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I am struggling to understand how to interpret a triple DiD with one continuous treatment. I found Olden and Møen (2020) useful, but it only considers the case of two binary treatment variables.

Consider this regression on individual $i$ in group $g$ in time $t$

$$ Y_{igt} = \beta_0 + \beta_1 D_{gt} + \beta_2 C_{gt} + \beta_3 Post + \beta_4 D_{gt}\times C_{gt} + \beta_5 D_{gt} \times Post + \beta_6 C_{gt}\times Post+ \beta_7 D_{gt} \times C_{gt} \times Post + \varepsilon_{igt} $$ where $Y_{igt}$ is a continuous outcome, $D_{gt}$ is the indicator treatment, $C_{gt}$ is the continuous treatment, and $Post$ an indicator function equal to one in post-treatment period $t \geq T$ (no staggered adoption etc.).

If $D_{gt}$ and $C_{gt}$ were both indicator treatments then, following the source above, $\beta_7$ gives the average treatment effect for the treated $D_{gt}=1$ and $C_{gt}=1$, see p. 8 of the Olden and Møen (2020):

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I am not sure how to interpret this in the case of continuous treatment. Can I get a weighted average of ATT at each level of the continuous treatment from $\beta_7$?

Sources:

Olden, A., & Møen, J. (2020). The triple difference estimator. NHH Dept. of Business and Management Science Discussion Paper, (2020/1).

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