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I wonder if you can help me to figure out how to rewrite basic difference-in-difference equation (pictured) so that it takes into account the fact that treatment has occurred at different times for different observations.

This is related to, for example, Diff-in-Diff with different treatment timings for different countries & Specifying a difference in differences model with multiple time periods. I didn't find the equations there quite intuitive (also maybe because my degree is not in economics), so I thought I might ask for a simple explanation for beginners like me.

I have a panel data set of 573 universities over period 2010-2019. Some of these universities got a top rector/president (most academically successful candidate) which is treatment, some didn't get one. The treatment happened in different years: some in 2011, others in 2014 etc. I am trying to figure out how the appointment of a top rector influenced the academic score of the university.

Your help would be so much appreciated! Thank you!

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I reproduced the canonical difference-in-differences (DiD) equation from your question below:

$$ y_{it} = \gamma Treat_{i} + \gamma Post_{t} + \delta(Treat_{i} \times Post_{t}) + \epsilon_{it}, $$

where, for example, we observe universities $i$ in years $t$. The subscript $i$ usually represents an aggregate unit (e.g., individuals, universities, counties, states, countries, etc.); some of these units receive some treatment/intervention, while others do not. The generalization of this equation, which allows for staggered treatment adoption, regresses your outcome $y$ (i.e., academic score) on a treatment indicator, and dummies for each university and each year. The following specification is a 'generalized' DiD equation which takes the following form:

$$ \text{Score}_{it} = \gamma_{i} + \lambda_{t} + \delta \text{President}_{it} + \epsilon_{it}, $$

where $\text{President}_{it}$ is equal to 1 for universities receiving the new president/rector—and only during years $t$ when the president/rector is actually serving in this position. $\gamma_{i}$ denotes university (unit) fixed effects; $\lambda_{t}$ denotes year (time) fixed effects. Your model will result in 572 separate “university” effects and 9 separate “year” effects. This may seem unwieldy, in practice, but functions exist in most software packages (e.g., R/Stata) to avoid extraneous output. Note, these fixed effects replace $Treat_{i}$ and $Post_{t}$, respectively, in the former equation. Your causal estimand of interest should be $\delta$.

The treatment variable $\text{President}_{it}$ is your interaction term $(Treat_{i} \times Post_{t})$. In the more general DiD setting, though, $Post_{t}$ is not well defined. Instead of specifying this interaction manually, we explicitly code a treatment dummy to reflect early/late adopter universities. Again, $\text{President}_{it}$ should be equal to 1 in only those university-year combinations when the treatment (i.e., president/rector assumes position) is in effect, 0 otherwise. For universities never receiving a new president/rector, it should be coded 0 for the entire observation period.

I encourage you to review this post which details the coding of the treatment dummy in greater detail.

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