In statistics, instrumental variables estimation (IVE) is an extension of linear regression analysis. IVE is used for examining the hypothesized relationship between an outcome variable and a "question" predictor, in cases when the question predictor is potentially fully or partially endogenous, a condition that leads to bias in ordinary least-squares estimates of the regression parameter associated with the question predictor.

Endogeneity in a question predictor occurs when the predictor is correlated with the residuals in the regression model that describes its relationship with the outcome. A correlation between a predictor and the residuals in a hypothesized regression model violates one of the fundamental assumptions of ordinary least squares (OLS) regression analysis.

In a regression model, endogeneity in a predictor can occur when:

• there is an "omitted variables" problem -- that is, when a critical predictor of the outcome that is also correlated with the current question predictor is omitted from the regression model,
• there is an "errors-in-variables" problem -- that is, when the question predictor has been measured fallibly (i.e., with error), or
• the system under investigation consists of a set of interlinked "simultaneous" statistical models but one or more of these required models has been ignored in the estimation process.

IVE uses the relationship between the potentially engodenous question predictor and an additional variable, called the "instrument," to tease out any exogenous variation actually present in the potentially engogenous predictor.

The IVE process is most often implemented using a two-stage least-squares (2SLS) approach. Under the 2SLS approach, in a first stage, the question predictor is regressed on the instrument using OLS methods and the predicted value of the question predictor is computed for each person in the dataset. If the instrument is exogenous, the these predicted values will contain only the exogenous part of the original endogenous question predictor, and can be used in place of the question predictor in a second stage model to examine the relationship between outcome and question predictor. Therefore, in the second stage of the estimation, the outcome is regressed not on the original question predictor but on the predicted values of the question predictor and the corresponding regression slope parameter is estimated by OLS methods. The slope estimate obtained is then an unbiased estimate of the hypothesized relationship between outcome and question predictor.

In IVE, covariates (i.e., additional predictors) are usually added to both the first and second stage models in order to improve the precision of the estimation. It is also usually recommended that the same covariates be added to both the first- and second-stage models in order to avoid simultaneous equations bias.

In using the 2SLS strategy to conduct IVE, the analyst must make a small correction to the sum-of-squared residuals in the second-stage fitted model in order that the associated standard errors be computed correctly.

There are three main assumptions that underpin the success of the instrumental variables estimation technique:

• The instrument must predict the question predictor (a fundamental requirement!).
• The instrument cannot be correlated with residuals in the second stage model (that is, the instrument cannot suffer from the same problem as the original question predictor).
• The instrument must act on the outcome only through the question predictor, and not directly.

The use of the instrumental variables estimation technique often provides a useful, convenient and ethical alternative to the classical randomized experiment. In the randomized experiment, exogenous variation in treatment is provided by the random assignment of participants to the treatment and control conditions, causing the investigator to deny the treatment to the control participants. Using IVE, participants can be permitted to self-select into treatment and control, and the investigator can subsequently tease out the exogenous component of the treatment variation using the instrument. Of course, one does not get anything for nothing -- the IVE technique is only as good as the instruments it employs.

Consequently, one particular problem in the use of IVE is in the selection and defense of suitable instruments. Good instruments are often created by exogenous policy changes (i.e., the cancellation of federal student aid scholarship program), geographic differences in the application of standards (i.e., different states implement different passing standards for a common exam) or generic randomness (e.g., the Vietnam Draft Lottery) have lead to exogenous disruptions in the values of the construct being measured by the selected instrument.

Another critical problem in IVE is caused by the selection of "weak" instruments. These are instruments that are very poor predictors of the endogenous question predictor in the first-stage equation. In this latter case, the prediction of the question predictor by the instrument will be poor and the obtained predicted values will have very little variation. Consequently, they are unlikely to have much success in predicting the ultimate outcome when they are used to replace the question predictor in the second-stage equation.