Causal inference
Related pages: Diagramming tools
Frameworks¶
See also:
1. Potential outcomes framework¶
Let \(D_i\) (for dummy) be a binary variable indicating whether individual \(i\) receives the treatment \((D_i=1)\) or not \((D_i=0)\).
Let \(Y_i(1)\) be the potential outcome for individual \(i\) under treatment, and \(Y_i(0)\) be the potential outcome for individual \(i\) without treatment.
The observed outcome \(Y_i\) can be expressed as
\[\begin{equation*}
Y_i
\equiv
Y_i(1) \, \mathbb{1}\{ D_i = 1 \} +
Y_i(0) \, \mathbb{1}\{ D_i = 0 \}
\end{equation*}\]
or
\[\begin{equation*}
Y_i
\equiv
\begin{cases}
Y_i(1) & \text{if } D_i = 1 \\
Y_i(0) & \text{if } D_i = 0
\end{cases}
% .
\end{equation*}\]
2. Graphical models (DAGs)¶
TBD
3. Nonparametric structural equation models (NPSEMs)¶
TBD
Parameters of interest¶
ITE and ATE¶
\begin{equation*}
TE_{i} := Y_{i}(1) - Y_{i}(0)
\end{equation*}
\begin{equation*}
ATE := \mathbb{E}\left[Y_{i}(1) - Y_{i}(0)\right]
\end{equation*}
ATT and ATU¶
\begin{equation*}
ATT := \mathbb{E}\left[Y_{i}(1) - Y_{i}(0) \mid D_{i} = 1 \right]
\end{equation*}
TBD
Selection¶
Selection on observables¶
Selection on unobservables¶
Mediation analysis¶
See the 2025 Methods Lecture.