Difference-in-Differences with a Continuous Treatment Variable

[Sam Bennett]

Hi,

I am estimating the effects of the proximity to a newly constructed highway on labor market outcomes at the district level. I am using the distance between the population-weighted district centroid and the highway as the treatment variable.

I thought about using the following regression equation where i and t subscripts denote district and year respectively, so y_it is the labor market outcome for district i and year t, Distance_it is the aforementioned distance variable, θ_i is the district fixed effects, δ_t is the year fixed effects, and ε_it is the idiosyncratic error term.

y_it = β₀ + β₁ Distance_it + θ_i + δ_t + ε_it

One tricky thing about this specification is that the Distance_it variable changes by district and year, taking the value of 0 before the highway stretch near the district is completed and taking the value of the continuous distance measure for years after the highway stretch was completed.

However, the issue of this model is that I cannot distinguish between the case when the distance to the highway is 0 km and the case when the distance to the highway is non-zero but since it is from pre-treatment time, the distance variable takes the 0 value.

I have two questions.
1. Is there a better and more sane way to model this?
2. I assume I can use something like the Callaway and Sant'Anna DID with continuous treatment with distance as the treatment dose and the year of treatment being the group (untreated years being 0). Is there a way to execute this in Stata with the continuous treatment variable?

Thanks!

[Clyde Schechter]

Using distance from the highway as a dose doesn't make sense, because it implies that the people most affected by the highway are those who are farthest away from it. And those who are immediately adjacent to the highway are completely unaffected by it. It should be the other way around. You need to use, not distance, but some decreasing function of distance. Exactly which decreasing function of distance might make the most sense really depends on your conceptual model of how the highway affects the labor market outcomes--a substantive question that I am not able to advise you on.

Please read the Forum FAQ for excellent advice about the best ways to share information here. Abbreviated references like Callaway and Sant'Anna DID may be instantly recognizable to people in your immediate field or niche. But this is an international, multidisciplinary forum. So full references are needed for most people to understand what you are talking about. Even better than a reference would be a link to a website containing a publicly available copy of the article itself.

[Sam Bennett]

What's your rationale behind those farthest from the highway being affected the most? They won't be able to use the highways as much and as frequently as those living closer to them. Think of a scenario in which there were no highways before and now highways are built. Maybe I wasn't clear about this in the OP.

[Clyde Schechter]

What's your rationale behind those farthest from the highway being affected the most?

I didn't say that--you did. If you use distance as your "dose," then those at greatest distance are being modeled as having the most intense exposure to the highway. It should be the opposite. So I'll turn that question back at you: what is your rationale for modeling those farthest from the highway as being the most affected?

[Sam Bennett]

Ah alright, my bad. I reread it. You're right, maybe I should use an inverse of the distance or something if I treat it as a "dose".