Week 12: Causal Forests for Heterogeneous Treatment Effects
DSAN 5650: Causal Inference for Computational Social Science
Summer 2026, Georgetown University
Wednesday, August 5, 2026
Sensitivity Analysis 2: How Sensitive Are My Results to Omitted/Included Variable Bias?
- You probably know about omitted variable bias from previous classes / intuition… but, included variable bias?
- That’s right folks… colliders can be agents of chaos, laying waste to our best, most meticulously-planned-out models
The Classical OVB Equation
- Remember that \(D\) is our treatment variable! (\(\mathbf{X}\) = covariates, \(Y\) = outcome)
- In a perfect world, we would estimate \(Y = \hat{\tau}D + \mathbf{X} \hat{\beta} + \hat{\gamma}Z+ \varepsilon_{\text{full}}\)
- But, \(Z\) is unobserved \(\leadsto\) “restricted” model \(Y = \hat{\tau}_{\text{res}}D + \mathbf{X} \hat{\beta}_{\text{res}} + \varepsilon_{\text{res}}\)
- What is the “gap” between \(\hat{\tau}\) and \(\hat{\tau}^{\text{res}}\)? \(\text{OVB} = \hat{\tau}_{\text{res}} - \hat{\tau}\)
\[ \begin{align*} \hat{\tau}_{\text{res}} &= \frac{ \text{Cov}[D^{\top \mathbf{X}}, Y^{\top \mathbf{X}}] }{ \text{Var}[D^{\top \mathbf{X}}] } \\ &= \frac{ \text{Cov}[D^{\top \mathbf{X}}, \hat{\tau}D^{\top \mathbf{X}} + \hat{\gamma}Z^{\top \mathbf{X}}] }{ \text{Var}[D^{\top \mathbf{X}}] } \\ &= \hat{\tau} + \hat{\gamma}\frac{ \text{Cov}[D^{\top \mathbf{X}}, Z^{\top \mathbf{X}}] }{ \text{Var}[D^{\top \mathbf{X}}] } \\ &= \hat{\tau} + \hat{\gamma}\hat{\delta} \\ \implies \text{OVB} &= \hat{\tau}_{\text{res}} - \hat{\tau} = \overbrace{ \boxed{\hat{\gamma} \times \hat{\delta}} }^{\mathclap{\text{Impact} \, \times \, \text{Imbalance}}} \end{align*} \]
More Generally
- What if we don’t have a specific omitted variable in mind? We just want to know the expected impact if there were omitted vars… Enter “Partial \(R^2\)”
