution of exper p in the PSID sample (assuming it is fixed and known) as the target counterfactual distribution, i.e. G = F Z 2 |R=0 . As depicted in Figure 2, we find that less-experienced workers tend to have even fewer years of experience in the counterfactual scenario than in the status quo, and the opposite is true for workers closer to the right tail of the distribution.
We report estimation results in Table 2 and Figure 3. A few remarks are in order.
First, our estimates suggest that the counterfactual effect of a marginal shift in the distribution of actual experience is heterogeneous across income groups. The effect is larger in magnitude for the lower-income groups as expected. When MDS and MQS are considered, the quantile effects are uniformly negative and the shapes of the effect curves are similar. The marginal shift could decrease the (log) earning by anything between 0.015 and 0.061 across income quantiles. For reference, when logistic link is assumed, the marginal effect of MDS amounts to a reduction of 4.8% in annual earnings for individuals the first quartile, 3.0% at the median, and 1.5% at the third quartile, respectively. For MLS, two-sample estimates are bounded between the two sets of one-sample estimates.
The marginal upward shift in the actual experience would increase a median worker's income by 1.9%.
Next, we consider the bias caused by using potential experience in lieu of the actual experience. We note that, one-sample UQE estimates based on IPUMS tend to be smaller in magnitude at lower income quantiles than that based on the combined data.
Eventually, the two estimators converge at higher income levels. In the context of our application, there does not seem to be sufficient evidence to support the claim that the bias is statistically significant. 7 All three plots contain the UQE of potential experience based on IPUMS (solid lines), the UQE of actual experience based on PSID (dotted lines), the two-sample UQE (dashed lines), and the two-sided 95% confidence intervals based on the improved variance estimator (shaded area).

Concluding Remarks
In this paper, we propose a framework to identify and estimate unconditional quantile policy effect under data combination. We establish the identification of UQE under two main conditions: a rank similarity assumption and a conditional independence assumption, based on which, we provide estimators for the identified UQE and derive their large sample properties.  Firpo et al. (2009b), under our two-sample structure, would also be an interesting direction for future research.

Appendix A Proofs of Lemmas and Theorems in Section 3
Proof of Lemma 1: We provide proof for the nonparametric identification here. The proof for the parametric case follows along exactly the same line and is omitted. We shall show (3) first, where the second equality is by Assumption 2, and the fourth line follows by Assumption 1(e). Likewise for (4), Thus, Lemma 1 follows immediately from (3) (or (4)) and Assumption 3.
Lemma A.1 Suppose (i) Λ(w; β) is measurable with respect to w for all β ∈ Θ β ; (ii) W is bounded complete for Z, relative to the auxiliary population; (iii) Λ(w; β) is differentiable with respect to β; and (iv) ∂Λ(·; β)/∂β is uniformly bounded and ∂Λ(·; β)/∂β ≡ 0 for all β ∈ Θ β . Then, under Assumptions 1 and 2, β 0 can be uniquely identified from (3) or (4). Proof of Theorem 2: We shall first prove the identification result for a fixed counterfactual distribution. Next, we take the derivative of the counterfactual experiments with respect to t. The result of Theorem 2 then follows by the fact that Hadamard derivative operator of the quantile functional is linear. For any t ≤ t 0 , fix t ∈ Φ * , and we have that

Proof of Lemma
where the second line follows by the definition of F Ys,t (q τ ), the third one comes from the definition of U s,t and a change of variable from x to u, the fourth equality follows by the construction of Φ * and Assumptions 4(a) and (b), the fifth line is again by Assumption 4(a), the eighth one follows by the definition of U s and standard change-of-variable argument, the tenth line is by Assumptions 1(a)-(c) and Bayes' Law.
To obtain the marginal distributional effect, we take derivative of F Gt Ys|R (q τ |1) with respect to t and evaluate it at t = 0. For the marginal distributional shift, ∂F Ys,t|R (q τ |1) Observe that | t=0 is the pathwise derivative of the inverse map H → H −1 at F Xs|R=1 in the direction of G(·)−F Xs|R=1 . By Lemma 3.9.23 in Van Der Vaart and Wellner (1996), the inverse map is Hadamard differentiable under the conditions specified in the theorem, with the derivative map given by, where h is the first-order derivative of H. Let φ(·) = G(·) − F Xs|R=1 (·) and H = F Xs|R=1 , it follows immediately that, for all u ∈ [0, 1], , and hence, for all x ∈ X , ∂G −1 t,p (F Xs|R (x|1)) ∂t Analogously, for marginal quantile shift, ∂F Ys,t|R (q τ |1) where the second equality follows from Lemma 1 and the definition of G −1 t,q (·).
To identify F Xs|R=1 , we exploit the following fact where the third line is due to Assumption 1 and Bayes' Law.
The right hand side of (16) is identified under the support condition in Assumption 1(a).
The proof for the lower bound follows by an analogous argument.

B Asymptotic Linear Representation of UQE Estimators
We specify additional regularity conditions in Theorem 5 and provide linear expansions for U QE p and U QE q in this section, the proofs of which are contained in the Supplementary Appendix.

Assumption B.1
(a) f X|ZR=0 is uniformly bounded, twice continuously differentiable with uniformly bounded first and second order derivatives on X Z.
(b) (i) K x (·) is a second order symmetric kernel function; (ii) the support of K x is continuous, bounded, with compact support, K x (·), and such that K x (x)dx = 1, xK x (x)dx = 0, x 2 K x (x)dx > 0, and K 2 x (x)dx < ∞.