Reduced form

Let Y be the vector of the variables to be explained (endogeneous variables) by a statistical model and X be the vector of explanatory (exogeneous) variables. In addition let ${\displaystyle \epsilon }$ be a vector of error terms. Then the general expression of a structural form is ${\displaystyle f(Y,X,\epsilon )=0}$, where f is a function, possibly from vectors to vectors in the case of a multiple-equation model. The reduced form of this model is given by ${\displaystyle Y=g(X,\epsilon )}$, with g a function.

Structural and reduced forms

Exogenous variables are variables which are not determined by the system. 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

supply: ${\displaystyle Q=a_S+b_{S}P+u_S,}$

demand: ${\displaystyle Q=a_D+b_{D}P+cZ+u_D,}$

where the terms ${\displaystyle u_i}$ 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:

${\displaystyle Q={\pi }_{10}+{\pi }_{11}Z+e_Q,}$

${\displaystyle P={\pi }_{20}+{\pi }_{21}Z+e_P,}$

where the parameters ${\displaystyle {\pi }_{ij}}$ depend on the parameters ${\displaystyle a_i,b_i,c}$ of the structural model, and where the reduced form errors ${\displaystyle e_i}$ each depend on the structural parameters and on both structural errors. Note that both endogenous variables depend on the exogenous variable Z.

If the reduced form model is estimated using empirical data, obtaining estimated values for the coefficients ${\displaystyle {\pi }_{ij},}$ 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 (${\displaystyle a_S}$ and ${\displaystyle b_S}$) can be derived:

${\displaystyle a_S=({\pi }_{10}{\pi }_{21}-{\pi }_{11}{\pi }_{20})/{\pi }_{21},}$

${\displaystyle b_S={\pi }_{11}/{\pi }_{21}.}$

Note however, that this still does not allow us to identify the structural parameters of the demand equation. For that, we would need an exogenous variable which is included in the supply equation of the structural model, but not in the demand equation.