but unit-invariant fixed effects that can be differenced out by taking time-dummies. _Zi_ are the observed variables that affect income per capita, while _μi_ are the unobserved variables that affect income per capita. _εit_ are the error terms with zero mean.

The synthetic controls method creates a control unit for each country by assigning a certain weight to each country in the donor pool. There can be many such combinations of weights, which is denoted by the vector _V_ = ( _w_ 2 _, w_ 3 _, ...., wJ_ +1) which includes many such synthetic controls.

Abadie et. al. show that if there are weights where the weighted income per capita before the intervention matches the real income per capita, and the weighted value of the observed covariates matches the real values of the covariates before the intervention, then the size of the bias in the difference between the post-intervention incomes for the real and the synthetic unit go to zero as pre-intervention period increases. They also show that this result holds when we have a close but imperfect match.

Mathematically, such weights are given by the set ( _w_ 2<sup>_∗, w_</sup> 3<sup>_∗, ...., w_</sup> _J_<sup>_∗_</sup> +1<sup>)where</sup> _J_ +1 � _j_ =2<sup>_w_</sup> _j_<sup>_∗Yj_1=</sup><sup>_Y_11</sup><sup>_,_�</sup><sup>_J_</sup> _j_ =2<sup>+1</sup><sup>_w_</sup> _j_<sup>_∗YjT_0=</sup><sup>_Y_1</sup><sup>_T_0</sup><sup>_,_and �</sup><sup>_J_</sup> _j_ =2<sup>_w_</sup> _j_<sup>_∗Zj_=</sup><sup>_Z_1holds.</sup> Abadie et. al. (2007) provide mathematically that in these cases, the biases created by time variant country characteristics tend towards zero, as long as the pre-intervention period is long enough, and the match between the pre-intervention trends are strong. We point the reader to Abadie et. al (2007) who prove this point mathematically in Appendix B to their paper.

Now, the key is to find weights such that the distance between the treated country and the synthetic control unit in the pre-reform period in terms of both GDP per capita and predictors of GDP per capita is as close to zero as possible. We use the Stata program provided by Abadie et. al. to find the best possible pre-reform matches for the country in each case study. For the method to yield accurate results, there are certain necessary conditions, which we describe next.

# **3.3 Feasibility of Synthentic Control Case Studies**

In order for an empirically rigorous case study that uses the synthetic controls method to effectively isolate the impact of certain interventions, the following broad criteria have to hold.

- Significant Intervention - The intervention has to be significant at the level of the unit of treatment, which, in this case, is the country itself. This might be an obvious point, but it is important. An anti-corruption program in a local municipality can be evaluated for its efficiency, but our goal with this research program is to identify and analyze the impact of various larger programs and policies. That can only be done if the intervention is done at the national level, or done at a local unit that makes up a large part of the national unit.

- Sustained Intervention - The intervention has to be sustained through a significant period of time. If the intervention is repealed after a year or two, then the possibility of it affecting variables of interest becomes lower, and the analysis cannot detect any effect.

- Absence of Significant Exclusive Shocks - The effect of the intervention can only be estimated accurately if the period of intervention was not characterized by significant shocks that affected the treated unit exclusively. In this case, let’s say for example that country X underwent a significant transparency reform program, while at the same time; a large civil war broke out inside the country. It is impossible to isolate the effects of the transparency program given that conflict affects production and allocation of resources. In a similar vein, global business cycle shocks are not problematic to this endeavor if they affect the treated unit and the donor pool similarly. This can be tested and the donor pool can be shrunk to meet this requirement as well.

- Non-Interference Across Units - The effect of the intervention cannot have an effect on the untreated unit. In this research project, an intervention cannot have a large effect on the

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