# Conditional expectation function and causal inference

It is my understanding that iff we have a model of the form $$Y = m(X) + e$$ and $$E[e|X] = 0$$ we know that $$m(X)$$ is the conditional expectation function thus: $$m(X) = E[Y|X]$$. If we observe $$X$$ and $$Y$$, their relationship is linear in the parameters and we know what this relationship looks like, we can estimate the CEF using OLS. And our estimand will be consistent and unbiased for the true CEF. If either of the assumptions is violated, we only estimate a linear projection model (that can or can not be close to the true CEF).

In texts on causal inference (e.g. Angrist and Pischke 2009), we define the average causal effect as $$E[Y|X, T = 1] - E[Y|X, T = 0] = E[Y(1) - Y(0)|X]$$ ($$T$$ denotes randomly assigned treatment). If $$T$$ is binary, I can see that we can fully saturate (include all possible interactions and polynomials, i.e. ensure we capture the correct functional form) the model (assuming $$X$$ only holds binary variables or is only a constant), which implies that any structural model of the form $$Y = \gamma T + X'\beta + e$$ is in fact the true CEF, and we can estimate our causal effect well (assuming large enough sample, finiteness of moments, and no perfect collinearity).

BUT what if $$T$$ is continuous? Or random assignment only holds conditional on $$X$$ and $$X$$ has continuous components? In that case, how can we know that we are actually estimating the true CEF? If we aren't, the definition of the causal effect above still holds in the population model, but we can't estimate it using OLS. So why do we still end up using it in cases like this in applied research (I am speaking for econometrics, I don't know how other fields deal with this)?

• from your description, why would it follow that we can't estimate the average treatment effect when $T$ or any of the $X$ are continuous? Commented Jan 18 at 16:01
• With continuous variables we can't saturate the model so we won't know what the true CEF looks like and/or if we have indeed found an unbiased estimator for it. AND the definition of the ACE is given in conditonal expectations
– ArOk
Commented Jan 18 at 16:16
• hm, I'm not familiar with this concept so I'm not sure I see the problem. Wouldn't a linear regression fitting $Y = \beta X$ to obtain $E(Y|X)$ have the same problem with continuous $X$ and $Y$ according to this reasoning? And yet the OLS estimator is known to be unbiased. Commented Jan 18 at 16:52
• What do you mean by "saturating the model"? Commented Jan 18 at 17:41
• @Scriddie Correct me if I am wrong, but OLS will be biased if $E[Y|X]$ is not a function that can be written as linear in its parameters (or if it can be but we don't fit the right model, e.g. the true CEF is $Y=\beta X^2 + e$ but we fit $Y=\gamma X + u$).
– ArOk
Commented Jan 18 at 21:06