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Identification when a regressor is entangled with the error17 min read
Endogeneity and Instrumental Variables
Understand endogeneity from omitted variables, simultaneity, and measurement error; IV relevance and exclusion; 2SLS, LATE, weak instruments, and financial identification.
Prepared by Mark · Primary sources below
Direct answer
Endogeneity means a regressor is correlated with the regression disturbance, so OLS mixes the target effect with other association. An instrument Z must have relevance by moving the endogenous X and must satisfy independence and exclusion so it does not affect outcome Y through another path. Two-stage least squares uses only variation in X predicted by Z, but a weak instrument or false exclusion restriction can produce results more unstable and misleading than OLS.
Endogeneity links a regressor to the error
The disturbance contains determinants of the outcome omitted from the equation; exogeneity fails when X moves with those determinants.
Omitted risk appetite, simultaneous price and demand, noisy exposure measurement, self-selection, and reverse causality are common routes.
A larger sample shrinks random uncertainty but does not remove systematic endogeneity, so an estimate can become a precisely measured wrong answer.
An instrument needs two distinct pillars
Relevance requires instrument Z to explain enough variation in endogenous X after conditioning on included exogenous controls.
Independence and exclusion require Z to be unrelated to hidden causes of potential outcomes and to affect Y only through X.
Relevance is partly diagnosable in data, but exclusion is generally not directly testable and needs an institutional, timing, and behavioral argument.
2SLS uses instrument-induced variation
The first stage regresses X on Z and exogenous controls to isolate the component of X predicted by the instrument.
The second stage connects Y to that predicted variation, while proper computation uses a 2SLS covariance estimator that accounts for first-stage estimation.
With one endogenous variable and one instrument, the simple IV estimand links the reduced-form effect of Z on Y to the first-stage effect of Z on X as a ratio.
The IV effect can be local
With binary treatment and conditions including monotonicity, IV can identify the local average treatment effect for compliers whose treatment changes because of the instrument.
This LATE need not equal the average effect in the whole population or the effect for people moved by a different policy instrument.
With continuous treatment or heterogeneous responses, interpretation requires care about which marginal effects receive which weights.
Weak instruments can break ordinary inference
If the first stage is weak, the instrument supplies little exogenous variation and the 2SLS distribution, coefficient, and interval can become highly unstable.
Adding many instruments can overfit endogenous noise in sample and pull 2SLS toward the OLS estimate.
Report first-stage coefficients, partial R², and suitable weak-identification statistics, using weak-IV-robust tests or intervals when needed.
Evaluate financial instruments by their pathways
Regulatory thresholds, index assignment rules, funding-flow shocks, or scheduled institutional changes may provide candidate variation in financing or prices.
Exclusion is threatened if regulation changes several firm behaviors or index inclusion simultaneously changes attention, liquidity, and information.
If traders anticipate the rule or manipulate a threshold, the instrument's independence and timing assumptions can also fail.
Make the identification claim auditable
Define the target effect and population, explain how the instrument moves X, and map every plausible side path from the instrument to Y.
Show the first stage and reduced form with balance, placebo outcomes, pre-effects, alternative controls, and weak-identification-robust inference.
Neither a failed overidentification test nor a Hausman test proves instrument validity, so an unrefuted assumption should not be stated as fact.
Common questions
What is an instrumental variable?
It is a source of variation that moves an endogenous regressor but, under stated assumptions, affects the outcome only through that regressor.
How does two-stage least squares work?
It first predicts the endogenous regressor with instruments, then relates the instrument-induced component to the outcome.
Does a large first-stage F statistic prove instrument validity?
It supports relevance but cannot prove independence or exclusion, which still require a credible account of the economic pathways.
Does passing an overidentification test validate every instrument?
No. The test depends on its assumptions and power and may miss instruments that share the same invalid pathway.
Sources and further reading
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