Difference in Differences: one treatment occurring at the same time, many years, 3 groups

[Simone Chinetti]

Dear all,

Probably the following question has been already asked, but I am not able to find a good questions.

I have estimated the following DID model:

y_{it} = \beta_{0} + \beta_{1} Treat_{i} + \beta_{2} Post_{t} + \beta_{3} Treat_{i} * Post_{t} + error

Or using individual and time fixed effects:

y_{it} = \lambda_{t} + \mu_{i} + \delta Treat_{i} * Post_{t} + error

where Treat=1 if the individual belongs to the treatment group, Post=1 in the post-treatment.

I have a panel data structure, with the same individuals over the period 2010-2015. In particular, 2010-2013 is the pre-treatment period and 2014-2015 is the period during which the treatment is effective.

I can distinguish individuals in 3 groups, and I suspect that there is some heterogeneity among these sub-groups of individuals. Hence, I would like to estimate a DiD model that allows for heterogeneity in the response to the treatment.

According to my understanding, in order to do so I should estimate the following (supposing that groups are: small, medium, large):

y_{it} = \beta_{0} + \beta_{1} Treat_{i}*small + \beta_{2} Treat_{i}*medium + \beta_{3} Treat_{i}*large + \beta_{4} Post_{t}*small + \beta_{5} Post_{t}*medium + \beta_{6} Post_{t}*large + \beta_{7} Treat_{i} * Post_{t}*small + \beta_8} Treat_{i} * Post_{t}*medium + \beta_{9} Treat_{i} * Post_{t}*large + error

or using fixed effects:

y_{it} = \lambda_{t} + \mu_{i} + \gamma_{g} +  \delta_{1} Treat_{i} * Post_{t}*small + \delta_{2} Treat_{i} * Post_{t}*medium + \delta_{3} Treat_{i} * Post_{t}*large + error

Am I right ? That is, have I understood the way to allow for heterogenous response in a DiD approach ?

Thank you very much in advance.

[Clyde Schechter]

Yes, this is the right approach. In Stata code, which is far more parsimonious, it looks like this:

panel_regression_command outcome_var i.group##i.treat##i.post i.time, fe

where group is your small/medium/large variable, and treat and post are as you defined them previously.