Double Machine Learning Density Estimation for Local Treatment Effects with Instruments

1 Introduction . Controlled experimentation is one powerful tool used throughout the empirical sciences to infer the effect of a certain treatment on a given outcome . The idea is to randomize the treatment assignment so as to neutralize the effect of unobserved confounders . However , in some practical settings , it may be challenging to ascertain that individuals who are selected for treatment will follow their recommendations . Issues of non-compliance and unmeasured confounding are quite common and lead to the non-identification of treatment effects in many real-world cases [ 29 , 50 , 32 , 56 ] . An approach known as instrumental variables ( IVs ) has been proposed to try to circumvent this issue [ 68 ] . The idea is to find a set of variables ( possibly singleton ) that are not the target of the analysis by itself but that will help to control for the unobserved confounding between the treatment and the outcome . In particular , IVs are special variables that ( i ) are correlated with the treatment , ( ii ) do not directly influence the outcome , and ( iii ) are not affected by certain unmeasured confounders . For concreteness , consider a study of the effect of 401 ( k ) participation ( X ) on the distribution of net financial assets ( Y ) [ 2 ] . This setting is represented in the causal graph in Fig . 1 . Note that a dashed-bidirected arrow exists between X and Y , which in graphical language represents unobserved confounding affecting both X and Y . The variable Z in this model represents the eligibility of 401 ( k ) . We note that Z qualifies as an instrument in this case – ( i ) it does affect the participation of 401 ( k ) ( X ) and ( ii ) has no direct influence on the net financial asset ( Y ) , ( iii ) is not affected by unmeasured confounders between X and Y . The variable W represents observed covariates ( e.g. , gender , age , ethnicity , income , family size ) . We are interested in the particular setting where only individuals who were offered the treatment may have access to it [ 31 ] . For instance , in the case of 401 ( k ) participation ( X = 1 ) , only eligible 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . individuals ( Z = 1 ) would be allowed to join the program . This assumption is known in the literature as monotonicity , which rules out the possibility that any units would respond contrary to the instrument . Under monotonicity , the causal effect in the subpopulation whose actual treatment X coincides with the assigned treatment Z ( called compliers ) is identifiable [ 31 , 2 ] . The average treatment effect ( ATE ) for the compliers is called ‘ Local ATE ’ ( LATE ) ( or Complier average causal effects , CACE ) [ 31 ] . The most common quantification of these effects in IV settings found in practice is the average ( e.g. , LATE ) . The average is certainly an informative summary ; however , it may fail to capture significant differences in the causal distributions of the outcome . For instance , consider Fig . 2 that shows the densities of outcomes Y under treatmentsX = 1 among compliers which are generated from samples drawn from four synthetic data generating processes represented by the IV graph in Fig . 1 ( further discussed in Sec . 5 ) . All of the four distributions have the same mean 0 and variance 2 . However , the difference in the LTE distributions is self-evident . Most of the prior work on quantifying distributions of treatment effects focuses on estimating cumulative distribution functions ( CDFs ) or quantiles , and little attention has been given to estimating densities ( refer to Sec . 1.1 for further comparison ) . As a complement to CDFs , densities have various advantages , including a more interpretable visualization of the distribution and generative capability of producing samples . One challenge with estimating densities is that while CDFs are pathwise-differentiable and enjoy √ n-rate estimators ( n is the size of data ) , densities are not ( i.e. , they are non-regular ) , and therefore possess no influence functions nor √ n-rate estimators without approximations [ 7 , Ch . 3 ] . In this paper , our goal is to provide methods to estimate densities of local treatment effects in IV settings under the monotonicity assumption . We develop two families of methods for this task based on kernel-smoothing and model-based approximations . The former smooths the density by convolution with a kernel function ; the latter projects the density onto a finite-dimensional density class based on a distributional distance measure . For both approaches , we construct double/debiased machine learning ( DML ) style density estimators [ 43 , 54 , 52 , 70 , 13 ] . We analyze the asymptotic convergence properties of the estimators , showing that they can converge fast ( i.e. , √ n-rate ) even when nuisance estimates converge slowly ( e.g. , n−1/4 rate ) ( a property called ‘ debiasedness ’ 1 ) . We illustrate the proposed methods on synthetic and real data . 