Synthetic Blip Effects Generalizing Synthetic Controls for the Dynamic Treatment Regime

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Synthetic Blip Effects:

Generalizing Synthetic Controls for the

Dynamic Treatment Regime

Anish Agarwal

Amazon, Core AI

anishaga@amazon.com

Vasilis Syrgkanis

Stanford University

vsyrgk@stanford.edu

Abstract

We propose a generalization of the synthetic control and synthetic interventions

methodology to the dynamic treatment regime. We consider the estimation of

unit-speciﬁc treatment effects from panel data collected via a dynamic treatment

regime and in the presence of unobserved confounding. That is, each unit receives

multiple treatments sequentially, based on an adaptive policy, which depends on

a latent endogenously time-varying confounding state of the treated unit. Under

a low-rank latent factor model assumption and a technical overlap assumption

we propose an identiﬁcation strategy for any unit-speciﬁc mean outcome under

any sequence of interventions. The latent factor model we propose admits linear

time-varying and time-invariant dynamical systems as special cases. Our approach

can be seen as an identiﬁcation strategy for structural nested mean models under

a low-rank latent factor assumption on the blip effects. Our method, which we

term “synthetic blip effects”, is a backwards induction process, where the blip

effect of a treatment at each period and for a target unit is recursively expressed as

linear combinations of blip effects of a carefully chosen group of other units that

received the designated treatment. Our work avoids the combinatorial explosion

in the number of units that would be required by a vanilla application of prior

synthetic control and synthetic intervention methods in such dynamic treatment

regime settings.

1 Introduction

In many observational studies, units undergo multiple treatments over a period of time; patients are

treated with multiple therapies, customers are exposed to multiple advertising campaigns, govern-

ments implement multiple policies, sequentially. Many times these interventions happen in a data

adaptive manner, where treatment assignment depends on the current state of the treated unit and

on past treatments. A common policy question that arises is what would have been the expected

outcome under an alternative policy or course of action. Performing counterfactual analysis from

such observational data with multiple sequentially and adaptively assigned treatments is the topic of

a long literature on causal inference, known as the dynamic treatment regime.

Dynamic treatment effects with unobserved confounding. Typical approaches for identiﬁcation in the

dynamic treatment regime require a strong sequential exogeneity assumption, where the treatment

decision at each period, essentially only depends on an observable state. This assumption is a

generalization of the standard conditional exogeneity assumption in the static treatment regime.

However, most observational datasets are plagued with unobserved confounding. Many techniques

exist for dealing with unobserved confounding in static treatment regimes (such as instrumental

variables, differences-in-differences, regression discontinuity designs, synthetic controls). However,

Preprint. Under review.

arXiv:2210.11003v1 [econ.EM] 20 Oct 2022

approaches to dealing with unobserved confounding in the dynamic treatment regime is a much less

explored area. Recent work, for instance, explores an extension of the differences-in-differences

approach to the dynamic treatment regime Shahn et al. (2022).

Synthetic controls/interventions in the dynamic treatment regime. In this work, we present the ﬁrst

extension of the synthetic controls literature to the dynamic treatment regime. Synthetic controls

method Abadie and Gardeazabal (2003); Abadie et al. (2010) —and its generalization to synthetic

interventions Agarwal et al. (2020b)—is a commonly used empirical approach to dealing with

unobserved confounding from observational panel data. However, the existing literature assumes that

units are treated only once or in a non-adaptive manner. This limits the applicability of the technique

to policy relevant settings where multiple interventions occur sequentially over a period of time.

