Difference-in-difference regression design

Question

I'm working with a large panel dataset tracking various units (i-dimension) over an extended period (t-dimension). These units are classified as either 'blue' or 'brown', and some (not all) of these units experience specific events at different points in time. Some of the units experience the event more than once.

I'm interested in assessing the impact on the dependent variable 'Y' around these events using a regression approach. While event studies are commonly used for this purpose, I'm curious if it's feasible to adapt a Difference-in-Differences (DD) model to examine effects around the event date.

Specifically, I'd like to analyze the 30 days following the event, as I expect the most significant reactions to occur during this period. However, I also suspect there may be structural changes beyond these 30 days.

Would the following regression model make sense?

$$Y_{i,t} = Intercept + Blue_i + Event\_Period_{i,t} + Post\_Event\_Period_{i,t} + Blue_i × Event\_Period_{i,t} + Blue_i × Post\_Event\_Period_{i,t} + ϵ_{i,t},$$

Where 'Event_Period' is a dummy variable indicating the [+1, +30] interval for units experiencing the event, and 'Post_Event_Period' indicates the [+31, inf) period for these units.

Is this setup appropriate for cases where units can experience multiple events, resulting in both 'Post_Event_Period' and 'Event_Period' dummies being 1? Additionally, what would be the interpretation of the coefficients in the model above?

I welcome any insights on the best approach to modeling this scenario. Thank you!

Answer 1

I intended to leave a comment but do not have the reputation. I have a set of comments for you and some suggestions.

Firstly, what exactly is the intended quantity of interest? You say you want to quantify the impact on $Y_{it}$ around these events, but the impact (or treatment) is not the event itself? If your theory is simply that the event allows for some other variable to affect $Y_{it}$, then it sounds like you have an interaction model: $$Y_{it}=\beta_0+\beta_1X_{it}+\beta_2 Z_{it}+\beta_3 X_{it}Z_{it}+\varepsilon_{it},$$ where $Z_{it}$ is a dummy variable indicating that the event happened in unit $i$ and time $t$ and $X_{it}$ is the variable you think has an effect on $Y_{it}$.

Secondly, if the event itself is the impact or treatment, you will have difficulty identifying the effect if the treatment is heterogenously applied to units and can occur multiple times. You can look at differences in differences (DiD) with multiple time periods but it is usually assumed that the treatment is applied similarly across units.

You can possibly look into using synthetic controls (e.g., Abadie 2021), where you essentially simulate control units (i.e., who did not experience event) that are similar to the treated units in all ways except the event(s) itself.

If you were insistent on DiD, you would need to include a term for time effects (e.g., $\gamma_t$) and be cautious that your standard errors will likely be incorrect if using more than two time periods (Bertrand et al. 2004). This vignette (Callaway and Sant'Anna 2023) seems particularly relevant for you and perhaps I would start there.

In general, $\underline{\text{Mostly Harmless Econometrics}}$ may be a useful source for you. If you can clarify your quantity of interest this book will help you estimate it and to be aware of the assumptions necessary for the estimate to be any good (Angrist and Pischke 2008).

Abadie, Alberto. 2021. "Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects." Journal of Economic Literature. https://www.aeaweb.org/articles?id=10.1257/jel.20191450

Angrist, Joshua and Jörn-Steffen Pischke. 2008. Mostly Harmless Econometrics. Princeton University Press. https://www.mostlyharmlesseconometrics.com/

Bertrand, Marianne, Esther Duflo, and Sendhil Mullainathan. 2004. "How Much Should We Trust Differences-in-Differences Estimates?" The Quarterly Journal of Econometrics. https://www.jstor.org/stable/25098683

Callaway, Brantly and Pedro H.C. Sant’Anna. 2023. "Introduction to DiD with Multiple Time Periods." https://bcallaway11.github.io/did/articles/multi-period-did.html