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documentation update w.r.t. continuous treatments
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thegland authored Aug 25, 2023
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Expand Up @@ -135,7 +135,7 @@ In a similar vein, event study style estimates that present treatment effects sp

In terms of implementing event study-style estimates, wooldid offers two approaches. The default pooled reference period approach is conceptually similar to estimating results like normal, but with the treatment onset date for all treated cohorts pulled earlier in time by just enough periods to estimate all requested relative time period effects. With this approach, we add no more (i,t) pairs to Z than is necessary and we allow the reference period to potentially pool multiple periods together for each cohort. Event study estimates derived from this approach always have a normalized effect of 0 in the earliest period reported (the pooled reference period). The alternative approach is to use a fixed reference period, in which case Z is modified to include all (i,t) pairs for treated cohorts, except the pair corresponding with the period immediately prior to treatment (or corresponding with some other user specified period). In this case, the regression is saturated with i-t indicator variables and all treatment effects are estimated relative to the single specified reference period.

With greater modification, the approach above can be extended to handle continuous treatment variables and treatments that come with an associated measure of treatment intensity or dosage. When cohorts are coarse -- i.e., there are multiple units within each cohort-period cell -- and when a continuous treatment variable varies within i-t cells, estimates of the effect of the continuous treatment variable can be computed within each treated i-t cell. To ensure treatment effect heterogeneity is adequately captured, a flexible function of the continuous treatment variable can be used to measure its effect within each i-t cell if needed. Note that this approach is a bit speculative, and is not proposed in Wooldridge (2021).
With greater modification, the approach above can be extended to handle continuous treatment variables and treatments that come with an associated measure of treatment intensity or dosage. When cohorts are coarse -- i.e., there are multiple units within each cohort-period cell -- and when a continuous treatment variable varies within i-t cells, estimates of the effect of the continuous treatment variable can be computed within each treated i-t cell. To ensure treatment effect heterogeneity is adequately captured, a flexible function of the continuous treatment variable can be used to measure its effect within each i-t cell if needed. Note that this approach is a bit speculative and is not covered in Wooldridge (2021), though Wooldridge [has informally suggested](https://twitter.com/jmwooldridge/status/1695115782739435922?s=61&t=pXC6R4euc7LaN6MMB5VApQ) handling continuous treatments using interactions between treated cell indicators and the continuoust reatment variable.

The particular modification to the underlying regression used for estimation in the continuous treatment context is as follows:
Step 1 (Continuous): Y_jt = fe_i + fe_t + sum_over_it_in_Z(beta_it * TreatedFlag_it * Tz_it + TreatedFlag_it * Tz_it * f(Contreat_jt)) + g(f(Contreat_jt))
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