Our work proposes an extension of the synthetic controls and synthetic interventions method, that

allows the identiﬁcation of mean counterfactual outcomes under any treatment sequence, even when

the observational data came from an adaptive dynamic treatment policy. Similar to the synthetic

interventions framework, our work assumes that the panel data stem from a low-rank data generation

model and that the latent factors capture the unobserved confounding signals. In the static regime, the

low rank assumption together with a technical overlap assumption allows one to express each unit’s

mean outcomes under any sequence of interventions as linear combinations of the observed outcomes

of a carefully chosen sub-group of other units. We extend this idea to the dynamic treatment regime

under a low-rank linear structural nested mean model assumption. Our work can also be viewed as

extending the g-estimation framework for structural nested mean models Robins (2004); Vansteelandt

and Joffe (2014); Lewis and Syrgkanis (2020) to handle unobserved confounding under a low-rank

structure. Thus our work helps connect the literature on synthetic controls with that of structural

nested mean models.

Overview of methodology. The key idea of our identiﬁcation strategy is to express the mean outcome

for a unit under a sequence of interventions as an additive function of “blip” effects for that sequence.

The blip effect of an intervention at a given period can be thought of as the treatment effect of

that intervention compared to a baseline intervention for that speciﬁc period, assuming a common

sequence of interventions for all other time periods. Subsequently, our low-rank assumption and a

recursive argument allows us to identify the blip effect of each treatment for each unit and time period.

Our procedure can be viewed as a dynamic programming method where a synthetic control type

procedure is used to compute “synthetic blip effects” at each step of the dynamic program to identify

time period speciﬁc causal quantities, which are subsequently used to build the overall counterfactual

quantity of estimating the outcome of any unit under any sequence of interventions.

1.1 Related Work

Panel data methods in econometrics.

This is a setting where one gets repeated measurements of

multiple heterogeneous units over say

time steps. Prominent frameworks include differences-

in-differences Ashenfelter and Card (1984); Bertrand et al. (2004); Angrist and Pischke (2009)

and synthetic controls Abadie and Gardeazabal (2003); Abadie et al. (2010); Hsiao et al. (2012);

Doudchenko and Imbens (2016); Athey et al. (2021); Li and Bell (2017); Xu (2017); Amjad et al.

(2018, 2019); Li (2018); Arkhangelsky et al. (2020); Bai and Ng (2020); Ben-Michael et al. (2020);

Chan and Kwok (2020); Chernozhukov et al. (2020); Fern

andez-Val et al. (2020); Agarwal et al.

(2021b, 2020a). These frameworks estimate what would have happened to a unit that undergoes

an intervention (i.e., a “treated” unit) if it had remained under control (i.e., no intervention), in the

potential presence of unobserved confounding. That is, they estimate the counterfactual if a treated

unit remains under control for all

time steps. This is a restricted case of what we consider in this

paper, which is to estimate what happens to a unit under any sequence of interventions over the

time steps. A critical aspect that enables the methods above is the structure between units and time

under control. One elegant encoding of this structure is through a latent factor model (also known as

an interactive ﬁxed effect model), Chamberlain (1984); Liang and Zeger (1986); Arellano and Honore

(2000); Bai (2003, 2009); Pesaran (2006); Moon and Weidner (2015, 2017). In such models, it is

posited that there exist low-dimensional latent unit and time factors that capture unit and time speciﬁc

heterogeneity, respectively, in the potential outcomes. Since the goal in these works is to estimate

outcomes under control, no structure is imposed on the potential outcomes under intervention.

In Agarwal et al. (2020b, 2021a), the authors extend this latent factor model to incorporate latent

factorization across interventions as well, which allows for identiﬁcation and estimation of coun-

terfactual mean outcomes under intervention rather than just under control. In Section 3, we do a

detailed comparison with the synthetic interventions framework introduced in Agarwal et al. (2020b).

This framework was designed for the static regime and has two key limitations for the dynamic

treatment regime: (i) The framework does not allow for adaptivity in how treatments are chosen

over time. (ii) If there are

possible interventions that can be chosen for each of the

time steps,

the synthetic interventions estimator has sample complexity scaling as

to estimate all possible

interventional sequences. The non-adaptivity requirement and the exponential dependence on

makes this estimator not well-suited for dynamic treatments, especially as

grows. We show that

by imposing that an intervention at a given time step has an additive effect on future outcomes, i.e.,

an additive latent factor model, it leads to signiﬁcant gain in what can be identiﬁed and estimated.

