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In a generalized Difference-in-Difference, Dasgupta, 2019 using this equation

$Y_{it}$ = $\alpha$ + $\beta$ $(Leniency Law)_{kt}$ + $\delta$$X_{ikt}$ + $\theta$$_t$ + $\gamma$$_i$ +$\epsilon$$_{it}$ (1)

The variable of interest here is $(Leniency Law)_{kt}$. This variable equals 0 before the passage of the leniency law in country $k$, and 1 afterward. Simplistically speaking, it is postxtreatment in DID setting. I am wondering why he did not add the variable post and treatment separately into equation (1) as the basic DID setting.

I read a clue here from a discussion but it is not totally clear to me.

The referenced post assumes you're estimating a generalized difference-in-differences equation. The model includes unit fixed effects, time fixed effects, and a binary treatment treatment. It is useful in settings with irregular exposure periods.

"irregular exposure periods" typically means any setting where the treatment starts at different times for different entities. It may start early for some and later for others. It may even switch 'on' and 'off' over time. Thus, this estimator may handle irregular treatment patterns.

What I want to focus on is how it is useful or how "this estimator may handle irregular treatment patterns"in this case.

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    $\begingroup$ I go into some detail here. The fixed effects will absorb "treat" and "post" in this setting. For example, a simple dummy variable indexing the "group" of treated/untreated firms is already captured by the firm fixed effects. Remember that the firm fixed effects adjust for all time-constant factors specific to your firms, which includes a firms's membership to the group. Try estimating a dummy which only tells you whether a firm is in a treated group or not. Software will usually drop it for you. $\endgroup$ Jun 2 at 6:33
  • $\begingroup$ @ThomasBilach Thank you so much for your further clarification $\endgroup$
    – Louise
    Jun 2 at 6:37
  • $\begingroup$ @ThomasBilach after a couple of days of asking and accumulate knowledge, now I understand clearly your statement as above. Just come back and say thank you very much! $\endgroup$
    – Louise
    Jun 8 at 5:37
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Basically it is because of difference in treatment timing. There is no single time $t^\star$ such that for any group who is at some time treated, it will be treated at any time after $t^\star$. When there is no single time of treatment initiation it is impossible to define a single post treatment variable.

In the staggered design several groups are treated but the are treated in different timeperiods. Look at these sequences - treatment histories - and ask yourself when do post-treatment time being?

Nevertreated ... $$(0,0,0,0,0,0,0,0)$$ Early treated $$(0,0,1,1,1,1,1,1)$$ Late treated $$(0,0,0,0,0,1,1,1)$$

Obviously, the answer depends. In some case you could then estimate different treatment effects for different groups. But in the case you are considering the different countries adopt the laws at very different times and anyway the interest may be in a single estimate.

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    $\begingroup$ thank you for your answer. (1) From my point of view, I want to argue about your question above. I think a country with a specific adoption year of law will have a binary variable called treat to denote whether it is before or after the event. For example, treat equals to 0 when it is before the event year, and 0 otherwise. After all, we can concatenate all the treatment and control together to run a regression on the extended equation (my equation above and treat and post variables). Please correct me if I fall into any fallacy $\endgroup$
    – Louise
    May 31 at 21:48
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    $\begingroup$ (2) And yes, I agree with you that the interest at the end is a single estimate to see the impact of the laws on a dependent variable in general. Is it what you said "econometricians are interested in causal estimates" here link $\endgroup$
    – Louise
    May 31 at 21:49
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    $\begingroup$ "before or after the event" will be swallowed by time fixed effects. If you make it country specific so 1 before event but only for the particular country then you are on the road towards making a model where you try to estimate one particulare treatment effect per country. $\endgroup$ May 31 at 21:55
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    $\begingroup$ I am not saying you cannot do it. With the sequences I illustrate you could do a 2x2 difference in difference comparing early treated with never treated and you could run another 2x2 difference in difference comparing late treated to never treated. But the you will have 2 estimates. And again in your case you do not only have 2 different times at which treatment are started but many. $\endgroup$ May 31 at 21:57

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