## Rank Condition Of Identification

In a model containing M equations in M endogenous variables, an equation is identified if and only if at least one nonzero determinant of order (M — 1)(M — 1) can be constructed from the coefficients of the variables (both endogenous and predetermined) excluded from that particular equation but included in the other equations of the model.

6The term rank refers to the rank of a matrix and is given by the largest-order square matrix (contained in the given matrix) whose determinant is nonzero. Alternatively, the rank of a matrix is the largest number of linearly independent rows or columns of that matrix. See App. B.

The Rank Condition of Identifiability*

CHAPTER NINETEEN: THE IDENTIFICATION PROBLEM 751

As an illustration of the rank condition of identification, consider the following hypothetical system of simultaneous equations in which the Y variables are endogenous and the X variables are predetermined.7

Y2t - P20 - P23Y3t - Y21 Xu - Y22X2t = u2t (19.3.3)

Y3t - P30 - P31Y1t - Y31 X1t - Y32X2t = u3t (19.3.4)

To facilitate identification, let us write the preceding system in Table 19.1, which is self-explanatory.

Let us first apply the order condition of identification, as shown in Table 19.2. By the order condition each equation is identified. Let us recheck with the rank condition. Consider the first equation, which excludes variables Y4, X2, and X3 (this is represented by zeros in the first row of Table 19.1). For this equation to be identified, we must obtain at least one nonzero determinant of order 3 x 3 from the coefficients of the variables excluded from this equation but included in other equations. To obtain the determinant we first obtain the relevant matrix of coefficients of variables Y4, X2, and X3 included in the other equations. In the present case there is