A Meta-learner for Heterogeneous Effects in Difference-in-Differences
Authors: Hui Lan
Haoge Chang
Eleanor Wiske Dillon
Vasilis Syrgkanis
Venue: ICML 2025 poster (2025)
Keywords: Difference-in-Differences
Heterogeneous Treatment Effect
Meta-Learners
Debiased Machine Learning
Conditional Functionals
Primary area: general_machine_learning->causality

Abstract: We address the problem of estimating heterogeneous treatment effects in panel data, adopting the popular Difference-in-Differences (DiD) framework under the conditional parallel trends assumption. We propose a novel doubly robust meta-learner for the Conditional Average Treatment Effect on the Treated (CATT), reducing the estimation to a convex risk minimization problem involving a set of auxiliary models. Our framework allows for the flexible estimation of the CATT, when conditioning on any subset of variables of interest using generic machine learning. Leveraging Neyman orthogonality, our proposed approach is robust to estimation errors in the auxiliary models. As a generalization to our main result, we develop a meta-learning approach for the estimation of general conditional functionals under covariate shift. We also provide an extension to the instrumented DiD setting with non-compliance. Empirical results demonstrate the superiority of our approach over existing baselines.

A Meta-learner for Heterogeneous Effects in Differeeneein-Difffeeness
Hui Lan't Haoge Chang?
Eleanor Dillon Vasilis Syrgkanis*
Abstract
estimating the causal effect of treatment intervention in observational studies often requires strong assumptions, such as conditional exogeneity, which rules out unobserved confounding. Panel data consist of repeated observations of the same units over time, which allows researchers to control for certain types of unobserved, time-invariant characteris- tics. Due to its flexibility and robustness in handling non- experimental data, the DiD approach has gained significant traction in empirical research, especially in the evaluation of policy interventions (e.g. Thome et al., 2024, etc.), labor market changes (e.g. Card & Krueger, 1994; Rossin-Slatte et al., 2013; Pierce & Schott, 2016, etc.), environmental regulations (e.g. Gao et al., 2020, etc.), and public health (e.g. Finkelstein et al., 2012; Dimick & Ryan, 2014, etc.).
We address the problem of estimating heteroge- neous treatment effects in panel data, adopting the (DiD) frame- work under the conditional parallel trends assump- tion. We propose a novel doubly robust meta- learner for the Conditional Average Treatment Effect the Treated (CATT), reducing the esti- mation to convex risk minimization problem in- volving a set of auxiliary models. Our framework allows for the flexible estimation of the CATT, when conditioning on any subset of variables of interest using generic machine learning. Leverag- ing Neyman orthogonality, our proposed approach is robust to estimation errors in the auxiliary mod- els. As generalization to our main result, we develop meta-learning approach for the estima- tion of general conditional functionals under co- variate shift. We also provide an extension to the instrumented DiD setting with non-compliance. Empirical results demonstrate the superiority of our approach over existing baselines.
popular Cenerrenne-i---ffffeeeeeee
Despite several recent methodological advances in the DiD literature (Roth et al., 2023; Chiu al., 2023), most state- of-the-art approaches are still only able to generate aver- age causal effects, or at best group average causal effects for predefined subpopulations. On the contrary, in many empirical applications, especially on large-scale datasets that stem from digital platforms, practitioners are interested in treatment effect heterogeneity for personalized decision making. The estimation of heterogeneous treatment effects has gained considerable attention in recent years due to its potential to uncover variation in how different subpopula- tions respond to an intervention. Motivated by the success of machine learning techniques in learning complex tasks, many studies have employed them in learning heteroge- neous treatment effects, see for instance Shalit et al., 2017; Shi al., 2019; Künzel al., 2019; Nie & Wager, 2021; Oprescu et al., 2019; Kennedy, 2023, etc. However, estimat- ing heterogeneous treatment effects for panel data remains relatively unexplored in literature.
1. Introduction
Difference-in Diffffrreeees estimators have become a foun- dational tool for causal inference in economies (Roth al., 2023), social sciences (Chiu et al., 2023) and healthcare (Wang et al., 2024) for evaluating causal effects of policy in- terventions or treatments when both pre- and post-treatment outcomes are observed. In contrast to cross-sectional data, having panel data enables researchers to work with different assumptions that are often considered more plausible ap- plication. Due to the non-random assignment of treatments,
In this paper, we explore the estimation of heterogencous treatment effects of a binary treatment using panel data under the canonical parallel trends condition used in DiD setups (e.g. Ashenfelter & Card, 1984; Card & Krueger, 1994, etc.). The parallel trends assumption posits that, in the absence of treatment, the treated and control units would have followed similar trends over time. Recent research has explored different approaches in addressing limitations of traditional methods (e.g. Roth et al., 2023). One line of work focuses on relaxing the unconditional parallel trends assumption by taking into account systematic differences
*Part of this work is done during an internship Microsoft Research. ** *Vasiils Syrgkanis and Hui Lan Supported by NSF Award 133793.. 'Inttitute Computational and Mathematical Engineering, Stanford University. Stanford, USA 'Department of Economies, Columbia University "Microsoft Research, New England 'Department of Management Science and Engineering, Stanford University, Stanford, USA. Correspondence to: Hui Lan <huilan stanford.edu>..
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).

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A Meta-learner for Heterogencous Effects in Differenee-in-Dfffereees
in the time trends due to other (observed) characteristies through the conditional parallel trends condition (e.g. Heck- man et al., 1997; Sant' Anna & Zhao, 2020, etc.). Another line of research tackles the challenges of estimating average treatment effects under treatment effect heterogeneity over time for multi-period settings (e.g. Sun & Abraham, 2021; Callaway & Sant' Anna, 2021, etc.). This paper synthesizes the insights from these two lines of works, and extends the framework to incorporate heterogeneous treatment effects across any dimension, in flexible manner.
helpful for decision making when trying to leverage the find- ings deploy a personalized policy on larger population for which only subset of covariates is available.
We demonstrate using synthetic and semi-synthetic exper- iments that the proposed meta-learner outperforms prior baselines. Finally, we applied our method on a real-world case study on the effects of raising minimum wage on teen employment. Our flexible doubly robust meta-learner au- tomatically identified dimensions and patterns of hetero- geneity that had not been highlighted in prior literature. In particular, our method uncovered that the county population plays a significant role on the magnitude of the treatment ef- fect of raising the minimum wage on teen employment and even though this effect can be quite large and negative for small counties, it becomes negligible and close zero on large counties. We developed an out-of-sample validation pipeline and showcased that the patterns of heterogeneity identified by our methodology are statistically significant.
We propose doubly robust estimation framework for the conditional average treatment effect on the treated (CATT), and show that the mean squared error (MSE) of the learned model robust to the estimation error of auxiliary models that need to be estimated. While there are doubly robust esti- mators proposed for unconditional ATT with panel data (e.g- Sant' Anna & Zhao, 2020; Callaway & Sant' Anna, 2021), ), there does not exist one for the heterogeneous effect. In contrast to the conditonal average treatment effect (CATE), the asymmetry of the CATT allows our proposed method to avoid estimating conditional outcome model under treat- ment, which can be hard learn given a unbalanced dataset with small number of treated units. We also draw the con- nection to the literature debiasing under covariate shift (Chernozhukov et al., 2023), and provide an extension of our main result to a unifying framework for general condi- tional functionals, encompassing many widely encountered empirical problems such as conditional prediction powered inference under co-variate shift, heterogeneous long-term effects via surrogates based on historical data and heteroge- neous treatment effects tailored to target cob-populations. Moreover, we extend our main result to the case of binary instrument (or exposure to treatment) with two-sided non- compliance, and provide a doubly robust estimator for the conditional local average treatment effect of the exposed.
2. Problem Statement
We consider the standard setup in the DiD framework. We observe balanced panel with units and T periods. We denote time by t = o,..., T be an i.i.d sample from superpopulation. For each unit i, we observe a time series of outcomes (Yu) a time se- ries of binary treatment status covariates W.. For simplicity, we restrict our discussion to T = 2 periods in this section and Section 3. We discuss extensions the multi time period setting in Section 5.
1. The units are assumed to
and time-invariant
We adopt the potential outcomes framework and assume for unit i at time t, the outcome is generated as;
Yr,-- = Vi. + (1 - (0,
where (d) denotes the potential outcome at time t under treatment d. For brevity of notation, we may drop the unit subscript i. We assume that both the treated and untreated groups are untreated at = 0, and the treated group becomes treated at t = 1, while the control group remains untreated. Our target estimand is the conditional average treatment effect on the treated (CATT), conditioning on any subset X of the covariates W:
Similar to Ogbur et al., 2015, Semenova & Chernozhukov, 2021 and Oprescu et al., 2019, consider framework that allows for the conditional parallel trends assumption to condition on a high dimensional set of observed covariates, denoted as W. This conditioning strengthens the plausi- bility of the assumptions and improves the robustness of the resulting estimators. Our focus on the estimation of the average treatment effect on the treated (ATT) while conditioning on any subset, X, of the covariates W. Es- timating the projection of heterogeneous treatment effects onto a subset of covariates is particularly advantageous for interpretation, when the goal is to uncover heterogeneity with respect to a set of key features that are of most interest. For instance, in medical applications, we might have high- dimensional imaging data that can be used to predict the outcome, while we are only interested in understanding how the treatment effect is modified by other features such as age, bone density, etc. Furthermore, this framework can be
(x(X) (1) (0)
2.1. Assumptions and Identification
Panel data allows us to disentangle unobserved confounding to some degree by leveraging both cross-sectional and time- series variations. In this section, we focus on the conditional parallel trends assumption that is commonly employed in the empirical literature to identify treatment effects for panel data. This assumption posits that the untreated outcome will
2

