Estimating Heterogeneous Treatment Effects by Combining Weak Instruments and Observational Data
Authors: Miruna Oprescu
Nathan Kallus
Venue: NeurIPS 2024 poster (2024)
Keywords: Causal inference
heterogeneous treatment effects
weak instrumental variables
unobserved confounding
data combination
Primary area: causal_inference
TL;DR: We propose a method to estimate heterogeneous treatment effects by combining weak instrumental variables with observational data, enabling reliable treatment effect estimation in the presence of confounding bias.

Abstract: Accurately predicting conditional average treatment effects (CATEs) is crucial in personalized medicine and digital platform analytics. 
Since the treatments of interest often cannot be directly randomized, observational data is leveraged to learn CATEs, but this approach can incur significant bias from unobserved confounding. One strategy to overcome these limitations is to leverage instrumental variables (IVs) as latent quasi-experiments, such as randomized intent-to-treat assignments or randomized product recommendations. This approach, on the other hand, can suffer from low compliance, i.e., IV weakness. Some subgroups may even exhibit zero compliance, meaning we cannot instrument for their CATEs at all. In this paper, we develop a novel approach to combine IV and observational data to enable reliable CATE estimation in the presence of unobserved confounding in the observational data and low compliance in the IV data, including no compliance for some subgroups. We propose a two-stage framework that first learns \textit{biased} CATEs from the observational data, and then applies a compliance-weighted correction using IV data, effectively leveraging IV strength variability across covariates. We characterize the convergence rates of our method and validate its effectiveness through a simulation study. Additionally, we demonstrate its utility with real data by analyzing the heterogeneous effects of 401(k) plan participation on wealth.

Estimating Heterogeneous Treatment Effects by Combining Weak Instruments and Observational Data
Miruna Oprescu Cornell University am0780cornel1. edu
Nathan Kallus Coreell University kallus@cornell.cm edu
Abstract
Accurately predicting conditional average treatment effects (CATEs) is crucial in personalized medicine and digital platform analytics. Since the treatments interest often cannot be directly randomized, observational data is leveraged learn CATEs, but this approach can incur significant bias from unobserved confounding. One strategy to overcome these limitations (IVs) as latent quasi-experiments, such as randomized intent-to-treal assignments or randomized product recommendations. This approach, on the other hand, can suffer from low compliance, i.e., IV weakness. Some subgroups may even exhibit zero compliance, meaning we cannot instrument for their CATEs all. In this paper we develop a novel approach to combine IV and observational data to enable reliable CATE estimation the presence of unobserved confounding in the observational data and low compliance in the IV data, including no compliance for some subgroups. We propose two-stage framework that first learns biased CATES from the observational data, and then applies compliance-weighe using data, effectively leveraging IV strength variability across covariates. We characterize the convergence rates of our method and validate its effectiveness through simulation study. Additionally, we demonstrate its utility with real data by analyzing the heterogeneous effects of 401(k) plan participation wealth.
leverage instrumental variables
correction
1 Introduction
The use of observational data for individual-level causal analyses becoming increasingly common in personalized medicine, online platforms, and any setting where understanding individualized responses is crucial and/or presents an opportunity for personalization. The key quantity for such analyses is the conditional average treatment effect (CATE), which captures how treatment effects vary according to baseline covariates (features). This measure provides insight into effect heterogeneity and enables personalization.
Using observational data can nonetheless introduce bias from unobserved confounding, where the observed relationship between outcomes and interventions is influenced not only by treatment effects but also by variables that influence both outcome and treatment, such as socioeconomic status, health, user mood, etc., which are not captured by baseline covariates. These biases can skew causal effect estimates, resulting in unreliable analyses or even harmful policy decisions.
Randomized trials are the gold standard for causal inference, but they are often infeasible. For instance, digital services cannot force users to view buy product, and clinical trials cannot require invasive treatments. A common alternative is to randomize the encouragement of certain actions, such as recommending product treatment. These encouragements can serve instrumental variables (IVs) which, under certain conditions, enable unbiased estimation of treatment effects [4]
Identification of CATEs using IVs crucially hinges on the premise that compliance the correlation between the treatment received and the intent/encouragemment is nonzero across all casvaine-covariaee
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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values. When compliance is nonzero but small, IV-based estimates tend to exhibit high variance, making them unreliable [3]. In practice, the assumption of strong compliance is often violated. For example, users on digital platforms may ignore recommendations entirely or reject certain types of content, while participants on mobile health platforms may disregard prompts (e.g. taking 250 steps per hour) due to time constraints or lack of interest.
To address the challenge of estimating unbiased CATES in the presence of unobserved confounding and low IV compliance, we introduce a two-stage framework. In the first stage, we estimate a biased, confounded CATE from observational data. Then, in the second stage, we utilize an IV learn the confounding bias by weighting the samples according to their compliance levels. By assuming only that the bias can be extrapolated, this approach extends treatment effect adjustments even to groups minimally influenced by the IV, employing transfer learning approach that leverages varying instrument strengths across covariate groups.
This framework mirrors strategies in causal inference that combine randomized trials with obser- vational data address low covariate overlap. Building on this body of work, we introduce two methodologies for extrapolating confounding bias within the observational dataset: a parametric estimation approach, assuming the confounding bias adheres a parametric form, and trans- fer learning strategy that assumes a shared representation between the true and biased CATE. We study the properties of our CATE estimators in finite samples and validate our approaches through comprehensive empirical studies.
Related Work
We briefly overview related work here; for a more comprehensive discussion, refer to Appendix ^
Heterogeneous treatment effect estimation from observational data: Recent advances in machine learning have expanded the use of observational data to estimate CATES using diverse techniques such as random forests [51], Bayesian algorithms [24], deep learning [48]. and meta-learners [33]. However, these methods often unrealistically assume an absence of confounding, limiting their real-world applicability. Efforts to account for unobserved confounding either construct bounds
on treatment effects [17 40 or use latent variable models and multiple/sequentia treatments to debias CATE estimates 19 36 533, but they frequently depend on unverifiable assumptions or require accurate proxy data, reducing their practical utility.
Heterogeneous treatment effect estimation using Ivs: Integrating machine learning with instru- mental variable (IV) methods enhances CATE estimation flexibility over traditional approaches. Techniques range from advanced two-stage least squares (2SLS) that incorporate complex feature mappings via kernel methods [49] and deep learning
(54) to neural networks for conditional density estimation 21 and moment conditions for IV estimation [. Yet, these rely on the consistent relevance of instruments across covariate groups, which is not guaranteed with weak instruments.
Treatment Effect Estimation with Weak Instruments:
Traditional IV methods like 2SLS can be unreliable when instruments are weak, leading to biased, high- variance estimates. Recent advance- ments include novel estimators such as bijs-adjusted 2SLS, limited information maximum likelihood, and jackknife IV estimators (see (25 and references therein). Other techniques attempt reduce variance by exploiting first-stage heterogeneity (variation in compliance) [ Some approaches also combine multiple weak instruments into robust composites, useful in settings like genetic studies [30. Our approach extends [ 33 by leveraging compliance weighting to estimate heterogeneous effects and address weak instruments using additional observational data.
Combining observational and randomized data: Increasing research focuses on integrating observational datasets with randomized control trial (RCT) data to mitigate observational bias. Strategies include imposing structural assumptions, such as strong parametric constraints (29]. or assuming shared structure between biased and unbiased CATE functions [23] as well as optimizing dual estimators from both data types for improved bias correction (55) Our work aligns with efforts to debias treatment effects using both observational and experimental data, but also addresses challenges such as low IV compliance, the need to debias the overall effect function rather than individual outcome functions, and the complexity of estimating CATES from IV data using a ratio estimator.
Where our work lies: To the best of our knowledge, no current estimation technique effectively com- bines an IV study, particularly one with weak instruments or low compliance, with an observational

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study to derive robust and unbiased CATE estimates. We bridge this gap by introducing two robust and consistent CATE estimation techniques, building upon previous work on combining RCT and observational data 23 29). as well as work that addresses the complexities associated with weak instruments
3 Background and Setup
We consider the standard setting of causal inference where the goal is to estimate the conditional average treatment effect of a binary treatment A E {0. 1 on an outcome Y E R in the presence of covariates X E X' S RR Our approach grounded in Rubin's potential outcomes framework, wherein each unit is associated with two potential outcomes (0), (1) of which only Y == (A) is observed (causal consistency). Our objective is to learn the CATE function, which is given by:
7(x) E|y (1) (() X
However, we only have access to no i.i.d samples from observational dataset O (x. :4 y0 ~ (X", A YO). Thus, we face the fundamental problem of causal inference: only the outcome under the administered treatment is observed, while the counterfactual remains unobserved. Without further assumptions, there exists the possibility of unobserved confounding. leading to situation where
7 (x) = E 1 = =
- x]
which indicates persistent bias in the observed treatment effects that does not diminish even with an increasing sample size. We denote this bias by b(x), that is:
b(x)) 7 (x)
Assuming this bias is induced by a set of unobserved confounders U C R the discrepancy arises because the selection into treatment in the observational population is influenced by U, which also impacts the outcome yo, Our goal is to mitigate this bias by leveraging additional data.
Alongside the observational dataset, we have ng i.i.d. samples from an experimental, intent-to-treat dataset E = (x 2 A E ~ (X ZE A²) where E is a binary instrument taking values in {0, 13. We X E X' and assume the px= (a) = px (x:), where px denotes the density of the random variable X. Moreover, we assume that the joint distribution of covariates and unobserved confounders (X, U) is consistent across both datasets. As before, we use Y (A, Z) to denote the potential outcome given treatment A and instrument Z. Additionally, let ( denote the potential treatment under instrument Z. and define the compliance and defiance indicators C and D by C == 1[A (1) > A (0) and D (I) < (0)) respectively. We assume that this dataset follows standard IV assumptions on the data generating process:
Assumption (Standard IV Assumptions). We assume the following properties hold: (Exclusion) (A, Z) = (A), i.e. the instrument affects the outcome only through the treatment; (Indepen- dence) Z U I for any unobserved confounder U; and (Relevance) there exists a subset X² C X with non-zero measure such that 2 1 A X for X E X".
Assumption 2 (Unconfounded Compliance 52 2. The individual treatment effect is independent of
the compliance status given covariates: Y F. (1) (0) (A ( AF XF.
We note that the relevance assumption in Assumption 1 is weaker version of the standard IV assumptions since we allow for arbitrarily weak instruments some regions of the covariate spaces. With Assumption and Assumption
2 we can identify the CATE for x E X" as:
Y A
E
z = 1, _ 2]
E E
Z 0, X 2]
8y 7(x)
-(x).
(3)
ZE 1, XE x
A a
7E 0, X = x]
We provide the proof of [Equation in Appendix B
Here, (xx denotes heterogeneous compliance
a measure of instrument strength, given by 7()) P(C = 1 x = x) P(D = 1 xx x)
under AAsumption 2 A strong instrument (7(z)= + 1) indicates high adherence to the recommended treatment, with 7(x) = 1 signifying perfect compliance, similar to true randomized controlled trial. Conversely, a weak instrument (7(z) - 0) suggests minimal influence on treatment uptake, with (z) 0 indicating compliance and a confounded selection into treatment. The relevance
3

