Parameter of Interest

2.3 Identification

2.3.1 Parameter of Interest

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

Definition 2.1 (Counterfactual Experiment) Letφ≡(Ufs,Ge_s,Z,e R,eeε_s,ge_s):Ω→K([0,1])×D(X)×Z× {0,1} × E ×l2(X,Z1,E),whereK([0,1])is the collection of all non-empty closed subsets of the unit interval, andD(X) denotes the space of distribution functions onX.We sayΦis the set of counterfactual experiments, if for allφ∈Φ, we have (i)Ge⁻¹_s (U_s) =Ge⁻¹_s (U_s⁰)almost surely for allU_s,U_s⁰∈Ufs; (ii)(eεs,Z,eR)e = (ε^d _s,Z,R); (iii)eg_s=g_s, (iv) for all U_s∈UsandUe_s∈Ufs, there existsUe_s⁰∈Ufs andU_s⁰∈Us, respectively, such that(Ue_s⁰|eZ₁,Re=1)= (U^d _s|Z₁,R=1)and (Ue_s|eZ₁,Re=1)= (U^d _s⁰|Z₁,R=1), whereUs={U˘∈U[0,1]:(F_X⁻¹

s|R(U˘_s|1)|Z₁,R=1)= (X^d _s|Z₁,R=1)}.

The definition of counterfactual experiments does not specify the counterfactural target covariateXe_sdirectly. It is implicitly defined through the first two elements ofφ. The first element,Ufs, is a set of rank variables associated

with the counterfactual target covariate,Xe_s. WhenXe_s is absolutely continuous,Ufs becomes a singleton set, but the set is generally not degenerate when the distribution ofXe_s contains a mass point. The second component,Ge_s, is the counterfactual distribution ofXe_s conditional on the study population. When the target covariate is continuously distributed,Ge_sis continuous and strictly increasing, and therefore,Xe_sis uniquely determined byXe_s=Ge⁻¹_s (Ue_s), where Ue_sis the only element inUfs. 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_sis changed, while the marginal distribution ofZand the dependence structure betweenX_sandZremain unaffected.

This notion of aceteris paribuschange is formally characterized by Conditions (ii)–(iv). Condition (ii) implies that the joint distribution of the observed variables(Z,R)and the latent variableεsremain unchanged across counterfactual experiments. Under Condition (iii), the structural production function, gis 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, bothUsandUfsare singleton sets. Hence, this condition reduces to(Xes|eZ1,Re=1)= (X^d _s|Z₁,R=1).

Each counterfactual experimentφ 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.

Remark 2.1 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 We_G= (Ge⁻¹_s (Ue_s),Ze₁⁰)⁰.The counterfactual outcome of the study population is then defined asYe_s=eg_s(We_G,eε_s), which fol- lows a marginal distribution,F

Yes,and a conditional distribution restricted to the principal population,F

Yes|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 pop- ulation in what follows. WhenX is discrete, a single counterfactual experiment is mapped to a set of counterfactual outcomes, and we denote the corresponding set of counterfactual distributions byF_Yes.

In our context, the sequence of counterfactual distributions is defined in terms of the “marginal” distribution of

the potential covariateX_s in the study population, rather than the true unconditional distribution of the observedX.

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 2.1 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 themarginal partial distributional policy effect(MPPE) by Rothe (2012).

Definition 2.2 (Marginal Partial Distributional Policy Effect) LetΦ^∗≡ {φ_t}_t≥0⊂Φ denote a sequence of coun- terfactual experiments, such thatGe_s,t →F_Xs|R=1, ast↓0. The MPPE for a given functionalν:D(Y)→Rand a sequence ofFe_s,t∈F_Yes,t is defined by,

MPPE(ν,{eYs,t}_t≥0)≡ ∂ ν(F

Yes,t|eR=1)

∂t _t=0

=lim

t↓0

ν(F

Yes,t|eR=1)−ν(F_Ys|R=1)

t .

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 ofX_s, in the direction ofG. MQS, on the other hand, considers a minuscule change in the quantiles ofX_s. This type of policy change includes the MLS, G⁻¹_t,ls(u)≡F_X⁻¹

s|R(u|1) +t, as a special case.

Definition 2.3 (Counterfactual Policy Distributions)

Marginal Distributional Shift (MDS):Gt,p(x) =F_Xs|R(x|1) +t(G(x)−F_Xs|R(x|1)).

Marginal Quantile Shift (MQS):G_t,q⁻¹(u) =F_X⁻¹

s|R(u|1) +t(G⁻¹(u)−F_X⁻¹

s|R(u|1)).

Remark 2.2 Figure 2.1 illustrates how the rates of change between the two types of counterfactuals are related.

Under the condition thatF_Xs|R=1is compactly supported with strictly positive density onX, MQS in a user-specified direction,q(x), can be approximated in the limit by MDS withG(x) =F_Xs|R(x|1)−f_Xs|R(x|1)q(x).

Turning to the case of quantiles, the quantile operator for a particularτis defined by,ντ(F_Ys|R=1) =F_Y⁻¹

s|R=1(τ). 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{eY_s,t}t≥0for notational convenience, and denote the MPPE with MDS, MPPE(ν_τ,{eY_s,t}t≥0), and MPPE with MQS, MPPE(ν_τ,{Ye_s,t}t≥0), byU QE_p(τ,G) andU 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.

Figure 2.1: Marginal distributional shift and marginal quantile shift

Notes: The blue curve depicts the data distribution of X. The red curve depicts a counterfac- tual distribution in the sequence {Gt}_t↓0, which can be induced by two equivalent policy changes.