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Two Sample Unconditional Quantile Effect
{\bf Keywords:} Counterfatual policy effect, unconditional quantile effect, data combination model, Mincer regression.
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\textemdash see, e.g. firpo2009unconditional, sasaki2020unconditional, etc.\textemdash which focuses primarily on the marginal location shift (MLS) of a covariate, we provide estimators of UQE under two general types of counterfactual distributional changes, namely the marginal distributional shift (MDS) and the marginal quantile shift (MQS),\footnote{The precise definitions of MLS, MDS, and MQS are given in Section (ref).} the latter of which includes MLS as a special case. To the best of our knowledge, large sample results for these two cases are new to the literature. We apply these results to study a variant of Mincer's earnings function. Using data from Integrated Public Use Microdata Sample (IPUMS) as our main data source and the Panel Study of Income Dynamics (PSID) as the auxiliary sample, we investigate the counterfactual income effect of actual work experience. The effect profiles with MDS and MQS are found to be similar in shape.
This paper belongs to the growing literature on the unconditional policy effect. Since firpo2009unconditional 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 koenker1978regression, 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.
rothe2012partial generalizes the method of firpo2009unconditional, 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 martinez2020identification, martinez2020sensitivity, and sasaki2020unconditional. For a comprehensive survey on counterfactual distributions and decomposition methods, see fortin2011decomposition.
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 klevmarken1982missing, angrist1992effect, arellano1992female, and is later extended by ridder2007econometrics, inoue2010two, among others. Conceptually, the semiparametric data combination model we consider here is different from the traditional missing data problem robins1994estimation. It is more closely related to the “verify-out-of-sample” model in chen2008semiparametric, and also to imbens1994combining, fan2014identifying, graham2016efficient, hirukawa2020yet, and buchinsky2021estimation, to name a few.
The paper is organized as follows. In the next section, we describe the model and assumptions on the data structure. In Section (ref) we introduce the parameter of interest, and then present identification results for continuously distributed and discrete target covariates, respectively. Section (ref) discusses the estimation strategy and large sample results. We apply the method to study the income effect of real labor market experience in Section (ref). Section (ref) concludes.
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 (ref). 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 firpo2009unconditional and rothe2012partial, or indirectly by recovering the structural function using standard identification results such as matzkin2003nonparametric and matzkin2007nonparametric. 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, $ \mathcal{S}_{s} = \{ Y_{i}, Z_{i} \}_{i = 1}^{n_{s}} $, from the study population, $\mathcal{P}_{s} $. The target covariate is missing completely from $ \mathcal{S}_{s} $. However, it is observed in the auxiliary sample, $ \mathcal{S}_{a} = \{ X_{i}, Z_{i} \}_{i = 1}^{n_{a}} $, from the auxiliary population, $ \mathcal{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} \in \mathcal{Y} \subset \operatorname{\mathbb{R}} $ is the potential outcome in the study population, $ \epsilon_{s} \in \mathcal{E} \subset \mathbb{R}^{d_{\epsilon}} $ is a vector of unobserved heterogeneity term. Equation (ref), links the target outcome, a scalar variable, $ X_{s} \in \mathcal{X} \subset \mathbb{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 (ref). We can think of (ref) as the reduced form relationship between $ X_{s} $ and $ Z $, where $ Z' := (Z_{1}', Z_{2}')' \in \operatorname{\mathcal{Z}}:= \operatorname{\mathcal{Z}}_{1} \times \operatorname{\mathcal{Z}}_{2} \subset \operatorname{\mathbb{R}}^{d_{z}} $ 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.\footnote{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.} The model in (ref) 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
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 $ \mathcal{S} $ using the two data sources as $ \mathcal{S} = \{ R_{i}, R_{i}Y_{i}, (1-R_{i})X_{i}, Z_{i}\}_{i= 1}^{n} $. 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 graham2016efficient.
Assumption (ref)(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 (ref)(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 $ \ell(\cdot) $ denote the conditional likelihood ratio of $ Z $ across two population, i.e. $ \ell(z) := f_{Z|R}(z | 1)/f_{Z|R}(z|0) $. Assumption (ref)(c) expresses this likelihood ratio as a function of $ r(\cdot) $, 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 (ref)(d) indicates that $ n_{s} $ grows at the same order of magnitude as $ n_{a} $. The second part of Assumption (ref)(d) ensures that the pseudo-true merged population is not a degenerate one conditional on all possible values of $ Z $. By Assumption (ref)(b)--(d) and Bayes' Law, we have $ r(z) = \operatorname{\mathbb{P}}(D = 1| Z = z), $ and thus, $ r(\cdot) $ can be viewed as the propensity score function.