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assumption in Assumption 1 Jensures y(x) * 0 for x' E X', validating the estimation procedure in Equation 3 However, small 7(x) values lead estimates of T(x) with high asymptotic variance. Moreover, we wish to extend the T (xx) estimation from X" to X, our population of interest.
Thus, relying solely observational data results in biased r(x) estimates, while experimental data alone can yield high variance or invalid estimates for a E X' with low compliance. This work addresses these challenges by strategically combining the strengths of both datasets to provide robust CATE estimation technique.
We denote the L2 norm of a function fsa || == (r and the L2 Euclidean norm of vector 0 E IRd as |0]|2. The notation f represents the estimated value of a parameter or function, where f is the true value. We omit the distribution subscript when clear from context; e.g., [[ and E[X²] denote expectations over experimental and observational samples, respectively.
Notation:
Estimation Method
To obtain robust estimates of the CATE function for the population of interest X, we propose two-step
1 1t-St
2dd Stage
Observationl Data
Experimental Data
framework that integrates information from both the
observational data and the IV study. First, esti- mate the confounded CATE function (2))) using the observational data (XP, A y ) This is well- established problem both causal inference and ma-
Estimate r"
Estimate
chine learning, and it can be addressed using various existing techniques, including meta-learners ((33), random forests (5I), and neural networks (48).
ararnin (t. R...)
where F. (t" (a)())))))
Next, we wish to approximate the bias function b(x) = T(x) - (z) using the learnd (z)). Without acle access to the true CATE function T(x), we in- stead rely samples from the experimental (IV) study (X.e 2 A. y?) ng for which we can estimate an unbiased, though potentially high variance, CATE for E X", as given in Equation Our approach hinges on the following lemma:
Output (+) (X)
Output F(x)
(2)))) b(x)
50
1(3)
Figure I: Illustration of our two-stage pro- cedure: the first stage learns a biased CATE from observational data, while the second stage uses IV data to correct the bias.
Lemma 1. [CATE Estimation with IVs] Let mz(x) = P(ZE 1 XE x)) be the instrument propensity. Then, the following identity holds for every I E X':
YEZE
( 2
X = x = T(x)
(z()((())
(1 -1- (1-xz(z)))aa)
We note that in the case of randomized instrument assignment, the instrument propensity is known and often given by a constant, i.e., 7 (x) T 0. By defining V2(z) == π2 (z) т (z)). Lemma 1 shows that the bias function b(x:) can be expressed in terms of observable quantities as
b(x)) = F (1-== -- (1- )=² - X
for æ E X". This formulation
Vz(X V=(X""(XX))
suggests that we can estimate ((x) and, if necessary, π2 (x) from data and utilize the pseudo-outcome
y 72 - Y 2 2
(XFE)(((
Vz (X
along with the estimated 70 (x). in a subsequent regression task to obtain an unbiased and consistent estimate of b(z) for X' (provided тz. 7 and 7o are estimated consistently). However, such an estimator only provides estimates for X" where 7(x) * O. Additionally, for small values of 7(x), T2 T2 (z). and (I), this method may result in high variance in the estimates b(z:), especially for certain parametric function classes. To address these challenges, we weight the data samples by the inverse variance of y /=<(((()
given by Var( | r) -133 (z). This approach is frequently used in generalized least squares methods (GLS, 2 to confer the algorithm asymptotic efficiency. While Var(Y | X a:) can be estimated from data using machine learning methods, is generally preferable to weight the estimator solely by compliance and instrument propensity to

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Algorithm CATE Estimation with Parametric Extrapolation
1: Input: Observational dataset 0 (XP; AP, Y IV dataset E (X,,, ZP, AP, Y 1=1 (x)
estimator T, (1)) estimator 9, Fz (I) estimator P, known mapping Φ -
2: Learn 7 (2) using T on O. Let R X R=xxd
3: for k = 1,,,..., K do
4: Set = E ([l...,,nE) :1
(mod K)}. 5: Use data {XX," Zi. A. Y, : $ Ix} to learn ((t) with 9 and 72 (z) with P.
6: for k (mod K) do
7: Set Y; =Y; 2,"
Tz (x
(1- (x
T0 (x.)
8: Set Xi, ((()) (X.² 10 (X,) for each E (I.....d)
9:
Output: 0 = (X x)
TT
avoid issues with small values of Var(Y x²). Assuming the bias function belongs to class of functions B, our proposed algorithm can be described by the following weighted empirical risk minimization (ERM) procedure.
b arg min BEB
where the factor (x)) was used for weighting the squared loss. This estimator automatically extrapolates to all X since we assign weights of 0 when 7(x)) = 0. Moreover, this method places thereby minimizing the risk of overfitting to data points with high variance. This weighting technique commonly employed in other IV estimation tasks, such local average treatment effect estimation (LATE), where weighting data points by compliance yields estimators with lower variance [3).
higher emphasis on lower-variance pseudo-outcomen
The weighting scheme in Equation 4
creates a weighted distribution, Px x (I)) for optimizing the ERM procedure. Since 5x* (z) differs from the target distribution (x), this introduces a transfer learning problem. Without additional constraints
the function class B, the minimization in Equation may yield many possible solutions. To ensure a unique or limited solution set, B must have low complexity require further structural assumptions. We explore two function classes B: a parametric class defined by b(x) = 8 ((x), E Rd with a known mapping Φ : X + Rd and a second parametric class where b(x:) = o² ((r), with v E IRd and Φ E being a learned representation common to both the observational and IV datasets.
4.1 Integrating Observational and Experimental Data via Parametric Extrapolation
We consider parametric class B3 (0)) E Rd) for known mapping ø X' + Rd. Since the compliance factor 7(x), instrument propensity 2 (z), and the parameter of interest HT are learned from the same dataset E, we propose the following K K-fold cross- foots-itteed estimation procedure:
V.F (1) (xF)x (xP) 8T S (xF))x(
= arg min
(5)
BERd 1 1ETE
where (x
(XF). and the compliance factor (R) and instrument propensity (k) bE [K] are trained on E' excluding the 1th fold containing indices Z K-fold cross-fitting is crucial because it ensures that the weights are learned from data distinct from that used in the ERM algorithm. This separation is essential for maintaining desirable theoretical properties as we remain methodologically agnostic the techniques used for learning y and π2.
The compliance factor y(x) = E[A F z = 1, XE = x] - E[A= 2 = 0, X = x] can
be estimated using standard machine learning classification al algorithms, either by training separate classifiers for A z 1, - & and A 2 0, X = x or by using one classifier with EE additional feature. Similarly, instrument propensity estimation is straightforward classification task with z as the target. Given estimates Go, 57(), and m(k) the result in Equation
is obtained by performing OLS procedure with the targets Y.5 (k) (x. FxF (x,5) and the
5

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design matrix X (xX*). Here, W) diag( (X,") (X ng
)), and
(xx (o(Xf)..... ((x The two-step procedure is detailed in Algorithm 1
Next, we provide theoretical guarantees for our parametric extrapolation approach. We begin by describing the regularity assumptions that enable the consistency of our estimator.
Assumption (Regularity Assumptions). The following claims are true:
1. (Treatment Positivity in O) P(A 1 = z) S 1 - for some 0.
2. (Insirument Positivity in E) S ( 7x S 1 - for some O.
3. (Boundedness) Y YO, | 2. 0 X 82)12 (z). (x) are uniformly bounded.
4. (Realizabillity of ((x) b(x) B i.e (.)) 7 (x) ft ((x)) for some 0 E CRd.
5. (Identifibillit of 0) ) is invertible.
The first two conditions in Assumption 3 aar standard in causal inference, ensuring that both treatments (or instruments) and controls are observable for every x E X. enabling CATE estimation. The third condition imposes common boundedness assumption control the growth of estimands. The fourth condition ensures our model for the bias function b(x) is well-specified given Bg. The final condition requires that the design matrix has rank d, ensuring we can learn the parameter @ from
data. Given
Assumption 3, we present the following theoretical result:
Theorem (Estimator Consistency for Parametric Extrapolation). Let T"y (n), Taz (nt) and r,o (n) be (1) functions of n. E N such that |/y - 1100)11, S (nn)) 7 (1k) I1. S Taz (ng). and 1700 - : IL & Tyo (no). Furthermore, assume the conditions of Assumption 1 Assumption 2 and Assumption 3 hold. Then, the parameter 0 returned by Algorithm is consistent and satisfies
|10 811 = Op (r,(ng) + TTx (ng) + rno((n)) 1//ng).
Moreover, f is consistent on X with convergence rate given by
| 1 Op (r- (ng) + (ng) + (noo + 1//ng).
We include the proof of Theorem 2 in Appendix B The core insight is that weighted OLS remains consistent as long as the estimates for 7, T2, and 70 are themselves consistent. However, the overall convergence rate is constrained by the slowest of these rates. In most cases, #z is assumed to be known, meaning the convergence rate is primarily dictated by the rates of 9 and f0, This result highlights the trade-off involved in leveraging both datasets achieve accurate effect estimation for the target population.
Remark 1 (Impact of Realizability Violations). When realizability does not hold, i.c. b(x) & B, our estimator may be inconsistent and exhibit asymptotic bias, proportional to the deviation of the true function from B. Nonetheless, conducting this analysis might still be valuable, as the resulting bias may be smaller than confounding bias in observational estimates the variance from low compliance in IV studies. Thus, even with uncertain realizability, our method may provide more accurate CATE estimates by effectively bulancing bias and variance.
4.2 Integrating Observational and Experimental Data via a Common Representation
Without expert knowledge, the mapping ((x) may not be known a priori. In this section, we introduce method to jointly learn both the unbiased CATE function and the mapping o6x:) (hereafter referred to as the representation), based on the assumption that the true CATE ((x) and the biased CATE O (x) share common representation. This approach leverages machine learning techniques that assume a common structure across tasks, such as multi-task and transfer learning. In causal inference, has been suggested that a shared representation can be assumed between treatment arms (47 48) or between randomized data and confounded observational data (23] This framework enables us to learn the bias function b(x) even when the mapping 6(x) E Φ is otherwise unknown.
We consider class & of representations ((x:) X + d and assume that there exists a shared representation & E & between the true and biased CATES. Specifically, there exist linear hypotheses h, ho E d such that r(x) hTT o(x) and 7 (x)) ( f0)). resulting in the bias function

