In many fields of study, including statistics and economics, researchers rely on valid exclusion restrictions when they are estimating outcomes using either instrumental variables (IV) or exogenous variables. Such calculations are often used to analyze the causal effect of a binary treatment.

### Variables and Exclusion Restrictions

Loosely defined, an exclusion restriction is considered valid so long as the independent variables do not directly affect the dependent variables in an equation. For example, researchers rely on randomization of the sample population in order to ensure comparability across the treatment and control groups. At times, however, randomization is not possible.

This may for any number of reasons, such as lack of access to suitable populations or budgetary restrictions. In such cases, the best practice or strategy is to rely on an instrumental variable. Simply put, the method of using instrumental variables is utilized to estimate causal relationships when a controlled experiment or study is simply not feasible. That's where valid exclusion restrictions come into play.

When researchers employ instrumental variables, they rely on two primary assumptions. The first is that the excluded instruments are distributed independently of the error process. The other is that the excluded instruments are sufficiently correlated with the included endogenous regressors. As such, the specification of an IV model states that the excluded instruments affect the independent variable only indirectly.

As a result, exclusion restrictions are considered observed variables that impact treatment assignment, but not the outcome of interest conditional on treatment assignment. If, on the other hand, an excluded instrument is shown to exert both direct and indirect influences on the dependent variable, the exclusion restriction should be rejected.

### The Importance of Exclusion Restrictions

In simultaneous equation systems or a system of equations, exclusion restrictions are critical. The simultaneous equation system is a finite set of equations in which certain assumptions are made. Despite its importance to the solution of the system of equations, the validity of an exclusion restriction cannot be tested as the condition involves an unobservable residual.

Exclusion restrictions are often imposed intuitively by the researcher who must then convince of the plausibility of those assumptions, meaning that the audience must believe the researcher’s theoretical arguments that support the exclusion restriction.

The concept of exclusion restrictions denotes that some of the exogenous variables are not in some of the equations. Often this idea is expressed by saying the coefficient next to that exogenous variable is zero. This explanation may make this restriction (hypothesis) testable and may make a simultaneous equation system identified.

Sources

- Schmidheiny, Kurt. "Short Guides to Microeconometrics: Instrumental Variables." Schmidheiny.name. Fall 2016.
- University of Manitoba Rady Faculty of Health Sciences staff. "Introduction to Instrumental Variables." UManitoba.ca.