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A Meta-learner for Heterogencous Effects in Differenne-in--fffereeee
evolve in parallel for both the treated and untreated group, for units with the same observed characteristics W. Assumption 2.1 (Conditional Parallel Trends).
where ga(W) (W) = E[ Yo | D d, Wj. However, under the conditional parallel trends assumption it is longer the case that E[Y, - Yo D 1, w] (1) (I() W). since the parallel trends assumption crucially does not make any restriction that the trends under treatment are condition- ally parallel between treated and control units. Therefore Elgi (W) X] + (1) (1) - Yo(1) | X]. which subsequently implies that (x( * (X)). This difference will be more pronounced for datasets where there is big difference in the covariate distribution between the treated and un-treated groups.
(0) - (oO))D; = W] = (0) - Yo(0) D1 = W]
Conditioning on covariates makes the assumption more plau- sible, as it allows the treatment assignment to depend on any baseline trends that are predictable from the observed covari- ates. A practical motivation comes from the abundance of pre-treatment outcome data (for time periods before = 0). It could be reasonable to condition the full outcome his- tory to try to account for cases where the magnitude of the growth (or decline) through time might depend on the base outcome level. For instance, employees with a higher salary usually receive higher pay raises through time.
Thus, when X C W. the statistical problem that we need to solve based on the identification formula in Proposition 2.3 is inherently different than the statistical problem of estimate CATE. Hence, we need develop novel meta-learners, specifically for the CATT, that enjoy local robustmess prop- erties analogous to the robustness properties of methods that have been developed for the CATE in prior work (Nie & Wa- ger, 2021; Foster & Syrgkanis, 2023; Oprescu et al., 2019; Kennedy, 2023). Our main result will be a doubly-robust meta learner for the CATT.
Assumption 2.2 (No-anticipatioo Assumption).
E[Y6(0) Yo(1) = 1, W]= =
The simplest plug-in meta-learning approach for the CATT is to construct estimate ğo of the baseline growth model Do using generic MI. techniques (since it corresponds to the regression problem of predicting the difference Yı - Yo from covariate W, using samples only from the control population, i.c., D = 0) and then estimate a CATT model by learning second- stage regression model that predicts the label Y Fi-Y: Yo- (W) from covariates X, using samples only from the treated population, i.e., D = 1.
In practical applications. Assumption 2.1 is imposed with the full set of covariates W for plausibility, we expect more covariates to able to capture more confounding. How- ever, we might only be interested in the heterogeneity of the treatment effect in a smaller and interpretable subset of the covariates X CCC W.
Proposition 2.3. Under Assumptions 2.1 and 2.2, the CATT, 0o(X), ) can be identifie as:
0o(X)= E[Yi - Yo -X0())DD- 11,
It is well-known (Chernozhukov et al., 2018) that using ML estimators in plug-in manner may cause large esti- mation bias due to, for example, regularization and model mis-speification. A doubly-robust estimator alleviates this concern as it less sensitive to errors in the baseline growth model go. and allows for consistent estimation under weaker statistical conditions.
where g0(x) = E Yo(0)D- w).
3. DR-Learner for CATT
In the special case when W = =X, the statistical problem that results from Proposition 2.3 is identical to the estimation of the conditional average treatment effect under conditional ignorability with outcomes Y, Yo (even though, the result- ing statistical model can only be interpreted as a CATT, due to the one-sided nature of the parallel trends assumption). For discussion, see Appendix C.
To present our main result, we need to present a set of preliminary definitions and assumptions. To avoid iol-posed extrapolations between the treated and untreated groups, we need the following overlap condition:
Assumption 3.1 (Sufficient Overlap). For W. there exist c <>0 such that c - P(D = 1|W) 1-c c.
However, when X C W, this equivalence no longer holds and prior approaches for CATE estimation under conditional exogeneity is no longer applicable and can lead to biased results even in the limit of infinite samples. For instance, the simplest identification formula for the CATE and its accompanying estimation estimation strategy, the TeLeareer, would estimate the statistical model:
A key concept related to robustness is that of Neyman or- thogonality:
Definition 3.2 (Conditional Neyman Othogonality). Let m(Z00, m)) be moment for the target estimand 0(-) with nuisance functions = (7) 72....). ). Such moment is Ney- man orthogonal if the directional derivatives with respect to all nuisance functions vj is zero when evaluated at the true
To (X) Elg - (W) |X],
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A Meta-learner for Heterogencous Effects in Snfference-in-ifffeeeeee
nuisances, i.e.
and allows it to be efficiently solved using any standard opti- mization solver. Another advantage of the loss minimization approach is that the out-of-sample loss can be used as met- ric for model selection over different function classes (Lan & Syrgkanis, 2024). Moreover, as we show next in our main estimation theorem, this loss-based estimator enjoys double robustness properties, in that leads to fast rates for the CATT if the product of the estimation rates for र and ğ decays fast enough.
O,EmmiZZ:0.. m)
0
Lemma 3.3 (Doubly Robust CATT on Subspace of Covari- ates). Under Assumptions 2.1, 2.2 and 3.1, the true CATT 8 is solution to the following conditional moment equation:
D #o(H) (AY - -oo(W)) - D x (a-
E
=0
In the theorem below, we use 0 to denote generic estimator that achieves small excess risk with respect to the plug-in loss L(;; n. g). where म,, 9 are nuisance estimates, con- structed from an auxiliary dataset (sample-splitiigg that the problem of achieving small excess risk with re- spect given loss a standard statistical learning theory problem and hence many ML techniques can be invoked to provide such a guarantee. Hence, our theorem accom- modates estimators resulting from variety of CATT ML estimators, such as empirical risk minimization on the em- pirical loss, gradient boosted forests or neural networks.
where AY = Vi Yo. (W) ELAYID- O, WT, (W) = P(D 1[W). Moreover, this moment is con- ditionally Neyman orthogonal with respect to all nuisance functions (i.e. (W) and g(W)).
Note
Remark 3.4. Comparing with the DR-learner (Kennedy, 2023; Chernozhukov et al., 2017) for conditional average treatment effect (CATE), we note that, by refocusing on the CATT, our proposed moment condition no longer requires the estimation of the conditional expectation of the outcome AY for the treated group w.r.t the high dimensional W. This can be especially advantageous in practical settings where there are only small number of treated units in the panel, making the estimation of the conditional expectation of the treated units difficult. Moreover, we show that simply regressing the CATE pscudo-outcome as in the DR-learner for CATE will give a biased estimate when the treated and control groups have very different distributions. For more details, please refer to Appendix C.
Theorem 3.6 (CATT Rates). Let ât, 9 be estimates of the nuisance functions, constructed using an auxiliary dataset. Let [|0|| D=1 = VE(o(X)²ID 1] denote the L2 norm over the treated population. Let be the result of any estimation process using n samples, satisfying w.p. 1
C(O: t. 9) - inf L(0: t. S R2, DEG
Suppose Assumptions 2.1, 2.2, and 3.1. If the hypothesis space θ is convex or is well specified (i.e. On E +), then Ô satisfies w.p. 1- 6:
The next key insight of our paper is that the Neyman or- thogonal moment restriction from Lemma 3.3 can be turned into a loss minimization problem and models that satisfy the conditional moment restrictions can be equivalently viewed as minimizers of a strongly convex loss function. This in- sight is crucial in order to turn the statistical problem into a statistical learning theory problem and subsequently into meta-learning estimation strategy, which will allow for the use of generic MI. methods for the estimation of 6g-
[|x( - (x))
4 P n.5 + 3E E (Q(W) - -9o -gn(W)
=o(W)-((W)) ) I-#(W)
]}]]
where p P(D 1) and 3 2 and
0. E arg min [I0(X) - 0o (x)111 DEG
Proposition 3.5. Consider the incomplete squared loss:
C(0; To. go) go) = [[D0(X2-220 -2Fexx)
Lagged Dependent Outcome Alternate Assumption: In Appendix B, we also provide extension of our approach under the lagged dependent outcome assumption, which posits that the past outcomes capture sufficient information to disentangle future outcomes and treatment assignment. This assumption is commonly used to model time series data and is also used in estimating treatment effects (c.g. Angrist & Pischke, 2009; Antonelli et al., 2024, etc.).
where (TT,, 9o) = - (W) (AY -9)()). Under the same assumptions as in Lemma 3.3, the minimizer of L(0; no, 90) over any hypothesis space is equivalent to the solution to the best-projection problem of the CATT among the treated:
min ([((0())00p))) 0o(x))3 D 1]
DiD with Instruments: As further extension, we con- sider the setting of estimating heterogencous effects from panel data with a binary instrument Z. This has applica- tions in policy evaluation where the exposure to the pol- icy does not perfectly determine treatment receipt due to
E
Note that this is a convex loss function, which suggests computational tractability and fast statistical learning rates
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A Meta-learner for Heterogencous Effects in Differeeneein-Dfffereeee
non-compliance (Gerber & Green, 2012), and we are only willing to assume parallel trends on the exposure and not the chosen treatment. Thus, policy exposure can be interpreted as the instrument. In Appendix A, we present a meta-learner for this IV-DID setup.
We further motivate this setup with several other empirically prevalent examples from the machine learning and causal inference literature.
Example 4.2 (Conditional prediction powered inference). In settings where prediction a central task, is often desir- able to leverage predictive models to improve the efficiency and accuracy of statistical inference. Consider some high- dimensional features or covariates W. some labels Y, and a simulation model g(W) for the predictive task EE( W]. An example of the prediction powered inference frame- work of (Angelopoulos et al., 2023), asks to estimate E, [Y]. However, we might only have labeled data on a smaller or slightly different sub-population D,. In this case, we can use the simulation model and instead target the statistical estimand [e])(W)], using the labeled data only for debias- ing the simulation model. Our work extends this setting to allow for the estimation of conditional means with respect to a subset of the covariates X, the target distribution, i.e. ((X) = Ee(g(W) X] and in setting where the covariate shift density ratio is unknown (prior work considers only the case of known covariate shift). In many applications, labels might be expensive to obtain and are only available for a small subpopulation which can be a different covariate distribution from the whole population. This setting fits into the framework with m (Z:g) = g(W).
4. General Conditional Functionals Under Covariate Shift
In this section, we show that the CATT estimation problem under conditional parallel trends can be viewed as a special case of much more broad statistical estimation problem which can capture many other empirical problems beyond heterogeneous effects in DiD analysis.
In particular, we consider the following estimation problem. Consider data consisting of Z, which contains covariates W drawn from a target distribution D,. Let Ea[-] denote the expectation with respect to the distribution De. The goal is to estimate a conditional linear functional E. [m(2: 90) [X] of the regression function go (W) = E| [W] where X is a subset of W, m is a linear moment functional of go and the expectation is taken with respect to the target distribution. On the other hand, labels for the target variable Y of the regression function are available only on data where the covariates are drawn from a different source distribution, i.e. (Y, W) D.. Let Ex[-] denote the expectation with respect to a.. We assume that there is only covariate drift and no concept drift, i.e.
Example 4.3 (Heterogeneous long-term effects from short- teem experiments using historical data). Here we consider settings where we have run a short-term experiment, where a treatment D was randomized over a population of users drawn from D, and our goal is estimate the effect of D on long-term outcome Y. However, we want to esti- mate that effect without the need to wait for the long-term effect to materialize, but solely based on short-term data. A typical technique used in this setting is the surrogate approach, where we assume that the long-term outcome Y, is not directly affected by the treatment D, but is af- fected indirectly through some short-term or "surrogate" post-treatment outcomes S, i.e. Y(d) (S(d)). Under this assumption, it can be shown that the long-term effect can be identified by measuring the effect of the treatment on the predicted long term outcome, based on the surro- gates and other potentially pre-treatment covariates X. For any set of pre-treatment co-variates X, we can identify the CATE as 0(X) = E (1) - (0) | X] Elg(W) D 1, X, - Elg(W) D = 0, where W = (S, and (W) = E|y W]. The function g(W) can be learned using historical data where we have access to short-term signals S, characteristies X and long term outcomes Y. However, the historical covariate distribution D. can po- tentially be different from the distribution D,. Since the treatment is randomized, we can write the target estimand
Assumption 4.1 (No concept drift). go (W) E. [Y[]] = E,[r[]] [[]
Let E denote the indicator variable of whether the sample stems from the target distribution environment. We can then rewrite the statistical estimand as:
((X) = E[m 9 x,
For instance, in the case of the CATT problem in the DiD set- ting, the moment is m(:gg) (1) - ([(1)))WW) and the outcome regression g(W) = (0) - Yo))) D 0] is learned based the covariate distribution of the un- treated units, while the estimand is the conditional func- tional E[mi(: Z: g)l X,
X D = 1] = E[ - Yo- xX, = 1. which is a conditional expectation taken over the covari- ate distribution of the treated units, conditioning on a subset X of W. Since the label for the regression function g(W) (0) (0), it is only available for the untreated group.
Debiasing techniques for unconditional functionals under covariate shift were analyzed in the prior work of Cher- nozhukov et al., 2023. In this paper, we substantially extend their analysis to the case of conditional functionals and pro- vide a doubly robust meta-learning strategy for any such conditional linear functional problem under covariate shift.
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A Meta-learner for Heterogencous Effects in Differeneeinn-iffeeeeee
Table MSE (mean standard deviation) Over 100 Simulations. Fach row represent a different meta-learner, and columns represent the
different nuisance function classes.
Linear Regression
Lasso (CV)
Ridge (CV)
Random Forest
Best
Neural Net (OR)
0.12 + 0.02
0.12 = 0.02
0.12 # 0.02
0.38 = 0.18
0.12 + 0.02
Neural Net (DR)
0.1 0.02
0.1 # 0.03
0.1 = 0.02
0.14 # 0.04
0.1 + 0.02
XGBoost (OR)
0.09 + 0.02
0.09 = 0.02
0.09 + 0.02
0.31 = 0.16
0.09 + 0.02
XGBoost (DR)
0.04 + 0.01
0.04 = 0.01
0.04 + 0.02
0.06 = 0.03
0.04 + 0.01
as:
nuisance functions T TWW), a(W) and g(W):
D 1- D
E E- (m (in(zig)-oo -x()))+
0(X) = E, 9(W)
X
π
π
x(W)
(1 (1-E) 1 (1) a -(W))
where π P(D P(D==))=(D==1]]) W.. This setting falls in the framework with m( - x ( 1-D
where T())= = P(E = 1 and a() is the conditional Riesz representer of E, [ (m: g) X].
Example 4.4 (CATE with covariate shift). Consider the case of estimating the CATE (x)) under conditional exo- geneity. Many times we want to understand the projection of the CATE on a subset of variables W and over some target population Dr over which we will deploy our per- sonalized policy. However, ger population D. to train our CATE model, so as to in- crease accuracy. In this setting, the target statistical esti- mand can be written as Ee[9(1, W) s(0, W) X]. where g(D W) = E D, W). This lies in the framework with m(Z g) = g1, W) - g(0, W).
In the CATT application the Riesz Representer is -1. In Example 4.2, the Riesz representer is 1. In Example 4.3 the Riesz representer is g(W) + 1- W. F = 1). In Example 4.4 the Riesz representer a(D, W) is F(D- 10-11EE O[
1-q(W), where q(W) = PP(D
might want to
big-
1-D
As in Proposition 3.5, this conditional moment can be turned into a convex doubly robust loss minimization problem.
[(0:T, = =E (E0(X)2280(37) 280(x)]
We provide a debiasing framework for this problem. Before presenting the main results, we state the necessary defini- tions and assumptions.
4-B7--W (W gW)). where P Em Z; + I-T(W) ar
Note that in this loss function the variable Y always mul- tiplied by E and therefore it respects the constraint that outcomes Y are only available in the source environment. The double robustness property of the loss will make the re- sulting estimand robust to estimation errors in the nuisance functions. Analogous to Theorem 3.6, we can prove fast statistical learning rates, for the resulting estimator based on this doubly robust loss. It is easy to verify that for the CATT setting this loss coincides with the loss in Section 3.
Definition 4.5 (Conditional Riesz Representer). The Con- ditional Riesz Representer of a continuous linear functional m(Z:g) g) on X, with respect some function (WW,, is the square-integrable random variable (xX) such that:
Eg[mm (4m(Zg)) [a(((())()W)))) Wg) s.t. Elg(W)) < oo
5. Extension to Multi-Period Setting
Assumption 4.6 (Sufficient Overlap Under Covariate Shift). For all W, there exist e such that < P 1|W) <
In the multiple time period setting. we observe the outcomes for each unit for time periods = 0, 1,..., T. assume that no unit is treated at period 0. Consider first the case where all treated units are treated at period G 1 and we assume the conditional parallel trends assumption that for all > 1, (() - Yo(O) D 1, W E|Y Yo()) D = 0. W). Note that in this case, we can treat the distance A E (O...., T - 1} of a target period f from the initial treatment time period 1 as a random variable. We can also denote with Ypoer(0) (0) as the random variable corresponding to the post-treatment period outcome we are
Moreover,
c
Theorem 4.7 (Neyman Orthogonal Moments for General Conditional Funcitonals under Covariate Shift). Suppose that Assumptions 4.1 and 4.6 hold. Consider nuisance regression function go (W) E.) and target estimand ((X) = E, (m( m [Z: g) is a continuous linear functional of g. The true solution (X) satisfies the following conditional moment restriction that is Neyman orthogonal with respect to all the
= E (mm Z; go) = 1, X), where
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A Meta-learner for Heterogencous Effects in Differeneeiin--ffferreee
Table MSE (mean standard deviation) Over 100 Simulations of Imhalanced Dataset. Fach row represent different meta-leamer, and
columns represent the different nuisance function classes.
Linear Regression
Lasso (CV)
Ridge (CV)
Random Forest
Best
Neural Net (OR)
0.22 + 0.06
0.21 = 0.06
0.21 0.06
0.4 0.15
0.21 0.05
Neural Net (DR)
0.18 = 0.07
0.18 = 0.05
0.18 = 0.05
0.24 = 0,07
0.18 0.05
Neural Net (CATE OR)
0.27 a: 0.08
0.27 = 0.08
0.27 = 0.08
0,51 0.16
0.27 - 0.08
Neural Net (CATE DR)
0.22 0.07
0.22 = 0.07
0.21 = 0.07
0.33 = 0.11
0.21 0.07
XGHoost (OR)
0.21 0.06
0.21 = 0.06
0.21 =
0.11
0.06
XGBoost (DR)
0.12 + 0.03
0.12 = 0.03
0.12 0.03
0
0.12 + 0.03
XGBoost (CATE OR)
0.27 + 0.08
0.27 = 0.08
0.27 = 0.08
0.11
0.27 = 0.08
XGBoost (CATE DR)
0.15 + 0.05
0.15 0.04
0.15 = 0.05
0.14
0.15 + 0.04
Table 3. MSE (mean + standard deviation) over 100 semi-synthetic datasets generated from the Minimum Wage dataset.
Linear Regression
Lasso (CV)
Ridge (CV)
Random Forest
Best
XGBoost (OR)
1.97 = 0.04
2.02 + 0.05
1,96 + 0.04
2.08 # 0.09
2.07 0.09
XGBoost (DR)
1.91 + 0.04
1.88 + 0.04
1.91 0.04
1.8 0.09
1.8 + 0.09
XGBoost (CATE OR)
2.72 0.06
2.71 # 0.07
2.73 4 0.06
3.4 0.36
2.69 0.07
XGBoost (CATE DR)
2.73 0.06
2.7 0.06
2.007
3.47 # 0.3
2.66 + 0.07
Linear (OR)
1.96 0.04
2.01 = 0.04
1.04
2.07 0.08
2.06 0.08
Linear (DR)
1.92 = 0.04
1.89 + 0.04
1.92 0.04
1.83 + 0.07
1.83 0.07
Linear (CATE OR)
2.78 + 0.05
2.76 0.05
2.78 0.05
3.14 0.38
2.76 + 0.05
Linear (CATE DR)
2.8 # 0.05
2.75 # 0.05
2.8 + 0.05
3.13 # 0.36
2.71 + 0.05
looking at. Then equivalently write:
work of Callaway & Sant' Anna, 2021 also considers doubly robust estimation and inference weighted averages of these heterogeneous effect models across different values of G and A, which we do not discuss in this work.
E[y (1) posst (0) | X = = E[Y((1 (7))) x
Thus we can treat the distance from treatment A, as yet another covariate in our framework and make it part of X. This way, we can flexibly estimate treatment effect heterogeneity as function of the distance from the initial treatment period and let ML methods select the best model on how distance from initial treatment changes the effect.
6. Experiments and Results
6.1. Fully Synthetic Data
First, to compare the results of our proposed method with other baselines, conducted fully simulated experiments where the datasets are generated from known data generat- ing processes that satisfy the identifying assumptions de- scribed in Section 2. The data has 20 covariates, and the CATT learners look at the projection onto covariates. We report the mean MSE (mean square error) between the predicted CATT and the true CATT covariates of the treated units of a held out test set. We compare our results with the following baseline models, here Ba(W) E|Y, Yo D d]. (W, D) = (W) (o) D), and n (W) = (DD = 1 W):
Next consider the more general setting of a staggered roll- out, i.e. each treated unit (D = 1) is treated at some period G G= [1 and remains treated after that period. We denote the never-treated group as G = co. this setting, we can make the parallel trends assumption that for all g E [1, and for all 9 (0) Yo()) D = W, G - g] EO) (o) D 0, W, oo). Similarly, incorporate heterogeneity as function of the initial treat- ment period G and the distance A to the treatment period, by making these variables as part of our heterogeneity set X: for all E [1 and E [].
can
E|ypos post (1) Ppoec(() (0) D= L,X,A C, (1( (0) D 1,
Outcome regression (OR) learner: 0(X) = ElY2
Yo - (W)(W))) X]
CATE outcome regression learner: 0(X) = Elg. - (W)) = 1,
Prior work of Callaway & Sant' Anna, 2021 considers het- erogeneity with respect to G and A, albeit in a fully non- parametric manner and does not provide a method for model selection, so as to uncover in a more data-driven manner the functional form of this heterogeneity. We note that the prior
CATE DR- 0(X) Elg(W, -9(,,) + D T[W I-m(W)) (Y (YW, D)) 1,
I-D

