Applying grf when the treatment variable is continuous

[ZO1DB3RG]

@swager @davidahirshberg @erikcs
Thank you for your long-term maintenance of this package. We have found answers to many questions in the numerous issues, including the one related to the “Assumptions required for continuous treatment” #256. We have a two follow up questions related to #256:

• 1). Whether we can apply causal_forest directly to an observational data with a continuous treatment variable W? In our empirical setting, we applied causal_forest using our data, and we do get estimates of tau. So here we want to confirm that what we get is actually "tau(x)" = Cov[Y, W | X = x] / Var[W | X = x], right (as mentioned by @swager in #256)?
• 2). If so, what specific assumptions are needed for “tau(x)” to have a causal interpretation? Existing theoretical literature (including references mentioned on the GRF webpage) primarily derives tau calculations based on binary treatment variables. There is little discussion on how to extend the GRF to continuous treatment case. Is there any directions/references to recommend? We are empirical researchers who mainly apply the method so it would be helpful to know how to interpret the output of causal_forest when W is continuous. And under what assumptions we can interpret as causal, similar to the binary case?

[erikcs]

Hi @ZO1DB3RG, from the causal forest docstring:

When W is continuous, we effectively estimate an average partial effect Cov[Y, W | X = x] / Var[W | X = x], and interpret it as a treatment effect given unconfoundedness