I am trying to find causal effects of a policy change in the UK that allows graduates with a student visa to stay in the country after their graduation to find work without having to return back home and its effect on the unemployment rate for the affected group.
I only have aggregated data on graduate labour market statistics stating the employment rate for different ethnicities of graduates in a certain year for two different age groups, meaning that the data is not on individual level. Since I don't exactly know which graduates required a visa I made the decision to put them into treatment and control group according to their ethnicity.
Now my question would be if I can still use diff-in-diff estimators to estimate the causal effect. So far I only have data on 2 years pre treatment and 2 years post treatment, which leaves me with a total of N = 16. Apart from the fact that the sample size is super small, will this have any implications other than the standard errors being different? I expect the coefficients to stay the same... Should I try to keep the amount of pre and post treatment periods balanced or should I try to find more data pre treatment? Is it feasible at all to include control variables with such a small number of samples?