st: Diff-in-diff: how to account for treatment intensity, rather than just a treatment dummy?

[John Carey <johncarey196@gmail.com>]

Hi everyone!

I have been working on a difference-in-differences strategy, and I was
hoping someone could clarify an important point for me.

In the beginning, the treatment I am working on was not a dummy. It is
a discrete variable ("D") which ranges from 1 to 10 when observations
are treated, and equals 0 otherwise. For the sake of simplicity, I
turned it into a dummy, equal to 1 when the discrete variable is
strictly positive, and equal to 0 otherwise. That way, I was able to
use a few common diff-in-diff models (OLS regression and psmatch2).
Also, I should specify that I only have 2 periods (pre-treatment, and
post-treatment).

However, I have been doing research about how to account for treatment
intensity, because I would like to take into account the fact that
being treated with 10 is not the same as being treated with 1.

For now, I have created 10 dummies for each of the possible values of
the treatment variable, and I have run 10 different regressions (1
against 0; 2 against 0; 3 against 0...). However, it is not easy to
get a full picture with that process. First, I have very few treated
observations for some of the treatment values, and therefore inference
is an issue. Second, I have not found an easy way to compare the
treatment effects to each other, since I have compared each of them to
getting 0 unit of treatment.

Therefore, here are two questions ;)

First, do you know of any way to account for treatment intensity
without breaking the treatment variable into 10 dummies? Ideally, I
would like to be able to run one regression which would take it all
into account. Some sort of weighted ATT.
For instance, do you think it is possible to use a regular OLS
diff-in-diff equation, plug the treatment variable as a discrete
variable (as opposed to a dummy), and include as many group fixed
effects and there are treatment values? I would be tempted to write it
like this:
Yit = a + b[T=t1] + c1[D=1] + c2[D=2] + ... + c10[D=10] + d[T=t1]*[D] + e
In that equaltion, I would let [D] range from 0 to 10, and d would be
the ATT. Do you think that makes sense?

Second, if I do not find another way than to break the treatment
variable D into 10 dummies, does anyone know how I could recover the
mean ATT and its standard error? I guess I would need to weight the 10
different ATTs that I got, but what should be the weights? How about
number of treated observations in each treatment group? I thought
about doing that, but I stopped because the fact that treatment was
not randomly allocated made me think otherwise.

Thank you everyone for your help, and I wish you a great week!
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