Synthetic Control Method I came across this journal http://www.hks.harvard.edu/fs/aabadie/ccsp.pdf which basically uses Synthetic Control Method (SCM) to estimate the difference between the impact on a variable when an event happens versus when it does not happen (well at least this is how i understand it). Another research paper that I found uses this method to estimate the impact of being a member of euro versus if it is not a member of euro (they use this on Greece if i'm not mistaken). But I'm a bit caught up on the explanation of the model (the equations, particularly..)
Hence I'm just wondering if anyone of you is familiar with SCM and could briefly explain it a bit in simpler words? To be specific, lets say that if I'm interested to know the impact of being a member of euro on growth rate vs. not being a member, what should I do if I want to use SCM? I notice that they are a few questions about SCM here, but they are all a bit advanced for me.  
Thanks in advance.
 A: Let's start with a standard regression setup where you are trying to estimate the effect in a fixed-effect model when you only have a single country that received "treatment" (entering the Euro). This strategy assumes that, conditional on observables, the average of the other countries serves as a reasonable counterfactual to what your treated country would have done if it hadn't received the treatment (ie stayed out of the euro). There is likely bias in this estimate as the treatment wasn't random, but probably related to factors that also affect the outcome you care about. 
SCM says that instead of using all the other units, let's find a weighted average of them that looks really similar during the pre-treatment. SCM defines "similar" as having a very similar path of your outcome variable pre-treatment while having similar pre-treatment values for covariates that appear to be related to your outcome variable. The "treatment effect" is then the difference between the actual path of your treated unit and the path of the weighted average post-treatment (the weights are fixed from the pre-treatment optimization). Inference is done by permutation tests (if we invented "fake" treatments and estimated SCM on the other units that weren't actually treated, what would be typical "effects" that we would see). SCM will consistently estimate effects in some settings where the FE model would yield biased estimates (Abadie et al. show it for a general factor model, which allows for non-parallel trends between treatment and non-treated units).
The main points SCM needs are:


*

*Lots of pre-treatment periods. The bias in the SCM estimate goes to zero as the number of pre-treatment period grows large in relation to the "error" in the model. Since usually SCM is used on aggregate data, the "error" is not so much sampling error but model mis-specification (inability of the synthetic control be a good counterfactual). This is sort of similar to root-N-consistency with other estimators.

*The unobserved factors that you worry about should have had some effect on the outcome during the pre-treatment period. This is an assumption that one needs to argue holds in your context.

*The synthetic control needs to match well the treated unit during the pre-treatment period. You can gauge this by looking at the fake permutations. Generally you need your treated unit to be in the convex hull of observations during pre-treatment.

A: The impact of being a member of euro on growth rate vs. not being a member is the difference of the growth rate from being a member and the growth rate if the country is not a member. This difference is calculated over different periods after joining the eurozone. The problem is that the growth rate if the country is not a member is not observed. It is a "counterfactual," so it has to be estimated.
For this estimation, the synthetic control algorithm builds a control as a weighted average of countries that did not join the eurozone (the "pool of donors"). How are the weights calculated? By minimizing a Euclidean distance between variables X of the synthetic control and the treated unit, subject to the constraints for the weights (each weight belongs to the interval [0,1] and the sum of all weights is 1). Variables X are presumably not affected by the treatment (joining the euro zone). This minimization problem has a quadratic objective function with linear constraints (linear quadratic minimization problem).
Thus we can say that the synthetic control is similar to the treated unit in the sense that both have similar variables X (in a Euclidean metric sense). But as the pool of donors did not receive the treatment (did not join the eurozone), the growth rate of the synthetic control after joining the euro is an estimation of the counterfactual.
