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 proposes two methods for estimating the density of the outcome variable for the compliers in the potential outcomes framework. The first is a kernel-smoothing based approach, while the second is a model-based approach, and in both cases, the double machine learning approach based on data splitting is employed. The nature of the contribution seems mostly theoretical, with a lot of effort into proving the convergence rates.