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Algorithm 2 CATE Estimation with Representation Learning
1: Input: Observational dataset O = (XP, AP ) IV dataset E (X,, 2P, AP, YP (6. estimator T, 7x))) estimator G, (x) estimator P.
2: Learn o(x) and h using T on O
3: Algoritmm with ø and (2) (ho) Let D be its output.
4: Output: V.
b(z) - -h ok) := v :>==<<>>>) For simplicity, we focus on linear-in-reprreenttiin classes, but [23 47). Thus, b(x) B for the unknown Ø, with B& defined in SScctinn 4.1 Suppose there exists an ERM algorithm T that can jointly learn <xx) and h from the observational data, O. Our learning algorithm proceeds as follows: first, we use T to learn o(x) and ho from O, alongside estimates s7(k) (x)) and (z2)(()) from E as described in Seccion 4.1 In the second stage, we apply the following ERM procedure to estimate the parameter
more complex hypotheses h with T(x) h(((=)) can be considered
17 arg min PER
F.F. - (1:)7 of (X. )(x)) - (x))
(6)
This procedure is detailed in Algorithm 2 Finally, we recover F(x) by setting f(x) = (ho + +)T Example 1 (Representation learning with neural networks). Let d be a class feed-oowarrd neural networks. Then é(x), To and (x) can be jointly learned by composing & with two linear output heads for Y A 1, X and y A = 0. X I. respectively, By taking the difference between the two output heads, we can reconstruct 70 (x)) assuming that E A = 1, ø] and E[ Ao 0 xi] are also linear in (see [47 48)). Without this assumption, can learn T (z) directly by composing & with one linear output layer and considering the pseudo-
40 (0-A19) Here, TA(XO)= P(A 1 x²) is the treatment propensity in O and can be learned using any black-box machine learning classifier.
outcome aA(X") (1=ma(X0)}'
With this setup, we obtain thcoretical results similar to those in Theorem 2
Theorem 3 (Estimator Consistency for Shared Representation Learning). Let r., (n), Tag (n), and (n) be functions of E N such that |l - 7,00 || S (ng)). || TZ 1l S Taz (ng), and |1ا 1 EllL, S (n())). Additionally, assume |0l|2 is bounded and IE (x))") is invertible. Let us also consider the conditions specified in Assumption 1 and Assumption 2 to be satisfied. Moreover, assume that 7 (x)) 0h ((x) for some & that realizable within the representation class to and let Assumption 3) hold for Φb Under these conditions, the parameter D returned by Algorithm 2 is consistent and satisfies
(k)
| - ul12 = Op (f,(1)) + TTz (ng) + (r()))) + 1//m/=)/////tt/////
Moreover, f is consistent on X with convergence rate given by
|| -F-v -TL,, 0p (ng) + Tx (mg) + (r(nn) + 1/ + L/O)
We provide the proof of Theorem 3 in Appendix B This result hinges on the realizability assumption in & and the inear-intreepresenttiion structure of both 7 and =0. Example 1, rn(n) bounds the generalization error for feed-forward neural networks. For ReLU activations and bounded outputs, (n) = CVWL /WL log W log n/ vn, where W is the total number of weights, L is the number of layers, and C' is constant independent of and W [15] 56). While this rate is parametric, it scales linearly with W, which becomes problematic for over wer-parameterized networks. For I-Lipschitz
activations and bounded weights, 18] derive a rate of (n) = C ,TTT,M((//// M(I)/n//1 where M(I)
bounds the Frobenius norm of layer I's weight matrix.
Practical Guidance in High- Dingen-nmniiinal Settings: When d(x) is high-dimensions controlling the complexity of B3 through regularization is crucial, especially since the bias function M(x) is used to extrapolate the CATE into low-variance regions where compliance is low and the risk of overfitting is high. In the parametric extrapolation approach (Section 4.1). applying L1 Le regularization via
7

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15-
(l(x) = SE (0X)
(F(x) + SF
T(x) + SF
Algorithm
10-
T(X)
100
Effect
-2
15
15-
15-
(9(x] =
f(x) # SE
Algorihm
Eflect
10-
T(x)
10-
Y(v)
1(0)
33
22
-3 -2 -1 0
(a) 1000
(b)
(c) F(z)
Figure 2: Means and standard errors of estimates from 100 simulated dataset pairs (O, E) using
Random Forest (top) or Neural Network (bottom) learners, (2 Biased observational CATE T (x).
(2b): High variance CATES from the IV dataset using Equation 3 (2c CATES from Algorithm
using parametric extrapolation (top) representation learning (bottom).
Lasso or Ridge regression in the final step is effective for controlling model complexity. In the shared representation approach (Example 1). regularization but also prevents ver-parammttrizttio in the neural network ø. The choice between Lı and L2 regularization, and how they are applied, should be aligned with the data-generating process and specific characteristics of the model.
only helps control the parameters ho and
Experimental Results
We apply method to both simulated and real-world data. First, use the confounded synthetic data example from [29]. along with a similar data generating process (DGP) to simulate an IV study, maintaining the same confounding structure and treatment effects. Using this DGP, evaluate Algorithm I and Algorithm in estimating the unbiased CATE by integrating these datasets. Next, we demonstrate estimators real-world dataset examining the impact of 401 participation financial wealth. Additional experiments, as well as details on model implementation, hyperparameter selection, and validation procedures are in Appendix The replication code is available at hhttps //github. Wewithuu.com/auuuuuu www.m//aaaaaa////....... Iwtakk--nnsttuuttteeemmm -baa Daaa--ataa-aaa CATE
5.1 Simulation Studies
We generate the observational dataset 0 (X", ,,,,, yO) follows (see 29
X NNO, 1), A (0.5), U X, N (X (A- 0.75)
==1+A+ 2AX + 0.5.² 0.75AX2 + U + 0.3ey, ey ((0,1)
(7)
In this DGP, the true CATE is given 7(r) 0.75x + 2z + 1, whereas the biased observational
CATE is represented by - (a)) = 0.75x 1. This results in a bias b(x) -r, which
linear in æ. We modify this DGP to generate the experimental IV dataset F = X ZE A B
as follows:
10,,011) ~~ Be.5() A ~ BermBBeB. (0.5)
-2X), C Bern(n = - (1-C)
U V.X,A,C~ -CAAC~---((OUUU + (1 (1-O) N(( 0.5)
where is the (unknown) compliance indicator, o is the logistic sigmoid and we keep the same outcome function in Eqquaiin In this modified DGP, the randomized instrument has compliance sharply deternined by X, with low X values indicating almost no compliance and high X values indicating near-perfect compliance.
For experimental results using higher-dimensiona. version of this DGP, refer to Appendix C

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Age:40, Income: $30,000, Single
Age:10, Income: $30,000, Married
14000
12000
(700
Etfect ($)
(F(x)
12000
70x)
10000
Eetrapolation
10000
8000
10
12
18
14
16
Education (years)
Education (years)
Figure 3: Impact of 401(k) participation on net worth by education level: Using f(x) from Algo rithm I] we fix age, income, and binary variables, varying education and marital status. The black line shows results from/Algorithm IJ and the dashed line indicates predictions in the no-compliance region. go (a) is the biased observational CATE, while & (x) is the IV CATE without non-compliance.
We generate 100 observational and IV datasets, each with 5,000 samples, from the proposed DGP. We first apply Algorithm 1 to each dataset. With randomized instrument, Tz (z) = 0.5. We estimate
((x) as the difference between Random Forest (RF) classifiers trained on (X <x A Z
and (x ") = = 1, i.e. one is trained the subset of data where the instrumental variable z 0 (using X and A as inputs), and another on the subset where 2 1. The biased observational CATE is modeled using the T-learner approach [B], with RF regressors trained on x yo A and X yo A
1. For comparison, we implement a CATE estimator for We compute (x) as the difference between RF regressors
the experimental data using Equation 3
trained on E Y | _ ===00am and X== YE z = 1, then divide by 7(x), clipping the compliance score at 0.1. We calculate 7(x), 70 (z),, and u (z) for each dataset pair and proceed with the second
step of Algorithm by setting <5(x) = x.
In Figure 2 (top row), we depict the means and standard errors of our estimators across 100 simulations.
The first two plots illustrate the learned observational CATE 70 (z) and the learned IV CATE 7 (z). As expected, (x) shows clear bias, while F ==(x) has high variance despite aggressive compliance score clipping- The third plot presents the results from Algorithm showing that the resulting f(x) is both unbiased and has low variance across X'. These findings demonstrate that our two-stage estimation procedure effectively leverages the strengths of both datasets to capture the true CATE and address the limitations of each individual study design.
We note that in our DGP, T(z), (xr), and b(xz) are linear in the polynomial representation (xr, Thus, we next apply Algorithm 2] with Example to learn the true CATE and the common repre- sentation from the generated dataset. For consistency, we employ feed-forward neural networks (NNs) to estimate all quantities. The estimator for 7 uses a NN with a sigmoid activation in the output layer, trained on XF with the pseudo-outcome Φ(x) and the biased CATE (x)x) are learned using a representation network with two output heads 1. A similar dual-head approach is used
2.A Z 2A 2²). The representation
for learning yo X .0 A to learn dy (r), by modeling Y X Z = 0 and Y X=, , z 1 simultaneously. When calculating dy (x)/ we clip the compliance score at 0.1. Unlike Unlike/Algorinn Algorithm 1| we don't guarantee the polynomial representation will be fully captured by the chosen representation class, but we expect a sufficiently flexible to adequately represent these relationships.
and yo A
The means and standard errors of our estimators from 100 simulations using neural networks and Algorithm 2 are shown in Figure (bottom row). As before, (z) shows bias, while 7 (I) has
high variance in low-complianee regions, despite compliance score clipping. However, Figure 2c shows that the f returned by Algorithm 2 remains unbiased with relatively low variance across X. This demonstrates that combining observational and IV data, where the biased and true CATE share a representation, allows us reliably learn both the representation and the unbiased CATE, overcoming the limitations of each individual study.
5.2 Impact of 401(k) Participation on Financial Wealth
We demonstrate our method's effectiveness with a real-world case study on the impact of 401(k) participation on financial wealth, using data from the 1991 Survey of Income and Program Partici- pation [1] The dataset includes 9,915 respondents with nine covariates: age, income, education, family size, marital status, two-earner status, pension status, IRA participation, and home ownership. The primary variable of interest is 401(k) participation (A), with eligibility (Z) as the instrumental

