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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.

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The professor's notation is a little non-standard, but in English, your professor's math is saying "conditional on $X$, the error, $U$ is mean-independent of $D$."

Thus, if we control for $X$ in the regression (condition on it), then we are taking $X$ out of the error, $U$, and, what remains in the error is uncorrelated with $D$, allowing us to estimate the treatment effect. The resulting equation is:

$$y_i =\beta_0 +\beta_1 D_i +\beta_2 X_i +U_i$$

$\beta_1$ estimates the change in $y_i$ associated with $D_i$ increasing by 1, holding fixed $X$. The key is that, by holding fixed $X$, we have removed the source of bias.

I'm trying to avoid technical math in my answer and use English sentences because I'm guessing that is what you want, but please follow up if you want a mathier answer.

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