Regression Analysis > Simultaneous Equations Model (SEM)

You may want to read this other article first: What is Sumultaneity?

## What is a Simultaneous Equations Model (SEM)?

A Simultaneous Equation Model (SEM) is a model in the form of **a set of linear simultaneous equations**. Where introductory regression analysis introduces models with a single equation (e.g. simple linear regression), **SEM models have two or more equations.** In a single-equation model, changes in the response variable (Y) happen because of changes in the explanatory variable (X); in an SEM model, other Y variables are among the explanatory variables in each SEM equation. The system is jointly determined by the equations in the system; In other words, the system exhibits some type of simultaneity or “back and forth” causation between the X and Y variables.

## SEM Example

The market for graduate nurses is influenced by:

**Demand behavior,****Supply behavior,****Equilibrium levels**for pay rate and employment.

Let’s say that the** simultaneous equations model** for this scenario is made up the following two equations:

- Demand: n
_{t}= β_{1}+ β_{2}g_{t}+ β_{3}p_{t}+ ε_{1t} - Supply: n
_{t}= β_{11}+ β_{12}m_{t}+ β_{13}p_{t}+ ε_{2t}

**Where**:

- n = number of employed nurses,
- p = earnings rate,
- g = graduate nursing school enrollment,
- m = median income for employed nurses.

**These formulas are just regression equations tailored for this specific model; β is the regression coefficient and ε is the error term — unexpected factors that can creep into the model.

## Complete Models and Structural Equation Models

When the total number of endogenous variables is equal to the number of equations, it is called a **complete SEM**. If earnings rate and number of employed nurses are the only two endogenous variables in the above example, then this SEM is complete. A complete SEM is called a **structural equations model**.

## Using the Model to Solve Problems

Remember those simultaneous equations from algebra? They can be solve together to find values for x and y. In the same way, the equations in SEM can also be solved. Using the above example, let’s say you wanted to find out the partial impact of median pay (m) on both the number of employed nurses (n) and the pay rate (p). You can model this by solving the equations for n and p: