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The Rubin Causal Model (RCM) is a framework that allows to define the causal effect of an intervention as a contrast of potential outcomes. In recent years, several methods have been developed under the RCM to estimate causal effects in time series settings. None of these makes use of ARIMA models, which are instead very common in the econometrics literature. In this paper, we propose a novel approach, named Causal-ARIMA (C-ARIMA), to define and estimate the causal effect of an intervention in observational time series settings under the RCM. We first formalize the assumptions enabling the definition, the estimation and the attribution of the effect to the intervention. We then check the validity of the proposed method with a simulation study. In the empirical application, we use C-ARIMA to assess the causal effect of a permanent price reduction on supermarket sales. The CausalArima R package provides an implementation of the proposed approach.

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Synthetic control methods have been widely used as an alternative to difference-in-difference methods to estimate causal effects in panel data where a subset of units receive a single persistent intervention, and the rest are unaffected by the change. In many applications, however, units that do not directly receive the intervention can still be impacted by the change because of interactions between units, also known as interference. For example, when a supermarket reduces the price of a store-brand of cookies, the policy change can impact the sales of both the store brand and its direct competitors. In this paper, we provide a framework for extending synthetic control methods to the setting of partial interference, which occurs when an intervention impacts units within predefined groups, but not across the different groups. We define three new classes of causal estimands that capture how a change impacts the focal unit as well as the other units within the group. To perform inference, we develop a multivariate Bayesian structural time series model that provides a flexible method for generating synthetic controls that would have occurred in the absence of an intervention. We further provide a Markov chain Monte Carlo algorithm for posterior inference and explain how to use the resulting draws for estimating our three new causal estimands. In a simulation study, we explore the empirical properties of our Bayesian procedure and show that it achieves good frequentists coverage even when the model is mildly misspecified. Our work is motivated by an analysis of the effectiveness of a marketing campaign by an Italian supermarket chain that permanently reduced the price of hundreds of store-brand products. We use our new methodology to make causal statements about the impact on sales of the affected store-brands as well as their direct competitors.