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Understanding Propensity Score Matching with Cross Sectional Data

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Hi,
Let me start with a simple summary of my understanding of matching with Panel data:
  • ​Usually when we use propensity score matching we have an event that happens at time T and we want to estimate the effect of that event (a new law or a training or whatever) on the participants.To do so we compare the people that receive the treatment (the treatment group) and the (control group) before and after the event. The difference between the treated group before and after the event and the the control group before and after the event, gives a good estimation of the effect of the event/treatment. However due to selection bias this estimation might be incorrect. So in order to better compare the treated group with the control group we match the observation that we have in each group based on the probability of participating given some PRE-treatment characteristics. This works better with panel data.
In my case I have pooled crossed sectional data and the treatment condition is being a US firm (treatment group) while the control group is NOT being a US firm. In order to look at the effect of my X on firm's Y I need to find firms that are identical. Someone suggested to use propensity score matching (where being a US firm is the treatment). However since I don't have a panel data and I can't observe before and after "becoming a US firm" event, I don't understand how matching would be any different than a regression where I control for some firm's specific characteristics.

Can anyone help?
Thank you

Marco

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