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. This gives the latter as functions of the exogenous variables, if any. 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.
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
\varepsilon
f(Y,X,\varepsilon)=0
Y=g(X,\varepsilon)
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:
Q=aS+bSP+uS,
demand:
Q=aD+bDP+cZ+uD,
where the terms
ui
Q=\pi10+\pi11Z+eQ,
P=\pi20+\pi21Z+eP,
where the parameters
\piij
ai,bi,c
ei
If the reduced form model is estimated using empirical data, obtaining estimated values for the coefficients
\piij,
aS
bS
aS=(\pi10\pi21-\pi11\pi20)/\pi21,
bS=\pi11/\pi21.
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.
Let y be a column vector of M endogenous variables. In the case above with Q and P, we had M = 2. Let z be a column vector of K exogenous variables; in the case above z consisted only of Z. The structural linear model is
Ay=Bz+v,
where
v
y=A-1Bz+A-1v=\Piz+w,
with vector
w
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. (This is again the parameter identification problem.) 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.