Reduced form of a Simultaneous Equation System

In a Simultaneous Equation System (SES), equations are interrelated, and the solution to one equation depends on the values of other equations in the system.

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The reduced form of a SES expresses each endogenous variable as a function of only exogenous variables and error terms, effectively breaking the system down into separate equations.

Consider a basic two-equation SES:

[ Y_1 = \alpha_1 + \beta_{11}X_1 + \beta_{12}X_2 + \varepsilon_1 ]
[ Y_2 = \alpha_2 + \beta_{21}X_1 + \beta_{22}X_2 + \varepsilon_2 ]

Here, ( Y_1 ) and ( Y_2 ) are endogenous variables, ( X_1 ) and ( X_2 ) are exogenous variables, and ( \varepsilon_1 ) and ( \varepsilon_2 ) are error terms.

To obtain the reduced form:

1. Isolate Endogenous Variables: Express each endogenous variable in terms of only exogenous variables and error terms. For example:
   [ Y_1 = \alpha_1 + \beta_{11}X_1 + \beta_{12}X_2 + \varepsilon_1 ]
   [ Y_2 = \alpha_2 + \beta_{21}X_1 + \beta_{22}X_2 + \varepsilon_2 ]
2. Substitute into Each Other: Substitute the expressions for endogenous variables into each other’s equations. This results in equations where each endogenous variable is expressed solely in terms of exogenous variables and error terms.
3. Final Reduced Form: The final reduced form expresses each endogenous variable as a function of exogenous variables and error terms without explicitly referring to other endogenous variables.

The reduced form is useful for estimating the parameters of the system without dealing with simultaneous equations. It simplifies the analysis by transforming the system into a set of single-equation models, making it amenable to standard econometric techniques like ordinary least squares (OLS) regression.