We study two variants, a time-varying and time-invariant version, which nest the classical linear

time-varying and linear time-invariant dynamical system models as special cases, respectively. We

establish an identiﬁcation result and a propose an associated estimator to infer all

counterfactual

trajectories per unit. Importantly, our identiﬁcation result allow the interventions to be selected

in an adaptive manner, and the sample complexity of the estimator no longer has an exponential

dependence on T.

Another extension of such factor models are “dynamic factor models”, originally proposed in Geweke

(1976). We refer the reader to Stock and Watson (2011); Chamberlain (2022) for extensive surveys,

and see Imbens et al. (2021) for a recent analysis of such time-varying factor models in the context

of synthetic controls. These models are similar in spirit to our setting in that they allow outcomes

for a given time period to be dependent on the outcome for lagged time periods in an autoregressive

manner. To model this phenomenon, dynamic factor models explicitly express the time-varying factor

as an autoregressive process. However, the target causal parameter in these works is signiﬁcantly

different—they focus on identifying the latent factors and/or forecasting. There is less emphasis on

estimating counterfactual mean outcomes for a given unit under different sequences of interventions.

Linear dynamical systems.

Linear dynamical systems are an extensively studied class of models

in control and systems theory, and are used as linear approximations to many non–linear systems

that nevertheless work well in practice. A seminar work in the study of linear dynamical systems is

Kalman (1960), which introduced the Kalman ﬁlter as a robust solution to identifying and estimating

the linear parameters that deﬁned the system. We refer the reader to the classic and more recent

survey of the analysis of such systems in Ljung (1999) and Hardt et al. (2016), respectively. Previous

works generally assume that (i) the system is driven through independent, and identically distributed

(i.i.d) mean-zero sub-Gaussian noise at each time step, and (ii) access to both the outcome variable

and a meaningful per-time step state, which are both used in estimation. In comparison, we allow for

confounding, i.e., the per time step actions chosen can be correlated with the state of the system in an

unknown manner, and we do not assume access to a per-time step state, just the outcome variable.

To tackle this setting, we show that linear dynamical systems, both time-varying and time-invariant

versions, are special cases of the latent factor model that we propose. The recursive “synthetic blip

effects” identiﬁcation strategy allows to estimate mean counterfactual outcomes under any sequence

of interventions without ﬁrst having to do system identiﬁcation, and despite unobserved confounding.

1.2 Setting & Notation

Notation. [R]

denotes

{1, . . . , R}

for

R∈N

[R1, R2]

denotes

{R1, . . . , R2}

for

R1, R2∈N

, with

R1< R2

[R]0

denotes

{0, . . . , R}

for

R∈N

. For vectors

a, b ∈Rd

ha, bi

denotes the inner product

of aand b. For a vector a, we deﬁne a0as its transpose.

Setup. Let there be Nheterogeneous units. We collect data over Ttime steps for each unit.

Observed outcomes. For each unit and time period

n, t

, we observe

Yn,t ∈R

, which is the outcome

of interest.

Actions. For each

n∈[N]

and

t∈[T]

, we observe actions

An,t ∈[A]

, where

A ∈ N

. Importantly,

we allow

An,t

to be categorical, i.e., it can simply serve as a unique identiﬁer for the action chosen.

We note that traditionally in dynamical systems, it is assumed we know the exact action vector in

where

is the dimension of the action space, rather than just a unique identiﬁer for it. For a sequence

of actions

(a1, . . . , at)

, denote it by

¯at∈[A]t

; denote

(at, . . . , aT)

at∈[A]T−t

. Deﬁne

n, At

analogously to ¯at, at, respectively, but now with respect to the observed sequence of actions An,t.