Assumption (ref)(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 (ref)(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 $ \operatorname{\mathcal{P}}_{s} $ and $ \operatorname{\mathcal{P}}_{a} $. This assumption is weaker than Assumption 1(ii) of graham2016efficient, as we do not impose a rank similarity condition on the outcome, which would imply $ F_{Y_{s}|ZR = 1}(\cdot | \cdot) = F_{Y_{a}| Z R = 0}(\cdot | \cdot)$.
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.
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 rothe2012partial, 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{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}')'. $ 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 (ref) that it can be recovered from the auxiliary data under the rank similarity assumption we impose in Assumption (ref).
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 rothe2012partial.
We consider two specific types of counterfactual distributional changes: MDS and MQS. The defintion of the former is due to firpo2009unconditional. 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}|Z R = 1} $ and $ F_{X_{a}|Z R = 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,\footnote{Information 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 (ref) and (ref) 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 (ref) 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 (ref) and (ref), the following moment-matching equation holds, for all $ z \in \mathcal{Z} $,
Equivalently, $ \Lambda $ can be identified based on a likelihood-ratio-weighting equation,
The next assumption is about the global identification of $ \Lambda $.
Assumption (ref) is a high level condition. It is implied by a bounded completeness condition.\footnote{For 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 newey2003instrumental and fan2014identifying. We refer readers to hoeffding1977some, blundell2007semi, and lehmann2012statistical for detailed discussions.
In Section (ref), 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 (ref), we provide a set of sufficient conditions which allow us to establish a global parametric identification condition analogous to Assumption (ref).
Lemma (ref) 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 (ref)(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 firpo2009unconditional, rothe2012partial, and chernozhukov2013inference, among others. Assumption (ref)(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 rothe2012partial, it is automatically implied by Assumption (ref)(a)(i). Assumption (ref)(b) ensures that the conditional distribution of $ Y_{s} $ given $ W $ is identified over the support of $ W $. Assumption (ref) 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 (ref)(a) is the counterpart of Assumption (ref)(a) for discrete covariates. Since the rank variables are no longer uniquely pinned down by strictly increasing quantile functions, we strengthen Assumption (ref)(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 (ref) 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 (ref) corresponds to the two-sample generalization of Theorem 5 in rothe2012partial, when $ \nu $ in that paper takes on the quantile functional.
In this section, we discuss estimation and inference for our two-sample UQE. First, we describe an estimation procedure for $ UQE_{q} $ and $ UQE_{p} $ as identified in Theorem (ref). We then show that our estimator is consistent and asymptotically normal in Theorem (ref).\footnote{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. chernozhukov2013intersection provide an extensive treatment of this topic. We omit discussion here and refer readers to Appendix D in rothe2012partial for a detailed discussion on how to apply their method. }
Following the discussion in Section (ref), 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 (ref) to estimate $ \beta $. The estimation of $ \beta $ consists of four-steps. In the first step, we estimate $ q_{\tau} $ by solving
The next three steps follow closely the Auxilliary-to-Study Tilting (AST) method proposed by graham2016efficient. Using the AST estimator, $ \beta $ and the propensity score can be jointly estimated from moment restrictions in (ref). 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)'\gamma) $, where $ L(\cdot) $ is any link function that satisfies Assumption (ref)(e). Using $ L(\cdot) $, $ \widehat{\gamma} $ can be obtained by solving the following problem,
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,
Using $ \widehat{ \lambda}_{s} $ and $ \widehat{ \lambda}_{a} $, we compute study and auxiliary sample tilts, which are defined as follows
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} r 1(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
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\lVertx\right\rVert^{2}_{\Omega_{n}} := x'\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)'\widehat{ \gamma } + t(z)' \widehat{\lambda}_{s} )}{ 1- L(k(z)'\widehat{ \gamma } + t(z)' \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. rothe2010nonparametric. 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.\footnote{ Trimming is widely adopted in the literature; see e.g. hardle1989investigating, powell1989semiparametric among others. This specific trimming function is inspired by guerre2000optimal and li2002structural. As an alternative, we can use a local polynomial density estimator that adjusts for the boundary bias adaptively; see cattaneo2020simple for details.} For a kernel $ K_{x} $ with compact support, and some bandwidth $ b_{x} $, we let
where $ K_{b_{x}}(\cdot ) := b_{x}^{-1} K_{x}(\cdot/b_{x} )$. $ I_{b_{x}} $ is a trimming indicator, which equals one for $x \in \{ [ \ubar{x} + \rho_{x} b_{x}/2, \bar{x} - \rho_{x} b_{x}/2 ] \}$, where $ \ubar{x} $, $ \bar{x} $, and $ \rho_{x} $ are the lower and upper bound, of $ \operatorname{\mathcal{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}(\cdot) $ that satisfies Assumption (ref)(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}) $.