1.1 Related work . Our work touches different areas , which we will discuss next . Double/Debiased Machine Learning ( DML ) [ 13 ] -based causal effect estimators . The DML framework has been adapted for estimating the average causal effect under the setting where the back-door criterion [ 50 , Sec . 3.3.1 ] ( also known as ignorability [ 57 ] ) holds ( e.g. , [ 12 , 19 ] ) . Recently , DML-based causal effect estimators have been developed for any identifiable causal functionals in a given causal graph and equivalence class thereof [ 33 , 34 ] . 1Also known as ‘ nonparametric doubly robust [ 37 ] or ‘ rate doubly robust ’ [ 59 ] . Local average & quantile treatment effect . The formal identification results for LATE under the monotonicity assumption in IV settings were developed by [ 31 , 3 ] . Building on these results , semiparametric estimation for LATE has received remarkable attention [ 2 , 60 , 23 , 62 , 48 ] , including robust LATE estimators that achieve debiasedness [ 47 , 40 , 38 , 64 ] . As shown in Fig . 2 , however , the average is sometimes insufficient to capture the effects of the treatment on the distributions of outcomes . To address this issue , the problem of estimating quantiles or CDFs has taken attention . A common approach to estimate quantiles or CDFs is based on the LATE estimation . Since the expectation of 1Y≤y ( Y ) , an indicator that outcome Y falls short of threshold y , reduces to the CDF ( i.e. , replacing Y in LATE with 1Y≤y ( Y ) ) , estimators for the LATE can be used to estimate quantiles or CDFs [ 1 , 2 , 15 , 24 , 16 , 30 , 45 , 18 , 69 ] . Non-regular target estimand . An estimand that possesses no influence functions nor √ n-rate estimators is called ‘ non-regular ’ . Densities are an example of non-regular target estimands [ 7 , Chap . 3 ] . One can approximate a non-regular target with a smooth one such that an influence function and√ n-rate estimators can be derived . Two broadly used approaches are kernel-smoothing-based ( e.g. , [ 52 , 6 , 42 , 19 , 35 ] ) and model-based ( e.g. , [ 46 , 52 , 21 , 41 , 40 , 39 ] ) . Causal density estimation . There is limited literature on estimating the density of treatment effects . Most of the results assume that the ignorability/backdoor admissibility holds [ 55 , 49 ] . [ 22 ] used the kernel-smoothing technique to estimate the density of a treatment effect , and [ 42 ] provided a kernelsmoothing-based density estimator that achieves doubly robustness and debiasedness building on top of the work in [ 53 ] . Recently , [ 39 ] investigated a model-based approach and developed estimators that achieve debiasedness properties . Under the IV setting , [ 10 ] provided a local polynomial regressionbased density estimator for local treatment effects ; We are not aware of any work studying debiased density estimators . As mentioned , this paper investigates both kernel-smoothing and model-based approaches for estimating local treatment effects under IV settings and develops DML-style density estimators for both . 2 LTE Estimation – Problem setup . In our analysis , each variable is represented with a capital letter ( X ) and its realized value with a small letter ( x ) . For a discrete ( e.g. , binary ) random variable X , we use 1x ( X ) to represent the indicator function such that 1x ( X ) = 1 if X = x ; 1x ( X ) = 0 otherwise . For a continuous variable X with a probability density p ( x ) of a distribution P and a function f ( x ) , EP [ f ( X ) ] ≡ ∫ X f ( x ) p ( x ) d [ x ] where X is the domain for X , and ∥f ( X ) ∥ ≡ √ EP [ ( f ( X ) ) 2 ] . f̂ is said to converge to f at rate rn if ∥f̂ ( x ) − f ( x ) ∥ = OP ( 1/rn ) . For a dataset D = { Vi } ni=1 , we use ED [ f ( V ) ] ≡ ( 1/n ) ∑n i=1 f ( Vi ) to denote the empirical mean of f ( V ) with D. Structural Causal Models ( SCMs ) . We use the language of SCMs as our basic semantic and inferential framework [ 50 , 4 ] . An SCMM is a quadrupleM = ⟨U , V , P ( U ) , F ⟩ where U is a set of exogenous ( latent ) variables following a joint distribution P ( u ) , and V is a set of endogenous ( observable ) variables whose values are determined by functions F = { fVi } Vi∈V such that Vi ← fVi ( pai , ui ) where PAi ⊆ V and Ui ⊆ U . Each SCMM induces a distribution P ( v ) and a causal graph G = G ( M ) over V in which there exists a directed edge from every variable in PAi to Vi and dashed-bidirected arrows encode common latent variables ( e.g. , see Fig . 