One popular technique for causal inference is difference-in-difference (DiD) estimation. Under DiD, there is one cohort of observations with both a pre and post intervention time period and another group that was not exposed to the intervention. The time trend of the group without an intervention in essence serves as a control group for the observation(s) where the intervention took place. However, the control cohort is only a valid comparison if the pre-intervention time trends in both groups were similar.
What can one do if the time trends of the intervention and control groups are not similar?
One approach is to use synthetic control method (SCM). The method was developed by Abadie and Gardeazabal (2003) and Abadie, Diamond and Hainmueller (2010). Rather than use comparable observations either individually or as a group to serve as the control group, the synthetic control method uses a weighted average of non-intervention observations to serve as a control group. The weights are chosen to minimize the difference between the pre-intervention control group characteristics and the pre-intervention treatment group characteristics. To implement SCM, data must be available for several periods prior to the treatment intervention. The SCM creates this hypothetical counterfactual region by taking the weighted average of preintervention outcomes from selected…regions. The…regions that combine to form the synthetic control are selected from a pool of potential candidates. Predictor variables that affect the outcome and the outcome variable itself before the policy is enacted determine the selection of donor regions and weights. The resulting synthetic closely matches the affected region’s outcome before policy enactment and is a control for the affected region following enactment. Note that this counterfactual control group is comprised of a a fixed combination of donor states; in other words, the weights do not change over time.
Mathematically, we want to minimize:
D(ω) = (x1 – X0 ω)’V (x1 – X0 ω)
where x1 is a (Kx1) vector is the intervention group’s pre-intervention K characteristics, X0 is a KxJ matrix of is the J control group observation’s K characteristics, ω is a Jx1 vector of weights. The weights must be nonnegative and sum to 1. Finally, V is a non-negative KxK diagonal reflecting the relative importance of the different predictors.
Let y be the outcome variable of interest where y1 is the outcomes for the intervention group and y0 is the outcomes for the control group. The synthetical control is just the weighted values that better match the treatment group. That is y*1=y0ω*, where ω* is the estimated weights.
Which predcitors should one pick? An Urban Institute paper provides some guidance:
Ideally, those predictors have a stable relationship with the outcome variable. The predictors’ ability to explain variation over the pretreatment years, however, is less important because only their time averages over pretreatment years are used when creating the synthetic state…. Including the lagged outcome variable for some pretreatment years is common, and Athey and Imbens (2006) even state that including the other covariates rarely matter.
Packages for Stata and R for implementing SCM are available on Dr. Hainmueller’s website. The Synth on Dr. Hainmueller’s website code uses the regression-based method to select predictor weights as the defaul, but using the “nested” option results in applying a fully nested optimization method that, while taking longer, generally results in a better fit.