In
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
, and particularly in
econometricsEconometrics is concerned with the tasks of developing and applying quantitative or statistical methods to the study and elucidation of economic principles. Econometrics combines economic theory with statistics to analyze and test economic relationships...
, the
reduced formIn 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 a function of the exogenous variables, if any....
of a system of equations is the result of solving the system for the
endogenousThe word endogenous means "proceeding from within", the opposite of exogenous.-Biology:Endogenous substances are those that originate from within an organism, tissue, or cell . Endogenous retrovirus are caused by ancient infections of germ cells in humans, mammals and other vertebrates...
variables. This gives the latter as a function of the
exogenousExogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....
variables, if any.
Let
Y and
X be random vectors.
Y is the vector of the variables to be explained (endogeneous variables) by a
statistical modelA statistical model is a set of mathematical equations which describe the behavior of an object of study in terms of random variables and their associated probability distributions...
and
X is the vector of explanatory (exogeneous) variables.
In
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
, and particularly in
econometricsEconometrics is concerned with the tasks of developing and applying quantitative or statistical methods to the study and elucidation of economic principles. Econometrics combines economic theory with statistics to analyze and test economic relationships...
, the
reduced formIn 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 a function of the exogenous variables, if any....
of a system of equations is the result of solving the system for the
endogenousThe word endogenous means "proceeding from within", the opposite of exogenous.-Biology:Endogenous substances are those that originate from within an organism, tissue, or cell . Endogenous retrovirus are caused by ancient infections of germ cells in humans, mammals and other vertebrates...
variables. This gives the latter as a function of the
exogenousExogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....
variables, if any.
Let
Y and
X be random vectors.
Y is the vector of the variables to be explained (endogeneous variables) by a
statistical modelA statistical model is a set of mathematical equations which describe the behavior of an object of study in terms of random variables and their associated probability distributions...
and
X is the vector of explanatory (exogeneous) variables. In addition let be a vector of error terms. Then the general expression of a
structural form is , 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 , with
g a function.
Structural form
As an example, we use a system of two equations. Both equations are linear. The system models the
supply and demandSupply and demand is an economic model based on price, utility and quantity in a market. It concludes that in a competitive market, price will function to equalize the quantity demanded by consumers, and the quantity supplied by producers, resulting in an economic equilibrium of price and...
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:
- supply:
- demand:
with positive
bS and negative
bD. 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 . This can be written as , or . Thus,
P is in fact a fixed number, independent of
Q. Below, this number is called , while the similar number for
Q is :
The structure of supply and demand has disappeared. The two 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 problemThe parameter identification problem is a problem which can occur in the estimation of multiple-equation econometric models where the equations have variables in common....
.)
It is easily verified that:
Structural and reduced forms with an exogenous variable
Exogenous variables are variables which are not defined by the system. If we assume that demand is influenced not only by price, but also by an exogenous variable,
Z. The structural form becomes:
- supply:
- demand:
In the above set of equations, 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.
This structural model can be rewritten in the reduced form:
As before, the four reduced form coefficients can be derived from the five structural form coefficients. Note that both endogenous variables depend on the exogenous variable
Z.
By combining the two reduced form equations to eliminate
Z, the structural coefficients of the supply side model ( and ) can be derived from the four reduced form coefficients :
Note however, that this still does not allow us to identify the structural parameters of the demand-side model. For that, we would need an exogenous variable which is included in the supply-side of the structural model, but not on the demand-side.
The general linear case
Let
y be a
column vectorIn linear algebra, a column vector or column matrix is an m × 1 matrix, i.e. a matrix consisting of a single column of elements.The transpose of a column vector is a row vector and vice versa....
of
M endogenous variables. In the case above with
Q and
P, we have
M = 2. Let
x be a column vector of exogenous variables; in the case above
x consists only of
Z. The structural linear model (without error terms, as above) is:
where
A and
B are
matricesIn mathematics, a matrix is a rectangular array of numbers, such asEntries of a matrix are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied...
;
A is a square
M ×
M matrix. The reduced form of the system is:
Again, each endogenous variable depends on each exogenous variable. It is easily verified that:
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 problemThe parameter identification problem is a problem which can occur in the estimation of multiple-equation econometric models where the equations have variables in common....
.) 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.
See also
- System of linear equations
In mathematics, a system of linear equations is a collection of linear equations involving the same set of variables. For example,...
- Simultaneous equations
In mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations...
- Parameter identification problem
The parameter identification problem is a problem which can occur in the estimation of multiple-equation econometric models where the equations have variables in common....