Some inquiries on staggered DID model specifications

[Saleh sharmah]

I am assessing the effect of a country level policy disclosure mandate on firms outcomes across 58 countries, some of which have policy mandatory disclosure in place starting from 2005 ( my sample's beginning period), other countries implemented the disclosure mandate at some point during my sample period, while other countries never implemented the disclosure mandate. With this in mind, I constructed this staggered DID model as follows:

Yi,t = α + β1 treated x mandate + δXi,t + μi + πt + εi.

Yi,t represents the firm's outcome during a given year (t), treated is a dummy equal to one if the firm is covered by the mandate, and zero otherwise, the mandate is an indicator equal to one capturing the year during which the disclosure mandate goes into effect and zero otherize, xi,t represents controls, μi represents firm FE, π represents year FE x industry FE, and i cluster at the firm level. Given my above model, I have the following concerns that are acknowledged by my supervisors:

1. the interaction term between treated and mandate will always be one, which might be an issue with being correlated firm FE in the mode.
2. the average treatment might be biased given that some countries have firm observations that are treated from the beginning of the sample period
3. I have three firm-level controls that are dummy variables and two country non variant controls, should I be concerned that they are non-variant variables and might be correlated with the firm FE specified in my model? I should note that I ran this regression model in Stata using: reghdfe Y treated x mandate, controls, absorb (Firm FE, year x industry FE), vce (cluster firm level) and there was no multicollinearity issue or dropped observations.
4. I still do not get why scholars need to use the interaction term between treated and post or mandate, I honestly feel treated should be enough to capture the effect of treatment
5. for your information, this study is related to the accounting and finance academic field .

I would appreciate your thoughts and suggestions on the above 4 concerns and whether or not my model specification and treatment identification is correct.

Much appreciation in advance for your valuable opinion.

[Clyde Schechter]

First, some problems with your model.

mandate is an indicator equal to one capturing the year during which the disclosure mandate goes into effect and zero otherize

This is not correct. If you are going to do it as an interaction, the value of mandate should be an indicator equal to 1 during the year when the disclosure mandate goes into effect and in all years after while it remains in effect. The variable mandate should be zero in any observation where the country does not have a mandate at all, or it has not yet gone into efect.

And εi should be εi,t.

the interaction term between treated and mandate will always be one, which might be an issue with being correlated firm FE in the mode.

No, it won't. It will be zero in all countries that never have a mandate, and it will be zero in observations before the mandate takes effect in countries that do get one. In fact, you should check this in your data before you run your analysis. If the interaction term does not distinguish 0 and 1 in this way, then you have calculated it wrong or your data are incorrect--either way you need to fix it.

the average treatment might be biased given that some countries have firm observations that are treated from the beginning of the sample period

I can't see any reason this would be the case. You should get an unbiased treatment effect estimate regardless. Can you explain what your concern is in more detail?

I have three firm-level controls that are dummy variables and two country non variant controls, should I be concerned that they are non-variant variables and might be correlated with the firm FE specified in my model? I should note that I ran this regression model in Stata using: reghdfe Y treated x mandate, controls, absorb (Firm FE, year x industry FE), vce (cluster firm level) and there was no multicollinearity issue or dropped observations.

The firm-level variables (by which, I assume, you mean variables that are time-invariant attributes of firms) should show up as coefficient 0 and standard error (omitted) in the output table because of their colinearity with the firm fixed effects. If that did not happen, then these variables are not, in fact, constant within firm and you should recheck your data to find the errors. As for observations dropped, that does not happen as a result of colinearity. Perhaps you are thinking about what happens with variables that have perfect prediction in logistic regression: observations with the perfectly predicting value(s) of such variables are dropped. But nothing similar happens with linear regressions.
The omission of the firm-level variables is not a problem. You have stated that you are including them as "control" variables, which means that there is no need to estimate their effects--you just want to adjust ("control") for their effects. The firm-level fixed effects do that automatically. So nothing to worry about.

I still do not get why scholars need to use the interaction term between treated and post or mandate, I honestly feel treated should be enough to capture the effect of treatment

Well, scholars will be scholars. I guess it is done this way to illustrate the correspondence with the interaction term in a simple DID model. It is just a matter of notation, nothing substantive. That said, you have it wrong in one respect: it is not the treated variable alone that could replace this interaction. It is the mandate variable alone (computed correctly as I described above) that would do just as well. In fact, in practice, I think most people coding these things will use the mandate variable alone for this purpose.

for your information, this study is related to the accounting and finance academic field .

Thank you for making that clear. I don't work in those fields and have no real knowledge of them. My comments above are based solely on general statistical considerations. Nevertheless, I do not believe there are any discipline-specific considerations that would override any of these points.

[Saleh sharmah]

Dear Clyde

Thank you so much for your valuable comments and time. If I may follow up on your comments above; I constructed the mandate dummy previously consistent with your explanation, so I was not clear enough in my previous first question. Concerning the average treatment bias, some countries start mandating their firms at the beginning sample, and some countries mandate their firms at some point in the sample period. Hence, my concern is that the treatment effect of firms that have been mandated for longer periods may affect the sample average treatment, given that some firms have only been mandated for a short period. Concerning the control variables, they are dummy variables that are equal to one and zero otherwise depending on a firm's output during a particular year. Regarding this, my previous concern since I have firm FE in place and these variables are dummy variables, is there a concern that they may correlate with the Firm FE? I truly appreciate your response and time so thank you.

[Clyde Schechter]

Hence, my concern is that the treatment effect of firms that have been mandated for longer periods may affect the sample average treatment, given that some firms have only been mandated for a short period.

This may well happen. It can happen if the effect of the treatment changes with duration of treatment. If there is reason to suspect this is the case, then a simple DID model cannot be used. You would need to change the mandate variable from a simple 0-1 variable to a variable whose values are proportional to the impact of the treatment according to the time from adoption of the mandate. That requires some knowledge or judgment about how much difference duration of treatment makes; this is something that somebody with experience in finance/accounting would have to advise you on.

Concerning the control variables, they are dummy variables that are equal to one and zero otherwise depending on a firm's output during a particular year. Regarding this, my previous concern since I have firm FE in place and these variables are dummy variables, is there a concern that they may correlate with the Firm FE?

They will correlate with the Firm FE. In fact they will be perfectly colinear with the Firm FE. But it is not something you should be concerned about. It doesn't matter at all because the Firm FE themselves will properly adjust ("control") for these firm-level variables. And it doesn't matter whether they are dummy variables or polytomous or continuous. As long as they are constant within firm, their effects on the outcome are automatically accounted for by the Firm FE themselves. So don't spend another second thinking about this. It will happen; it's not a problem. Move on.

[Saleh sharmah]

Thanks, Clyde for your follow-up response. it is noted with many thanks.