Dynamic treatment timing in a panel-DiD framework

Question

I have a question regarding the timing of treatment effects and how one could use the difference-in-difference estimator on a panel data set.

Let me begin by saying that I have a big firm level unbalanced panel dataset with large N (7000-ish), small T (varies from 3 to 28). I'm interested in analysing the effects of a particular policy intervention, however, the problem is that the treatment timing is different for different firms and I'm wondering how (and if) one can account for this in the DiD framework.

As I understand it, the general DiD panel data setup with simultaneous treatment would look something like this: $$ y_{it} = \alpha_0 + \alpha_1 \text{Treat}_{i} + \alpha_2 \text{After}_{t} + \delta (\text{Treat*After})_{it}+ x_{it}'\beta + \text{FFE}+\text{TFE} + \varepsilon_{it}, $$ where:

• Treat = 1 if in the treatment group, 0 otherwise
• After = 1 if after policy intervention, 0 otherwise
• $x$ is a vector of controls, $\alpha_i$ are the parameters/constants and $\delta$ is the treatment effect
• FFE are the firm fixed effects and TFE are the time fixed effects

After searching for the site I haven't been able to find an answer I fully understand, however, this post as I undstand it suggests running a model along the lines of:

$$ y_{it} = \alpha_0 + \alpha_1 \text{Treat}_{i} + \delta \text{Policy}_{it}+ \sum^T_{t=2} \alpha_t \text{year}_t+x_{it}'\beta + \text{FFE} + \varepsilon_{it}, $$ where:

• "...policy is a dummy for each individual that equals 1 if the individual is in the treatment group after the policy intervention/treatment..." (from the post linked above)
• year are a set of time dummies

I guess I have two things I don't understand with this approach;

1. The construction of the dummy(s) $\text{Policy}_{it}$. Is this one dummy variable just taking the value one for each treated firm after their time varying treatment takes place? Or does the author of the post mean one dummy for each firm indicating the timing of treatment?
2. My second question relates to the first but is more conceptual. To my understanding the difference-in-difference approach is about using the non-treated as a counterfactual outcome for the treated (assuming parallel trends) - in absence of treatment. However, when the treatment timing is different for different firms in this case, there is no clear "after period" for the control group and I beleive this is the cause of my confusion here. What is the conceptual idea behind this approach suggested in the previous link? Is this approach even remotely possible or should one apply some different identification strategy? In that case, what would be appropriate given the circumstances?

Any answers, references to papers or books working with panel data sets with different treatment timings (preferably of econometric nature) would be greatly appreciated.

//Billy

Answer 1

You construct the policy dummy the way you first describe it, i.e. create a column of zeroes. Then for each firm you replace this with ones if a firm is in the treatment group AND it is in the post-treatment period. Something like this

$$ \begin{array}{ccccc} \text{firm} & \text{time} & \text{treated} & \text{post} & \text{policy} \\ \hline 1 & 1 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 & 0 \\ 1 & 3 & 0 & 1 & 0 \\ 1 & 4 & 0 & 1 & 0 \\ \hline 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 0 & 0 \\ 2 & 3 & 1 & 1 & 1 \\ 2 & 4 & 1 & 1 & 1 \\ \hline 3 & 1 & 1 & 0 & 0 \\ 3 & 2 & 1 & 0 & 0 \\ 3 & 3 & 1 & 0 & 0 \\ 3 & 4 & 1 & 1 & 1 \\ \end{array} $$

where $\text{post}$ is an indicator for the post treatment period. In your equation above, the $\alpha_0$ and $\text{Treat}_i$ are going to be absorbed in the firm fixed effects.

Regarding the interpretation, this setting makes an assumption which I probably did not state in the previous answer. The assumption is that the treatment effect is the same across all periods. This means that if a firm is treated yesterday and has a gain of 2, then a firm which is treated today also has a gain of 2 (relative to firms which are never treated). I made a graph to show what this assumption means

In case you would like a reference for this, you can check out Jeff Wooldridge's notes on difference in differences and the section on extensions for multiple groups and time periods: http://www.nber.org/WNE/Slides7-31-07/slides_10_diffindiffs.pdf (What’s New in Econometrics? Lecture 10 Difference-in-Differences Estimation, Wooldridge 2007).