Interpretation difference in differences

[Sacha Elinemez]

Hi all,

I know there have been many previous posts about interpreting coefficients of difference in differences, but I am still not sure about my interpretation.

My dependent variable "performance" is calculated as operating income divided by lagged total assets. It's a proportion, it ranges between 0 and 1.
My treatment variable is the tax amount divided by total revenue of the firm. This is also a proportion, it ranges between 0 and 1.
The variable "post" is a dummy variable with the value 0 for the period without the tax and the value 1 for the period with the tax.

The coefficient of the treatment variable on the dependent variable is -0.111. How do I interpret this variable?

The coefficient of the interaction variable of the treatment variable and the "post" variable, so treatment*post, is 0.140.
Does this imply that a 1 percentage point increase in the interaction variable, increases the dependent variable "performance" with 0.14 percentage point?

Thanks in advance!

[Clyde Schechter]

Does this imply that a 1 percentage point increase in the interaction variable, increases the dependent variable "performance" with 0.14 percentage point?

Well, sort of. It's technically correct (except that the use of causal language is not justified), but not the right way to think about it.

The useful interpretation is as a difference-in-slopes. The slope of the performance:tax treatment variable relationship is 0.14 greater after the tax was implemented than beforehand.

It is not meaningful to speak of a unit increase in the interaction variable, because a unit increase in the interaction arise from combinations of differences in the treatment variable and differences in the post variable that multiply to 1. In your case, since I assume that the tax never exceeds total revenue of the firm, your treatment variable is confined to values between 0 and 1 (you say as much yourself in #1). And your post variable is always either 0 or 1, I presume. So in this context, no real world situation ever corresponds to a unit change in the interaction term. (And in real world situations where a unit change in the interaction term is a possibility, there are usually many, perhaps infinitely many, different combinations of changes in the constituent variables of the interaction that would give rise to a unit difference in the interaction variable--so that's not very useful either.)

[Sacha Elinemez]

Dear Clyde,

Thank you for your response.

I understand now that for the interaction term it is not meaningful to speak of a unit increase.

Now the question arises, can we interpret the coefficients of the Treatment variable and the Interaction variable together to speak of unit increases?
The regression is as follows:
Y = α + β1(treatment) + β2(post) + β3(treatment∗post)

The coefficient for treatment = -0.111
The coefficient for the interaction term = 0.140

Can we now conclude that a one unit increase in the treatment variable, after the event, increases the independent variable with 0,029 (-0.111+0.140)?

Thank you very much for your help!

[Clyde Schechter]

Can we now conclude that a one unit increase in the treatment variable, after the event, increases the independent variable with 0,029 (-0.111+0.140)?

I assume you meant to ask about whether it increases the dependent variable.

Your question is ambiguous. Under one interpretation, the condition you describe involves both going from pre to post intervention and a unit increase in the treatment variable. So the net effect would be β1+ β2+β3. Under the other interpretation, if entirely during the post-intervention interval, the treatment variable were to increase by one unit, then your calculation would be correct.

[Sacha Elinemez]

Thank you for clarifying!