---

variable. Although (k) eligibility is not randomly assigned, it is argued to maintain conditional independence given observed features [T] 43 We assume 401 eligibility influences net worth only through (k) participation, characterizing this as an IV study with one-sided non-compliance, where non-eligible individuals cannot participate (A = 0). The target variable (Y) is net financial assets, calculated as the total of (k) balance, bank account balances, and interest-earning assets, minus mon-mortgage debt.
To replicate the scenario in this paper, we split the dataset into two halves: one for the IV study and the other for the observational study (where we intentionally remove the instrument information). Our goal is to use these datasets, along with the parametric extension approach in in/oogorithm to recover the unbiased CATES. Due to one-sided non-compliance, the estimated compliance factor 7() is high (0.49 0.90, see Appendix Cj. To show the utilityy of our method, we introduce artificial non-compliance by setting ((x) to for individuals with less than 12 years of education (13% of the population). In the first stage of Algorithm I] we use RF regressors and classifiers to estimate 70 (a),, (ar), and (x)) with hyperparameters set based on other related work on this dataset [12]. In the second stage, we define the mapping (xx) with an intercept term, the 9 covariates, and their interactions (46 features total). We apply a mild L1 regularization in the final linear regression due to
the large number of resulting features.
we study how the CATE function from Algorithm I varies with education. We focus on education as it is selected as a top feature by the compliance model, while the outcome models do not rank it as highly significant (see (see|Appent) Appendix C). To explore this relationship, we vary education and marital status, holding age and income at their median values and setting all binary variables to zero. Since compliance in the IV study is high, we consider the estimate 7B without the artificial non-compliance as the ground truth for comparison. Our analysis shows that observational data treatment effects are upwardly biased, likely due to unobserved confounders such as financial literacy. The F(x:) from Algorithm 1 shown with a dashed line for non-compliance regions, closely aligns with PD (x) (excluding the artificial non-compliance). This demonstrates that combining IV and observational data can effectively estimate unbiased CATES in real-world settings, offering a robust solution for causal inference even in the presence of low compliance and unobserved confounding.
In Figure 3]
6 Conclusion
This study introduces a method that combines observational and instrumental variable (IV) data to address unobserved confounders in observational studies and low compliance in IV studies. Our two-stage framework first estimates biased CATES from the observational data, then corrects them using compliance-weighte IV samples. We explore two variations of our procedure: one that models confounding bias parametrically, and another that leverages a shared representation between the true and biased CATES. Both methods are shown to be consistent, validated through simulations and real-world applications. Our approach holds significant promise for applications in digital platforms, personalized medicine. and economics, offering robust tool for deriving reliable, actionable insights from complex data. Limitations of our work are discussed in Appendix D)
Acknowledgements
We thank the anonymous reviewers for their valuable feedback and insightful suggestions. This material is based upon work supported by the National Science Foundation under Grant No. 1846210 and by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under Award Number DE-SC002311.
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---

A Extended Literature Review
Heterogeneous treatment effect estimation from observational data: Recently, there has been a significant interest in applying machine learning estimate CATEs using observational data. This field has seen adaptations of a wide range of machine learning techniques, from random forests and Bayesian algorithms e.g. [20, 24 to deep learning [ 27 48] and blackbox meta- 33 38 that utilize efficient influence functions [14 31 and Neyman orthogonality [12 16].
B9 [1
learners
Despite these advancements, significant challenge remains these methods typically assume the absence of confounding in observational data (ignorabilityy) - an often unrealistic and unverifiable assumption limiting their real-world applicability. Without ignorability, point identification of effects is impossible, although some studies propose methods to construct bounds on treatment effect estimates under assumptions about the structure of unobserved confounding 17 28 451
Nonetheless, these bounds often have limited practical utility. Other efforts to address confounding bias CATE estimation rely on latent variable models to recover unobserved confounders from noisy proxies [4 36 or utilize multiple or sequential treatments (9 22 53]. However, these methods also have limited practical impact, as they depend on either the availability of additional accurate proxy data or unverifiable assumptions such as no unobserved single-cause confounders.
Heterogeneous treatment effect estimation using IVs: Machine learning techniques have recently been integrated with instrumental variable methods, offering significant advantages over traditional approaches, including the flexible estimation of CATEs. [49] and [54] expand on two-stage least squares (2SLS) to incorporate complex feature mappings via kernel methods and deep learning. In the same vein, 21 introduced a two-stage neural network for conditional density estimation, while (8] applied moment conditions for IV estimation. [50] propose novel IV estimators that exhibit Neyman orthogonality. However, these techniques rely on the assumption that instruments are relevant across all covariate groups, a condition that is not consistently met with weak instruments.
Treatment effect estimation with weak instruments: Weak instruments compromise the reli- ability of traditional IV methods like 2SLS, often producing biased, high-variance estimates and undermining causal claims. To mitigate these issues, several approaches have been developed, includ- ing bias-adjusted 2SLS estimators, limited information maximum likelihood (LIML), and jackknife instrumental variable (JIVE) estimators (see (25 and references therein). Recent methods reduce 2SLS estimator variance by exploiting first-stage heterogeneity (variation in compliance) through a weighting scheme, as detailed in [ 32). However, these methods do not extend to estimating conditional average treatment effects. Another strand of research focuses on combining multiple weak instruments into robust composite, showing promise in genetic studies using Mendelian randomization (B3), 355), ). These approaches require access to multiple weak instruments for the same treatment. Our work aligns most closely with II 13 32 in that we leverage compliance heterogeneity and employ compliance weighting to merge IV studies with observational data for efficient confounding bias estimation. Unlike these studies, however, our approach distinctively estimates heterogeneous effects and leverages additional observational data to address challenges posed by weak instruments.
Combining observational and randomized data: There has been a proliferation of research in combining observational datasets with randomized control trials experimental data with perfect compliance- - to mitigate bias from observational studies. One of the strategies is to impose structural assumptions such strong parametric assumptions for the confounding bias (29] a shared structure between the biased and unbiased CATE functions that can be estimated from the two datasets (23) Other studies advocate for dual estimators from both data types, optimizing bias correction through a weighted average [10 46) (5). Additionally, approaches like those by [6] and (26) leverage outcomes from different time-steps, such as short-term and long-term effects, to enhance estimation accuracy. Our work is closest to 22 and 23) However, our study faces additional complexities firstly, the CATE estimation techniques differ between the datasets, requiring us to debias the overall effect function rather than just individual outcome functions. Secondly, RCTs may not represent the target population due to their narrow scope, our instrumental variable (IV) study faces representation issues due to minimal or absent compliance in strata that are not known priori. Thirdly, the CATE estimation in our IV study uses ratio estimator, which is highly sensitive to changes in the compliance denominator, adding a layer of complexity to our analysis.
14

---

B Proofs of Theorems and Lemmas
B.1 Proof of Equation
The exclusion and independence conditions in Assumption equation holds:
imply that the following identification
E y (A) - (A) x x] = E 1 =, = =x] -e-EE -E 1 =0 = z²..
(8)
By noting that
Y (A) - Y (A = -Y()) when A (1) = 1, 1d (0) = 0 (compliers)
YE (A) - YE (AR(0) = ((0) - -Y(),
when A (1(1) 0, A- (0) 1 (defiers)
YE (1) - YE (AR(0) =,
when AF(1) 1, (0) =1 (always-takers)
YE (AB(1)) -YE (A8(0) = 0,
when A A (1) 45 (0) (never-takers)
the left-hand side of this equation can further be written as:
EA - Y (² x = x] =
( x xx==]] =
X =2|- (1) (0) x = x|
Assumption 2
-(xx) (EEAA) 8 1, =
= 1 = 0 =x)
Assumption I
Since the claim of Equation 3 holds for X', have that EIAE 2 1, X z E = 0, x = x F 0 by the relevance condition in Assumption
E
From Eqs.
8
and
we obtain:
E
E 1, X x| -E E
0, X
Txx)
A
ZE = XE = x EE
7B = 0, XE =
for x E X"
B.2 Proof of Lemma! 1
Recall that for any x E X', we have that ((x) * 0 by Assumption 2 we use the law of total expectation as follows:
Then, assuming that xz(xx) > 0.
ZE
( - 2E
E
X
(x(x)(x)
(1 (1-T2 (x))))(()
YEZB
YE(1 22)
EE
2 1, X. = P 1| = =x)
TZ (z))((²)
(1 - (2())((x))
YEZE
yB ZF)
E
z 0, = P(Z =0 01 xx== = x)
(2(xx(())
(1 - -TT2(x)))
Y
EE
E 1, XB TT(I)
(+)
YE
E
F 0, X
(1 (z()))
(1 π2 (2)))))(2)
E F
F X
= - E F
F R =
7(x)
_ = Y
1, = x - E
Y
E 0, x = x
T(x)
Equation 3)
E
A
1,
A
7 = 0 XE 0,
where the intermediate steps follow from the definitions of xz(x) and 7(x) and the last equality comes from the identification result in Equation 3
15

---

B.3 Proof of Theorem 2
For simplicity, omit the E subscripts from X _ AB, Y throughout this proof. Furthermore, assume that ng an integer multiple the number of folds K. Let fZ) E Eiez, f(Z)). recalling that In = {i E ,I,...,,y nE} : = k - (mod K)}, which indexes the subset of data in the hthh fold. Then, we write the estimated parameter 8 as:
K
0-
10
K
1=1
K
1
E5
(vZ(1-=22
-XI (x)
=
2) a0 (x)+ (x)(())
K
k=
We also define the following quantities:
ONE = Eng [u(X))o(X)x(x)]"
Ene 1 Y Z)Tz((X) w(X)7 (X)) w/ w(X)((X)]
Ø E (x {x{x()(())))
E (2 Y(1 Z)z2 - (X) (x)) w (
We note that these quantities are well defined because E no( x 2 (X)) (X))))) is invertible. This follows from the first and last conditions of Assumptiin 3] along with the stipulation in Assumption 1 that (xx) f 0 for all x a set non-zero measure. Using these definitions, we can write
0
Ong
Ong
0 + 0
2
Onn
8
+
(Triangle Inequality)
A
Aa
We study these terms separately. We notice that X just linear regression of the modified outcome Y YZ(1 (X)) Y(1 (X) w(X)r (X) ((X) using weights w(X). Given the regularity conditions in inAAsuumpiinn 3 (which subsume the standard regularity conditions of linear regression), we have that 22 (1(1/ /ng). Then, consider the 0 term. We have:
i E [xxXX))((X)((x)))
[[(Y((1-72X)) Y(1 (()))) w(X)((X)]
([(YY((1=2((E) Y(1 (X) (((x)--w(x((()
w(X)7(X) w/ (x (X)0 ((X)) (Realizability of b(X))
x())0=((X)) wi (X))
=E[w(x3(((x)6x)] 7(X) #0 P(>(X) 0)
(e((VZ(1- - Y(1 (X) - ((X))((X) (X)))))))) 1x(x) +
+E(u(X))0(x))(7)) (x))
#0] ( #0)
P()(X)
E [xxX) (x() 0]
((()2(1-7(XX)) - -Y(I - Z))z(X) ((())))((((xx))x) ((x))))))))))) (xx) 00
=E (xXx)X((XX))x 1x()) # 0
YZ
Y(1 2)
(1- (1-xz(X))(3 - -(x) w(X)2(()) ((X) * +
(x(X))(x)
(Since 7(X) * 0 implies w(X)
* 0 by Assumption 3 Lemma
8.
16