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A Meta-learner for Heterogencous Effects in Difference-nn-ifffeeeee
We considered three different final-stage models for the CATT: neural net, XGBoost, and linear models, to fit the meta-learners. Simulation results are presented in Table 1. The results of linear models can be found in the Appendix 6. The columns represent different MI. methods that are used to learn the outcome regression. The propensity function, i.e P(D = [ always fitted using logistic regression. The "Best" column, represents using the ML method that achieved the lowest out-of-sample MSE for the outcome regression. In the Appendix, investigate the performance of our DiD CATT method and the baselines even when the parallel trends assumption is violated. The results in Appendix E suggest that the dou- bly robust estimator reduces the MSE, as compared to the baselines, even under the violation of parallel trends.
county's minimum wage exceeds the federal minimum wage. The dataset includes covariates such as county pop- ulation and average annual pay, which serve controls to account for differences in local economic conditions. For the ease of interpretation, we focus only on the raise in minium wage at year 2004 as the treatment. In our analysis, we treat years after treatment assignment time (2004) as an additional covariate to control for, as discussed in Section 5, Employment rates as well as other covariates from before 2003 are also used as the covariates.
also provide results that
XGBoost CATT leamner
f+
0.0
Doubly GATT cstmate
Moreover, also consider unbalanced datasets, where the size of the control group is much larger than that of the treated group. In particular, the propensities of each unit was lowered by a factor of 10. The results are presented in Table We see that while all models suffered performance, our proposed doubly robust model still outperforms the other meta-learners it leverages the asymmetry of the CATT definition be robust unbalanced settings. Notably, the proposed learner out- terrperforms the doubly robust CATE learner as discussed in Remark 3.4.
-0.2
-0.3
-0.12
-0:10
-0.00
-0.06
-0.04
-0.02
0.00
Predicted
6.2. Minimum Wage Case Study
Figure 2. Calibration plot for CATT of minimum wage with respect to log county population.
Predicted CATT of Raising Minimum Wage on Teen Employment
As a preliminary evaluation, we tested the performance of our methods on semi- semi-synthetic data generated from this dataset. The semi-synthetic data was generated by boot- strapping the samples in the dataset and applying a chosen funetion for treatment assignment and treatment effect to compute the post-treatment outcomes. Since this dataset is rather low dimensional, we did not experiment with neural networks. The mean MSE with respect to the true CATT function is reported in Table 3. The results show that the proposed method out-performs outcome regression learners as well as the CATE learners.
linear
0.15
kernel
xgb
0.10
CATT Log Employment
0.00
-0.05
-0.10
-0.15
-0.20
We then applied our method the original real dataset. Fig- ure 1 shows the CATT prediction over different values of log county population. The three models used (linear re- gression, XGBoost, and kernel ridge regression) all showed some extend of positive trends. This suggests that raising minimum wage might have a smaller negative effect on teen employment for counties with a larger population.
10
12
14
Log Countyy Population
Figure Predicted CATT with respect to log county population.
We applied our proposed approach the minimum wage dataset that is also studied in Callaway & Sant' Anna, 2021 and Callaway, 2023. This dataset studies the effect of mini- mum wage changes on teen employment during the period 2001-2007. The outcome variable of interest is the log of county-level teen employment, while the treatment vari- able is defined as a binary indicator representing whether
Since do not have access to ground truth treatment effects for real datasets, we need a way to validate the heterogeneity that is picked up by the model is not due to noise. One approach is through calibration. The first step the procedure is quantile binning the CATT predictions on held out validation set. Next, the CATT
Validating CATT:
8

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A Meta-learner for Heterogencous Effects in Differeneeiin--fffereeess
predictions held out test will be put into the bins according to the tresholds. The group average treatment effect the treated (GATT) for each bin is calculated as the mean of the heterogeneous model predictions, as well as calculating the unconditional ATT for each group (i.e. conditioning on the empty set, we get 0 E E|)]/E[D)) the heterogeneity is indeed significant, we expect the calibration plots line up in the 45 line with non-overlapping confidence intervals.
Review, 84(4):772-793, 1994. ISSN 00028282. URL
http:/ wwww://wwwwww jww.w.ssrrrrrgoo httt:://ww.sstorr.orrg http://rrrrs.aaae///1//.0.0/ 22180000
Chemoozukov, V., Chetverikov, D., Demirer, M., Dutlo, E., Hansen, C., and Newey, W. Doublerdebiaseddneymmal ma- chine learning of treatment effects. American Economic Review, 107(5))2112265, 2017.
Chemoozukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., and Robins, Double/debiased machine learning for treatment and structural parameters, 2018.
Figure 2 presents the calibration plot for the doubly robust CATT learner realized using XGBoost. While we see that the GATT for the lowest quantile has a larger confidence in- tervals, the highest most two quantiles have non-overlapping confidence intervals. This suggests that there significant heterogeneity between high and low populations. Together with Figure 1, the results seem suggest that the treatment effect for counties with large populations is close to zero.
Chernozhukov, V., Newey, M., Newey, W. K., Singh, R., and Srygkanis, V. Automatic debiased machine learning for covariate shifts. arXiv preprint mrXiv:2307.04577 2023.
Chiu, A., Lan, X., Liu, Z., and Xu, Y. What to do (and not do) with causal panel analysis under parallel trends: Lessons from a large reanalysis study. arXiv preprint arXiv:2309.1599 2023.
Impact Statement
This paper presents work whose goal is to advance the field of Causal Inference and Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
Daw, J. R. and Hatfield, L. A. Matching and regression to the mean in hifferenceein-diffeeene analysis. Health services research, 53(6):4138-411) 2018.
References
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A Meta-learner for Heterogencous Effects in Difference-nn-ifffeeeee
A. DiD with Instruments
As an extension, we consider a widely encountered setting estimating heterogeneous treatment effects from panel data with a binary instrument Z, with two sided concoomllaece. In this setting, the target estimand the conditional local average treatment effect among the exposed (CLATT) in the second (post-treatment time period:
(x(X)= = ([) - (1(0) Di(1) > (0),
Z =,
First, we consider the natural conditional extensions, which allows for more heterogeneity and flexibility, of the parallel trends assumptions stated in Miyaji, 2024.
Assumption A.1 (No carryover assumption). Let d (do, d1) denote the treatment path, then Yo(d, z) = Yo(do: 2) and (i(d, =) Y
(1
This assumption requires that the outcome is only affected by the current treatment, in other words, there is no carry effects from previous treatments.
Assumption A.2 (Exclusion restriction for potential outcomes), For all t, Y;(d, 2) = Yí(d)
This is the standard exclusion restriction assumption for instrumental variables that the instrument only affects the outcome through the treatment.
Assumption A.3 (Monotonicity Assumption). (1) 2 Di(0) = or s Di()))
This assumption requires that the effect of the instrument is monotone that it either increases treatment adoption decreases treatment adoption, but not both. This assumption is needed for identification under two-sided non-compliance. Assumption A.4 (No anticipation in treatment). Do(1) = Do( for all units with Z =
Similar to the standard no-anticipation assumption that the treatment assignment in the second period should not have an affect on the outcome the first period, here we assume that the exposure event that happens at the second period should have an anticipatory effect on the treatment adoption in the first period.
Assumption A.S (CPTA in Treatment).
(() - Do(0) = W] (0) Do W]
Here we no longer require the instrument to be independent with the potential outcome of the treatments, but instead require that the trend under no exposure is (mean) independent the exposure. Assumption A.6 (CPTA in Outcome).
- (62(0))) (D) (o(P))))) = W]
Assumption A.7 (Sufficient Overlap in Instrument). For all W, there exist > 0 such that e S P(2 = 1(W) < -
Moreover if the instrument does not have any effects on the treatment, the local average treatment effect will also not be identified. Hence, we need the following assumption low-dound the effects of the instrument on the treatment.
Assumption A.8 (Strong Instrument under PTA). There exist C₂ > 0 such that
EiD1 Do E E[D: Do 2 W] 1, 2 c
Proposition A.9. Under Assumptions A.2, A.3, A.4, A.5, A.6, and A.8, the CLATE, (w)W), can be identified as:
E|y - Y, - E| Y, 0, W| 0 L,
(o(W)
E E.D; Do E D. Do Z W 2 = 1,X
In the special case that W = X under the parallel trends assumption, the problem is again equivalent to the standard IV problem, and Syrgkanis et al., 2019 proposed a doubly robust algorithm for estimating the heterogeneous LATE, For more discussion, see Appendix C.