Control & interventional period. For each unit

, we assume there exists

t∗

n∈[1, T ]

before which it

is in “control”. We denote the control action at time step

0t∈[A]

. Note

and

for

`6=t

, do

not necessarily have to equal each other. For

t∈[T]

, denote

0t= (01,...,0t)

and

0t= (0t,...,0T)

For

t < t∗

, we assume

An,t = 0t

, i.e.,

At∗

n−1

n=¯

0t∗

n−1

. That is, during the control period all units

are under a common sequence of actions, but for

t≥t∗

, each unit

can undergo a possibly different

sequence of actions from all other units, denoted by

At∗

. Note that if

t∗

n= 1

, then unit

is never in

the control period.

Counterfactual outcomes. As stated earlier, for each unit and time period

n, t

, we observe

Yn,t ∈R

which is the outcome of interest. We denote the potential outcome if unit

had instead undergone

¯at

Y(¯at)

n,t

. More generally, we denote the potential outcome

Y(¯

n,a`+1)

n,t

if unit

receives the observed

sequence of actions

till time step

, and then instead undergoes

a`+1

for the remaining

t−`

time

steps. 1

We make the standard “stable unit treatment value assumption” (SUTVA) as follows.

Assumption 1 (Sequential Action SUTVA).For all n∈[N], t ∈[T], ` ∈[t],¯at∈[A]t:

Y(¯

n,a`+1)

n,t =X

¯α`∈[A]`

Y(¯α`,a`+1 )

n,t ·1(¯

n= ¯α`)

Further, for all ¯

n∈[A]t:

Y(¯

n,t =Yn,t.

As an immediate implication,

Y(¯α`,¯a`+1 )

n,t |¯

n= ¯α`

equals

Y(¯

n,a`+1)

n,t |¯

n= ¯a`

, and

Y(¯at)

n,t |¯

¯atequals Yn,t |¯

n= ¯at.

Goal.

Our goal is to accurately estimate the potential outcome if a given unit

had instead undergone

¯aT

(instead of the actual observed sequence

), for any given sequence of actions

¯aT

over

times

steps. That is, for all

n∈[N] ¯aT∈[A]T

, our goal is to estimate

Y(¯aT)

n,T .

We more formally deﬁne

the target causal parameter in Section 2.

2 Latent Factor Model in the Dynamic Treatment Regime

We now present a novel latent factor model for causal inference with dynamic treatments. Towards

that, we ﬁrst deﬁne the collection of latent factors that are of interest.

Deﬁnition 1

(Latent factors)

For a given unit

and time step

, denote its latent factor as

vn,t

. For

a given sequence of actions over

time steps,

¯at

, denote its associated latent factor as

w¯at

. Denote

the collection of latent factors as

LF :={vn,t}n∈[N],t∈[T]∪ {w¯at}¯at∈[A]t, t∈[T].

Here vn,t, w¯at∈Rd(t), where d(t)is allowed to depend on t.

Assumption 2 (General factor model).Assume ∀n∈[N],t∈[T],¯at∈[A]t,

Y(¯at)

n,t =hvn,t, w¯ati+ε(¯at)

n,t .(1)

Further,

E[ε(¯at)

n,t | LF]=0 (2)

(1)

, the key assumption made is that

vn,t

does not depend on the action sequence

¯at

, while

w¯at

does not depend on unit

. That is,

vn,t

captures the unit

speciﬁc latent heterogeneity in determining

the expected conditional potential outcome

E[Y(¯at)

n,t | LF]

;

w¯at

follows a similar intuition but with

We are slightly abusing notation as the potential outcome

Y(¯

n,a`+1)

n,t

is only a function of the ﬁrst

t−`

components of a`+1, which is actually a vector of length T−`.

respect to the action sequence

¯at

. This latent factorization will be key in all our identiﬁcation and

estimation algorithms, and the associated theoretical results. An interpretation of

ε(¯at)

n,t

is that it

represents the component of the potential outcome

Y(¯aT)

n,T

that is not factorizable into the latent factors

represented by

; alternatively, it helps model the inherent randomness in the potential outcomes

Y(¯aT)

n,T

. In Sections 4 and 5 below, we show how various standard models of dynamical systems are a

special case of our proposed factor model in Assumption 2.

