In statistics, econometrics, epidemiology and related disciplines, the quasi-experimental 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 (also known as independent or predictor) variable of interest is correlated with the error term (endogenous), in which case ordinary least squares and ANOVA give biased results. When used, a valid instrument changes the explanatory variable (the variable correlated with the endogenous variable) but has no independent effect on the dependent variable and is not correlated with the error term, thus allowing a researcher or analyst to uncover the true 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 when:
changes in the dependent variable change the value of at least one of the covariates ("reverse" causation), there are omitted variables that affect both the dependent and explanatory variables, or the covariates are subject to measurement error. Explanatory variables that 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, conditionally on the value of other covariates. In linear models, there are two main requirements for using IVs:
The instrument must be correlated with the endogenous explanatory variables, conditionally on the other covariates. If this correlation is strong, then the instrument is said to have a strong first stage. A weak correlation may provide misleading inferences about parameter estimates and cause the standard errors in the second stage to be larger than the ordinary least squares estimates. The instrument cannot be correlated with the error term in the explanatory equation, conditionally on the other covariates. In other words, the instrument cannot suffer from the same problem as the original predicting variable. If this condition is met, then the instrument is said to satisfy the exclusion restriction.
Example Informally, in attempting to estimate the causal effect of some variable X ("covariate" or "explanatory variable") on another Y ("dependent variable"), an instrument is a third variable Z which affects Y only through its effect on X. For example, suppose a researcher wishes to estimate the causal effect of smoking (X) on general health (Y). Correlation between smoking and health does not imply that smoking causes poor health because other variables, such as depression, may affect both health and smoking, or because health may affect smoking. It is not possible to conduct controlled experiments on smoking status in the general population. The researcher may attempt to estimate the causal effect of smoking on health from observational data by using the tax rate for tobacco products (Z) as an instrument for smoking. The tax rate for tobacco products is a reasonable choice for an instrument because the researcher assumes that it can only be correlated with health through its effect on smoking. If the researcher then finds tobacco taxes and state of health to be correlated, this may be viewed as evidence that smoking causes changes in health.
History The first use of an instrument variable occurred in a 1928 book by Philip G. Wright, best known for his excellent description of the production, transport and sale of vegetable and animal oils in the early 1900s in the United States. In 1945, Olav Reiersøl applied the same approach in the context of errors-in-variables models in his dissertation, giving the method its name. Wright attempted to determine the supply and demand for butter using panel data on prices and quantities sold in the United States. The idea was that a regression analysis could produce a demand or supply curve because they are formed by the path between prices and quantities demanded or supplied. The problem was that the observational data did not form a demand or supply curve as such, but rather a cloud of point observations that took different shapes under varying market conditions. It seemed that making deductions from the data remained elusive. The problem was that price affected both supply and demand so that a function describing only one of the two could not be constructed directly from the observational data. Wright correctly concluded that he needed a variable that correlated with either demand or supply but not both – that is, an instrumental variable. After much deliberation, Wright decided to use regional rainfall as his instrumental variable: he concluded that rainfall affected grass production and hence milk production and ultimately butter supply, but not butter demand. In this way he was able to construct a regression equation with only the instrumental variable of price and supply. Formal definitions of instrumental variables, using counterfactuals and graphical criteria, were given by Judea Pearl in 2000. Angrist and Krueger (2001) present a survey of the history and uses of instrumental variable techniques. Notions of causality in econometrics, and their relationship with instrumental variables and other methods, are discussed by Heckman (2008).
Theory While the ideas behind IV extend to a broad class of models, a very common context for IV is in linear regression. Traditionally, an instrumental variable is defined as a variable Z {\displaystyle Z} that is correlated with the independent variable
X {\displaystyle X} and uncorrelated with the "error term" U {\displaystyle U} in the linear equation
Y = X β + U {\displaystyle Y=X\beta +U}
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