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A Meta-learner for Heterogencous Effects in Difference-nn-iffferenee
Lemma A.10 (Doubly Robust Conditional Moment Restriction for CLATE). Under Assumptions A.2, A.3, A.4, A.5, and A.6, A.7, the true CLATE solution to the following conditional moment equation:
E 2 ((AY (r()W) (AY -nn(W)))(X)) x =0
where AS = for Y or D, gs(W) = E[S] SolZ = 0, W and = 1-P[Z=I Moreover, this moment is Neyman orthogonal with respect to all nuisance functions. Proposition A.11. Consider the incomplete squared loss:
3-P(Z-1
Liv 56v 90.y B 9..
E 2 [(AD 90. (W))( 2 90,r(W)))(X))
S S = or D. go.s (W))s(W- E[Si Do WI, and Z-
1- for To rroW)= P 1(W).
Under the same assumptions as in Lemma A. 10, the minimizer of [(0; nu, 9o.Y 9o.D) over any hypothesis space θ equivalent to the solution to the best- pest-prooection problem of the CATT among the treated:
min ( (xxx)) Z 1, D(O) 8EB
Theorem A.12 (CLATE Rates), Let f.. åD. gy be estimates of the nuisance functions, constructed using an auxiliary dataset. Let ||0]] D=1.CM = ( = 1, D(1) ) > D(0) denote the L2 norm over the compliers among the treated population.
Let f be the result of any estimation process using samples, satisfying w.p. 1 6:: 6:
E [iv(O: inf Liv $ R2
dee
Define the nuisance errors be:
xx= 27
E To(W) T(W) ))W
Error(n, (T,, (T, gD) ==
(W) - (90)
x(W
IF (
To (To(W) T(W)
(g0.r (w) gr (W)) x x]
Error(T,,y
E
1 (W)
Suppose Assumptions A.2, A.3, A.4, A.5, and A.6, A.7 are satisfied. Moreover, assume that there exist finite constant B such that (0(X) S B for all X with positive measure and all f E O. If the hypothesis space θ is convex or is well specified (i.e. chk Oo (), and Error(T, gD) is sufficiently small (Error(T, (9))) 8B
), then 0 satisfies, w.p. 1 8:
4
max 2)
[[(x) - (x))) D- &
R +
(Error(T, (Tt, gv) + Error(mT,g)
hk &B Error(T, (D)
c(hk 8B2 Error(T,,9) gD))
where h P = 1), k= P(D(1) > D(O) 1), and
0, arg min [|((X) (x)) lie
B. Alternate Assumptions: Lagged Dependent Outcome
The parallel trends assumption guards against linear, time invariant, additive confounding. However, this may be unrealistic in practice. For instance, it might be sensible for the increment with respect to time to depend on the initial level of the outcome, i.e. (0) - (0) x (0) One may also be interested in the natural extension to the parallel trends assumptions that also accounts for more complicating confounding pattern. An popular alternative to model panel data through the lagged dependent variable assumption.
12

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A Meta-learner for Heterogencous Effects in Differenceein-Dfffrreeee
Definition B.1 (Outcome Support). Let d denote the support of the outcome at time for the cohort with treatment assignment d.
Assumption B.2 (Lagged Dependent Outcome with Covariates). ([Y]()))X(()) = y, = 1, 1, = (|){()))X(0)) = y. D 0, W] for all y E You, and æ E X.
Note that this assumption may be seen as a special case of Assumption 2.1, where the conditioning variable also includes the pre-treatment outcome as well as the observed covariates. This assumption might be more convincing in some practical applications. For instance, in wage studies, current income generally believed to he highly dependent on past income. However, in cases where the distribution of outcome is significantly different between the treated and untreated groups, this assumption might lead increase in bias due matching the pre- te-tteeammet outcomes, shown in (Daw & Hatfield, 2018).
Assumption B.3 (Overlap in Pre-treatment Outcome). io C Yoo
Note this is testable assumption, and one can also perform diagnostic test to ensure that the treated and control outcome distributions have sufficient overlap.
Proposition B.4. Under Assumptions B.2 and B.3, the CATT, (W)) can be identified as:
(x = (1Y Y10)) - 1, E|y D = 1, x - E(g(Yo, D) 1,
where glu. W) = E|Yi = Y. y, W]
Proof.
(|)] = 1,
E[Y v, D, - 1,W - v, W = Xdy
Yı0
[ E|Y1 = y, 0, W/p = y. W = X,yy
= [ y. 0, W.W = y. W D = 1, XX
= =
V
= E))(
Similarly, we may also replace the parallel assumptions for IV-CATT by the conditional lagged dependent variable assumptions for both the outcome and treatment.
Assumption B.5 (Lagged Treatment).
E =
Assumption B.6 (Lagged Outcome).
E| = 0, 1 1,
Under the lagged outcome framework, need slightly different notion of strong instruments in the PTA framework. Assumption B.7 (Strong Instrument under Lagged Outcome). There exist e, > 0 such that
E E[Di 1, W, Doll 1, X]] cs
Proposition B.8. Under Assumptions A.2, A.3, A.4, B.5, and B.6, the CLATE, e, (W), can be identified as:
E[Y1 E|y Z = 0, Yo. W
Z = 1,
(x)
E E E Z Do,
2 1
13

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A Meta-learner for Heterogencous Effects in Difference-in-ifffereeee
Unifying the two assumptions: First we observe that both Proposition 2.3 and B.4 shares the general form:
(x( = - -
where S is an observed outcome random variable and g(V) is a nuisance function that is the conditional expectation E|S| = 0, VJ on some covariates V, which is superset of X. Under the parallel trends assumption, $ = Yi Yo and V = w. and under the lagged outcome assumption, = Y1 and V [W, Yol- Thus, we see that the results in Section can generalized to the lagged dependent variable assumption. For IV-DID, similarly to the standard DiD case, when considering X W. we can rewrite the identification in Proposition A.9 so that the CLATE is the solution as:
E[Sy gy (Vy) Z = 1, E[Sp (DDV)
(x(
Z = X,
where S, an observed outcome random variable for y= Y, D. and gr is nuisance function that is the conditional expectation [[S,|2 0, V] on some covariates V, which a superset of X.
C. Conditioning on the full set of W
C.1. Standard DID
Here, we consider the case where X W, and are interested in estimating:
(w( (1) W.
Leveraging the no-anticipation and (conditional) parallel trends assumption, we can identify this as:
(w) E(Y D1 = w] E[Y Yo 0, W]
Note that this shares the same form identification with the conditional average treatment effect (CATE) under conditional ignorability, but with the differences as the outcome:
CATE = - - -E
In other words, the leta-learners of CATT can be constructed the same way that was constructed for CATE using the difference in outcome. For instance, the doubly-robust pscudo-outcome can be constructed as:
( ((W)
1 1- - 1-D (
Y DR = g(1, -) g(0, W)
(M- WW)
where g(D, W) is estimator for the conditional expectation E|Y, Yo/W, P(D = -. The nuisance functions g(D, W) and r(W) may be estimated using any ML methods.
(SWWD) DJ. and π (WW is estimator of the propensity
As in (Kennedy, 2023), the doubly robust learner (DR-learner) can be constructed as ((W) = E.Y Note that due to the asymmetry of the parallel trends assumption, this estimator gives the conditional average treatment effect of the treated. we further assume that the treatment effects are also mean independent of the treatment conditional on W (i.e. (1) y (0) 1, W] E[Yi(L) Yi (O) = o, WI), then the CATE estimator on the difference in the outcomes identifies the conditional average treatment effects.
However, this CATE pseudo-outcome will give the biased estimate of the CATT when projecting subset of covariates, i.e. @CATE X = D X where X C. W. Here we see that GCATE (X) = E|EeY [i] = E - (O) = 1. X * (1) Y, (0) = 1, W]|D = 1, = E[Y (1) - (0) = x. This difference will be more pronounced for datasets where there a big difference in the covariate distribution between the treated and un -reatee groups
14

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A Mcta-learner for Heterogencous Effects in Differeneeiin--fffereees
C.2. IV-DID
In this section, we show that when conditioning on the full set of variables W, the CLATE can be estimated by the DR-IV leamner in (Syrgkanis et al., 2019). By the standard LATE identification argument write:
(1) (0) (1(1) (0), 2 = 1, W = ((( - <D1())) Di(0)) 2 - 1,
P(Di Di(0) Z=1,W)
- - (D(0)))))))) Di(0)) == = 1.
[[D(())-D(()]
([D - (0) 2 1,
Under the parallel trends assumption in the outcome, the numerator is identified as:
- -V((
z 1, w] = E Yo 1, W] E EI)((D(0)) Yo == =2=11 W]
E Yo Z 1, w] E[ Yo 2 0, W]
Moreover, under the parallel trends assumption in the treatment, the denominator is identified as:
([) - (i( z 1, E(D1 Do 2 1, W E(D((0) Do 2 1]
E|D: Do Z 1, W] - E[D: Do Z W.
For brevity of notation, let Y denote Y1 - Yo and let D denote D1 DoThus, under the PTA assumptions the effect is identified as:
E|Y Z = 1, Y Z = W
(W) = E[D Z 1. E|D Z 0, W]
Moreover, note that we can also write these quantities as conditional covariances.
In particular, let a(W)= Y Z 1, W E 0, W and (W) E Z 0, W). Without loss of generality we can write:
E|Y| Z, Z(E 2 = 1, 1, W) E|Y Z = 0, W]) + E Z W Za(W) + xWW
Thus we can write:
== Za(W) (WW +
Z, W] = 0 E[c
Then we have:
Cov(Y, 2 W) = E| 2 W| = E|Y 2 W]
where Ý = Y E|y | W] and Ž Z - To( Z - EiZ W].
Moreover, note that:
E[Y 2 W = E(a(W) + E W + E W]
=a(W)Var(2 (2 W) + EW E|Ee Z, W]ž W]
=a(W)Var(2 7 W)
Thus have:
Cov(Y, Z W) Ely 2 w| (E 2 = W| E|Y 2 0, W.) Var(Z W)
15

