Order & Rank Conditions of Identification

The identification problem is a mathematical (as opposed to statistical) problem associated with simultaneous equation systems. It is concerned with the question of the possibility or impossibility of obtaining meaningful estimates of the structural parameters. The identification problem can be solved if economic theory and extraneous information can be used to place restrictions on the set of simultaneous equations. These restrictions can take a variety forms (such as use of extraneous estimates of parameters, knowledge  of exact relationship among parameters, knowledge of the relative variances of disturbances, knowledge of zero correlation between disturbances in different equations, etc.), but the restrictions usually employed, called Zero Restrictions, take the form of specifying that certain structural parameters are zero, i.e., that certain endogenous variables and certain exogenous variables do not appear in certain equations. Mathematical investigation has shown that in the case of Zero Restrictions on structural parameters each equation can be checked for identification by using a rule called the Rank Condition. It turns out, however, that this rule is quite awkward to employ, and as a result a simpler rule, called the Order Condition, is used in its stead. This rule only requires counting included and excluded variables in each equation.

Here is a brief illustration of order and rank conditions of identification in simultaneous equation system:

M = number of endogenous variables in the model
K = number of exogenous variables in the model
m = number of endogenous variable in an equation
k = number of exogenous variables in a given equation
Rank condition is defined by the rank of the matrix, which should have a dimension (M-1), where m is the number of endogenous variables. This matrix is formed from the coefficients of the variables (both endogenous and exogenous) excluded from that particular equation but included in the other equations in the model.
The rank condition tells us whether the equation under consideration is identified or not, whereas the order condition tells us if it is exactly identified or overidentified.
1. If K-k>m-1 and the rank of the ρ(A) is M-1 then the equation is overidentified.
2. If K-k=m-1 and the rank of the ρ(A) is M-1 then the equation is exactly identified.
3. If K-k>=m-1 and the rank of the ρ(A) is less than M-1 then the equation is underidentified.
4. If K-k<=m-1 the structural equation is unidentified. The rank of the ρ(A) is less M-1 in this case.

From these rules, we can tell that, the order condition is only a necessary condition, not a sufficient one. So that, technically speaking, the rank condition must also be checked. Many econometricians do not bother doing this, however, gambling that the rank condition will be satisfied (as it usually is) if the order condition is satisfied. This procedure is hence not recommended.