Doubly Robust Proximal Synthetic Controls Hongxiang Qiu1 Xu Shi2 Wang Miao3 Edgar Dobriban4 and Eric Tchetgen Tchetgen4

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Doubly Robust Proximal Synthetic Controls

Hongxiang Qiu1, Xu Shi2, Wang Miao3, Edgar Dobriban4, and Eric

Tchetgen Tchetgen∗4

1Department of Epidemiology and Biostatistics, Michigan State

University

2Department of Biostatistics, University of Michigan

3Department of Probability and Statistics, Peking University

4Department of Statistics and Data Science, University of Pennsylvania

Abstract

To infer the treatment eﬀect for a single treated unit using panel data, syn-

thetic control methods construct a linear combination of control units’ outcomes

that mimics the treated unit’s pre-treatment outcome trajectory. This linear com-

bination is subsequently used to impute the counterfactual outcomes of the treated

unit had it not been treated in the post-treatment period, and used to estimate

the treatment eﬀect. Existing synthetic control methods rely on correctly modeling

certain aspects of the counterfactual outcome generating mechanism and may re-

quire near-perfect matching of the pre-treatment trajectory. Inspired by proximal

causal inference, we obtain two novel nonparametric identifying formulas for the

average treatment eﬀect for the treated unit: one is based on weighting, and the

other combines models for the counterfactual outcome and the weighting function.

We introduce the concept of covariate shift to synthetic controls to obtain these

identiﬁcation results conditional on the treatment assignment. We also develop

two treatment eﬀect estimators based on these two formulas and the generalized

method of moments. One new estimator is doubly robust: it is consistent and

asymptotically normal if at least one of the outcome and weighting models is cor-

rectly speciﬁed. We demonstrate the performance of the methods via simulations

and apply them to evaluate the eﬀectiveness of a Pneumococcal conjugate vaccine

on the risk of all-cause pneumonia in Brazil.

∗Author e-mail addresses: qiuhongx@msu.edu,shixu@umich.edu,mwfy@pku.edu.cn,

dobriban@wharton.upenn.edu,ett@wharton.upenn.edu

arXiv:2210.02014v7 [stat.ME] 6 May 2024

1 Introduction

1.1 Background

Interventions such as policies are often implemented in a single unit such as a state, a

city, or a school. Causal inference in these cases is challenging due to the small number

of treated units, and due to the lack of randomization and independence. In various

ﬁelds including economics, public health, and biometry, synthetic control (SC) methods

[Abadie and Gardeazabal, 2003, Abadie et al., 2010, 2015, Doudchenko and Imbens, 2016]

are a common tool to estimate the intervention (or treatment) eﬀect for the treated unit

in time series from a single treated unit and multiple untreated units in both pre- and

post-treatment periods. For example, SC methods have been used to estimate the eﬀects

of terrorist conﬂicts on GDP [Abadie and Gardeazabal, 2003], tobacco control program

on tobacco consumption [Abadie et al., 2010], Kansas’s tax cut on GDP [Ben-Michael

et al., 2021b, Rickman and Wang, 2018], Florida’s “stand your ground” law on homicide

rates [Bonander et al., 2021], and pneumococcal conjugate vaccines on pneumonia [Bruhn

et al., 2017].

Classical SCs are linear combinations of control units that mimic the treated unit

before the treatment. Outcome diﬀerences between the treated unit and the SC in the

post-treatment period are used to make inferences about the treatment eﬀect for the

treated unit. In Abadie and Gardeazabal [2003] and Abadie et al. [2010], a SC is a

weighted average of a pool of control units, called the donors. The weights are obtained

by minimizing a distance between the SC and the treated unit in the pre-treatment period,

under the constraint that the weights are non-negative and sum to unity. Many extensions

have been proposed. For example, Abadie and L’Hour [2021] proposed methods for

multiple treated units, and Ben-Michael et al. [2021a] further considered the case where

these treated units initiate treatment at diﬀerent time points; Doudchenko and Imbens

[2016] and Ben-Michael et al. [2021a,b] introduced penalization to improve performance;

Athey et al. [2021] and Bai and Ng [2021] used techniques from matrix completion;

Li [2020] studied statistical inference for SC methods; Chernozhukov et al. [2021] and

Cattaneo et al. [2021] considered prediction intervals for treatment eﬀects. Among these

extensions, some also incorporate the idea that, similarly to the control units’ outcomes

in the post-treatment period, the treated unit’s outcomes in the pre-treatment period

can be used to impute the counterfactual outcome had it not been treated [Ben-Michael

et al., 2021b, Arkhangelsky et al., 2021].