Now, plugging in the estimators of nuisance quantities, $ UQE_{j}(\tau, G) $ can thus be estimated by,
We summarize the estimation procedure in the following algorithm.
In this section, we present inference results for the estimators introduced in the previous section. We first establish large sample properties of $ \widehat{\beta} $, for which purpose, some additional regularity conditions are in order.
Assumption (ref)(a) is standard in the microeconometric literature. Assumption (ref)(b) requires the conditional density $ f_{Y|ZR}(\cdot | \cdot, 1) $ be bounded uniformly for all $ (y,z) \in \operatorname{\mathcal{Y}}_{0} \operatorname{\mathcal{Z}} $. Assumption (ref)(c) imposes mild smoothness conditions on the parametric function $ \Lambda(\cdot, \cdot ;\cdot) $, requiring it to be bounded between the unit interval, thus behaving like a distribution function. Assumption (ref)(d) states that $ \Omega_{n} $ is consistent for $ \Omega $, which is positive definite. Assumption (ref)(e) implies that the true “propensity score” is known up to finite dimensional $ \gamma_{0} $. It also specifies smoothness and boundedness conditions on the parametric propensity score. Finally, due to the estimation of $ q_{\tau} $, we impose a finite fourth moment condition in Assumption (ref)(f), which is stronger than the usual square-integrability condition.
Lemma (ref) shows that the parameters of $ F_{Y_{s}|X_{s}Z R = 1} $ are consistently estimated by $ \widehat{\beta} $. Furthermore, it admits an asymptotic linear representation with influence function given by $ \psi_{\beta}(A; \theta_{0}, q_{\tau}) $, which plays a key role in establishing the large sample properties of UQE. Towards this ends, we need the following set of assumptions.
Assumption (ref)(a) strengthens Assumption (ref) and imposes stronger smoothness conditions on the distribution of $ Y_{s} $. Assumption (ref)(b) states several regularity conditions on kernel functions, which is standard in the literature. Assumption (ref)(c) specifies admissible rate for the bandwidth parameter. We can choose $ b_{y}= O( n_{s}^{-\kappa}) $, for $ \kappa \in [1/5, 1/2) $. Assumption (ref)(d) imposes support and smoothness conditions for the counterfactual target covariate.
Asymptotic properties of $ \widehat{UQE}$ are formally characterized in the next theorem.
From the linear expansions in Theorem (ref), we conclude that $ UQE $ 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(\tau, G, b_{y}) $. If we assume, as in firpo2009supplement, $ n b^{5}_{y} \to 0 $ or $ \kappa <1/5 $, the bias vanishes asymptotically.
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. regan2009work. 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 (ref), we need a dataset where the actual work experience is available. For that purpose, we use the 1972 wave of PSID as cleaned by hirukawa2020yet. 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_{Y_{s}|X_{s}Z_{1} R = 1} $, we consider the following specification,
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 (ref) 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 $ \Lambda(\cdot) $ 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 using $ b_{y} = n_{s}^{-0.01} b_{n,0} $, where $ b_{n,0} := 1.06 \min\{\sigma(Y_{s}), interquartile(Y_{s})\} n_{1}^{-0.2} $ is the usual “rule-of-thumb” bandwidth.
Table (ref) 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 (ref), 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 (ref) and Figure (ref). 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.\footnote{Our analysis is local to the direction of counterfactual change, and therefore, does not allow the insignificance result to be extrapolated globally. For the same reason, the comparison between $ UQE_{2s} $ and $ UQE_{psid} $ is not meaningful.}
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.
Our current approach can be extended in the following directions. First, although we have restricted our attention to the quantile effect throughout this paper, our results can be easily extended to other statistical functionals such as mean, interquartile, and inequality measures. It would be interesting to see how the identification requirements change with respect to the functional we adopt. Second, we have focused exclusively on the pointwise identification and inference. While extension to uniform results seem straightforward, it comes at a cost of stronger cross-sample restrictions. Under such assumptions, conditional quantile regression is likely feasible. Comparing conditional and unconditional quantile effects, as in firpo2009unconditional, under our two-sample structure, would also be an interesting direction for future research.