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A Meta-learner for Heterogencous Effects in Difference-in-Dfffeeeeee
Similarly, we derive:
Cov(D, W) = - W| (B 11 Z =, WJ) Var(Z W)
Thus we have deduced that we can equivalently identify the conditional LATE among the exposed as:
E Ž
W] Cov(Y, Z W) E 2 = 1, W| y Z = 0, W Var Z W)
(o(W) =
E
W Cov(D, Z W)
( 2 = 1, D D
Z 0, W Var W)
E|Y Z 1| E[Y Z = O.
D Z W| E|D Z 0, W]
Let be an estimate of:
o (w(W) = E 2 W| (E(Y 2 1, EyY Z 0, WJ) Var(Z W)
and B an estimate of:
(W) = E(D W| (E(D) 1, E = 0, WI) Var(Z W)
and let 0 = =a/B. Then we can construct the random variable
Y()) ((W) + (Y -(W)D)
B(W)
and the moment equation for the conditional LATT is:
Φ = -
Similar to the standard DiD case, projecting onto lower dimensional subset of covariates will give a biased estimated of the CLATE.
D. Proofs
D.1. Identification
Proof of Proposition 2.3.
(1) - 11
([ - 1,
1,
E[Y1 -Y|D=1,,X (0) --00)
(By Assumption 2.2)
E|Y; Y/|D= =1,] ({E((Y(() -Y(0) = 1,, = 1,
E[YY T, ((E(Y)(0) - YY(D) 0, D 1,
(By Assumption 2.
= E|Y Yo - (0) Yo(0) D = 0. W. X
Proof of Proposition A.9. Here we want to show that the CLATE, (x(X), can be identified as:
E Yn Y Y, 7 0, W Z X
(x(X)= E - Dy - E D Do
Z = 0, W
= 1,X
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A Meta-learner for Heterogencous Effects in Dineerence-i--Dffeeeecce
We first analyze the denominator:
E[D; - D, E Do Z X] =
-1]D:(1) Do(1) E(Di(0) Do(0) Z 0, W] Z =
=I[Di(1) DI(0) + D. Do(1) (0)))) ((())) 0, W/ / = 1,1,X] - (
EI) - +
Z = 0, -
(By Assumption A.4)
= EI) - Di(0) +E/((0))) 1,,
- - (0) Do(0) 0 W]
Z = X]
= =E[D((1) - (1 [,
(By Assumption A.5)
(1) (i(0) 1,
(By Assumption A.3)
Now analyze the numerator:
E|y; Yo E[Vi Yo Z 0, |
E[))(D(L)) -DX 1,,
E[L)()(LL) - 00000000000000000000000000000000
-
Z 00 2 = X] (By Assumption A.4)
= E()((()) Yo((O))) Z = 0, W] Z 1,
(D)((D(L)) - X((D(0))) -
E[(((D(1)) - 1,
(By Assumption A.6)
12
(By Assumption A.3)
(1) - Xi( 2 I, D(0) =
Thus, combining them, we get:
y Yo E[y Y, 2 0, W] / 1,-]
= 1, (1) (0) == 1(1) > D[
E Di Do - E Dn 0, W
D.2. Orthogonal Moments
Proof of Lemma 3.3. First, we show that the true CATT function lx(X) (L Y100 1, is the solution the moment:
(0- To))) E [mm]Z:00:00 9o: x] [=) (AY - -oo()) - D06(X) x =0
where 7 (W, AY = Yo. (W) D 0, W]. no ( PPD 1 First, since this moment is conditioned on X, we can multiply by any functions of X. Thus, can divide by the propensity with X, i.c. (x( PDD 1|X), which is bounded away from zero:
D (W) (AY so(W)) DOo(X) x =0 ( (
E
$ D
D - To)
- (1))
(AY go())) xx( 8o(x) x =0
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A Mcta-learner for Heterogencous Effects in Differeneeinn-Dfffereees
The latter term is E
D xo[
0o (X)
x = E[0o(X)) = x| = (x)). Now we consider the first term:
D - TO(W)
(AY -00 xx]
E
(o(
-
D
- D (WT)
E
(AY - (w) X
()(
(1-#o(W) =o( (70
D
(1 - D)TO
E
(AY -go(W)) x
(AY 9o((W)
X
(X)
(1 To To( 70
(1 D)
E
[AY Go(W) D 1, E
(AY (0(W))
X
(1 - () (X)
(W)
(X))
(AY go((W)
D 0-
0, w
X
TO (X)
(W)
(X() E
JAY
w| x
(X)
(X)
Thus, the moment condition is satisfied for the true CATT (X)) Now, we show that the moment is Neyman orthogonal with respect to all nuisance functions. It suffices to show that the directional derivative with respect to all the nuisance functionn are 0 when evaluated at the true nuisance and target functions. Recall that the directional derivative of functional m Z: f) with respect the function f(W) in the direction of (WW defined as: ByE[m f)[[A]] _ E[m(z; f + A
First, we look the directional derivative with respect the outcome regression gWW):
D
(W)
0zE|m(Z: 9,
(.9g,)))38 x||x1|1 go.
E
WW
x
1 - To(
D To (W)
E
E
W
Ag(W)
X
1 - To(WW
To(W) - (W)
no
Ag(w)) x
E
- ToW
Now, we look at the the directional derivative with respect to the outcome regression x(W):
- (1 - no AT) + +D To (m)) (AY - so(W)) x
(2:0,(200, = E
(W))2
(D - 1)
E
(AY (AY- (AY-90(W))) go(W)) x
(1 [W])2
AT(W)(D - 1)
- (oX--)(W)) w X
E
E
To())))
AT(W) E E (o
E
D 0, W]
x =
no(W))
Thus, we have shown that this moment Neyman orthogonal with respect all nuisances.
Proof of Theorem 4.7. First, we show that the true estimand (x) Esource (m: satisfies the following condi- tional moment restriction
E m (2; So. no, ao) |x] =E Emm( go)) (x(x)) + (W) (o)) - ([WW)())g((W)) x =0
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A Meta-learner for Heterogencous Effects in Differenneein-Dfffeeeees
where m P(E 1 and a(W) the Riesz representer of E,g E.(m(Z:g)) Similar the earlier the proof of
Lemma 3.3, divide both sides of the moment equation by "xo(X) = P(F 1|X) since is bounded away from O. So it is equivalent show:
E (X
(1 - -90(w)) x 0
E
(m(Z: -oo))))))))o)))) (1 To(W)) 00
First, let's look at the first term:
E
E'
(m( - (x()))
x = =E [m - -xx))) 1, =
(x())
Thus, it remains to show that the second term also has conditional expectation of O.
()))(W)
E
No mo(W))ool 000 (Y so((w) x
(1 E) ITo (W)
E
(1 (o0o0)) (O( (o((W)
W
X
To(W)
E
(ooWW(( - 9o((W)) E w
x
xx(
E
(W)
(W)E [(Y 9o(W)) 0, 0,
x
(X)
Now, we proceed to show that the moment m (2; o.) is Neyman orthogonal. First, we look at the directional derivative with respect to the nuisance g(W).
3 DR (Z: (Z ( No.90.M..00
(1 (W)
0,E[Em((:)))
IE
To(W) ar (W) (W))g(W))
x
We first look at the first term:
( 1,
(By the Definition of on (W))
[[ = X|
Putting this back, we get:
0
E m D (2: T) ()] |
N0.000
(1 (W)
E
Eos Eon(W))g(W))
and (W) (W) x
1 (W)
(1 (W))
E
00
(W)))((WW)
E
X
(W)
(1 E) (W)
E
E
W
X
1 (W)
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A Meta-learner for Heterogencous Effects in Difference-nn-ifffereee
Next, we look the derivative with respect m(W):
O,Elm PR 2, (, ()) 00.90000000000000
E) (WW)) (W) +11 (1- B)T(W)Ax((M) on (( x
E
(1-To(()) [o))2
(1 (W)
(1 mo())) ao (W) - so(())) -800 x
(W)
E
E
(1 - TT()))² ao(W)(X - -9o(W))
W
X
AT(W)
1- ToW) no (V 90)())) F: = 0, W
AT(W) (Y ( 0, w| x
E
(W)
Lastly, we show that the directional derivative with respect to (W)) is equal to O.
0 EL (R; g, n,, a: x)|X][ 80.00 50.00
(1 (W) (a( - (o(W)) x
To(W)
(1 ()
E
To(W) Aa(W)() - (o))) W
[E (W) - F. =0 0, W] X
(=( -oo()] 0, W] X 0
Proof of Lemma 10 First we show that the true CLATE, (x(X) (1) Yi()) 2 D(1) D(O), X] is the solution to the following moment equation:
EE [ DR 9o.
(o = =E ((AY - (o)) - (AD go.D (0))
x =
where AS = S So for S = Y or D. (W) = ElS - So = 0, W). and & = Z-zo(W)
1-5o(W] with To To = PPZ = 1/W).
We apply same trick in the other orthogonality proofs divide by 7x( P(Z 1[X). We first consider the first term:
E
(AV (W)) X
(x(
Z - To(W) (AV go,r (W)) X
E
11 To(W))oo
Z
(1- - (W)
E
(DY - (W)) X
(x()
(WW
xx(X)) (1x())
Z
no (W)
[( ([]))(
1-
(AY (orr(W)) [/] x
E
(AY go.r
x
IE
E To (X)
E |Y (W) 7= 1,
EE
E ([AY to.y -oo) 2 0, 0, x
(X)
E EA
- (w) 1,
20

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A Meta-learner for Heterogencous Effects in Difference-in-Dfffeeecc
Similarly, for the second term:
Z (X)
E
(AD - (o,)(
x = (x J
(AD - (o.) X Yia (X
( IDD ( = 1,
By the definition of ( X), this shows that it is a solution to the doubly robust moment equation. Now, we proceed to show that the moment m DR [Z; 0, gy, 9D. T) is Neyman orthogonal. First, we look at the directional derivative with respect to the nuisance yr (W).
Ogr F(m PR (Z; .r..9r999.0 7)) [grr
E
A (W)
x]
00-90.y -.0
Z To(W)
E
Agr (W) x
1 (W)
2 (W)
w/ Aor(W)) x x}= =0
E
E
(o(W)
Similarly,
0g0 PR Emm (2:0, (y:0,gr.9D. 9D. {|Dg]
E
x]
00.00,Y Di
7 (W)
(X(X) E
W Agn x =0
(W)
Lastly, we check the directional derivative with respect to r(W):
0,E m DR br [, 00.90.r.90.0 D.50
(1 To (W) + (Z- (T)))
{(AY - (orr(W) (AD ()) x
(1 (W)2
(Z- ())
11 (W))2 (AY - (w,r (AD - -0.) x
(Z- ()
(1 To (W))² ((AY - (W)) -ADD-ADD----
- (0) W
(AT(W)
ToW ((AY - (w)r - -(DD go. (0.
- 0, w x
(AT(W)
(W) {E[AY 90. (W) 2 = = W] E - 90.0 =V) -, x =0
D.3. Losses
Proof of Proposition 3.5. Note that the true CATT 0g satisfics the conditional moment restrictions in Lemma 3.3, which imply that:
E[D(,(X) X] E x]
Hence, the loss C(0; no, go) at any function 0 can be simplified as:
[(0; no. go) = E [D(x)22 2
E [De(x)] - 2E|2 x10(x))
=E(DO(X)² - 2E( x]0(X)]
[DDO(X -X)0(())(()
21

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A Meta-learner for Heterogencous Effects in Differenne-in--ifferreees
Note that when the loss evaluated Og then takes the value (l-D0((X))) Moreover, note that minimizing L(0; no: 9o) equivalent to minimizing the difference L(O; no, 90) [(0p; No. 9o), which in turn simplifies to:
E DD(X) - 2D0o(X)0(X) + DO, = E [D((X) 0o(X))
Hence, minimizing C(;; To. 90) over any space θ is equivalent minimizing over θ the loss function:
E (0(X) - (u(x))) D-
Proof of Proposition A.II. Note that the true CATT Og satisfies the conditional moment restrictions in Lemma A.10, which imply that:
E ELD (D(W))) x] EIŽ(AY -9 (W)) x]
Let t/u denote the set of nuisance functions. The loss LIv (6; no) at any function 0 can be simplified as:
L0v no)
[(AD gD (W
=E Í(AD - gD ())) - 2E (x) (
= 2(AD - (W)) (0(x)² -20(())(()
Note that when the loss is evaluated at 8u. then it takes the value E[-Ž(AD- - 9D (W)))))o(X))) Moreover, note that minimizing LIv (0; no) is equivalent to minimizing the difference LI (0; mo) - Liv(Os no). which in turn simplifies to:
E
7(D gD(W)) (0(x)²
+
Í(AD (D())) (x(2)))
E (z (1-Z)mo (W)
- To(W) (AD -
E [Z(DD ( (o()))
- - ( (D 12 0, W]]
= E [Z(AD (0(W))(((X) - (o()))
= [AD (V)) (x(x)) 1] P
E [E][AD (W)) 1, (0(X) - (o())) = P P| 1)
{P( (2(0) 1, x)((()) (x P(Z 1)
(See the proof of Proposition A.9)
=E[(D(()) > ()()))(()) - (o(x)) = 11] P]
=E [((())) - -60x)) DI) - > D(O) P(Z = L)D(DD(I > D(O) = 1)
Hence, minimizing Liv(8: qu) over any space θ is equivalent minimizing over θ the loss function:
[(((X) (x()) = 1, D(1) > D(O)]
D.4. Rates
Before proving Theorem 3.6, we first present some auxiliary Lemmas.
22