Existing methods often rely on assuming linear models and on the existence of near-

perfectly matching weights in the observed data. Under such assumptions, valid SCs are

linear combinations, often weighted averages, of donors. However, if such assumptions do

not hold, these methods may not produce a valid SC. This may happen if the outcomes

in the donors have a diﬀerent measuring scale from the treated unit, or if the treated

unit’s and the donors’ outcomes have a nonlinear relationship.

To relax these assumptions, Shi et al. [2023] viewed SCs from the proximal causal

inference perspective. For independent and identically distributed (i.i.d.) observations,

Miao et al. [2018], Deaner [2018, 2021], Cui et al. [2020], Tchetgen Tchetgen et al. [2020]

derived nonparametric identiﬁcation using proxies, variables capturing the eﬀect of the

unmeasured confounders. Shi et al. [2023] viewed control units’ outcomes as proxies and

obtained nonparametric identiﬁcation results for the potential outcome of the treated

unit had it not been treated as well as the treatment eﬀect in a general setting, beyond

the common linear factor model [e.g., Abadie et al., 2010]. They assumed the existence

of a function of these proxies, termed confounding bridge function, that captures the

(possibly nonlinear) eﬀects of unobserved confounders. With this function, they imputed

the expected counterfactual outcomes for the treated unit. Estimation of, and inference

about, the average treatment eﬀect for the treated unit (ATT) followed from this iden-

tiﬁcation result. Instrumental variables have also been used. For example, Holtz-Eakin

et al. [1988] considered a linear model with interactive ﬁxed eﬀects and showed how to

identify it using appropriate instruments. The solution to this problem relies on a partic-

ular diﬀerencing strategy, which may be viewed as an application of a confounding bridge

function. Cunha et al. [2010], Freyberger [2018] and references therein considered general

nonparametric models with interactive eﬀects, showing how to identify it using appropri-

ate instruments. While treatment confounding proxies in proximal causal inference are

sometimes described as instruments, it is crucial to note that they are more general than

instrumental variables (IVs), in the sense that valid IVs are valid treatment confound-

ing proxies, but invalid IVs dependent on hidden confounders are also valid treatment

confounding proxies [Tchetgen Tchetgen et al., 2020]. In addition, while IVs require a

form of homogeneity condition for nonparametric identiﬁcation (e.g., separable errors or

monotonicity), proxies do not require such a condition.

1.2 Our contribution

Existing methods rely on correctly specifying an outcome model, based on which one can

impute the counterfactual outcome trajectory of the treated unit, had it not been treated,

after treatment. This outcome bridge function model may be diﬃcult to specify correctly,

or may not exist. In this paper, we relax this requirement by leveraging the proximal

causal inference framework as in Shi et al. [2023]. We develop two novel methods to

estimate the ATT. One method relies on weighting and is a building block to a second

method which we rigorously prove is doubly robust [Bang and Robins, 2005, Scharfstein

et al., 1999]. It is consistent and asymptotically normal if either the outcome model

or the weighting function is correctly speciﬁed, without requiring that both are. An

advantage of the doubly robust method compared to existing methods is that it allows

for misspeciﬁng one of the two models, without the user necessarily knowing which might

be misspeciﬁed.

We observed that our estimand of interest, the ATT, is closely related to the average

treatment eﬀect on the treated for i.i.d. data [e.g., Hahn, 1998, Imbens, 2004, Chen

et al., 2008, Shu and Tan, 2018]. The method in Shi et al. [2023] corresponds to using

an outcome confounding bridge function [Miao et al., 2018], which is the proximal causal

inference counterpart of G-computation, or an outcome regression-based approach in

causal inference under unconfoundedness [Robins, 1986]. Our proposed methods are

motivated by the existing identiﬁcation results in proximal causal inference in the i.i.d.

setting [Cui et al., 2020]: one result is based on weighting and the other is based on the

inﬂuence function.

Despite these similarities, it remains challenging to adapt these ideas from the i.i.d.

setting to panel data. Since treatment assignment is often viewed as ﬁxed in SC prob-

lems, a key concept from the i.i.d. setting, the propensity score [Rosenbaum and Rubin,

1983], is undeﬁned. Thus, existing results for the i.i.d. setting cannot be directly applied

to SC problems. We leverage the notion of covariate shift [e.g., Qui˜nonero-Candela et al.,

2009] to circumvent this issue. We also ﬁnd a relaxed version of the i.i.d. assumption

to allow for serial correlation, while still obtaining identiﬁcation via weighting. We illus-

trate our proposed methods in simulations and three empirical examples: two examples

concern public health outcomes, one studying the eﬀect of the PCV10 vaccine in Brazil

on pneumonia [Bruhn et al., 2017], and the other studying the eﬀect of Florida’s “stand

your ground” law on homicide rates [Bonander et al., 2021]; the third example concerns

economic outcomes, studying the eﬀect of Kansas’s tax cut on GDP [Rickman and Wang,

2018].