---

Thus, 0- = A which implies A = O. We now tackle the Y term. To streamline the exposition, let us introduce the following shorthand notation:
(h)) Y2 - (X)) - Y(1 2)7 (x)
y = YZ - -z - YY(1 - (x)
K
1
Σx =
R (X)
K A=1
Σx E [ (X))((X)(X)]
Σ = ENN [u(X))(XX))(
Σ = E (xxx)))(x))xx)]
We can then write the 0- Ong as follows:
①
Ong
K
= 5 K
Bb
(P(0) [1910) (x)F0()) (i (x)(x)
K
k=1
[ (v-w(x))00)) [[)))(X)
K
1 (x)=o) (x)) Dol (x))())
=Ext 1ー
1
Σ
Ex -
(111))
K
k=1
K
+ +
[[((((N) ( (x)= (X)((X)
K
- BE( -
(112)
+ 5
EE
(x (x)7(x)) (5xh) (x)x(()
K
( w(X) (x)) w/ w(X)
(11))
By Cauchy-Schwarte we can bound the X term as
A = 0
Ong
where we used the A notation introduced in the preceding equation. We bound each of the A separately. We let || l denote the Frobenius norm of the matrix A. Then, consider A1.1:
|A,,1l2
K Ex
<
S K
111--11
(x)+ (0 (X)(x)]
F
K
k=1
K
=
| (S - -
1
(S/(0) - -600 (x)= ( (x)
K
k=
< <|<>>' | || |0 $x 11-
|--1 |
1
Σ R2
- wي - (x)=0(x) w (x)0()]
F
K
1=11
= o, (1 Ex] |+)
(By the boundedness conditions in Assumption 3)
17

---

Furthermore,
K
5- Sx = =Ene [u(x))6(X))(x))
- 1
Es
K
1 k
Σ
F2
B) (mx())=- w. (x)}) (())(x))
K
(xx) T (x)p (x))
+ E
K
3 Σx K
Σ
(Bx E)) [w(x) [w
-000 (x))) (12)))))(()((1)) 6(x))
-
K
1=1
LE
(( (x()) - (x)²) o(x)6(x))]11
K
(Ex ED) | [xx))² - T0th (x)P) (())
-
$
+==1
1/2
w きt l1,, E (x)) 1
(Holder's inequality)
By our boundedness assumptions, the second term yields an Op (Iw $ (11)))) = Op(r,(ng) + Taz (næ) term in the expression for Op (15 Exl1r) To analyze the first term, let Eg. represent
the samples in the htth fold of the E dataset. Then, 6x F& has mean 0 since ((()) is independent from Eg due to the K-fold sample splitting. Then, we can apply Chebyshev's inequality to obtain
1/2 E n
[w(x} (000 (x)=)) (X)
1/2
6x Ex = Op
Ex
from the consistency assumptions for 9(0), #2 which translate into a consistency assumption for (K)) By the bounded convergence theorem, this implics that 6g is also (1/1/ /ng). Putting everything together, we obtain
||r.1 (p(r,(ng) + (ne) + (1/ /ne)
We now tackle 2
A1.2
K
()(k) - (x))
K
E((Ý - (x)
||1,2112
$ |
K 1=1
(X) (X4)))0))))) (xx))))xx)) o(X),]
1
[5
11
K
( (x) w(xi (x)x)))))))+))()))0(00)))))))333 | 12
1=1
18

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Since the [Ix(X) s are bounded by assumption and 3-1 P, E-1 from the continuous mapping theo-
rem, it suffices study the term E (()))(())) w( w(X) ( (x)
w/ (x))x
| (b) ( -
11
S ||B|Y Z 1, 111101/000000000000000((10)10001
+ ||||Y z= 0, x] - (x) - (xx])
+ (1 (x) (X) -(X))r00X)l1
< [|0((0) w|11, + 117(0) - -7 + 11 Tz 1.2 + 1170 - - 1
(Boundedness assumptions)
S [17(()] - 7alea + 11 æzlloa + 1170 - - S (m)) + (1z( + (mo)
(Definition of wx()) ())
where X absorbs constants. Thus, |112212 is Op (ry(m)) + TTz (ng) + (no)) Lastly, we note that 11,3 is the empirical process equivalent of λ1, and thus, by leveraging sample splitting through arguments similar those used for the AI.I term, we have that [||r,a|2 is (,(1 (/ll//////.//// (.E.. Putting all A terms together, we have that λ is Op (m=) + Txz (ug) (no)) + op ()(E)) Recall that A2
is (1/ /ng) and A3 = 0, we obtain the desired result:
||0 - 011 = Op (r (mE) Tag (nE) + (no) + 1/ /E))
Given that 1 = [ 8) (X) + (T (X) - (x we further have
||F - - و|ا,, LG Op (1F) + Trz (Næ) + r.o (no) + 1/VOg)
by using the derived i rates, the Cauchy-Schwartz inequality and the boundedness of ()o(X)l2 assumption. Our proof now complete.
B.4 Proof of Theorem 3
We first study the convergence rate of 7O using the conditions of Theorem 3 (ar) solve the following joint optimization problem:
Assume that h and
no
YY - Tx(X)
(19FO(x9))
To, - arg onggiin.ge...o dEb
1 1- 1-
Then, (x) = ((0))((z) Thus, we write:
||70 - s [1(19) - (()))
< ||(19))((x) - (19)) ox)) + (xxx)
x 1 10 1
tr(nn))
(Boundedness assumptions)
We further expand the first term:
| - 11
[ - Eno (x())(()) "Eno ( [F0X)]||
(Y== = yo y-
[x(xT (-)x(X)
<
- P.n ( -
-'êmo [V(())]|
+
(mo) + 1/
19

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Table I: Hyperparameters of models in simulated data experiments.
Method
Model(s)
Algorithm
Hyperparameter
Value
Algorithm
Compliance
Random Forest
max t_xxp_epthhhhhhh
(scikit-learn)
min_sane mii__saplees. Samples_leaf
50
Algorithm
Outcomes
Random Forest
max_deptl
(scikit-learn)
min giin_samlees_eee _eaff
Algorithm 2
Representation
Neural Network
activation
ELU
CATE
(PyTorch)
hidden units network depth
Compliance
weight decay
0.02
optimizer
Adam
learning rate
0.01
batch size
2000
epochs
1000
Thus, ||00- |L,, is 1s0g(ra(no)+1//ng) Next, we build upon the insights provided by the Proof of Theorem We note that we can apply the same analysis as the Proof of Theorem 2 by using & in- stead of @ and everything goes through except the A term which is not since y depends on á and not é. Thus, the convergence rate of || | v112 will be Op (rg(ng) Tax (ng) + rrr (no) 1/ 1//nz) Op (ry(ng) +xzz (ng) (n)) + 1/VnE + 1/ no) plus term that depends on the deviation between Φ and Φb. This term is given by:
As = (xxx)))
E (YZ( - Y(1 -2)=x(X)xxx))) -w w(X)= (X)) w(x)((x)
-v1
= (xX)) (x)) + of (
(XXXXXXX()))(x *0 -
Lemma
E (xX) (())) (X) * 0] (xX( ((x)) / * 0] Or(r((no))
However, this term simply gets absorbed into Op (r, (mg) + raz (mg) + +r((1o) Thus, we obtain the desired results:
+ 1/ / + 1/ /no)
1 2112 Op (r-(mg) TTx (ng) (1ng)+rra (r() 1/ + 1/
and
|| 1 T|l,, = Op (1y( + (rzz(nE) + +ro(no) + 1/ 1/ \o)
C Additional Experimental Details
C.1 Simulation Studies
The results for the parametric extension from (SomSncction.. 5.1 were generated on a consumer laptop equipped with 13th Gen Intel Core i7 CPU. The execution took approximately 1.5 minutes using 20 concurrent workers. In contrast, the representation Icarning outcomes were derived using NVIDIA Tesla T4 GPU on Google Colab [19]. The execution took roughly 1.5 hours, with half the time spent on Algorithm and the other half on learning 7 (x: over 100 iterations.
Implementation Details:
The Random Forest (RF) models used in Algorithm 1 emmloy the RandomForest algorithms from the scikit-leat 42 Python library. For the feed- forward neural networks within the representation learning component, we utilize the nn module from the PyTorch package 4T Details regarding the hyperparameters for these models are provided
and
RandomF wrwetClassssisee.cooocrsssss
in Table [
20

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Table 2: MSE + SD for estimators in nigh-dimensional DGP
(1)x)
(x)
f(æ)
d= 5
1.40
3.97 1.21
0.40 0.07
d d= 10
3.25 +
7.70 1150 1.54
1.25 0.20
d= 20
9.32 + 0.51
19.2 2.58
4.05 + 0.68
d= 50
37.1 0.94
43.2 + 2.89
9.39 + 1.64
Table 3: 401(k) dataset description
Name
Description
Type
age
age
continuous covariate
inc
income
continuous covariate
educ
years of completed education
continuous covariate
fsize
family size
continuous covariate
marr
marital status
binary covariate
two_earn
whether dual-earning household
binary covariate
db
defined benefit pension status
binary covariate
pira
IRA participation
binary covariate
hown
home ownership
binary covariate
e401
(k) eligibility
binary instrument
p401
(k) participation
binary treatment
net_tfa tfa
net financial assets
continuous outcome
We configured the parameters for the Random Forest (RF) models based on the theoretical guidance outlined in [44]. For the neural networks, implemented early stopping using validation dataset that constituted 20% of the total generated datasets.
Result for High--Dimesiionl DGP: We perform additional experiments to highlight the effectiveness of our method nigher-dimensions settings. To this aim, we modify the DGP in Section 5.1 include d features X E Rd, with both baselines and bias depending on all features as follows:
Y 1 X 2A5 X + 0.5 + 0.75.XX + U + 0.5ey
U X, A N (7² X (A - 0.5), 0.75)
where the coefficients 3.7 E [-1, 1]d are set at random at the beginning of the simulation. this setting, the bias function given by b(x)) -y-'. We leave all other settings and parameters (including no = ng = 5, 000) unchanged and perform parametric extrapolation using Algorithm
In Table 2
we report the mean squared error (MSE) and standard deviation (SD) of predictions on a fixed sample of 1,000 points drawn from the same distribution as X, over 100 iterations and for various dimensions (d E 5, 10, 20, 50). The high MSE of the IV estimator (z(z) reflects the challenges of estimating compliance high-dimensional settings. Likewise, the observational data
estimator f0(x) shows clear bias. In contrast, the combined data estimator f(x) from Algorithm 1 significantly outperforms both, demonstrating improved accuracy in this high high-dimensional context.
C.2 Impact of 401(k) Participation on Financial Wealth
The dataset from [T] is comprised 9,915 observations with covariates: age, income, education, family size, marital status, two-earner household status, defined benefit pension status, IRA participation, and home ownership indicators. We describe the features of the 4K)(k) dataset in Table
Implementation Details:
Given the heavy-tailed distribution of net worth measures, we perform a pre-processing step to remove outliers. Specifically, we eliminate the top and bottom 2.5% of observations, effectively narrowing the range of potential outcomes from [-0.5 x 10 1.5 10 to [-1.4 x 10 1.34 x 109] This adjustment leaves us with 9,419 observations, which are then evenly distributed between the observational and experimental datasets. We find that this procedure improves the stability of regression and classification awwiiionnnggiiii algorithms across different random data splits.
21

