Reduced form

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. 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 ε {displaystyle varepsilon } be a vector of error terms. Then the general expression of a structural form is f ( Y , X , ε ) = 0 {displaystyle f(Y,X,varepsilon )=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 Y = g ( X , ε ) {displaystyle Y=g(X,varepsilon )} , with g a function. 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 where the terms u i {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: where the parameters π i j {displaystyle pi _{ij}} depend on the parameters a i , b i , c {displaystyle a_{i},b_{i},c} of the structural model, and where the reduced form errors e i {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 π i j , {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 ( a S {displaystyle a_{S}} and b S {displaystyle b_{S}} ) can be derived: