I am trying to analyse the impact of a cash stipend program, onto child learning outcomes. I have first modeled the conditional probability of each student receiving this program. So basically I have generated the propensity score of each student of receiving (or not receiving) the program based on their observed characteristics. Now I want like to estimate the impact of the treatment onto child learning outcomes. I want to use the diff-in-diff strategy for this.
My question is that do I need to use matching methods before I can use the diff-in-diff strategy to estimate the treatment effect? Or can I directly use diff-in-diff where I use the propensity scores in lieu of the treat variable? Which would imply that my treat*time interaction term would give me the required treatment effect (where treat would be the propensity score values).
In the conventional diff-in-diff set up, the treat (and also treat*time) are binary variables (where treat=1 if group receives treatment, and is 0 otherwise; time=1 if treatment is provided in time period 2, and is 0 otherwise). In my set up the treat variable would be the propensity score, which is a range of values from 0-1. I.e: the propensity score is not dichotomous.
Can I still apply diff-in-diff with this treat variable? Or do I need to match treatment and control groups (and if so, which method would be the best way to go about it) before I can estimate the treatment effect of through diff-in-diff?
Your help would be much appreciated! Thanks!