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High
R00
pira
800
age
Inc
educ
age
Counts
filze
Choccme
400
Complience
educc
para
twosam
200
twoearn
fsiie
bown
marr
db
0.6
0.8
1.0
-01
0.1
Low
Compliance
SHAP value (impact on model output)
SHAP (impact model ouigut)
(a) (z)) Histogram
(b) 7(x)) Shap Plot
E A x Shap Plot
Figure 4: Characteristics of the (k) dataset derived from the first stage of Algorithm (4a): Histogram compliance scores for æ E X (4b): Shapley plot [37 for the compliance model in the IV dataset with features arranged in decreasing order by feature importance. 44ch: Shapley plot
for the estimated outcome model Ely A 1, Xo in the observational dataset with features arranged in decreasing order by feature importance.
Table 4: MSE + SD across different (k)(k) data splits. Age: 40, Income: $30,000, Single
Educ
70 (in 1,000S)
7 (in 1,000S)
f (in 1,000$)
8
2.18
2.20
9.83 + 2.22
10
11.8 + 2.17
10.2 1 2.42
9.99 + 2.18
12
11.8 2.22
9.88 = 2.36
10.2 + 2.20
Table 5: MSE SD across different (k) data splits. Age: 40, Income: $30,000, Married
Educ
7O (in 1,000S)
7 (in 1,000S)
F(in 1,0008)
8
11.3 2.40
9.49 + 2.23
9.54 2.50
10
1
9,63 2.40
9.59 + 2.38
12
11.2 2.41
9.93 = 2.39
9.65 + 2.29
This dataset has previously been analyzed using Random Forest algorithms in 122.. Con- sistent with this earlier work, we employ the same models SonddooFooretReggeesee and RandomForess SrnddooForestClassiiiiee from scikit-learn) and use identical hyperparameters (n_ estimators 100, max foox___pph-aae = 6, max features = 3, min saansssmmllss__ leaf 10) for various regression and classifi- cation tasks outlined [Algorithm I] For the second stage Algorithm we use a Lasso regressor from scikit-lears with a penalty of a 0.07 selected via 5-fold cross-validation.
display several characteristics of the 4k) dataset derived from the first stages of Algorithm T] In particular, we illustrate the spread in compliance scores in IV dataset, as well the impact of important features the predictions of the compliance and outcome models, respectively. As noted in the main text, the compliance scores are relatively large and range between 0.49 and 0.90 (mean=0.70%, 70). Furthermore, the primary features influeneing the compliance score model include income, age, and education. In contrast, the features impacting the outcome model B|y 1 = 1, X are IRA participation, income, and age, with education having a significantly lesser effect. This motivated us to investigate how education influences the derived CATEs.
In Figure 4
Quantifying Uncertainty Across Data Splits: We quantify our claims for the 401(k) study by repeating the experiment across 100 different (O, E)) splits of the original data. We calculate the means and standard deviations of the treatment effects by years of education for the two examples described in the paper, with the results shown in TTall and Table 5] We note that the original trend (biased observational estimates, accurate extrapolation to the no-compliance region) is largely preserved, and our method demonstrates the ability to interpolate well on average in the artifically introduced non-compliance region. However, the uncertainty, deviations, is substantial enough that the results are not statistically significant, which limits the strength of the conclusions we can draw from this experiment (unfortunately")) This is most likely
reflected by the large standard
22

---

due to the prevalence of outliers, net worth follows a heavy-tailed distribution, and RF regressors tend to overfit these extreme values.
D Limitations and Societal Impacts of Our Work
Our methodology hinges several key assumptions, and violations can significantly affect the accuracy and reliability of our estimates. First, the standard IV assumptions (Assumption 1) must hold. If the instrument directly affects the outcome, is correlated with unobserved confounders, is weak across all strata of covariates, our estimates may be biased and unreliable. Some of these issues can be mitigated in experimental settings where the instrument is fully randomized. Additionally, the unconfounded compliance assumption requires that compliance is independent of potential outcomes given the covariates. Violations here can also lead biased estimates unrecorded explanatory variables affect both outcomes and compliance. Lastly, our method relies on realizability assumptions regarding the bias function. If these assumptions do not hold, our estimates might be biased.
The societal impacts of our method stem from potential inaccuracies in treatment effect estimates and Inaccurate treatment effect estimates could lead to range of adverse outcomes,
their subsequent from a diminished user experience on online platforms to less effective healthcare recommendations, economic and public policies. Furthermore, while accurate estimates can provide substantial benefits, they must be used responsibly to avoid unintended consequences such as privacy concerns or potential biases in decision-making, It is thus crucial apply these methods with careful consideration of ethical implications and societal impacts.
23

