The Encouragement Problem
In many field experiments, we can't force people to take a treatment. We can only randomize an *encouragement*. Some people we encourage won't take the treatment, and some people we *don't* encourage will get it anyway. This is **two-sided noncompliance**, and it complicates how we measure cause and effect.
Understanding the Population
To solve this, we classify everyone into four hidden groups, or **Principal Strata**, based on how they would act under any assignment. An individual's group is a fixed characteristic, just like their age.
Compliers
Take treatment if & only if encouraged.
Observed Action: Gets Treated
Always-Takers
Get treatment regardless of encouragement.
Observed Action: Gets Treated
Never-Takers
Never get treatment, regardless.
Observed Action: Not Treated
Defiers
Do the opposite of their assignment.
Observed Action: Not Treated
The core analytical challenge: in the treatment group, we see a mix of Compliers and Always-Takers getting treated. We can't tell them apart just by looking at the data.
The Pillars of Identification
To estimate the true causal effect for Compliers (the CACE), we must rely on a specific set of assumptions that link our random encouragement to the outcome.
Relevance (The "First Stage")
The encouragement must actually work. It needs to have a real effect on whether people take the treatment. If nobody changes their behavior because of the encouragement, we can't learn anything. This is testable from the data.
The Exclusion Restriction
This is the most critical assumption. It states that the encouragement itself has no direct effect on the outcome, except by getting more people to take the treatment. For Always-Takers and Never-Takers, the encouragement should not change their outcome at all. This is not testable and must be defended based on the experimental design.
Monotonicity (No Defiers)
We must assume there are no "Defiers" — people who would take the treatment only if they were *not* encouraged. This ensures that the encouragement only pushes people in one direction (towards treatment). In two-sided designs, this is a strong behavioral assumption.
Estimating the Effect: The Draft Lottery
A classic example is the Vietnam draft lottery. A random lottery number acted as an "encouragement" (instrument) for military service. Let's see how to estimate the causal effect of serving in the military on mortality for those who were induced to serve by the draft.
The CACE Formula (Wald Estimator)
The Complier Average Causal Effect (CACE) is a simple ratio: the effect of the encouragement on the outcome (ITT_Y), divided by the effect of the encouragement on treatment take-up (ITT_D).
CACE = Effect on Outcome The Intention-to-Treat (ITT) effect on mortality. How much did draft eligibility change the death rate? / Effect on Participation The ITT effect on serving. How much did eligibility increase enlistment? (This is the compliance rate).
Calculated Effects
Effect of Eligibility on Mortality
Effect of Eligibility on Military Service
Designing Better Experiments
While IV is a powerful analytical tool, the best strategy is to design experiments that maximize compliance and make assumptions as plausible as possible.
Maximize Compliance
- Simplify the intervention: Reduce administrative burdens and "hassle factors".
- Leverage behavioral science: Use reminders, clear messaging, and social norms.
- Pilot test: Identify barriers to compliance before a full-scale launch.
Strengthen Assumptions
- Design a good instrument: The ideal encouragement is strong enough to be relevant but "content-free" enough for the exclusion restriction to hold.
- Use placebo encouragements: Give the control group a placebo message to isolate the effect of the encouragement's content.
- Profile the Compliers: Use baseline data to understand who the compliers are. The CACE is the effect for *them*, and they may not be representative of the whole population.