Difference-in-Differences

I have been reading up on the diff-in-diff estimator (following MHE). However, my lecturer said something that confused me:

Say I have panel data of hospitals in low-income neighbourhoods and a variable for patient's health outcome. Now a new policy is introduced such that some hospitals are privatised during the sample period while the rest of the hospitals are privatised at some later date. My initial thought was to consider the hospitals who received converted in the sample period as the 'treatment group' and hospitals who converted later as the 'control group'. And then follow the canonical approach and compares pre- and post differences in health outcomes. However, the problem is that not every hospital in the 'treatment group' converted at the same time. And apparently this means that diff-in-diff is not suitable.

However, I came across this post which makes it sound as if it is possible to use diff-in-diff. In the above link the suggested answer is to estimate:

$Y_{st} = \alpha + \gamma_s\text{Treat}_s + \lambda (\text{year dummy}_t) + \delta (\text{Treat}_s \times d_t ) + \epsilon_{st}$

In my case $\text{Treat}_s$ would be 1 if the hospital was privatised in the sample period. $d_{t} = 1$ if the observation occurs after conversion. Thus the interaction term is 1 for the treatment-units in the post-treatment period.

Although the OP in the above link doesn't have units receiving treatment at different times. My main source of confusion is how to think about the 'post-treament' dummy in cases such as this.

• Welcome CV, Kronecker! You have the start of a good question here, but it would be very much improved if you could related the pertinent details from your link. You can make such improvements by clicking the "edit" link in the lower left. As a rule, questions and answers on CV should not require visitors to go 'read up' elsewhere in order to be fully understood (although links to further or supporting info, or even simply citations are welcome). Dec 6, 2017 at 19:46
• Thank you Alexis. I have made the recommended changes, hope it clarifies things. Dec 6, 2017 at 19:58