ASSESSING CAUSAL EFFECTS OF INTERVENTIONS IN TIME USING GAUSSIAN PROCESSES Gianluca Giudice Sara Geneletti and Konstantinos Kalogeropoulos

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ASSESSING CAUSAL EFFECTS OF INTERVENTIONS IN TIME

USING GAUSSIAN PROCESSES

Gianluca Giudice, Sara Geneletti and Konstantinos Kalogeropoulos

Department of Statistics

LSE

g.giudice@lse.ac.uk;k.kalogeropoulos@lse.ac.uk;s.geneletti@lse.ac.uk;

October 7, 2022

1 Introduction

Recently, many applications have been devoted to understanding and revealing causal rather than associative relations

among variables. One approach in the context of time series is that of synthetic controls (Abadie and Gardeazabal,

2003) and various extensions. This is based on the idea of recovering the counterfactual outcome that would have been

observed had an intervention not taken place.

This article contributes to expanding and generalizing this class of models, allowing for non-linearity in a non-

parametric manner through Gaussian Processes. These models have high degree of ﬂexibility in building the counter-

factual outcome, using all types of information and without any limitations on the functional form. They also make

it possible to assess the robustness of the synthetic controls, as we can use the posterior distributions of the Gaussian

Processes to quantify uncertainty stemming from the functional form estimation. Lastly, as the models learn the rela-

tionships which prevail amongst all associated variables, there is no need to match the time series on a calendar basis,

making the most of the available data.

To our best knowledge the only paper that uses Gaussian process in the context of potential outcomes is Alaa and

van der Schaar (2017). The purpose of the article was to infer individualized treatment effects across a series of

cross sectional experiments. However, the bivariate setting arises from the use of the treated and control group as

dependent variables and no time component is exploited. There exists very recent literature that explores multi task

causal learning using Gaussian process exploiting Judea Pearl’s Do-Calculus (Pearl, 1995). These papers (Aglietti

et al., 2020) however are mainly focused on understanding the main correlation structure of multiple continuous

intervention functions - deﬁned with a directed acyclic graph (DAG) - as opposed to a single discrete intervention.

Furthermore, the main data domain consist of cross sectional experiments, not time series data.

This paper is structured as follows. In section 2 we brieﬂy introduce the causal framework and the synthetic

arXiv:2210.02850v1 [stat.ME] 6 Oct 2022

APREPRINT - OCTOBER 7, 2022

control approach, presenting our main assumptions and the causal effect estimands. In Section 3 we deﬁne the

proposed models based on Gaussian Processes. In Section 4, we present the estimation procedure. In Section 5

we present an illustrative empirical analysis of our approach to to obtain estimates of the causal effect of the UK’s

effective vaccination programme, introduced in January 2021, on deaths and infection rate. Section 6 describe the

main results of the analysis. Finally, we conclude in section 7.

2 Background

The application we refer to throughout the paper serves as an illustrative example and is analysed in Section 5. It

attempts to understand whether the early and intense vaccination campaign introduced in the UK affected the number

of deaths and level of contagiousness of Covid-19 in the ﬁrst semester of 2021.Formally, the treated unit is the UK and

the treatment is the substantially accelerated vaccination schedule. Other EU countries, with other slower vaccination

campaigns will be used to construct the synthetic control for the UK. We would like to note here that we are comparing

the UK with a non-treated counterfactual version of itself, using other European countries to create this counterfactual.

Each observation is denoted with yi,t ∈ Y, where i= 1,··· , m is reserved for countries and t= 1,··· , Tifor

observation times, is associated with a set of dpotentially time-varying predictors xi,t ∈ Xdsuch that

yi,t =f(xi,t) + i,t, i,t ∼ N(0, ω2

i),

where f(·)is a generic function which express the input-output relationship and i,t is the error term, having mean 0

and variance ω2

i. The d-dimensional feature vector xi,t is a set of time series speciﬁc to each unit i. In our application

this includes mobility data and number of tests for each country. The data span Tiperiods and the ﬁrst t0periods

correspond to the data before the intervention, i.e. when the vaccination campaign began in the UK.

2.1 Synthetic Control Methods

Synthetic control methods have gained traction as methods to estimate causal effects from variables that were subject

to a single intervention or treatment in time. A traditional approach is the one based on the difference-in-difference

(DD) method, a static linear regression model where the causal effect is estimated as the difference between the

regression coefﬁcient in the treated and the control group. This is often implemented in a linear regression setting,

with the quantity of interest being the interaction term of the dependent variable and the treatment group dummy

variable. In this case, f(xi,t) = α+x0

i,tβ+γD(t0), where D(t0)is a dummy variable which take values 1 for

t > t0and α, β, γ are the ordinary least square coefﬁcients. However, DD methods suffer from two main drawbacks

