Causal Catalog

Assumptions

Consistency

Definition	The observed outcome \(Y\) for a unit is the same as the potential outcome \(Y(T=t)\) for that unit under the treatment that was actually observed \(T=t\).
Mathematical definition	\(Y = Y(T)\) For binary treatment, that means: \(Y = Y(1) \text{ if } T=1\) \(Y = Y(0) \text{ if } T=0\)
Intuition/Examples	Activity 3 consistency scenarios

Exchangeability

Definition	The distribution of \(Y(0)\) and \(Y(1)\) for the \(T=1\) and \(T=0\) groups are the same. Also known as ignorability.
Mathematical definition	\(Y(0), Y(1) \perp T\)
Intuition/Examples	Lecture 3 (slide 12): Caffeine and exam performance exchangeability brainstorm

Conditional Exchangeability

Definition	The distribution of \(Y(0)\) and \(Y(1)\) for the \(T=1\) and \(T=0\) groups are the same, conditional on some variables \(X\). Also known as unconfoundedness. We determine which variables \(X\) to condition on based on the backdoor criterion.
Mathematical definition	\(Y(0), Y(1) \perp T \mid X\)
Intuition/Examples	Activity 8 on how to determine which variables satisfy the backdoor criterion

Positivity

Definition	Every unit has a non-zero probability of receiving the treatment, and every unit has a non-zero probability of not receiving the treatment. We can also view it as: every subgroup \(x\) in our sample has to have some units that received the treatment, and some units that did not receive the treatment. Also known as overlap.
Mathematical definition	For all values of covariates \(x \in X\):          \(0 < P(T = 1 \mid X = x) < 1\)
Intuition/Examples	See Lecture 8, beginning at slide 25 for intuition and PollEverywhere example, as well as Activity 9 for seeing bins of covariates in Yeager et al. (2019).

Study Designs

Randomized Experiments

Assumptions needed	Consistency No interference
Assumptions ensured	Exchangeability
Causal quantities identified	Average treatment effect (ATE) Average treatment effect on the treated (ATT) Average treatment effect on the untreated (ATU)

Pros/cons

Advantages	Disadvantages
Mitigating the impact of confounding variables Exchangeability is ensured Ensure that the results are causal	Cost a lot Some experiments cannot be randomized because of ethics Potential biases in random assignment It is not always possible to design randomized experiments for certain circumstances

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Observational Studies

Assumptions needed	Consistency No interference Conditional exchangeability / unconfoundedness Positivity
Assumptions ensured	None 😞
Causal quantities identified	Average treatment effect (ATE) Average treatment effect on the treated (ATT) Average treatment effect on the untreated (ATU)

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Instrumental Variables

Assumptions needed	Consistency No interference Relevance Exclusion restriction Instrument unconfoundedness Linear outcome or monotonicity
Assumptions ensured by design	None, but does not need conditional exchangeability / unconfoundedness
Causal quantities identified	Average treatment effect (ATE) Local average treatment effect (LATE)

Case studies

See Activity 13 and slide 16 of Lecture 15 for our case studies!

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Regression Discontinuity

Sharp RDD

Description	Treatment is completely deterministic based on the running variable Treatment is "forced" once the running variable crosses the cutoff c
Assumptions needed	Consistency No interference Continuity
Assumptions ensured by design	None, but does not need conditional exchangeability / unconfoundedness
Causal quantities identified	Average treatment effect (ATE) at the cutoff

Fuzzy RDD

Description	Treatment is not deterministically forced, but it is discontinuous
Assumptions needed	Consistency No interference Continuity Monotonicity (no defiers)
Assumptions ensured by design	None, but does not need conditional exchangeability / unconfoundedness
Causal quantities identified	Local average treatment effect (LATE) at the cutoff

Pros/cons

Advantages	Disadvantages
Useful when randomization is not possible Helps examine specific trends surrounding the cutoff of interest Can visually see the assumption needed for both types of RDDs We can check for manipulation by doing the histogram against the threshold, and identify any spots that would make our RDD invalid	Data inefficent (we need a lot of data), there isn't a guaranteed sweet spot for the bandwidth value b Participants may change their behavior based on knowledge of the cutoff Have to assume/guess who is a complier (same problem as with IVs) Can only look at the effects at the cutoff The choice of the bandwidth really matters