In statistics, and particularly in econometrics, the reduced form of a system of equations is the result of solving the system for the endogenous variables.
In econometrics, the equations of a structural form model are estimated in their theoretically given form, while an alternative approach to estimation is to first solve the theoretical equations for the endogenous variables to obtain reduced form equations, and then to estimate the reduced form equations.
If we assume that demand is influenced not only by price, but also by an exogenous variable, Z, we can consider the structural supply and demand model where the terms
are random errors (deviations of the quantities supplied and demanded from those implied by the rest of each equation).
By solving for the unknowns (endogenous variables) P and Q, this structural model can be rewritten in the reduced form: where the parameters
of the structural model, and where the reduced form errors
If the reduced form model is estimated using empirical data, obtaining estimated values for the coefficients
some of the structural parameters can be recovered: By combining the two reduced form equations to eliminate Z, the structural coefficients of the supply side model (
) can be derived: Note however, that this still does not allow us to identify the structural parameters of the demand equation.
Let z be a column vector of K exogenous variables; in the case above z consisted only of Z.
is a vector of structural shocks, and A and B are matrices; A is a square M × M matrix, while B is M × K. The reduced form of the system is: with vector
of reduced form errors that each depends on all structural errors, where the matrix A must be nonsingular for the reduced form to exist and be unique.
Without restrictions on the A and B, the coefficients of A and B cannot be identified from data on y and z: each row of the structural model is just a linear relation between y and z with unknown coefficients.
The M reduced form equations (the rows of the matrix equation y = Π z above) can be identified from the data because each of them contains only one endogenous variable.