(Brodersen et al., 2014): the ﬁrst one is that it assumes that the data are independent and identically distributed,

thus disregarding the temporal component; secondly, the pre and post intervention periods are captured solely by

two time points Abadie et al. (2010) Abadie and Gardeazabal (2003) proposed models generalizing the DD as they

allowed the effect of unit-speciﬁc unobserved variables to vary with time. In particular, they recover the counterfactual

outcome by developing a control group that has a similar pattern in the pre-intervention period as the treated unit. To

do so they ﬁnd a vector of weights {W1,··· , Wm−1}0, Wj>0,PWj= 1 which minimize the squared distance

between the pre-intervention features (not time series) of the exposed region xiand the the features for the unaffected

regions {xj}j6=i. Then, the counterfactual outcome becomes yi,t =Pi6=jWjyj,t. However, this method has its

own limitations. Indeed, it focuses only on possible convex combinations of control time series to match the treated

APREPRINT - OCTOBER 7, 2022

variable. Furthermore, there is a non-negligible data loss in regards to the temporal component. First of all, only data

in the pre-treatment period is used to ﬁt the model and ﬁnd the optimal weights of the counterfactual unit. Second, time

series evolution and interaction over time is neglected, as data is aggregated over time or treated individually for each

time period. An alternative class of models is identiﬁed by Brodersen et al. (2014), whose approach addresses many

of the previous methods limitations. The authors approach relies on Bayesian state space models which encompass

the outcome’s temporal evolution with exogenous regression components to efﬁciently build a counterfactual model.

State space models allow for ﬂexibility when modelling a variable that is affected by external noise, distinguishing

between a state equation which describe the transition of the latent variable from one point in time to the next one,

and a measurement equation, which describes the accuracy of the signal1. Being fully Bayesian makes it possible

to (i) incorporate prior information about the model structures and parameters and (ii) have a posterior distribution,

and thus a probabilistic uncertainty quantiﬁcation of the causal impact of the intervention. Although the models focus

on one outcome variable and multiple controls, an extension to a multivariate setting has been implemented using

Multivariate Bayesian Structural Time Series (Menchetti and Bojinov, 2020), which is limited to linear relationships

between outcomes and controls and subject to the Markovian assumption of the variables.

2.2 Assumptions

In this subsection, we set up the framework to estimate the causal effect of an intervention on the treated subject. Each

subject yi,t, i.e. each country in our application, is associated with a binary potential outcome yi,t(wi,t)∈Rwhere

wi,t ∈ {0,1}is a treatment assignment indicator with ‘1’ referring to the variable being treated (the UK) and ‘0’ to

the controls (other European countries). Furthermore, deﬁne w1:m,1:T={w1,1:T,··· ,wm,1:T}as the assignment

path up to time Tof all units i= 1, ..., m and denote w1:m,1:Ta realization of this path. As in Menchetti et al.

(2021), throughout the paper we make a set of assumptions to guarantee that the differences in the potential outcome

trajectories are a direct statistical consequence of the intervention. Since some of them can not be directly tested, we

rely on the concept of plausibility in our empirical settings. In particular, we assume the following:

Assumption 1 (Single intervention) Unit ireceived a single intervention if there exist a t0∈ {1, ..., T }such that

wi,t = 0 for all t<t0and wi,t =wi,s =wifor all t, s > t0.

This says that the treatment is single, i.e. it occur at one point in time, and is persistent, i.e. has no disruptions. Then,

we can ease the notation and drop the tsubscript for t > t0

Assumption 2 (Temporal no-interference) For all for all i∈ {1, ..., m}and all t>t0, the outcome of unit iat time

tdepends only on its own treatment path

yi,t(w1:m,t0+1:T) = yi,t(wi,t0+1:T)

If it holds, one can drop also the subscript ifrom wias this assumption asserts that whether or not other units receive

the treatment at time t0, this has no impact on other units’ potential outcome. Units do not interfere with each other

at any point in time. This is the time series equivalent of the cross-sectional Stable Unit Treatment Value Assumption

(SUTVA) Rubin (1974) also known as Temporal SUVTVA from the work of Bojinov and Shephard (2019). In our

1Formally, dropping the subscript i,yt=Z

αt+γxt+trepresent the observation equation and αt+1 = Φαt+Rηtis the

state equation, where t∼ N (0, σ2

)and ηt∼ N (0, σ2

). See Brodersen et al. (2014) for more details

APREPRINT - OCTOBER 7, 2022

empirical study, each countries vaccination plan is conﬁned by the country’s border and does not affect other countries

rate of contagion or number of deaths. The main underlying observation around this assumption is that, during the

period analysed, each country was isolated due to government restriction. Thus, people mobility was prohibited or at

least signiﬁcantly limited. Even after the intervention, we likely expect other European countries mobility not to affect

the UK number of deaths or contagion rate.

Assumption 1 and 2, allow as to simplify the notation so that we can use yi,t(w)to indicate the potential outcome

of a generic unit iat time t. Thus, the observed outcome for t>t0is yi,t(1) while yi,t(0) is the unobserved or

counterfactual potential outcome which has to be estimated to measure the causal impact of the intervention.

