In the fields of statistics and econometrics, the term **instrumental variables** can refer to either of two definitions. Instrumental variables can refer to:

- An estimation technique (often abbreviated as IV)
- The exogenous variables used in the IV estimation technique

As a method of estimation, instrumental variables (IV) are used in many economic applications often when a controlled experiment to test the existence of a causal relationship is not feasible and some correlation between the original explanatory variables and the error term is suspected. When the explanatory variables correlate or show some form of dependence with the error terms in a regression relationship, instrumental variables can provide a consistent estimation.

The theory of instrumental variables was first introduced by Philip G. Wright in his 1928 publication titled *The Tariff on Animal and Vegetable Oils *but has since evolved in its applications in economics.

### When Instrumental Variables are Used

There are several circumstances under which explanatory variables show a correlation with the error terms and an instrumental variable may be used. First, the dependent variables may actually cause one of the explanatory variables (also known as the covariates). Or, relevant explanatory variables are simply omitted or overlooked in the model. It may even be that the explanatory variables suffered some error of measurement. The problem with any of these situations is that the traditional linear regression that might normally be employed in the analysis may produce inconsistent or biased estimates, which is where instrumental variables (IV) would then be used and the second definition of instrumental variables becomes more important.

In addition to being the name of the method, instrumental variables are also the very variables used to obtain consistent estimates using this method. They are exogenous, meaning that they exist outside of the explanatory equation, but as instrumental variables, they are correlated with the equation's endogenous variables. Beyond this definition, there is one other primary requirement for using an instrumental variable in a linear model: the instrumental variable must not be correlated with the error term of the explanatory equation. That is to say that the instrumental variable cannot pose the same issue as the original variable for which it is attempting to resolve.

### Instrumental Variables in Econometrics Terms

For a deeper understanding of instrumental variables, let's review an example. Suppose one has a model:

y = Xb + e

Here y is a T x 1 vector of dependent variables, X is a T x k matrix of independent variables, b is a k x 1 vector of parameters to estimate, and e is a k x 1 vector of errors. OLS can be imagined, but suppose in the environment being modeled that the matrix of independent variables X may be correlated to the e's. Then using a T x k matrix of independent variables Z, correlated to the X's but uncorrelated to the e's one can construct an IV estimator that will be consistent:

b_{IV} = (Z'X)^{-1}Z'y

The two-stage least squares estimator is an important extension of this idea.

In that discussion above, the exogenous variables Z are called instrumental variables and the instruments (Z'Z)^{-1}(Z'X) are estimates of the part of X that is not correlated to the e's.