Difference-in-difference multiple time periods

[Sander Voldsund]

Dear all

For my research, I want to estimate the effect of introducing direct election of mayors in Norway. These reforms were introduced in different years. The first election with direct election was I 1999-2003 where 20 municipalities introduced direct election. In 2003-2004, 34 municipalities introduced direct election, but some of the municipality from 1999 did not continue to elect their mayor directly anymore. For the period 2007-2011, 50 municipalities elected their mayor directly, but again some of the municipality from 2003 didn’t continue to elect their mayor directly anymore.

I want to use a Difference-in-Difference model using panel data. The treatment is a binary variable “directmayor” taking value 1 if municipality have direct election and 0 otherwise. I have used election period 1995-1999 as the years before any treatment occurred. I have used Fixed effects and the following stata command:

xtreg lnADmCost directmayor i.year if year>=1995&year<2012, fe cluster(knr)


Do I get a difference-in-difference estimate and is this the correct way to do this?

Does it matter that some of the municipality doesn’t have treatment the whole period after introducing it?

Is it possible to separate between election periods? Can I use a diff-in-diff for the period 1999-2003 and 2003-2007 and 2007-2011? Can someone explain how and correct the stata commands?

[Clyde Schechter]

Do I get a difference-in-difference estimate and is this the correct way to do this?

The validity of your model depends on exactly what you mean by "The treatment is a binary variable “directmayor” taking value 1 if municipality have direct election and 0 otherwise. "
If your variable directmayor takes the value 1 in any observation where that city had direct mayor election in that year, and 0 otherwise. Then it is fine, and the coefficient of directmayor is your generalized difference-in-differences effect estimator.
If, however, directmayor is 1 in any observation where the city at some point directly elected its mayor, but is also 1 even in years where that city didn't directly elect (before or after), then it is wrong.

Does it matter that some of the municipality doesn’t have treatment the whole period after introducing it?

Well, it depends. You don't say what outcome you are looking at, other than that it's lnAdmCost, whatever that might be. (logarithm of some kind of administrative cost?) The key issue here is whether reverting to non-direct-mayor-election after having direct mayor election is really going back to the same state as before having direct mayor election in the first place, or whether it is better thought of as a new state. This is sometimes known as "path-dependence." Clearly if I get divorced or my spouse dies, I am unmarried once again, but in many ways that is different from the initial single state before I ever got married, but for other purposes it would be reasonable to consider them equivalent. So that is the question you face here. Is it reasonable to model the pre-direct-mayor state and the post-direct-mayor state as the same thing for your purposes? I know nothing about this content area, so I can't help you answer that question, only point out that this is the question you need to ask.

If you conclude that the two states should be thought of as different, then the solution would be to change your treatment variable into a three level variable: 0 = before, 1 = during, and 2 = after direct mayor election.