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A Meta-learner for Heterogencous Effects in
Lemma D.I. 1 (T, g) denote the of nuisance functions, and let no be the true nuisance functions. Consider the loss defined in Proposition 3.5. Then, we have that for all 01, 02 yn and 172.
(Pmn)-((m))(( [(())mm) L((): [(02; m1) + 1202:122) s 2// E E (m)) - / 1|0 - 021l
Proof of Lemma D.1.
[(01:m)) L(62: m)) [(02: + (()2
E [x) (x)) + +28(mm)))))))) (m))(2(()))))))))))))))))))))))) -1 (009(X)- 03(X) 28 -
EE [2 (f(m)) (12)) (Xi(X) (x())
E E [(P(m) P(1z)) (0, - 0x(x)) x]]
V E E (f((m)) 8 x]]] 10 (x)
We then show that the bias in the pseudo-outcome Y is equal to the product of the biases the nuisance functions.
Lemma D.2. Let = (T, g) denote the set of nuisance functions, and let be the true nuisance functions. Consider the pseudo-oulcome defined in Proposition 3.5. Then we have:
E|y (mo) ( (n)) E (o(W)- ((WW)-9(WW)))) To(W) +(W)
X
#(W
Proof of Lemma D.2.
E Y (mo) - (
D (W)
D - #(W)
E
To (W) -AY -goo))
(AY ((W)) x
#(W)
D (1 (--)) WW) (W)
(D- (1 - ())((W) 11-Dxt()) (AY (D--S(W)) [ x
E
E
(AY go(W))
1 (W)
1- #(W)
(1- - (W)
(AY -Wo) + (1 (1-D)a(W)
EE
E
D(@(W)- (w))
(AY ((W)))
W x
(W)
#(W)
[11 (1- (1=))+(W)
E
E Ia (W)(O(W)) (o))) W - E ITAY (o 0, w1 + E
(AY ((M))
[ X
1 1- #(W)
1 - D (1 (oW)))+(W)
E
To (WW)(W(W) (o)W) E
(AY -AY- ((W)) W
X
(W) 1- #(W)
(1 (W)) (AY - D 0, W #(W)
E
<OWW(W(W) - so())) E
X
(1 No (WW)))((W (90(W)- -
E
(o(W) (w)) E
D O, W
X
#(W)
(o(W)- ())) To( +(W)
E
X
1 ((W)
23

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A Meta-learner for Heterogencous Effects in iifferreeeein-Dfffereeee
The rates in Theorem 3.6 is an application of Theorem in Foster & Syrgkanis, 2023. We reproduce the theorem in our notation for completeness. Let d(t), Mo) denote distance metric for the function space of the nuisance functions F, and Il())le denote norm for O We denote Star ( 0) to be the star hull, i...tta((,, 0) [tt (1 t)0" v0' E e t E [0, 1|}. Moreover, let 6' be an arbitrary element in θ.
Assumption D.3 (First Order Optimality). l' satisfies the first-order optimality condition for C(0:)))
(p 0' 0 A 0 Star(O, 0*)
Assumption D.4 (Higher Order Smoothness). There exist constant 31 such that:
((0, 0', 07) < 3 0 - 0'118
for all 0 E θ and all 0 E Star(O,0').
Assumption D.5 (Strong Convexity). The population loss is strongly convex with respect to 0, i.e. there exist constants X, > and 2 0, such that for all 0 E O, 0' E Star(O, 0'), and F:
(0, 0¹, - 01 2 A|1o -011 - rd(n) = mo) 14+
Assumption D.6. There exist [0, 1) and constant B2 such that for all 0, 0' E Star(O, 0*) and all m. 7/2 in F:
[[((0m) L(0'; 12) ((0) 32 0' 0'1 d(m,
Theorem D.7 (Theorem I from (Foster Syrgkanis, 2023)). Suppose Assumptions D.3, D.4, D.5, and D.6 are satisfied for some 0' O Then for any 0 θ the following holds:
||0 113 S A
4 (C(0, -(0
(() +
82
It
+
d(nu, TTr
A
We are finally ready prove Theorem 3.6.
Proof of Theorem 3.6. Since results follow from Theorem D.7, we first show that the minimizer of the loss in the function class O, i.e. 02, satisfies Assumptions D.3, D.4, D.5, and D.6 for the proposed loss L(0; n) with [[()]le I1OI)l==
Assumption D.3 is satisfidd when is convex or when 0g E O. Assumptions D.4 and D.5 requires us to bound:
c(, 0.,0 0_] = E[D(e((X) 8, (x)) = (e - 8, (x)))
Thus Assumptions D.4 and D.5 are satisfied with 31 À = and = 0. To show Assumption D.6, we need to convert the [l())]2 D.2 into ||()) |==
[(((X) (x(x))
((((X) - -x(x))
(0(X) (x(X)) 11X) P(D 1|x) D(yxx
= 1xx)x(X)dx S / (0(()) - -
1 -10( (x)
Thus, Lemmas D.I and D.2 imply Assumption D.6 with r = 0. B2 = 3
and
a.
1/2
((W) ( -
(oo (TO(W)-
12) = E E (0(W) -
-9o (((W)-90W)))
(W))
x
24

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A Meta-learner for Heterogencous Effects in infference-in-ifffreeee
Thus invoking Theorem D.7, get that:
4 |0 - 0' v R(n,5) p
+ p²
E E (Q(W) so())
- - (MO(W))) #(W)
x
27
Analogously, we can prove the rates in the case with instrument. Consider Crv 17) from Proposition A.11, where we let 17 denote the of nuisances (W), gD (W) and gr (W). We first present an auxiliary lemma bound (Orv LIv ( m) (02v Liv (02; 7-))l-
Lemma D.8. L (T,, gy gD) denote the set of nuisance functions, and let тjo (mo, 9o,Y 90, ) be the true muisance functions. Consider the loss defined in Proposition A.11. Assume there exist finite constant B such that 10(X) S B for all X with positive measure, and all A E θ. Then, we have that for all 02. O2
1Liv (01:n) - Liv(02)))) - (CIv no) ) LIv (02; n7))
27
To(W)-x(1W) - x (0o,D(W) (w()))
< A
E
1- 1-=(WW) T(W)
x
[10(x) - ((X)
x 27
- (To(W)) - -- x(W)
+ 2E
E
(W) - gxr)
[0((X) O(X)
(-x(W)
Proof of Lemma D.8.
(iv(0i:)) - (12 [2(m) ((AD- (o(W)) +3(X)) 2(AY - (W -
Liv (61;7)) Liv = ((m) ((AD- - -
- - 2(AY 90.Y (x)) -x(()))
Let's first consider the E - 2200000-00000010000000000000)0)) term:
((D)) - (0) - -
(1- (1-2)molll (W) (2-1-))m(1)))
E (z 1 1-7o(11) (W)
(AD- (0.0 -
o3
EIZAD(O((X)) - (%(())) ( -
-E[EI(AD - (, ( -((())
(X -((1))))1E23000011111111 0((X))]
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A Meta-learner for Heterogencous Effects in Dnffereneeiin-Dffereeees
Now, for the E (D)) - -M))) term:
E ((n)(AD - -
(z- (1 (1-2) [-]))((W) (AD - ( -(((x))
=E
1 -()
= EIAD - 0 x)) E[Zgn
- -
1 1-2
(1- (LTT(WW))TO(H) -xxx)))9(x))
-E
E
(W) (AD 9D(())
1- (W)
= =E(ADZ((((X) (x)) E (W) - -x3(1))
E(AD 9D 9D (W))2 0, W] (1 (-mo(WW))TO(H (S(x))00((X))
-E
1 (W)
-((())
-
(1 (1-70 (W) (90.D(W) (00 -([()
E
(W)
Putting them together, we get:
( - gDD (2(X))
9o, (V)))))))))))(())-)-))))))))))))
E[ (W)) - (D 03(x))] E [11 (1-7o
(TO
To
1 - xWW (90 ()))))0(())
E [3 (1 To (o) (o, (X)
- क (W)
(o) +TW)
(W) (D(W))(0((X) (B(X)))
- x(W)
Similarly,
E 2 - (O 02(x)) 2(m)) -xx) -
E To( 1-=(W)
(W() (W) - -
-2201))
Thus, we have shown that:
(11(i))) - (Iv( (LIv (12:
TO (W) z(W) (Go.D(W) (0(W))((((X) (x(x) - To(W)
( T(W)
(0o,r(W) or (x)))(()) ox(x))
E
x(W)
1 x(W)
-
(W) ggD xx]
E [(( 03(x)))1 1/2
E
E
1 xWW
27 1
- -x()
2E
- (
X
E [(0,(x)-
1- (W)
27 $
xW) (o(W)-m(W)
To - T
< 4
E
(go.D(W)-so) - (D(W))
x
110(X) 0(X)1l
- (WW
-
1
To (W) x(W) -
x]
21E
(0) - (
|I((X) ((X)I
1 1- TWW)
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A Meta-learner for Heterogencous Effects in iifiereeee-i---iffreneee
We prove Theorem A.12.
Proof of Theorem 12. Since results follows from Theorem D.7, we first show that the minimizer of the loss in the function class θ i.e 8, satisfies Assumptions D.3, D.4, D.5, and D.6 for the proposed loss (rv(,; with II(-) = Il(-)llz=I.cmm First, Assumption D.3 is satisfied when θ is convex or when 8g E e. Now, we look at the second order directional derivative with respect to 0. Following the same steps as the proof Porposition A. we get:
(0. 10) 0., 0.]
E(Z(mo)(AD - (o). - (x)))
=E =E((((X) - D(1) > D(0)) P(Z = L)P( > D(O) =
-klll(()) - O. (.))]llz=,ccm
Thus Assumption D.4 is statisfied with 31 = hk
However, for Assumption D.5, we need to bound the second directional derivative for any 1). Therefore, we consider the distance between (v,
(0, 1) 0.,0 0_] - fg (0. n1) 0.,0 0 0.]:
8. 0., 0.] 03 Cv n0 00.00000000--000
-2E(2(n)
- -(2))) g0, -o,,(WW))((() - (x))²
TW) (W)
2E
(W) 9D ())))) (x))
(By the same reasoning in the proof of Lemma D.8)
- x(W)
27 1/2
No (W( - )
s 2E
(oD(W) -wob(WW)
1 1- -WW)
[I((()) - 0, (x)))I
(By Cauchy-Schware
<8B²
(Ôn(W) (w))
(o())) (
1- - -
X
27
1/2
II((()) (x()))1
#(W)
1/2
8B2
To(W) - +(W)
E
E
(gn(W) (W))
X
[I((()) (x))
(W)
Thus, for sufficiently small nuisance error, Assumption D.5 is satisfied with 0. and
8B² Error(T, (9)
A hk
Error(T, (p) + Error(T,, )v). Lemma D.8 implies Assumption D.6 with r 0, 33 = max (1B" 11B2,22 2). and d(n, no) Thus invoking Theorem D.7, we get that:
4
max(4B2.2 2)
[[0(X)- - 0x())) <
R², c(hhk-
(Error (T, gy) + Error (T, 9D))
hk 85
Error(T, n, gD)
8B2 Error( n, gD))
E. Additional Experiment Details and Results
E. Experiment Setup
Here we describe the data generating processes (DGP) for the fully synthetic experimens. We consider soome observed covariates W with dimension dw, and some unobserved confounding U, of dimension dy. Let pww Py be the mean of W
27

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A Meta-learner for Heterogencous Effects in Difference-nn-ifffeeenee
and U, where each entry is sampled from uniform ditsribution ranging from 0 to 1. Let Ia denote the identity matrix with dimension d.
W N (pw, Idx)
W masked Half of the dimensions of W are randomly set to 0
U Npe. Idg)
(p is clipped s.t. E (0.9,
1 exp(p 2app(-20100 (W pw) * (a - mo) )
D Binomial(p))
0o = Wi * 1(2 > 0)
2
For experiments with DGP that satisfies the conditional parallel trends assumptions:
Yo 5U m=)) W/ + 2/ + 6o 0.5)
Y1 5(a p))) 2w 1( 0) + By masked W3 + D t e ((000)))
The results in Table 1 and 4 are generated using this process with dw = 20 and de 5. We also ran experiments with higher dimensional covariates (dw 100), and the results are presented in Table 6. The results in Table 2 is generated using the same setup, but with 0.1 as the treatment probabilities. These results all showcase that our proposed doubly robust CATT learner out performs the baseline methods. addition to this DGP, we also experimented with a DGP that does not satisfy the conditional parallel trends assumptions.
Y ~ Uniform([-1,1]
-wo)) Yo (aT(U - )u))² X6 + X2 + co. cu N(0.05))
m = Yol
Yi (U )²))²² X + my X
0)X2 D Oo t1, E1 ~ (10,0.))
Experiment results for this DGP is presented in Table 7. We that in this case, the conditional parallel trends is violated so the learner that assumes conditional parallel trends has a higher MSE than the those that assume lagged dependent outcome (as this DGP has a lagged outcome component). Moreover, we see that even when the assumptions are violated, the proposed learner still more robust than the baseline outcome regression learner.
For the semi-sythetic experiments on the minimum wage dataset, each dataset constructed by first sampling 10000 units with replacement from the original dataset. We keep the covariate and pre-treatment outcome information, and generate the treatment assignment and the outcome in the post-treatment time period. The probability of receiving treatment is generated from the logitistic transformation of a linear transformation of linear function of 2 "region" variables that are binary, and the log average payment information for year 2001 (i.e. 2 (region 3) 2 (region 4) + ((log average pay) 10)). The time trends, i.e. Y.e.Y, posse(( (0) - Ypre(()) is generated by 0.1 * (log average pay) + 0.1 (region 3) + 0.1 * (years after treatment) + (region (years after treatment)* + (log average pay) (log average population). The treatment effect is defined as 0.1 * (log average population) + 0.1 (log average population)*.
E.2. Additional Results
28