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Reviews:
[{"review_id": "hjv2NtGWCp", "reviewer": "Reviewer_znEs", "summary": "The paper tackles the problem of estimating CATE when unobserved confounding is present in an observation study but an IV experiments is accessible, though the instrument could be weak. The paper proposes a two stage framework to first learn a biased CATE from observational data and makes a bias correction using complican weighted IV samples. The paper then demonstrate the effectiveness using simulation studies and a real world example on 401k.", "questions": "1. Do you have any results on violation of different assumptions? For example the realizibility. \n2. Also see weakness on questions on experiments.", "limitations": "The authors list the limitations in the appendix.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Originality: There is a lot of work on combining observational data with experimental data to better estimate causal estimands. This paper is the first to consider combining with an IV study with potentially weak instruments. \n\nQuality: The paper has clear theoretical results covering two cases and both simulation study and real world data example. \n\nClarity: I found the paper easy to follow, with clear related work, contributions of the paper, motivation, theoretical results and experiments.", "weaknesses": "Experiments:\n1. It would be better to have some baseline comparisons, for example some debiased CATE estimation methods;\n2. The simulation study and the real world example both use an equal size IV/observational data. What would happen if we only have a much smaller sample size IV study? I believe this is more common in real life since observational data is cheaper to get. \n3. It would be interesting to see what happens in high-dimensional settings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Strengths**\n\nThank you for your insightful feedback. We appreciate your recognition of our novel approach to combining an observational dataset with an IV study with weak instruments. We are glad that our efforts to ensure the paper is both rigorous and accessible are apparent.\n\n**Re: More Experiments**\n\n* Baseline comparisons: Debiased CATE estimation methods are not ideal comparisons as they address finite sample bias rather than confounding bias. The confounding bias, typically found in observational studies, is an asymptotically irreducible bias:\n$$\nb(x) = \\mathbb{E}\\left[(\\mathbb{E}[Y^O\\mid A^O=1, X^O, U] - \\mathbb{E}[Y^O\\mid A^O=0, X^O, U])\\mid X^O=x\\right] - (\\mathbb{E}[Y^O\\mid A^O=1, X^O=x]- \\mathbb{E}[Y^O\\mid A^O=0, X^O=x]) \n$$\nwhere U denotes the unobserved confounders. To the best of our knowledge, the only types of methods that address confounding bias either (1) use latent variable models to recover unobserved confounders, often from noisy proxies or multiple/sequential treatments (see [1,2]), or (2) combine observational and randomized data with perfect compliance (see [3,4]). For (1), additional data (e.g. multiple treatments, nosy proxies) that enable the latent modeling might not be available, and even if available, the unconfoundedness condition given the additional data cannot be tested in practice. For (2), randomized data with perfect compliance is difficult to obtain in practice in the settings we discuss, either due to implementation bottlenecks, ethical considerations, financial constraints, ets, which is what our paper addresses by considering encouragement/IV designs. We will add this discussion to the literature review section. \n\n* Dataset sizes: The dataset sizes (denoted by $n_E$​ for the experimental/IV dataset and $n_O$​ for the observational dataset) do not need to be equal. In Theorems 2 and 3, we show how both sizes influence the algorithm's convergence rate. The theorems detail the dependency of the convergence rate on $n_E$​ and $n_O$, illustrating how a smaller IV dataset ($n_E$​) affects the convergence rate. Since this is not currently reflected in the experimental section, we have added additional experimental results in the rebuttal PDF. Specifically, in Figure 1, we performed parametric extrapolation simulations by varying the ratio $n_E/n_O$​ while keeping $n_O=10,000$ fixed throughout the experiments. We display the mean squared error $\\pm$ standard deviation across 100 iterations. As expected, the error in the $\\widehat{\\tau}^E$ estimation increases significantly with smaller $n_E$​ (due to low compliance in the finite sample). However, the corrected $\\widehat{\\tau}$ still shows a smaller mean error than the observational $\\widehat{\\tau}^O$, and this error steadily decreases as the $n_E/n_O$ ratio increases. We will include these additional results in the experimental section of the appendix. \n\n* High-dimensional settings: We conducted additional experiments by modifying the data-generating process (DGP) to include $d=10$ features, with both baselines and bias depending on all features:\n$$\nY = 1 + A + X + 2A\\beta^T X + 0.5X^2 + 0.75AX^2 + U + 0.5\\epsilon_Y $$\n$$ \nU \\mid X=x, A=a \\sim N\\left(\\gamma^T x\\left(a-\\frac{1}{2}\\right), 1-\\left(a-\\frac{1}{2}\\right)^2\\right)\n$$\nwhere the coefficients $\\beta, \\gamma\\in [-1, 1]^{d}$ are set at random at the beginning of the experiment. In this scenario, the bias is given by $b(x)=-\\gamma^T x$. We leave all other settings and parameters (including $n_O=n_E=5000$) unchanged and perform the parametric extrapolation described in the paper. For this high-dimensional setting, we obtain the following results for the mean squared error (MSE) and standard deviation (SD) across 100 iterations:\n\n| |$\\widehat{\\tau}^O(x)$ | $\\widehat{\\tau}^E(x)$| $\\widehat{\\tau}(x)$|\n|--|--|--|--|\n|MSE$\\pm$SD|$3.25\\pm 0.15$|$7.70\\pm 1.54$|$1.25\\pm 0.20$|\n\nThe high MSE of the IV estimator ($\\widehat{\\tau}^E(x)$) indicates the difficulty in estimating compliance in high-dimensional settings. Similarly, the observational data estimator ($\\widehat{\\tau}^O(x)$) is clearly biased. However, the combined data estimator ($\\widehat{\\tau}(x)$) performs much better in this high-dimensional setting. We will include more results (such as varying $n_E$​ and $n_O$​) for the high-dimensional setting in the experimental section appendix.\n\n\n**Re: Other Questions**\n\nWhen realizability does not hold, our estimator may be inconsistent, exhibiting a bias that persists asymptotically. The magnitude of this bias depends on the extent to which realizability is violated, i.e., how far the true representation deviates from the assumed function class. In some cases, it may still be beneficial to perform this analysis despite uncertainty about realizability, as the bias from this violation could be substantially smaller than the confounding bias.\n\nReferences:\n\n[1] Kuzmanovic, M., Hatt, T. and Feuerriegel, S., 2021, November. Deconfounding Temporal Autoencoder: estimating treatment effects over time using noisy proxies. In Machine Learning for Health (pp. 143-155). PMLR.\n\n[2] Wang, Y. and Blei, D.M., 2019. The blessings of multiple causes. Journal of the American Statistical Association, 114(528), pp.1574-1596.\n\n[3] Kallus, N., Puli, A.M. and Shalit, U., 2018. Removing hidden confounding by experimental grounding. Advances in neural information processing systems, 31.\n\n[4] Hatt, T., Berrevoets, J., Curth, A., Feuerriegel, S. and van der Schaar, M., 2022. Combining observational and randomized data for estimating heterogeneous treatment effects. arXiv preprint arXiv:2202.12891.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6jOEmcZgfr", "reviewer": "Reviewer_xHgN", "summary": "The authors propose an approach for conditional average treatment effect estimation using instrumental variables, extending existing work in IVA to settings with weak instruments (i.e., low treatment compliance in some population subgroups). In particular, the approach leverages a two-stage estimation setup: first, a biased CATE estimate is computed from the observational data, then corrected via reweighting based on compliance to obtain a final estimate. \n\nOverall, the paper provides good exposition and intuition for a novel approach, and backs it up with convincing theoretical analysis. The theoretical results are fairly insightful, but can be improved with a little more clarification (details below). The empirical results are not quite as well-motivated, and could also be improved with some clarification of the motivation and differentiation from past work. In particular, I felt that better justification of why these datasets/evaluations are the correct ones to test the proposed approach are needed for me to appreciate the paper’s contributions. Furthermore, I’m a little bit unsure if there are sufficient comparisons to baselines in inference under weak IV in the paper as-is (citations below).", "questions": "I think my largest concerns were expressed above in the Weaknesses. I’d be happy to raise my score with a thorough and precise response to especially 1-2. Addressing the following might help strengthen the paper even more, but I don’t consider them as high-priority. \n1. I’m slightly confused about the interpretation of Lemma 1 — to me, it seems like we have simply chosen to use IPW to estimate the numerator in Eq. 3. Is this understanding correct? If so, why not consider alternative estimators for the numerator (e.g., doubly-robust methods such as [1])? This is not a huge issue — just making sure I parsed the equation correctly. \n2. Did the authors consider/evaluate the sensitivity of the proposed approach to different instantiations of the base learners (e.g., something besides RF/T-learner)? This is not a dealbreaker but would be a nice result to have.\n3. Re: “The weighting scheme in Equation 4 creates a weighted distribution…” (L172) — what is this a weighted distribution of? The paper goes on to claim that the difference between the weighted distribution and the target distribution creates a transfer learning problem, but I’m having a bit of trouble understanding what these distributions are. Are these distributions defined over the covariates? \n4. Re: Assumption 3-4 (realizability of $b(x)$) — could the authors expand on why this assumption is a reasonable one to make, or point to some works that have made similar assumptions? I think I’ve seen versions of the other assumptions, so I buy those, but I’m not sure if assuming $b(x)$ is linear in the representation $\\phi$ is too limiting. Similarly, in Sec. 4.2, there’s an assumption that $\\tau^O$ and $\\tau$ — one of which is biased — have some shared representation. Is there a more concrete reason for why this is reasonable?\n\n[1] Kennedy, E. H. (2023). Towards optimal doubly robust estimation of heterogeneous causal effects. Electronic Journal of Statistics, 17(2), 3008-3049.", "limitations": "The authors address their limitations in the Appendix. As a tiny suggestion, I think it is important to note that the lack of unobserved confounding is statistically unverifiable yet necessary for many causal inference approaches. Overall, I find the discussion of the limitations to be complete and well-written. As a very, very minor nitpick, I do believe it important to acknowledge the limitations of an approach more up-front (often in the conclusion) — it does not detract from my appreciation of the method and relegating discussion of limitations to the Appendix does not feel quite right to me.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The exposition is clearly written and provides good intuition on IVA (though I am writing from the perspective of someone that has working knowledge of IVA already).\n2. The proposed approach is intuitive and well-motivated, with good theoretical properties. Weak instruments are an inevitable problem in instrumental variable methods, so proposals to reduce their downstream impact are a salient area of research.\n3. The paper itself is quite clearly written and mostly easy to understand.", "weaknesses": "I'd love to see the following points addressed to clear up any misunderstandings on my part: \n1. My biggest criticism is about the motivation and coverage of the empirical results. Coverage-wise, while I appreciate the comparison with a vanilla LATE estimator ($\\tau^E$ if I followed correctly?), additional comparisons to baselines in learning with weak instruments would help strengthen the paper, such as the ones cited by the authors as closely related [1, 2] — it’s not clear to me the empirical lift provided by the proposed approach compared to these methods. For the rebuttal phase, in lieu of new results, perhaps a precise explanation of how the authors’ proposed approach shares similarities and differs with [1, 2] would be most helpful.\n2. The analysis of the 401k dataset results is slightly imprecise — I’m not sure I buy that $\\hat{\\tau}(x)$ closely tracks with $\\tau^E$; I don’t have a prior for what’s “close enough.” Could the authors clarify more precisely (1) the motivation behind experiments on the 401k dataset (beyond extension to real-world data), (2) and why the results show proof-of-concept for the proposed approach?\n3. I have a couple of overarching concerns about the theoretical results. As a first-order comment, how do the generalization bounds of the proposed approach compare to similar bounds for IVA/what are the trade-offs compared to past bounds? As a second-order comment, if I know that $\\tau^O$ will be biased (i.e., due to unobserved confounding), why is it imperative for estimation error w.r.t. $\\tau^O$ to be low as well (maybe this is so that, given a good estimate of $\\theta$ — we get a good estimate of $b(x)$, and therefore correct for the bias)?\n\nNits (points that would improve the paper in my opinion, but are not urgent):\n1. There are a few critical derivations where the clarity could be improved somewhat. It took me quite some time to follow how Eq. 4 was derived — while it’s probably okay to reserve most details to the Appendix, some intuition about which terms are being substituted where would be helpful, and making it clear in Eq. 4 that you’re taking the squared difference (example-wise) of the pseudo-outcome of Eq. 3 (as fitted on the intention-to-treat dataset) and a reweighted version of $b(x) + \\tau^O(x)$ as fitted on the observational dataset (since it is equal to $\\tau(x)$ by definition).\n2. I notice that $\\hat{\\tau}^E$ and $\\hat{\\tau}^O$ are fitted on different subsets of the data — somewhat reminiscent of cross-fitting based estimators (e.g., [3]). Out of curiosity, is it necessary to fit the two estimators on different data splits for the theoretical guarantees to hold?\n\n[1] Abadie, A., Gu, J., & Shen, S. (2024). Instrumental variable estimation with first-stage heterogeneity. Journal of Econometrics, 240(2), 105425. \\\n[2] Coussens, S., & Spiess, J. (2021). Improving inference from simple instruments through compliance estimation. arXiv preprint arXiv:2108.03726. \\\n[3] Kennedy, E. H. (2023). Towards optimal doubly robust estimation of heterogeneous causal effects. Electronic Journal of Statistics, 17(2), 3008-3049.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Strengths** \n\nThank you for your encouraging feedback. We are pleased that you recognize the novelty of our approach for estimating conditional average treatment effects using IVs, particularly in scenarios with weak instruments. Your positive feedback on the clarity of our exposition, the intuition of our methods, and the robustness of our theoretical results is appreciated and encourages us to improve our work further.\n\n**Re: Comparison with Coussens & Spiess, J. (2021) and Abadie et al. (2024)**\n\nThese works focus on improving the efficiency of the 2SLS estimator for the local average treatment effect (LATE) by incorporating weights based on heterogeneous compliance. Our work, however, aims to estimate the CATE $\\tau(x)$, which coincides with the conditional LATE (CLATE) $\\tau^E(x)$ under our assumptions. The compliance-weighting strategy in their works cannot be directly applied here and does not improve the efficiency of our CATE ratio estimator in Eq. (3). We build on the concept of compliance weighting to improve the estimator in Eq. (3) in scenarios of low or no compliance by weighting samples for bias correction using observational data, diverging significantly from these studies.