Both our doubly robust method and the method in Ben-Michael et al. [2021b] take

the form of augmented weighted moment equations, but the known robustness properties

of these methods diﬀer. In Ben-Michael et al. [2021b], the treated unit’s counterfactual

outcome had it not been treated after treatment can be imputed with two approaches.

One is a weighted average of control units’ outcomes, identical to classical SC methods

[Abadie et al., 2010]; the other is a prediction model obtained using the treated unit’s out-

comes before treatment. By combining these two approaches, Ben-Michael et al. [2021b]

developed a SC method with improved performance. Arkhangelsky et al. [2021] proposed

a method combining two imputation approaches based on similar ideas. Nevertheless,

to date, neither method has formally been shown to be doubly robust. In contrast, we

formally establish double robustness, inherited from the inﬂuence function of the ATT in

i.i.d cases.

2 Problem setup

We observe data over Ttime periods. The ﬁrst T0time periods are the pre-treatment

periods, and the last T−T0time periods are the post-treatment periods. For the treated

unit, at each time period t= 1, . . . , T , let Yt(0), Yt(1) ∈Rbe the counterfactual outcome

corresponding to no treatment and treatment, respectively, and Yt= (t≤T0)Yt(0) +

(t > T0)Yt(1) be the observed outcome. At each time period t, other variables such as

other control units’ outcomes are observed. We provide more details about these variables

below. We treat T,T0and the treated unit as deterministic, and treat other variables

such as the treated unit’s potential outcomes Yt(1) and Yt(0) as random. In other words,

our proposed methods are conditional on the study design. We study the ATT causal

estimand, that is,

ϕ∗(t) = E{Yt(1) −Yt(0)}

in a post-treatment period t>T0. We treat times, namely Tand T0, and all units as

deterministic, and treat the potential outcomes as stochastic [Greenland, 1987, Robins

and Greenland, 1989, 2000, VanderWeele and Robins, 2012]; that is, Yt(0) and Yt(1) are

both stochastic processes indexed by tthat are randomly generated over time, rather

than ﬁxed unknown scalar sequences. In the frequentist interpretation, under repeated

sampling, the times and units are all ﬁxed and hence identical for all samples, but the

outcomes are randomly generated from a ﬁxed unknown joint distribution and hence

may diﬀer across samples. Stochastic counterfactuals are commonly assumed in the SC

literature, implicitly in the random noise or residuals of a linear latent factor model or

an autoregressive model [e.g., Abadie et al., 2010, Abadie and L’Hour, 2021, Ben-Michael

et al., 2021b,a, Athey et al., 2021]. This notion of stochastic counterfactuals Yt(0) and

Yt(1) as time series is required in our paper because the expectation in ϕ∗(t) is taken over

the joint distribution of (Yt(0), Yt(1)).

In the main text, we focus on the case without covariates, and discuss using covariates

in Web Appendix S5. We assume that all unmeasured confounding is captured by a

latent factor Ut, with assumptions stated in later sections. We use t−and t+to denote

generic times before and after treatment, respectively; that is, t−≤T0and t+> T0.

When stating asymptotic results, we consider the asymptotic regime where T→ ∞

with T0/T →γ∈(0,1). This asymptotic regime may be interpreted as the number of

observations in both pre- and post-treatment periods growing to inﬁnity, collecting more

data before and after the treatment time. Beyond this asymptotic regime, ﬁnite-sample

results concerning the error in pre-treatment ﬁtting or treatment eﬀect estimation have

been established in previous works [e.g., Abadie and L’Hour, 2021, Athey et al., 2021,

Ben-Michael et al., 2021a,b].

3 Review of identiﬁcation via outcome modeling

In classical SC methods, a weighted average of a pool of control units called donors forms

the SC [Abadie et al., 2010]. The motivation for using control units’ outcomes to learn

about Yt+(0), despite the presence of potential unmeasured confounder Ut, is that these

control units may be aﬀected by, and thus contains information about, Ut. This charac-

teristic resembles that of proxies in proximal causal inference. In an i.i.d. setting, proxies

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DoublyRobustProximalSyntheticControlsHongxiangQiu1,XuShi2,WangMiao3,EdgarDobriban4,andEricTchetgenTchetgen∗41DepartmentofEpidemiologyandBiostatistics,MichiganStateUniversity2DepartmentofBiostatistics,UniversityofMichigan3DepartmentofProbabilityandStatistics,PekingUniversity4DepartmentofStatisticsand...

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Doubly Robust Proximal Synthetic Controls Hongxiang Qiu1 Xu Shi2 Wang Miao3 Edgar Dobriban4 and Eric Tchetgen Tchetgen4

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