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A Meta-learner for Heterogencous Effects in Difference-nn-ifffeeeee
Table MSE (mean standard deviation) over 100 simulations following the conditional parallel trends condition. Each row represent a different meta-learner, and columms represent the different nuisance function classes.
Basic
Lasso (CV)
Ridge (CV)
Random Forest
Best
Neural Net (CPTA OR)
0.12 = 0.02
0.12 0.02
0.12 0.02
0.38 0.18
0.12 + 0.02
Neural Net (CPTA DR)
0.1 0.02
0.1 # 0.03
0.1 0.02
0.14 0.04
0.1 0.02
Neural Net (Lagged OR)
0.12 0.02
0.14 0.04
0.12
1.27 0.65
0.12 = 0.02
Neural Net (Lagged DR)
0.02
0.1 0.03
0.1
0.63 = 0.4
0.1 = 0.02
XGBoost (OR)
0.09 = 0.02
0.09 0.02
0.09
0.31 # 0.16
0.09 = 0.02
XGBoost (DR)
0.04 = 0.01
0.00
0.04
0.06 = 0.03
0,04 = 0.01
XGBoost (Lagged OR)
0.09 = 0.02
0.11
0.09 0.02
1.15 + 0.69
0.09 + 0.02
XGBoost (Lagged DR)
0.04
0.04 0.05
0.54 0.45
0.04 1.00
Linear (OR)
0.26
0.26
0.26 0.07
0.51 0.18
0.26 0.26
Linear (DR)
0.26 0.07
0.26
0.26 0.07
0.26 0.07
0
Linear (Lagged OR)
0.26 0.07
0.26 0.28
1.18 +
0
Linear (Lagged DR)
0.26 = 0.07
0.27
0.26 0.07
0.42 # 0.19
0.26 0.07
Table 5. MSE (mean + standard deviation) Over 100 Simulations of Imbalanced Dataset. Each row represent different meta-learner, and columns represent the different nuisance function classes.
Linear Regression
No Controls
Lasso (CV)
Ridge (CV)
Random Forest
Best
Neural Net (OR)
1.53 0.74
0,22 = 0.06
0.21 0.06
0.21 0.06
0.4 + 0.15
0.21 + 0.05
Neural Net (DR)
0.52 0.31
0.18 # 0.07
0.18 # 0.05
0.18 0.05
0.24 0.07
0.18 + 0.05
Neural Net (CATE OR)
0,66 0.27
0.27 0.08
0.27 0.08
0.27 = 0.08
0.51 0.16
0.27 = 0.08
Neural Net (CATE DR)
+
0.22 0.07
0.21 + 0.07
0.33 + 0.11
0.21 + 0.07
XGBoost (OR)
1.22 0.58
0.21 0.06
0.21 2 0.06
0.21 0.06
0.11
0.21 = 0.06
XGBoost (DR)
0,4 0.14
0.12 0.03
0.12 + 0.03
0.12 + 0.03
0.18
0.12 # 0.03
XGBoost (CATE OR)
0.66 + 0.27
0.27 = 0.08
0.27 + 0.08
0.27 + 0.08
0.51 0.16
0.27 + 0.08
XGBoost (CATE DR)
0.49 + 0.22
0.15 + 0.05
0.15 + 0.04
0.15 + 0.05
0.34 + 0.13
0.15 # 0.04
Table MSE (mean standard deviation) over 100 simulations following the conditional parallel trends condition, with 100 covariates.
Linear Regression
Lasso (CV)
Ridge (CV)
Random Forest
Best
Neural Net OR
0.21 + 0.05
0.21 0.05
0.2 + 0.06
1.27 0.69
0.21 = 0.06
Neural Net DR
0.18 + 0.05
0.18 0.06
0.18 0.06
0.64 0.38
0.18 = 0.06
Neural Net CATE OR
+
0.28 + 0.08
0.28 0.08
0.65 + 0.25
0.28 + 0.08
Neural Net CATE DR
0.1
0.29 + 0.09
0.29 + 0.1
1.08 + 0.71
0.2
Linear OR
0.27
0.27 # 0.08
0.27 + 0.08
1.3 0.63
0.27 # 0.08
Linear DR
0.27
0.27 + 0.08
0.27
0.44
0.27 = 0.08
Linear CATE OR
0.28 0.08
0.27 0.08
0.28 0.08
0.65 0.25
0.27 + 0.08
Linear CATE DR
0.29
0.29 0.08
0.29 0.08
0.82 0.33
0.28 = 0.08
XGBoost OR
0.21 0.06
0.05
0.21 221
0.45
0.21 = 0.05
XGBoost DR
0.13
0.12 0.12
0.61 0.25
0.12 0.27 + 0.08
XGBoost CATE OR
0.28 + 0.08
0.27 + 0.08
0.28 + 0.08
0.65 + 0.25
XGBoost CATE DR
0.26 + 0.09
0.24 + 0.08
0.25 + 0.09
1.18 + 0.85
0.15 + 0.05
29

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A Meta-learner for Heterogencous Effects in Difference-in-Die Diffeerrncss
Table 7. MSE (mean standard deviation) over 100 simulations that does not satisfy the conditional parallel trends assumption. Each row represent different meta-leamer, columns represent the different nuisance function classes.
Linear Regression
Lasso (CV)
Ridge (CV)
Random Forest
Best
Neural Net
76.93 135.25
76.34 + 129.71
74.87 127.94
35.49 + 91.98
36.85 87.85
(CPTA OR)
Neural Net
17.07 + 74.16
15.54 + 54.25
20.41 86.35
17.24 + 63.37
18.38 65.13
(CPTA DR)
Neural Net
70.31 # 98.19
70.07 + 93.93
69.98 100.44
26.21 + 39.93
24.81 = 32.88
(Lagged OR)
Neural Net
4.37 + 4.66
4.93 5.65
4.94 + 5.89
4.99 14.46
5.09 + 10.25
(Lagged DR)
XGBoost
65.67 122.65
63.5 127.89
63.82 113.59
29.39 + 69.84
30.15 85.27
(CPTA OR)
XGBoost
20.49 58.55
22.99 + 81.52
23.31 74.74
26.9
128.52
31.11 149.05
(CPTA DR)
XGBoost
55.62 + 82.27
56.87 + 83.95
53.95 + 77.19
21.29 + 32.85
22.39 + 38.38
(Lagged OR)
XGBoost
9.88 + 14.34
9.34 13.13
10.33 21.81
10.76 + 40.39
8.14 + 13.38
(Lagged DR)
Linear
18.41 62.54
18.0 61.84
18.41 62.68
17.61 + 65.51
17.61 65.51
(CPTA OR)
Linear
14.56 57.69
14.7 58.43
14.56 57.7
15.84 + 64.12
15.84 64.12
(CPTA DR)
Linear
12.08 # 28.93
11.64 + 27.55
12.07 28.9
9.78 + 21.18
9.78 + 21.18
(Lagged OR)
Linear
4.78 + 5.81
4.85 5.97
4.78 + 5.81
3.99 7.18
3.99 + 7.18
(Lagged DR)
30

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A Meta-learner for Heterogencous Effects in Snfferencein-Differee
Kernel Ridge CATT learner
0.1
0.0
Doubly Robust GATT estimate
-0.1
-0.2
-0.3
-0.4
-0.5
-0.00
-0.08
-0.06
00.04
-0.02
0.00
Predicted GATT based on CATT model
Figure 3. Calibration plot for CATT w.r. log county population for the XGBoost doubly robust learner.
31

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A Meta-learner for Heterogencous Effects in Snfferencein-Diffeeee
Linear CATT learner
0.0
Doubly Robust GATT estimate
-0.1
-0.2
-0.3
-0.4
-0.5
-0.08
00.06
-0.04
-0.02
0.00
Predicted GATT based on CATT model
Figure 4. Calibration plot for CATT w.r log county population for the linear doubly robust learnrr
32


