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In some Difference-in-Difference analyses, is there any common rule for how many "lead" and "lag" years around the event that we should use? For example, in some papers, I saw the author using [-2;+2] but in some papers, I also see the authors using the event window [-2;+5]. From the end of this discussion, I saw a paragraph documenting that in a specific case

You could estimate a finite number of leads and/or lags, or trace out the full dynamics of exposure by saturating the model. If the treatment is transient, then incorporating a full series of time indicators is very demanding. Moreover, some units may opt out of the treatment over time. As you move farther and farther away from the first adoption year, you may have less and less data to estimate your lags. This isn't usually a problem in practice, but you may find that the fifth, sixth, and seventh lags are less precisely estimated.

Louise
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1 Answers1

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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.

In some Difference-in-Difference analysis, is there any common rule for how many "lead" and "lag" years around the event that we should use?

No.

It depends upon the policy under evaluation. Suppose a crime bill was passed in 2016 and the number of visible police on patrol doubled. The goal of the legislation was to reduce the violent crime rate. Only a subset of counties were treated. There is no hard-and-fast rule to determine the number of leads and/or lags to incorporate. Here are some questions to consider:

  • Did individuals living in treated counties anticipate a surge in guardianship? Before the bill was passed, did it receive a lot of coverage in the media? Would law abiding residents in non-treated jurisdictions move to treated jurisdictions in response to the crime bill, or vice versa?

If media coverage was consistent throughout most of 2015, then one lead indicator is worth including. Assuming individuals "perceived" the increased coverage and actually changed their behavior, then it seems warranted to assess anticipatory behavior within treated counties.

  • Did prospective offenders actually "perceive" the surge in police presence? Suppose violent crime decreased in treated jurisdictions. Was the reduction immediate, or did it set in with a lag? If it was immediate, did effects last for a couple of years or was it permanent? If it set in with a lag, in what year did effects emerge?

Suppose theory tells me the effects of the bill will be perceived immediately and then quickly dissipate after the first year. I may include the immediate effect and then the first and second lag. Note the "first lag" as I have defined it is a dummy equal to 1 if the county is a treated jurisdiction and is in the second year post-shock. This is the first full period after the initial adoption year. Likewise, the "second lag" is equal to 1 if the county is a treated jurisdiction and is in the third year post-shock. Including a third, fourth, or even a fifth lag may be appropriate, assuming you're agnostic about how long it takes for effects to dissipate. Note how if you only observed counties up until 2020, then you can't go beyond four lags.

This is just one of many different ways to proceed in practice. I would argue that there is no optimal lead/lag structure. Rather, you should be guided by what theory tells you about how the treatment affects your outcome.

Thomas Bilach
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  • Thank you @Thomas Bilach for your answer, I have some curiousities as below: (1) what does "**irregular exposure periods**" mean ? (2) "**If media coverage was consistent throughout most of 2015, then one lead indicator is worth including**" ? I am wondering why you use "**one lead indicator**" here, I assume you are talking one year for pre-event period.(3) I am quite ambiguous about the "**first lag**", for example, New York passed the law in 2003, so is the first lag 2004 while the immediate effect is 2003? – Louise May 28 '21 at 04:25
  • But regarding my question 3 above, it seems not to be the case, because if the first lag is 1 year after the enforcement year, so, in your last sentence of the penultimate paragraph "Note how....can't go beyond four lags" . it should be five lags rather than four lags assuming you mentioned the number 2015 before that. – Louise May 28 '21 at 04:37
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    (1) It 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. (2) Yes. I mean including a lead indicator for the year *before* treatment. (3) Yes. Suppose New York adopts a policy in 2003. The effect in 2003 is often referred to as the *immediate effect* of treatment. If you observed effects start to emerge in 2004 or later, then we'd often say the treatment kicked in with a lag. – Thomas Bilach May 28 '21 at 23:07
  • Thank you, Thomas, is there any reference for this one "Thus, this estimator may handle irregular treatment patterns", I also create a separate about this curiosity, could you please have a look if it is convenient to you, please? [link](https://stats.stackexchange.com/questions/526619/why-do-not-we-include-post-and-treatment-separarately-in-generalized-difference) – Louise May 30 '21 at 23:02
  • Apart from that, " I mean including a lead indicator for the year before treatment". Do you mean that I set one year before the real event date as the fake event day and run the regression to see if the results still significant? – Louise May 30 '21 at 23:11
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    This estimator *must* be used once we depart from the classical difference-in-differences setting. Once the exposure period isn't well-defined we must regress the outcome on unit fixed effects, time fixed effects, and a treatment dummy which 'turns on' in any unit-time combination where the policy is in effect, 0 otherwise. A nice companion is *Introductory Econometrics: A Modern Approach* by Jeffrey M. Wooldridge. – Thomas Bilach Jun 01 '21 at 22:04
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    A "lead" as I have defined it is a dummy equal to 1 in a year(s) *before* the exposure of interest. You correctly note that if it's indexing a period *before* the event date then it's, technically, a "fake" treatment period. Think about it as purposefully including a pre-period indicator to see if effects emerge *before* the actual event starts. A strong, non-zero effect in the pre-policy epoch may be interpreted as selection bias, unless there's a theoretical justification to suspect units "anticipated" the policy. – Thomas Bilach Jun 01 '21 at 22:15
  • thank you for your answers, they are very clear. However, I am wondering about this sentence " Once the exposure period isn't well-defined we must regress the outcome on unit fixed effects, time fixed effects, and a treatment dummy which 'turns on' in any unit-time combination where the policy is in effect, 0 otherwise". I understand what you mean, but it seems that we cannot plot a chart based on this description, can you have a look at a topic [here](https://stats.stackexchange.com/questions/526787/how-to-plot-the-graph-for-testing-paralell-trend-for-staggered-did) – Louise Jun 01 '21 at 23:28