My professor provided the following when introducing control variables
Suppose we have the causal model
Y = Beta_0 + Beta_1*D + U
Suppose E[UD] not 0
then, if we have extra observable characteristics X such that
E[U|D,X] = E[U|X] = f(x)
then we can use this to estimate Beta_1
Can someone explain the meaning behind E[U|D,X] = E[U|X] = f(x)? My professor says this allows us to make an "apples to apples" comparison of the causal effect we are trying to study. I don't have an issue running a regression and interpreting control variables, I just have trouble understanding what this specific expression tells us.