Instrumental Variables, explained.
An instrumental variable changes treatment uptake and identifies a causal response only under additional assumptions about assignment, exclusion, and behaviour.
Why it happens
The encouragement raises uptake by ΔD between 0 and 0.8. Each treatment unit adds two outcome units, so ΔY = 2ΔD plus an optional direct encouragement bonus. The Wald ratio divides ΔY by ΔD; at zero uptake change it is undefined.
An instrumental variable changes treatment uptake and identifies a causal response only under additional assumptions about assignment, exclusion, and behaviour.
Read the result
Follow the encouragement through uptake before reading the ratio. The same direct bonus causes more ratio bias when the first stage is weaker.
A worked example
An invitation that also supplies a reward
Start with 40% extra uptake and no direct bonus. An outcome difference of 0.8 divided by 0.4 recovers two.
Add a one-unit direct encouragement bonus. The ratio becomes 1.8 divided by 0.4, or 4.5.
The treatment effect has not changed. The instrument now affects outcomes through a second route, breaking exclusion.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
The encouragement raises uptake by ΔD between 0 and 0.8. Each treatment unit adds two outcome units, so ΔY = 2ΔD plus an optional direct encouragement bonus. The Wald ratio divides ΔY by ΔD; at zero uptake change it is undefined.
Where this idea is useful
A practical use
A random invitation to a course can affect attendance without directly changing skills, if the exclusion assumption is credible.
A common misconception
“Any variable correlated with treatment is an instrument.”
Relevance is only one requirement. An instrument must also support the necessary assignment and exclusion assumptions, with additional conditions for local-effect interpretation.
What this explanation leaves out
- This homogeneous deterministic model omits sampling noise and heterogeneous responses. In general, a valid instrument may identify a local effect for compliers rather than a population average. The direct path is exposed by design, not discovered statistically.
What if encouragement changes no one's uptake?
The first stage is zero, so the displayed ratio is undefined. A strong outcome association cannot repair the missing treatment variation.
What route from your proposed instrument to the outcome would make exclusion implausible?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.