\n\n**Re: 401(k) Dataset Analysis** \n\nWe agree that the language around this analysis needs precision and will revise it in the final version. The motivation behind these experiments is to demonstrate that our method can interpolate accurately in regions of no compliance, which we artificially introduce. We chose the 401(k) dataset due to its high estimated compliance ($0.49-0.90$), allowing us to establish a reasonably accurate ground truth CATE using Eq. (3). The figures show projection onto one axis of heterogeneity (years of education), but the procedure involves learning the biased CATE using the full sample and bias-correcting across all covariates and their interactions (46 features total for $\\phi(x)$). The statement that the learned CATE extension \"tracks\" the true CATE refers to the close alignment in this projection. To quantify our claims, we repeated the experiment over 100 data splits, calculating the means and standard deviations of the treatment effects by years of education for the two examples in the paper. These tables are included in the rebuttal PDF and will be in the final version.\n\n**Re: Theoretical Results**\n\nTo our knowledge, this is the first work to estimate CATEs from IV data with weak or no compliance by leveraging an additional observational dataset. Thus, comparing our estimator's bounds with IV-only estimators is challenging/inappropriate. Our estimator reduces variance in Eq. (3) when compliance is low and provides valid estimates when compliance is zero, where Eq. (3) fails. This reduction in variance and handling of zero-compliance cases is evident in our simulations.\nYou are correct that a good estimator for $\\theta$ (and therefore $b(x)$) requires a reliable estimator for $\\tau^O(x)$. We use the learned $\\widehat{\\tau}^O(x)$ with the IV dataset to determine $b(x)$, crucial for correcting the unobserved confounding bias in $O$. We will discuss these points in the paper.\n\n**Re: Nits**\n\nEq. 4 follows from Lemma 1 by noting (as you also pointed out) $\\tau(x)$ as $\\tau^O(x)+b(x)$ from our assumptions. We will make this connection explicit. We emphasize that $E$ and $O$ are fundamentally different datasets. $E$ includes an instrument from an IV study or randomized experiment with non-compliance, while $O$ is an observational dataset that may have been collected passively. Depending on the collection method, $O$ could be considered part of an IV study with $Z=0$, but this is not necessarily the case.\n\n**Re: Questions**\n\n* Yes, we could consider other estimating equations that might reduce the variance of the estimator in Eq. (3) at the cost of estimating 4 additional nuisances (see [1] below). However, this approach will not address the variance caused by low compliance, our primary concern. It would only reduce variance from potentially poor compliance estimation.\n* Our method is agnostic to the types of estimators used for $\\tau^O(x)$. We can include additional evaluations in the experimental section using other estimators, such as the doubly robust estimator. It is important to note that we do not expect these alternative estimators to reduce confounding bias, as it is asymptotically irreducible.\n* The distribution over covariates $X$ becomes the distribution over compliers’ covariates when weighted by compliance.\n* Yes, several works assume a joint representation between causal inference learning tasks to enhance generalization across tasks such as counterfactual learning and bias correction in observational samples (e.g., see [2, 3, 4, 5] below). We discuss [2] and [3] in the related works section of our paper. Additionally, [4] and [5] are foundational works in joint representation learning for causal inference. While it is possible to generalize beyond linear representations and consider $b(x) = h \\circ \\phi(x)$, where $h$ is a hypothesis class (see [3] for an example), this is beyond the scope of our current work.\n\nReferences:\n\n[1] Frauen, D. and Feuerriegel, S., 2022. Estimating individual treatment effects under unobserved confounding using binary instruments. \n\n[2] Kallus, N., Puli, A.M. and Shalit, U., 2018. Removing hidden confounding by experimental grounding. Advances in neural information processing systems, 31.\n\n[3] Hatt, T., Berrevoets, J., Curth, A., Feuerriegel, S. and van der Schaar, M., 2022. Combining observational and randomized data for estimating heterogeneous treatment effects. \n\n[4] Shalit, U., Johansson, F.D. and Sontag, D., 2017, July. Estimating individual treatment effect: generalization bounds and algorithms. In International conference on machine learning (pp. 3076-3085). PMLR.\n\n[5] Shi, C., Blei, D. and Veitch, V., 2019. Adapting neural networks for the estimation of treatment effects. Advances in neural information processing systems, 32.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "e6gsOA2r52", "reviewer": "Reviewer_Gwhh", "summary": "The paper presents a novel method for estimating Conditional Average Treatment Effects (CATEs) by integrating weak instrumental variables (IV) and observational data. This method addresses the challenges of unobserved confounding and low compliance often encountered in causal inference studies. The proposed framework involves a two-stage process: initially, biased CATEs are estimated using observational data, and subsequently, a compliance-weighted correction is applied using IV data. This correction leverages the variability in IV strength across covariates to improve the accuracy of CATE estimation. The method's efficacy is validated through simulations and real-world applications, such as assessing the impact of 401(k) participation on financial wealth.", "questions": "See weakness", "limitations": "Yes", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper introduces a two-stage framework designed to estimate CATEs by effectively combining observational data with weak instrumental variables.\n\nThe proposed method is adept at handling unobserved confounding in observational data and low compliance in IV data, including scenarios with zero compliance for some subgroups.\n\nThe effectiveness of the method is demonstrated through extensive simulations and real-world applications, such as evaluating the effect of 401(k) participation on financial wealth.", "weaknesses": "How to understand assumption 2? Is it correct about $Y^{E}(1) - Y^{E}(0) \\perp\\!\\!\\!\\perp C \\mid X^{E}$?\n\nWhat is IV datasets?\n\nHow could there be IV in the experimental data? IV has a correlation with treatment, but according to Lemma 1, this instrumental propensity does not have any correlation with treatment.\n\n\nIn the experimental data, was the treatment not randomly assigned?\n\nExperimental data are usually small samples or difficult to collect, which makes the proposed method difficult to apply.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Strengths**\n\nThank you for your thoughtful review. We appreciate your positive feedback on our method for estimating CATEs by integrating weak instrumental variables and observational data, and your acknowledgment of its novelty and effectiveness. \n\n**Re: Questions**\n\n* Assumption 2 states that the treatment effect is independent of compliance given the observed features, meaning there are no unobserved confounders affecting both compliance status and potential outcomes. This standard assumption (see [1, 2]) is crucial for identifying the CATE, rather than the CLATE (conditional local average treatment effect), from data with instrumental variables. We will include further explanations about this in the final version.\n* An IV dataset is a dataset that includes an instrumental variable along with the treatment, features, and outcome. We refer to this as experimental data or an experimental dataset, as we assume the instrument is randomized (conditionally or marginally). Our focus is on an intent-to-treat/encouragement design (e.g., movie recommendation, exercise encouragement) that can be easily deployed.\n* From Assumption 1, the instrument is relevant in some covariate strata, meaning it affects treatment uptake. The instrument propensity, defined as $\\pi_Z(x) = P(Z^E = 1 \\mid X^E = x)$, is independent of the treatment and can be estimated directly from the data.\n* Since the experimental data refers to intent-to-treat data with an instrument, the actual observed treatment is not randomized and is influenced by each unit's compliance. In short, the instrument is randomized, but the treatment is not. As you correctly pointed out, experimental samples with fully randomized treatments are challenging to obtain due to ethical considerations, financial constraints, and the infeasibility of enforcing specific treatments. This challenge is precisely why we propose our method.\n\nBased on these suggestions, we will include more clarifying discussions in our final version.\n\n[1] Wang, L. and Tchetgen Tchetgen, E., 2018. Bounded, efficient and multiply robust estimation of average treatment effects using instrumental variables. Journal of the Royal Statistical Society Series B: Statistical Methodology, 80(3), pp.531-550.\n\n[2] Frauen, D. and Feuerriegel, S., 2022. Estimating individual treatment effects under unobserved confounding using binary instruments. arXiv preprint arXiv:2208.08544.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XtMIsvzTIH", "reviewer": "Reviewer_UxMs", "summary": "This paper introduces a robust two-stage framework leveraging observational and instrumental variable data to predict conditional average treatment effects (CATEs), addressing biases from unobserved confounders and low compliance.", "questions": "- Does the method proposed in this paper require the additive noise assumption?\n    \n- Do the observed variables X and latent U exhibit distribution shifts between Observational data and Experimental data? Are the distributions of X and U consistent across both datasets?\n    \n- Is it feasible to set all Z=1 in the experimental data? Given that observational data effectively corresponds to the part where Z=0.", "limitations": "N/A", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This paper studies leveraging observational data and encouragement data with low compliance to predict conditional average treatment effects (CATEs) accurately.\n\n- This paper proposes a two-stage framework that first learns biased CATEs from observational data and subsequently applies a compliance-weighted correction using weak IVs.\n\n- This paper demonstrates its utility on real data 401(k) participation on financial wealth.", "weaknesses": "**About the identification of CATE.**\n\n- The relevance assumption in Assumption 1 cannot ensure $\\gamma(x)>0$.\n    \n- The identification of CATE (Eq.(3) on line 121) would be violated when $P(A^E(1)|x) < P(A^E(0)|x)$ for some $x$. In such cases, even if Assumptions 1 and 2 hold, and $\\mathbb{E}[A^E \\mid Z^E=1, X^E=x] - \\mathbb{E}[A^E \\mid Z^E=0, X^E=x] < 0$,  $\\gamma(x)$ would be zero and Eq.(3) with $\\gamma(x)=0$ no longer holds.\n    \n- In lines 39-41, the paper provides examples: “certain users on digital platforms may disregard recommendations either altogether or of certain undesired content, and on mobile health platforms certain participants may ignore recommendations (e.g., taking 250 steps per hour) due to time constraints or disinterest.” However, in Eq.(3), the paper does not consider the possibility that $\\mathbb{E}[A^E \\mid Z^E=1, X^E=x] - \\mathbb{E}[A^E \\mid Z^E=0, X^E=x] = 0$. This raises concerns about the soundness of the paper regarding the assumptions and theorems.\n    \n\n- The paper seems to implicitly assume the monotonicity of instrumental variables. It is necessary for the authors to clearly state the assumptions required for the theory and provide a complete proof of identifiability (Eq.(3)).\n    \n\n**I will re-evaluate the soundness of this paper according to the responses of the authors.**\n\n**About the related work.** In the presence of unmeasured confounding, there are numerous methods that use proxy variables to estimate heterogeneous treatment effects, including VAE-based methods [Causal effect inference with deep latent-variable models] and negative control methods [A selective review of negative control methods in epidemiology].\n\n**Types:** Line 97 has an extra space. Line 103 should be revised to 'b(x), i.e.'.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Strengths**\n\nThank you for your positive feedback. We appreciate your recognition of our work on leveraging observational and encouragement data with low compliance to predict CATEs, and for acknowledging the utility of our two-stage framework.\n\n**Re: CATE Identification**\n\nYou are absolutely correct that in its current form, our work requires that monotonicity is added to Assumption 1. The issue stems from the fact that we noticed that monotonicity is not actually needed, but did not propagate the required changes to the rest of the paper. In order for the analysis to be fully sound without monotonicity, the following small and local changes need to be implemented in the current draft (and will be added to the final version):\n\n* Add defiance indicator to Assumption 2:\n\n*Assumption 2 (Unconfounded Compliance). Define the compliance and defiance indicators $C$ and $D$ as $C:=\\mathbb{I}[A^E(1)>A^E(0)]$ and $D:=\\mathbb{I}[A^E(1)<A^E(0)]$, respectively. Then, $Y^E(1)-Y^E(0)\\perp C \\mid X^E$ and $Y^E(1)-Y^E(0)\\perp D\\mid X^E$.*\n\n* Revise definition of $\\gamma(x)$ right after Eq. (3)\n$$\\gamma(x)=P(C=1 \\mid X^E=x)-P(D=1\\mid X^E=x)$$\n\n* Proof of Lemma 1 in Appendix B.1:\n      Replace $\\gamma(x)> 0$ with $\\gamma(x)\\neq 0$.\n\nWe will also add a derivation of Eq. 3 to the appendix. We believe that removing the monotonicity assumption strengthens our work without significantly altering the analysis. \n\n**Re: Related Work**\n\nThere are several other works that tackle confounding in observational studies. The ones that target point estimation (rather than bounds) usually assume the existence of auxiliary data and/or structure (e.g. proxy variables, negative controls). We will extend our literature review discussion to include these works as well. \n\n**Re: Questions**\n\n* The paper does not require the additive noise assumption, although additive noise is one way in which we can ensure the validity of Assumption 1. \n* Yes, the distributions of X and U are the same across the two datasets as the datasets are assumed to be from the same population/environment. We will make this explicit. \n* Excellent observation! Under certain assumptions about the collection of the IV and observational data, the observational data can be treated as the $Z=0$ component of the IV study. This would allow us to set $Z=1$ in the experimental study, thereby potentially increasing the data available for analysis.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all our reviewers for their thoughtful comments and constructive feedback. We are encouraged by the consensus on the novelty and effectiveness of our method, as well as its theoretical and empirical contributions. We have addressed additional questions and concerns in individual responses to each reviewer. If there are any remaining or new questions, please let us know, and we will address them promptly.

Meta-review: The paper develops a new method for instrumental variable (IV) regression in a setting with weak instruments. All reviewers agreed about the novelty and relevance of this work to the NeurIPS community. The authors managed to iron out a few initial misunderstandings and unclear points during the rebuttal. For example, we concur with the authors that the comparison with standard IV methods is not meaningful as the underlying setting is inherently different. Lastly, we strongly encourage them to include the promised action points (e.g., the clarifications) in the final version. One option is to, for example, include a table but I leave the eventual decision to the authors as to what is appropriate.
Decision: Accept (poster)