Weak Instruments and LATE
Weak Instruments and LATE
A ratio becomes unstable when its denominator is small
The Wald estimator is
If the first stage is 2 percentage points and the reduced form is 1 point, the ratio is 0.50. A one-point sampling change in the first stage moves it to 0.33 or 1.00. Conventional normal approximations can become misleading.
Compare strong and weak IV ratios
Both designs use a valid simulated instrument. The weak design produces a much wider and more erratic ratio even with the same structural effect.
The first-stage F statistic is a warning, not a certificate
Rules such as “F above 10” are context-dependent. Instrument count, heteroskedasticity, clustering, multiple endogenous variables and the desired inferential guarantee matter.
Andrews, Stock and Sun (2019) review weak-instrument diagnostics and robust inference, including non-homoskedastic settings. Report the first-stage specification, relevant statistic and weak-IV-robust confidence set rather than one threshold alone.
LATE belongs to compliers
For binary and , principal strata are:
| Type | Meaning | ||
|---|---|---|---|
| never-taker | 0 | 0 | never receives scholarship |
| complier | 0 | 1 | receipt changed by offer |
| always-taker | 1 | 1 | receives regardless |
| defier | 1 | 0 | does opposite of offer |
Under monotonicity, the Wald ratio identifies
the effect for compliers. Different instruments can move different people and identify different LATEs.
A strong instrument can answer a narrow question
If the offer affects only applicants uncertain about participation, the LATE may not describe always-takers, never-takers or applicants outside the eligibility band. Characterise compliers through first-stage heterogeneity and institutional knowledge, without pretending individual types are observed.
Sensitivity goes beyond strength
Even a strong instrument can violate exclusion or independence. Cinelli and Hazlett (2025) develop an omitted-variable-bias framework for IV sensitivity, separating possible side effects of the instrument and confounding of the instrument.
The teaching lesson is practical: report how strong an omitted pathway would need to be to change the conclusion, benchmarked against observed variables. Sensitivity analysis quantifies a violation; it does not prove the violation absent.
Quick check
Two valid instruments produce estimates of 0.10 and 0.30 with precise first stages. Must one be wrong?