Synthetic Control Weight Selection

Question

I have implemented synthetic controls in two of my dissertation chapters. In a recent seminar, someone asked me a question I had never encountered. A cursory scan of the literature also seemingly does not address this issue.

I understand mathematically that the weights are chosen to minimize the weighted norm of the difference between the pre-treatment predictor variables. I also get the fact the weighting matrix is the positive-definite diagonal matrix that minimizes pre-intervention root mean square predicted errors.

That said, how are initial weights chosen by synthetic controls? Secondly, how does the command converge to sets of weights that satisfy the regularity conditions? In other words, how do Stata or R arrive at the set in terms of mechanical implementation?

Answer 1

Here is a snippet from a paper my co-author discovered.

It seems like SCM constructs these predictor weights every period prior to intervention. I’m assuming they are regressing the outcome on itself, too. That would make their argument in the paper clear that other predictors don’t matter.

That makes sense why the search in Stata is so quick. Note, there is a grid search. I'm assuming it uses the predictor weight values as the starting point for the linear optimization problem.