Instrumental variable

In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term, in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.Developing the β G M M {displaystyle eta _{GMM}} expression:Note that the usual OLS estimator is: ( X ^ T X ^ ) − 1 X ^ T Y {displaystyle ({widehat {X}}^{mathrm {T} }{widehat {X}})^{-1}{widehat {X}}^{mathrm {T} }Y} .Replacing X ^ = P Z X {displaystyle {widehat {X}}=P_{Z}X} and noting that P Z {displaystyle P_{Z}} is a symmetric and idempotent matrix, so that P Z T P Z = P Z P Z = P Z {displaystyle P_{Z}^{mathrm {T} }P_{Z}=P_{Z}P_{Z}=P_{Z}} In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term, in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable. Instrumental variable methods allow for consistent estimation when the explanatory variables (covariates) are correlated with the error terms in a regression model. Such correlation may occur 1) when changes in the dependent variable change the value of at least one of the covariates ('reverse' causation), 2) when there are omitted variables that affect both the dependent and independent variables, or 3) when the covariates are subject to non-random measurement error. Explanatory variables which suffer from one or more of these issues in the context of a regression are sometimes referred to as endogenous. In this situation, ordinary least squares produces biased and inconsistent estimates. However, if an instrument is available, consistent estimates may still be obtained. An instrument is a variable that does not itself belong in the explanatory equation but is correlated with the endogenous explanatory variables, conditional on the value of other covariates.