Causal assumptions

Establishing causal relationships requires making assumptions.

Causal inference does not rely purely on data and statistics. It requires more assumptions for causal relationships to be established.

DAGs are a visual way of expressing our beliefs about how different quantities influence each other.

As it turns out, DAGs are not enough. Even with infinite data, some causal relationships cannot be estimated.

Exchangeability: assuming if the control and treatment groups were swapped, the results would have been the same.

Formally, exchangeability reads

\[Y(0) \perp T, \quad Y(1) \perp T\]

where \(T\) is the treatment assignment, \(Y(0)\) is the outcome for when not receiving the treatment, and \(Y(1)\) is the outcome when receiving the treatment.

A less strict version of this assumption would be

\[E[ Y(1) \, | \, T = 0 ] = E[ Y(1) \, | \, T = 1 ]\]

which asks for the expected value of having received the treatment (a counterfactual in the left-hand side case) to be the same under being assigned to the control group (\(T=0\)) or the treatment group (\(T=1\)).