Assumption 3 (Covariates-treatment independence) Denote xi,t the vector of exogenous variable that are predic-

tive of yi,t. For t>t0those covariates are not affected by the intervention

xi,t(1) = xi,t(0)

These covariates help improve the outcome prediction but they produce an estimation bias if they are inﬂuenced by the

treatment. For the analysis, we don’t expect that a earlier vaccination, during the period considered, neither alters the

the number of tests taken nor people mobility, during the same period. We must remark that out treatment is indeed

the quicker vaccination program and not the program just by itself. As a consequence, people could anticipate a mass

vaccination taking place in the future months and adjust their mobility patterns, but we can assume that they would

not travel more because of just being in a country with higher vaccination rate.

Assumption 4 (Non-anticipating potential outcomes) for all i∈ {1, ..., m}, the outcome of the unit iat time t<t0

is independent of the treatment that occur in t0.

yi,t(w1:m,1:T) = yi,t(w1:m,1:t0)

This assumption is usually made in the literature (Bojinov and Shephard, 2019; Callaway and Sant’Anna, 2021)

to afﬁrm that the future intervention has no inﬂuence on pre-intervention statistical units, implying that there is no

anticipation of the treatment effect before t0. In the empirical application, although the government advertised the

forthcoming vaccination campaign, the outcomes, such as the number of deaths, did not shift before the program

took place. Furthermore, people had no way to anticipate that the UK program would have been signiﬁcantly faster

compared to other countries.

Assumption 5 (Non-anticipating Treatment) The assignment mechanism at time t0for the unit idepends only on

past covariates and past outcomes

p(wi,t0=wi,t0|w1:t0−1,yi,1:T, Xi,1:T) = p(wi,t0=wi,t0|yi,1:t0−1, Xi,1:t0−1)

This assumption is the analogous of the unconfounded assignment mechanism (Imbens and Rubin, 2015) in a time

series framework and ensures that conditionally on past yi,1:t0−1, Xi,1:t0−1, any variation in the outcomes are to be

attributed to the intervention. In our setting, the UK set up a faster vaccination campaign just looking at previous

number of deaths and rate of reproduction.

APREPRINT - OCTOBER 7, 2022

2.3 Causal Estimands

Let δi,t =yi,t(1) −yi,t(0) be the individual level (UK) causal effect at time t, then the additive causal effect on the

subject iat time tis the population average treatment effect and it is given by

τi,t =E(δi,t|xi,t).

We are also interested in the uncertainty surrounding the treatment effect. This can be either be measured through the

variance

i,t =V(δi,t|xi,t),

or directly applying quantile functions to calculate credible regions

qδ

i,t(α) = F−1

α(δi,t|xi,t),

with levels of conﬁdence generally set to α= 95%. We aim to estimate these values from a dataset D={X, y,w},

which involves T=Pm

i=1 Tisamples of different time series. The main challenge is that we only observe one of the

potential outcomes for every subject i, which implies that the treatment effect is unobserved, so we cannot directly

estimate τi,t.

In addition to its point-wise impact, we are interested in the cumulative effect of the intervention over-time

Ti=

t=t0+1

τi,t (1)

where t0represents the time in which the intervention takes place. The cumulative sum is a suitable measure when

yi,t is a ﬂow variable which is measured over an interval of time (e.g number of deaths in a country). This quantity

however loses its interpretability when yi,t is a stock variable, i.e. a quantity measured at a speciﬁc time, representing a

quantity existing at that point in time (e.g rate of infectiousness). In this case, it is more meaningful to use the average

treatment effect of the intervention

τi=1

Ti−t0

t=t0+1

τi,t =Ti

Ti−t0

(2)

This measure extends S¨

avje et al. (2019) to the time series framework of the average distributional shift effect since

here it is averaged across time as opposed to units.

Within a GP framework, expected values and variances are straightforward to derive and we have that

δi,t ∼ N(τi,t, %2

i,t). Sometimes however, Gaussian likelihoods may not be appropriate and some mathemati-

cal transformations may be needed. For example a random variable which takes only non-negative values (e.g. counts,

lengths) would be better represented using a log-normal distribution. Thus, if a random variable δi,t = log(δ∗

i,t)has a

Gaussian distribution, then δ∗

i,t ∼log N(τ∗

i,t, %∗2

i,t)is log-normally distributed. By directly modelling the transformed

variable one obtains that the causal effect given by

δ∗

i,t = exp (log yi,t(1) −log yi,t(0)) = yi,t(1)

yi,t(0)

Then, taking the expectation E[δ∗

i,t|xi,t] = τ∗

i,t and using the fact that δ∗

i,t is log-normal

τ∗

i,t =Eyi,t(1)

yi,t(0)xi,t= exp τi,t +%2

i,t

2!,

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ASSESSINGCAUSALEFFECTSOFINTERVENTIONSINTIMEUSINGGAUSSIANPROCESSESGianlucaGiudice,SaraGenelettiandKonstantinosKalogeropoulosDepartmentofStatisticsLSEg.giudice@lse.ac.uk;k.kalogeropoulos@lse.ac.uk;s.geneletti@lse.ac.uk;October7,20221IntroductionRecently,manyapplicationshavebeendevotedtounderstandingan...

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ASSESSING CAUSAL EFFECTS OF INTERVENTIONS IN TIME USING GAUSSIAN PROCESSES Gianluca Giudice Sara Geneletti and Konstantinos Kalogeropoulos

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