Target Causal Parameter

Our target causal parameter is to estimate for all units

n∈[N]

and any

action sequence ¯aT∈[A]T,

E[Y(¯aT)

n,T | LF],(target causal parameter)

i.e., the expected potential outcome conditional on the latent factors,

. In total this amounts to

estimating

N× |A|T

different (expected) potential outcomes, which we note grows exponentially in

3 Limitations of Synthetic Interventions Approach

Given that our goal is to bring to bear a novel factor model perspective to the dynamic treatment

effects literature, we ﬁrst exposit on some of the limitations of the current methods from the factor

model literature that were designed for the static interventions regime, i.e., where an intervention

is done only once at a particular time step. We focus on the synthetic interventions (SI) framework

Agarwal et al. (2020b), which is a recent generalization of the popular synthetic controls framework

from econometrics. In particular, we provide an identiﬁcation argument which builds upon the SI

framework Agarwal et al. (2020b) and then discuss its limitations.

3.1 Identiﬁcation Strategy via SI Framework

3.1.1 Notation and Assumptions

Donor units.

To explain the identiﬁcation strategy, we ﬁrst need to deﬁne a collection of subsets

of units based on: (i) the action sequence they receive; (ii) the correlation between their potential

outcomes and the chosen actions. These subsets are deﬁned as follows.

Deﬁnition 2 (SI donor units).For ¯aT∈[A]T,

I¯aT:={j∈[N] : (i)¯

j= ¯aT,(ii)∀¯αT∈[A]T,E[ε(¯αT)

j,T |¯

j,LF]=0}.(3)

I¯aT

refers to units that receive exactly the sequence

¯aT

. Further, we require that for these particular

units, the action sequence was chosen such that

∀¯αT∈[A]T,E[ε(¯αT)

j,T |¯

j,LF] = E[ε(¯αT)

j,T |

LF]=0

, i.e.,

ε(¯αT)

j,T

is conditionally mean independent of the action sequence

unit

receives.

Note a sufﬁcient condition for property (ii) above is that

∀¯αT∈[A]T, Y (¯αT)

j,T ⊥¯

j| LF

. That is,

for these units, the action sequence for the entire time period

is chosen at

t= 0

conditional on the

latent factors, i.e., the policy for these units is not adaptive (cannot depend on observed outcomes

Yj,t for t∈[T]).

Assumption 3. ∀n∈[N],¯aT∈[A]T

suppose that

vn,T

satisﬁes a well-supported condition, i.e.,

there exists linear weights βn,I¯aT

∈R|I¯aT|such that:

vn,T =X

j∈I¯aT

βn,I¯aT

jvj,T .(well-supported factors)

Assumption 3 essentially states that for a given sequence of interventions

¯aT∈[A]T

, the latent

factor for the target unit

vn,T

lies in the linear span of the latent factors

vj,T

associated with the

“donor” units in

I¯aT

. Note by Theorem 4.6.1 of Vershynin (2018), if the

{vj,T }j∈[N]

are sampled

as independent, mean zero, sub-Gaussian vectors,then

span({vj:j∈ I¯aT)}) = Rd(T)

with high

probability as |I¯aT|grows, and if |I¯aT|  d(T)(recall d(T)is the dimension of vn,T ).

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SyntheticBlipEffects:GeneralizingSyntheticControlsfortheDynamicTreatmentRegimeAnishAgarwalAmazon,CoreAIanishaga@amazon.comVasilisSyrgkanisStanfordUniversityvsyrgk@stanford.eduAbstractWeproposeageneralizationofthesyntheticcontrolandsyntheticinterventionsmethodologytothedynamictreatmentregime.Weconsid...

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