A.1 Conditional MSE of the treatment effect estimator The expression for the conditional mean squared error used in Section 2 can be derived as follows. 's as the only source of randomness in the above expression and assuming that they are Abadie et al., 2010), or the assumption that treatment periods are themselves chosen at random and In this section we present the exact mixed-integer programming formulations that can be used for solving the proposed models in one of the available academic or commercial solvers. SCIP (Gamrath et al., 2020) which can handle mixed-integer nonlinear programs (MINLP's) with We need two additional observations to formulate the problem as a quadratic objective with linear constraints. 's can be carried inside the The problem becomes more complicated when there is no constraint on the number of treated units. In this section we provide a proof of Theorem 1. A B null which is independent of the index l .

Estimating treatment effects in time series data presents a significant challenge, especially when the control group is always unobservable. For example, in analyzing the effects of Christmas on retail sales, we lack direct observation of what would have occurred in late December without the Christmas's impact. To address this, we try to recover the control group in the event period while accounting for confounders and temporal dependencies. Experimental results on the M5 Walmart retail sales data demonstrate robust estimation of the potential outcome of the control group as well as accurate predicted holiday effect. Furthermore, we provided theoretical guarantees for the estimated treatment effect, proving its consistency and asymptotic normality. The proposed methodology is applicable not only to this always-missing control scenario but also in other conventional time series causal inference settings.

Airlines have been making use of highly complex Revenue Management Systems to maximize revenue for decades. Estimating the impact of changing one component of those systems on an important outcome such as revenue is crucial, yet very challenging. It is indeed the difference between the generated value and the value that would have been generated keeping business as usual, which is not observable. We provide a comprehensive overview of counterfactual prediction models and use them in an extensive computational study based on data from Air Canada to estimate such impact. We focus on predicting the counterfactual revenue and compare it to the observed revenue subject to the impact. Our microeconomic application and small expected treatment impact stand out from the usual synthetic control applications. We present accurate linear and deep-learning counterfactual prediction models which achieve respectively 1.1% and 1% of error and allow to estimate a simulated effect quite accurately.

Appropriate treatment regimens play a vital role in improving patient health status. Although some achievements have been made, few of the recent studies of learning treatment regimens have exploited different kinds of patient information due to the difficulty in adopting heterogeneous data to many data mining methods. Moreover, current studies seem too rigid with fixed intervals of treatment periods corresponding to the varying lengths of hospital stay. To this end, this work proposes a generic data-driven framework which can derive group-treatment regimens from electronic medical records by utilizing a mixed-variate restricted Boltzmann machine and incorporating medical domain knowledge. We conducted experiments on coronary artery disease as a case study. The obtained results show that the framework is promising and capable of assisting physicians in making clinical decisions.