Causal inference
Frameworks¶
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 \(Y^1_i\) be the potential outcome for individual \(i\) under treatment, and \(Y^0_i\) 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
\[\begin{equation}
Y_i
\equiv
\begin{cases}
Y_i^1 & \text{if } D_i = 1 \\
Y_i^0 & \text{if } D_i = 0
\end{cases}
% .
\end{equation}\]
See also:
Graphical models (DAGs)¶
Structural equation models¶
Parameters of interest¶
ITE¶
TBD
ATE¶
TBD
ATT¶
TBD
TBD
LATE¶
TBD
MTE¶
TBD