Chapter 10 Outcome Regression

The most straightforward way in which to obtain a causal effect of a variable \(A\) on another \(Y\) in the presence of observed confounders \({\boldsymbol X}\) is simply to estimate the conditional expectation of \(Y\) given \(A\) and \({\boldsymbol X}\), and then read off the effect of \(A\). This is the method of performing an outcome regression, which in the linear case means \[\begin{align} Y = \beta A+ \sum_{i=1}^{d_X} \alpha_i X_i + \varepsilon, \tag{10.1} \end{align}\] where \(\varepsilon\) is from some zero mean distribution with finite variance. In this case, if \(X\) is causally sufficient and the model is correctly specified, then \(\mathop{\mathrm{ATE}}= \beta\).

The key phrase here is if the model is correctly specified; it may or not be the case that the correct structural specification is given by (10.1), but this is something that should be carefully considered—if the model is misspecified then (at least in this case) we can fall back upon \(\beta\) representing the best linear approximation to the causal effect in a Kullback-Leibler sense. More generally, model misspecification is a significant problem in causal inference.