Reviews:
[{"review_id": "cuKwTziFME", "reviewer": "Reviewer_q3LX", "summary": "The paper develops a Neyman-orthogonal meta learner for estimating a conditional average treatment effect on the treated (CATT) in the framework of difference-in-differences (DiD). This framework casts the problem of CATT estimation as a convex risk minimisation that involves auxiliary (nuisance) models. The authors leveraged Neyman-orthogonality to further make the framework robust to the misspecification of the auxiliary models. Additionally, the authors provided several extensions of the method to (1) general functionals under covariate shift and (2) an instrumental DiD setting with non-compliance. At the end, the authors empirically demonstrated the superiority of their method.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the helpful reviews! We will fix the inconsistent notations and restructure the paper for the camera-ready version. We introduced “auxiliary models” in the abstract to also accommodate for audiences that are not very familiar with the field. That being said, given the target audience, we might change it to “nuisance functions” for clarity. \n\nOn the clarification of conditional Neyman orthogonality: in [1,2], neyman-orthogonality is referenced as a property of the loss, and is defined by:\n\n$$D_g D_{\\theta} \\mathcal{L} (\\theta^*, g_0)[\\theta - \\theta^*, g - g_0] = 0 \\quad \\forall \\quad \\theta \\in \\Theta, \\quad g \\in \\mathcal{G}$$\n\nIn Definition 3.2 of our paper, Neyman orthogonality is introduced as the property of the moment (see also [3]). Since it is a conditional moment restriction, it is natural that the directional derivatives vanish even conditioning on $W$. A conditionally Neyman orthogonal moment implies a Neyman orthogonal loss, if the gradient of the loss is associated with the moment. Thus conditional Neyman orthogonality of the moment suffices for a Neyman orthogonal loss, but as the reviewer suggests it is not necessary. However, if the target model space Theta is completely unstructured, then the two become equivalent.\n\n$$D_g D_{\\theta} \\mathcal{L} (\\theta^*, g_0)[\\theta - \\theta^*, g - g_0] \n= 2D_g \\mathbb{E} [m(\\theta, g_0) (\\theta - \\theta^*)] [g - g_0]\n= 2D_g \\mathbb{E}\\left[\\mathbb{E} [m(\\theta, g_0)|W] (\\theta - \\theta^*)\\right] [g - g_0]$$\nwhere $m(\\theta, g) = \\widehat{Y}(g) - D\\theta(W)$ is the score function for the conditional moment restriction that we proposed. \n\nSince this has to hold for an arbitrary $\\theta$, we have $D_g D_{\\theta} \\mathcal{L} (\\theta^*, g_0)[\\theta - \\theta^*, g - g_0] =0$ if and only if $D_g \\mathbb{E}[m(\\theta, g_0)|W] [g - g_0]=0 $ for all $g$ with measure 1 (since we can take $\\theta=\\theta^* + \\delta$, where $\\delta$ is the Dirac-delta function at a particular $W$). Similarly, since we are working with a conditional moment restriction (since the estimand is the CATT instead of the ATT), we also need the conditional Riesz Representer. \n\nConstructing Neyman orthogonal losses by first constructing a conditional Neyman orthogonal moment and finding the loss whose gradient is associated with this moment is a typical way of constructing an orthogonal loss. The meta-learners that the reviewer references can also be viewed through this lens. They correspond to Neyman orthogonal losses, because their gradient is associated with a moment restriction that satisfies conditional Neyman orthogonality (see e.g. the [4] for this fact, where also conditional Neyman orthogonality is used for both DR and R Learner style estimators of CATEs via forests). Moreover, this sufficient condition is satisfied for all the applications that our work is motivated by (e.g. DiD CATTs and all the applications of the general framework of conditional linear functionals with covariate shift).\n\nFinally, we really appreciate the time you put into reviewing our paper, and the constructive feedback that will help us improve our work!\n\n**References**\n- [1] Foster, Dylan J., and Vasilis Syrgkanis. \"Orthogonal statistical learning.\" The Annals of Statistics 51.3 (2023): 879-908.\n- [2] Morzywolek, Pawel, Johan Decruyenaere, and Stijn Vansteelandt. \"On a general class of orthogonal learners for the estimation of heterogeneous treatment effects.\" arXiv preprint arXiv:2303.12687 (2023).\n- [3] Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., & Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters.\n- [4] Oprescu, Miruna, Vasilis Syrgkanis, and Zhiwei Steven Wu. \"Orthogonal random forest for causal inference.\" International Conference on Machine Learning. PMLR, 2019.", "claims_and_evidence": "The claims in the paper seem valid and are well supported. The authors provided the proofs for all the theoretical statements in the paper. Also, they provided the empirical evidence that the proposed approach is superior to existing methods.", "methods_and_evaluation_criteria": "The contribution of the paper is rather theoretical. Yet, as a causal ML paper, the work follows a well-established practice of (semi-)synthetic evaluation.", "theoretical_claims": "I did not carefully check the proofs, but the overall approach follows the Neyman-orthogonality theory from [1].  \n\nReferences:\n- [1] Foster, Dylan J., and Vasilis Syrgkanis. \"Orthogonal statistical learning.\" The Annals of Statistics 51.3 (2023): 879-908.", "experimental_designs_or_analyses": "See Methods And Evaluation Criteria.", "relation_to_broader_scientific_literature": "The proposed approach fills the gap of meta learners in the DiD setting. The authors also extended it further to the covariate shift problem and the instrumental DiD.", "essential_references_not_discussed": "It is not clear, why the conditional Neyman-orthogonality is required for estimating CATT and the regular (marginal) Neyman-orthogonality wouldn’t suffice (it seems to me that the conditional Neyman-orthogonality is a strictly stronger condition). Many existing learners like DR-/R-learners for CATE only rely on the marginal Neyman-orthogonality [1, 2], where in [2], any subset $W \\subseteq X$ can be considered as the second-stage model input. Hence, I strongly encourage the authors to provide the discussion on why the conditional Neyman-orthogonality (and conditional Riesz Representer) are essential for this work.\n\nReferences:\n- [1] Foster, Dylan J., and Vasilis Syrgkanis. \"Orthogonal statistical learning.\" The Annals of Statistics 51.3 (2023): 879-908.\n- [2] Morzywolek, Pawel, Johan Decruyenaere, and Stijn Vansteelandt. \"On a general class of orthogonal learners for the estimation of heterogeneous treatment effects.\" arXiv preprint arXiv:2303.12687 (2023).", "other_strengths_and_weaknesses": "Strength:\n- I really like the scope of the paper and its extensions to the covariate shifts and the instrumental DiD settings.\n\nWeaknesses (can be fixed during the rebuttal):\n- I found the structure of the paper slightly confusing. The authors often switch back and forth between (i) assumptions, (ii) identification formulas, and (iii) estimation approaches (e.g. in Sec. 3 and 5). I would suggest having a clear cut between all three and present the corresponding sections in the sequential manner.", "other_comments_or_suggestions": "I found some notational inconsistencies (e.g., the lower indices of $D$ in all the sections). I encourage the authors to carefully check the notation and fix it during the rebuttal.", "questions_for_authors": "- Do we need a new term for the nuisance functions: auxiliary models?", "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": "k6uTIAbsr0", "reviewer": "Reviewer_QLbJ", "summary": "The paper proposes a doubly robust metalearner for the Conditional Average Treatment Effect on the Treated for panel data. The method is targeted for binary \ntreatments and builds upon the parallel trends assumptions common in Difference-in-Differences settings.\nFurthermore, the paper proposes\na robust meta-learning framework for the estimation of general conditional functionals under covariate shift.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the helpful reviews! We apologize for any typos and will definitely fix them for the camera-ready version. As pointed out by many reviewers, we were not able to include a detailed discussion and literature review sections in the initial submission due to length constraints. In the camera-ready version, we will expand on the related literature on CATE meta-learners to provide more background, and also provide more discussions on the limitations including the assumption that the covariates are time-invariant. First, we acknowledge that this is often not the case in real world applications. In such cases, one solution is to only use the pre-treatment covariates as in [1]. Alternatively, if there are other time varying variables that are also believed to be important for the time trends, one solution is to concatenate the covariates at the two different time points, and include all of them in the conditioning set for the conditional parallel trends conditions. But one must be careful to remove any covariates that are downstream from the treatment (i.e. descendents of the treatment in the context of a causal graph, these can be potential mediators) for the validity of the causal interpretation of the CATT. \n\nRegarding the Lagged Dependent Outcome assumption - this is also a common assumption that is widely used in economics when analysing panel data [2]. Depending on the context, this assumption might be more plausible than the parallel trends assumption, e.g. salary increase might be strongly correlated with the baseline salary level. While the parallel trends assumption might be seen as a special case of this assumption, many think of them as different assumptions, and the difference in the identification will result in different doubly robust estimation procedures when learning the conditional ATT on the full set of covariates $X$. In the case where the covariates of interest are a strict subset of the full set $X$, they can be estimated using the same framework. Hence, we include this setting in the appendix for comprehensiveness.\n\nIn Section 5 of the paper, we discussed how to extend the learner to the multiple-time period settings. One solution is to learn a separate model for each lag period as in [1]. In this case, looking at more time periods is essentially the same task as in the two time period setting. An alternative would be to treat the lag period as an additional covariate to include in the conditioning set for the conditional parallel trends assumption. Practically this is what people can do. Qualitatively the insights from our current work should extend to this setting, even though the data are no longer i.i.d.; we believe that this also allows one to study the heterogeneity of the ATT with respect to the lag-periods. In this framework, the lag period may be seen as any other covariate used in the model. For this reason, we believe that results from simulation experiments in the two time-period setting should also generalize to the multiple time period setting. Note that our main theorems 3.6 and 4.7 do not really depend on the i.i.d. nature of the data. They are meta-theorems that state that as long as we have an ML algorithm that achieves a plug-in excess risk bound, then this translates to a mean-squared-error bound for the target causal estimand. Whether data are i.i.d. or not would be important when proving plug-in excess risk bounds, which would be easier for i.i.d., while for time-series or panel datasets, one would need to invoke martingale analysis to prove such excess risk bounds. Irrespective, our robustness conclusion will remain valid and is versatile, allowing for an arbitrary argument for the plug-in excess risk.\n\nOn the clarification of conditional Neyman orthogonality we point out our response to Reviewer q3LX, who raised the same question. \n\nFinally, we really appreciate the time you put into reviewing our paper, and the constructive feedback that will help us improve our work!\n\n**References**\n- [1] Callaway, B., & Sant’Anna, P. H. (2021). Difference-in-differences with multiple time periods. Journal of econometrics, 225(2), 200-230.\n- [2] Angrist, J. D., & Pischke, J. S. (2009). Mostly harmless econometrics: An empiricist's companion. Princeton university press.\n- [3] Foster, Dylan J., and Vasilis Syrgkanis. \"Orthogonal statistical learning.\" The Annals of Statistics 51.3 (2023): 879-908.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes. However, experiments are only conducted on the setting with a specific treatment timestep, i.e., T=2. Empirical evaluation of the method for greater\ntime horizons would be of interest.", "theoretical_claims": "The proofs of the claims in the main paper.", "experimental_designs_or_analyses": "Yes", "relation_to_broader_scientific_literature": "The paper is related to a recent literature stream on Neyman-orthogonal learning for various causal effects (e.g., the popular CATE DR- or R-learner). It extends\nthe idea of constructing such an orthogonal meta-learner to estimate the conditional average treatment effect on the treated on panel data.", "essential_references_not_discussed": "/", "other_strengths_and_weaknesses": "**Strengths:**\n- The idea is novel and relevant.\n- The paper is well written and, in most parts, easy to follow for readers with sufficient background.\n\n**Weaknesses:**\n- The method assumes that all covariates are time-invariant. However, this is barely the case in real-world settings.\n- The paper does not discuss the proposed method in terms of limitations or future work\n- The paper does not provide a general literature review, making it difficult for readers not very familiar with orthogonal (CATE) meta-learners to assess\nthe differences to existing works and the novelty of the proposed method", "other_comments_or_suggestions": "- Line 121: Typo \"problem of estimating\"\n- Theorem 3.6: R should be defined\n- Theorem 4.7: typo in title\n- Please format the paper in a revised version: The running title is still the one from the template; Figures and tables are not were they are supposed to be\n- The supplementary material has many typos", "questions_for_authors": "- Definition 3.1: What is the difference to Neyman-orthogonality and why is conditional Neyman-orthogonality necessary here? In my opinion, the text in the\ndefinition describes normal NO, whereas the formula describes conditional NO.\n- What is the benefit of the Lagged Dependent Outcome Alternate Assumption? Why is this introduced here?", "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": "auE5EdJHaV", "reviewer": "Reviewer_r9fP", "summary": "The paper introduces novel orthogonal meta-learners for estimating the conditional average treatment effect of the treated (CATT) in a difference-in differences (DID) setting. The main contributions include the proposed orthogonal loss, theoretical guarantees in form of doubly robust rates, and extensions to generalized settings. The proposed learner is evaluated using synthetic and real-world data.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the helpful reviews! The result of Proposition 2.3 follows as a simple extension of the ATT identification strategy under conditional parallel trends. The reviewer is right that we should not claim this as a result of this work. We will reference for instance, in the preamble of the theorem the recent survey of [1], where a variant of this formula (when $X=W$) also appears (e.g. Equation (11) in [1]). The extra projection step on $W$ is a simple adaptation of that formula.  Regarding to the comment that there are limited new ideas, while we acknowledge that our methods stems from the idea of debiasing the nuisance functions, we want to highlight that our method provides a framework that allows one to debias nuisance functions that is learned on a different distribution (i.e. under covariate shift) from the target distribution for the final estimand, and this allows us to generalized the framework to many other estimands as described in Section 4. \n\nAs pointed out by many reviewers, we were not able to include a detailed discussion section in the initial submission due to length constraints. In the camera-ready version, we will provide more discussions on the limitations of the proposed method including the unstable performance in low-overlap regimes, which is common for doubly robust learners. As you have mentioned, it is an interesting future direction to looking into overlap-weighted orthogonal learners. If one is interested in estimating overlap weighted projections of the true CATT, then we suspect that a simple adaptation of our loss, where one weights each sample by an appropriate weighting function of $W$, will cancel out the large variance of the density ratio, while maintaining Neyman orthogonality of the loss and will lead to a weighted MSE projection guarantee, weighted by the function of W that was used to reweight the loss at the sample level.  \n\nFinally, we really appreciate the time you put into reviewing our paper, and the constructive feedback that will help us improve our work!\n\n**References**\n- [1] Roth, Jonathan, et al. \"What’s trending in difference-in-differences? A synthesis of the recent econometrics literature.\" Journal of Econometrics 235.2 (2023): 2218-2244.", "claims_and_evidence": "The claims (orthogonality, double robustness) are clearly formulated and backed up with both theoretical and empirical evidence.\n\nOnly one question: is the identifiability result from Proposition 2.3. novel? If not, a clear reference would be needed.", "methods_and_evaluation_criteria": "Evaluation criteria follow standard practices in the causal inference literature.", "theoretical_claims": "I did not verify the correctness of the theoretical results in detail. However, I am quite familiar with similar work on orthogonal learning and the results look reasonable to me.", "experimental_designs_or_analyses": "Results are mostly on synthetic data as common in causal inference (evaluation on real-world data is often impossible). The case-study on real-world data is in my opinion not entirely necessary (could be moved to the appendix).", "relation_to_broader_scientific_literature": "Key literature seems to be cited (even though I am not an expert when it comes to DID).", "essential_references_not_discussed": "See above.", "other_strengths_and_weaknesses": "Strengths:\n- Versatile framework that allows the use of arbitrary machine learning models for causal inference DID settings with nice theoretical guarantees\n\nWeaknesses:\n- While the meta-learner is novel, the derivation follows an established framework that has been leveraged for many other causal estimands already. New ideas remain relatively limited.\n- As is common for doubly robust learners, the proposed learner may become unstable in low-overlap settings. It might be interesting to extend the methodology to overlap-weighted orthogonal learners (e.g., R-learner for DID).\n\nIn sum, a nice paper leveraging an existing framework for a new causal estimand. The result is a solid methodology that fills a gap in the literature.", "other_comments_or_suggestions": "It would be nice if the authors could add a discussion section to the camera-ready version of the paper, discussion limitations and future work (e.g., as above).", "questions_for_authors": "No further comments", "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:

Meta-review: The paper introduces a meta-learner for estimating heterogeneous effects in difference-in-differences designs. 

I agree with the reviewers that additional discussions on the necessity of conditional Neyman orthogonality would improve the paper (e.g., including a condensed version of your reply to q3LX's review would be very helpful). Similarly, a short discussion of the low-overlap setting (raised by r9fP) would be useful.

I also tend to agree with Reviewer r9fP's assessment of the novelty ("the derivation follows an established framework that has been leveraged for many other causal estimands already. New ideas remain relatively limited"). I believe that careful consideration of the multiple time point design and more complex designs (e.g., staggered difference-in-differences) would enhance the overall significance/impact of the work. Your response to r9fP makes me hopeful that such extensions are possible and I would encourage you to explore them a bit further.
Decision: Accept (poster)