1 ) . Within the structural semantics , performing an intervention and setting X = x is represented through the do-operator , do ( X = x ) , which encodes the operation of replacing the original equations of X ( i.e. , fX ( pax , ux ) ) by the constant x and induces a submodelMx and an interventional distribution P ( v|do ( x ) ) . For any variable Y ∈ V , the potential response Yx ( u ) is defined as the solution of Y in the submodel Mx given U = u , which induces a counterfactual variable Yx . Local Treatment Effect ( LTE ) with IV . We consider the IV setting represented by the causal graph G in Fig . 12 , where Z is a binary instrument with domain { 0 , 1 } , X is a binary treatment with domain { 0 , 1 } , and Y is a ( set of ) continuous outcomes with bounded domain Y ⊂ Rd , and W is a set of covariates ( continuous , discrete , or mixed ) . G satisfies the IV assumption that Z has no direct influence on outcome Y and is not affected by unmeasured confounders between X and Y . 2It is common in the literature to define IV assumptions in terms of conditional independences among counterfactual [ 51 , 9 , 8 , 2 , 60 , 47 , 64 ] , whose connection with the causal graph in Fig . 1 is discussed in Assumption A.1 The causal density p ( yx ) is not identifiable from the observed density p ( x , y , z , w ) due to the unobserved confounders between X and Y . However , the effect is possibly recovered for certain subpopulation under additional assumptions . Formally , a unit in the population is an always-taker if XZ=1 = XZ=0 = 1 , a never-taker if XZ=1 = XZ=0 = 0 , a complier if XZ=1 = 1 , XZ=0 = 0 , and a defier if XZ=1 = 0 , XZ=0 = 1 [ 3 , 2 ] . We will make the following assumptions based on literature . Assumption 1 ( Monotonicity ) . There are no defiers : XZ=1 ≥ XZ=0 . Assumption 2 ( Positivity ) . P ( x|z , w ) > 0 , P ( z|w ) > 0 for any x , z , w. Let C denote the event that a unit is a complier ( i.e. , a unit such that XZ=0 = 0 and XZ=1 = 1 ) . For a given constant a and a variable X , let xa denote the event X = a . The LTE p ( yx|C ) is identifiable under monotonicity and is given by [ 31 , 2 ] : p ( yx|C ) = EP [ p ( y|x , zx , W ) P ( x|zx , W ) − p ( y|x , z1−x , W ) P ( x|z1−x , W ) ] EP [ P ( x1|z1 , W ) − P ( x1|z0 , W ) ] , ( 1 ) where the expectation is over W . In this paper , we aim to estimate the LTE density p ( yx|C ) in Eq . ( 1 ) . We will make the following mild assumption on some densities , popularly employed in density estimation literature ( e.g. , [ 44 , 25 , 27 , 61 , 26 , 42 ] ) . Assumption 3 . For any x , z , w , y , densities p ( y|w , z , x ) , p ( y|z , x ) and p ( yx|C ) are bounded , and p ( yx|C ) is twice differentiable . DML method . Let ψ ≡ ψP ′ denote a functional of an arbitrary distribution P ′ . We use P to denote the true distribution such that D ∼ P . Let ψ0 ≡ ψP denote the true parameter to be estimated . To estimate ψ0 , DML-based estimators use a Neyman Orthogonal score φ ( V ; ψ0 , η ) ( where η ≡ ηP ′ is a set of nuisance parameters and η0 ≡ ηP denotes the true nuisances ) , a function such that EP [ φ ( V ; ψ0 , η0 ) ] = 0 , ( ∂/∂η ) |η=η0EP [ φ ( V ; ψ0 , η ) ] = 0 . Given φ , an DML estimator is constructed using the cross-fitting technique as follows : For randomly split halves of D denoted { D0 , D1 } , let η̂p for p ∈ { 0 , 1 } denote the estimates for η fromDp . Let Tp denote a solution such that ED1−p [ φ ( V ; Tp , η̂p ) ] = oP ( N−1/2 ) . Then , T ≡ ( T0 + T1 ) /2 is an DML estimator [ 13 , Def . 3.1 ] . In addition to being consistent , the estimator T exhibits a robustness property called debiasedness : T converges to ψ0 in the root-N rate even when η̂ converges to η0 in slower N−1/4 rate [ 13 , Thm . 3.1 ] . A Neyman Orthogonal Score can be derived by adding ψ to its influence function ϕ [ 14 , Thm . 1 ] . An influence function of the functional ψP is defined as a solution satisfying EP [ ϕ ] = 0 , EP [ ϕ2 ] < ∞ , and ( ∂/∂t ) ψPt |t=0 = EP [ ϕ ( V ; ψ , η ) St ( V ; t = 0 ) ] where Pt ≡ P ( v ) ( 1 + tg ( v ) ) for t ∈ R and any bounded mean-zero functions g ( · ) over V , and St ( v ; t = 0 ) ≡ ( ∂/∂t ) logPt ( v ) |t=0 [ 63 , Chap . 25 ] . Due to space constraints , all the proofs are provided in Appendix B in suppl . material .

Summary: This paper deals with the problem of the estimation of the local treatment effect (LTE) which is measure of the causal impact of a treatment. To this extent the authors develop kernel-smoothing and model-based methodologies to recover this quantity from the data.  The theoretical properties of the machinery are developed and the authors show that the estimator of the LTE is consistent. Two applications of the proposed methodology showcase the technique.