# Triple difference-in-difference with continuous treatment

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):

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).