vided that the number of pre-intervention periods is large relative to the scale of the transitory shocks. The intuition behind this result is that if conditions (a) and (b) are true, then it must be that the weighted values of the relevant unobserved characteristics of the control group best resemble those of the treated unit.

In practice, suppose _X_ is a set of variables that includes some of the observed covariates ( _Z_ ) as well as the outcome _Y_ for some of the pre-intervention years. These are the “predictor” variables. The synthetic control weights _wj_<sup>_∗_are selected to minimize the distance between the predictor vari-</sup> able values of the treatment unit (that is, _X_ 1) and the synthetic control (that is, the weighted mean _Xs_ of the control group).<sup>22</sup> We implement the method in Stata using the synth package. The synth package implements the procedures outlined in Abadie and Gardeazabal (2003) and Abadie et al. (2010), and optimizes the choice of weights using a nested procedure.<sup>23</sup> Following the standard practice in synthetic control studies, we also conduct placebo tests to address the uncertainty about how well the synthetic control reflects the counterfactual (Abadie et al. (2010); Abadie et al. (2015)). It would be difficult to argue that the synthetic control estimates truly reflect the impact of the intervention if it is common to obtain estimates of similar or larger magnitudes in placebo treatments.

Note that a significant difference in post-intervention outcome between the treated unit and its synthetic control – as measured by size of the post-intervention Mean Squared Errors (MSE) – would convince us that the intervention had an impact only if the synthetic control matched the treated unit’s pre-intervention outcome closely; that is, if the pre-intervention MSE was low. Hence, it is the ratio of the post and pre-intervention MSEs that matters for inference, with a higher MSE ratio in the true treatment case adding to our confidence in the estimate. The inferential

> 22This distance is typically measured as the squared sum of deviations, wherein the squared difference between the treatment unit and synthetic control value is calculated for every predictor variable and summed up.When calculating this squared sum of deviations, the predictor variables are weighted according to their predictive power on the outcome variables. These predictor variable weights should be chosen so that the resulting synthetic control closely tracks the relevant outcome of the treatment unit in the pretreatment period.

> 23The procedure first calculates starting points for the predictor variable weights using three approaches - a) data-driven regression based method; b) equal weights; and c) weights calculated using Stata’s Maximum Likelihood search procedure. Given a particular set of predictor variable weights, it then calculates the corresponding synthetic control _wj_ weights, and the resulting mean squared prediction error in the outcome variable over the pre-intervention period. It then uses a nested optimization procedure that searches among predictor variable weights (and corresponding _wj_<sup>_∗_weights) for the best fitting convex combination</sup> of the control units. Among the three approaches for selecting a starting point, the approach that gives the best result is then adopted.

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