Reduced form Totally Explained

Everything about Reduced Form totally explained

In social science and statistics, particularly 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 a function of the exogenous variables, if any.

Structural form

As an example, we use a system of two equations. Both equations are linear. The system models the supply and demand of some specific good. The quantity of the demand varies inversely with the price: a higher price decreases demand. The quantity of the supply varies directly with the price: a higher price makes supply more profitable. In formulas: ť supply:

Q = a_S +   b_S P ,

ť demand:

Q = a_D +  b_D P ,

with positive b_S and negative b_D. This is the structural form of the equation system: the equations as derived from the theory. (In this case, the economic theory of supply and demand.)
The two endogenous variables are the traded quantity Q and the price P, defined by the two equations of the system. Of course there are always as many endogenous variables as there are equations.

Reduced form

To find the reduced form, one must solve the equations for the endogenous variables. This reduces the system considerably. For instance, we know that the two right-hand sides of the equations are the same (both equal to Q), and hence

a_S + b_S P = a_D +  b_D P

. This can be written as

P (b_S-b_D) = a_D-a_S

, or

P = (a_D-a_S ) / (b_S-b_D)

. Thus, P is in fact a fixed number, independent of Q. Below, this number is called

pi_2

, while the similar number for Q is

pi_1

: ť

Q = pi_1 ,

P = pi_2  ,

The structure of supply and demand has disappeared. The two

pi

coefficients are the reduced form coefficients. They are easily identified from data on Q and P. (However, the four structural form coefficients above can not be identified from data: the parameter identification problem.)
It is easily verified that:
ť

pi_1 = (a_D b_S - a_S b_D) /(b_S - b_D) ,

pi_2 = (a_D-a_S ) / (b_S-b_D ) ,

Structural form with exogenous variable

Exogenous variables are variables which are not defined by the system. In the following structural system, Z is an exogenous variable: ť supply:

Q = a_S +   b_S P ,

ť demand:

Q = a_D +  b_D P  + c Z,

(Note that the choice of the endogenous variables can not be derived from the equations themselves; the modeller might alternatively have chosen for instance Q and Z as endogenous variables, which would make P the exogenous variable.)

Reduced form with exogenous variable

The reduced form is now slightly more complicated: ť

Q = pi_B ,

Without restrictions on the A and B, the coefficients of A and B can not be identified from data on y and x: each row of the structural model is just a linear relation between y and z with unknown coefficients. (Again the parameter identification problem.) The M reduced form equations (the rows of the matrix equation y = Π x above) can be identied from the data because each of them contains only one endogenous variable.

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