Interpretation of ATT & ACRT in a binary DiD but where the true treatment is continous

[Anna Olivia Jones]

Dear all,

I am working with a panel dataset at the municipal level, where treatment is defined by exposure to a reform. I estimate several difference-in-differences (DiD) models.

In the first model, I use a binary treatment variable. Specifically, I split municipalities into two groups based on the median value of treatment intensity:

• Treated group: municipalities above the median, with an average treatment dose of 15.4 pct.-points.
• Control group: municipalities below the median, with an average treatment dose of 2.4 pct.-points.

I want to compare the results from the Binary DiD model to estimates from DiD models using a continuous treatment variable (i.e., treatment intensity).

I tried the following as a start:

reg labor_avg treat post treat_post pop_density age_com
margins, dydx(treat_post) at(treat_post = (0 1))

In this case, the coefficient on treat_post is equivalent to the marginal effect reported by margins, which is expected since treat_post is binary (0 or 1).

If I assume a constant average marginal effect across the treatment distribution, is it valid to scale the ATT=ACRT (Average causal response of the treated) from the binary model by the difference in treatment intensity between the two groups (15.4 - 2.4)? i.e. ACRT_bin/(15.4-2.4)=ACRT_con

Can I interpret the ATT from the binary specification as representing the effect of a treatment intensity difference of 13 pct. points (15.4 - 2.4)? Or should we interpret it differently — for example, as the difference between the effect of receiving 15.4 units versus 0, minus the effect of receiving 2.4 units versus 0?

To summarize:
How should I interpret and compare the ATT=ACRT from the binary DiD specification to the ACRT from a continuous treatment DiD model?

Any clarification or references on how to align these two approaches would be greatly appreciated.

Best regards,
Anna

[Carlo Lazzaro]

Anna:
welcome to this forum.
You may want to use the -continuos- option available from -xtdidregress-, as in the following toy-example:

. use https://www.stata-press.com/data/r19/patents.dta
(Excerpt from Moser and Voena (2012))

. xtset classid

. gen alfa=runiform()

. xtdidregress (uspatents fpatents) ( alfa , continuous ), group(classid) time( year )


Difference-in-differences regression                   Number of obs = 471,120
Data type: Longitudinal

                            (Std. err. adjusted for 7,248 clusters in classid)
------------------------------------------------------------------------------
             |               Robust
   uspatents | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
ATET         |
        alfa |  -.0048612   .0044875    -1.08   0.279     -.013658    .0039356
------------------------------------------------------------------------------
Note: ATET estimate adjusted for covariates, panel effects, and time effects.

.