I am reading lot of material regarding Causal Inference using Regression Analysis but I am unable to resolve my doubt.
Suppose I have a data with Outcome Y, Treatment Tr and covariates X1, X2, X3, X4, ....
I need to find Average Treatment Effect using Regression Analysis for my data. with three model.
First with only outcome and treatment
model1-> lm(Y~Tr, data)
Second with outcome, treatment, and covariates
model2-> lm(Y~Tr+X1+X2+X3+X4+...., data)
Third with outcome, treatment, covariates and Interaction between covariates and treatment
model3-> lm(Y~Tr+X1+X2+X3+X4+....+X1*Tr + X2*Tr + X3*Tr + X4*Tr +......, data)
I know for
model1 Average treatment effect(ATE) is coefficient of Tr in the
model2 I think ATE is still coefficient of Tr in the
model2. But I am not sure.
I am really confused what will be the ATE in our third model i.e.