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  "body": "\n## Summary\n\nInstrumental variables estimation recovers the coefficients of a linear model when some regressors are correlated with the error term, a situation called endogeneity.\nEndogeneity arises from omitted variables, simultaneous equations or [[measurement error|2XX0A85WSF]], and it makes [[ordinary least squares|YM708KHV2W]] inconsistent.\nAn instrument is a variable that is correlated with the endogenous regressor but uncorrelated with the error.\nIt isolates the part of the regressor's variation that is free of the confounding.\nTwo-stage least squares is the standard instrumental variables estimator, and it is a special case of the [[generalized method of moments|ZPVMCCDYZ5]].\n\n## Explanations\n\n### L1 human\n\nSuppose one wants the effect of years of schooling on wages, but ability raises both, and ability is not observed.\nLeast squares then mixes the effect of schooling with the effect of ability.\nAn instrument is something that shifts schooling but has no direct link to wages or to ability.\nAn example is a rule that changes the age at which pupils may leave school.\nComparing how wages and schooling both move with the instrument gives the effect of schooling alone.\nWith one regressor and one instrument $z$, the estimate is the ratio $\\operatorname{Cov}(z, y) / \\operatorname{Cov}(z, x)$ of sample covariances.\n",
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