It is useful to summarize what we have determined thus far. The unknown structural parameters consist of
Simply counting parameters in the structure and reduced forms yields an excess of which is, as might be expected from the earlier results, the number of unknown elements in T. Without further information, identification is clearly impossible. The additional information comes in several forms.
1. Normalizations. In each equation, one variable has a coefficient of 1. This normalization is a necessary scaling of the equation that is logically equivalent to putting one variable on the left-hand side of a regression. For purposes of identification (and some estimation methods), the choice among the endogenous variables is arbitrary. But at the time the model is formulated, each equation will usually have some natural dependent variable. The normalization does not identify the dependent variable in any formal or causal sense. For example, in a model of supply and demand, both the "demand"
r = an M x M nonsingular matrix, B = a K x M parameter matrix, E — an M x M symmetric positive definite matrix. The known, reduced-form parameters are
II = a K x M reduced-form coefficients matrix, SI = an M x M reduced-form covariance matrix.
12For other interpretations, see Amemiya (1985, p. 230) and Gabrielsen (1978). Some deeper theoretical results on identification of parameters in econometric models are given by Bekker and Wansbeek (2001).
equation, Q = f(P, x), and the "inverse demand" equation, P = g(Q, x), are appropriate specifications of the relationship between price and quantity. We note, though, the following:
With the normalizations, there are M(M— 1), not M2, undetermined values in T and this many indeterminacies in the model to be resolved through nonsample information.
2. Identities. In some models, variable definitions or equilibrium conditions imply that all the coefficients in a particular equation are known. In the preceding market example, there are three equations, but the third is the equilibrium condition Qd = Qs. Klein's Model I (Example 15.3) contains six equations, including two accounting identities and the equilibrium condition. There is no question of identification with respect to identities. They may be carried as additional equations in the model, as we do with Klein's Model I in several later examples, or built into the model a priori, as is typical in models of supply and demand.
The substantive nonsample information that will be used in identifying the model will consist of the following:
3. Exclusions. The omission of variables from an equation places zeros in B and I\ In Example 15.5, the exclusion of income from the supply equation served to identify its parameters.
4. Linear restrictions. Restrictions on the structural parameters may also serve to rule out false structures. For example, a long-standing problem in the estimation of production models using time-series data is the inability to disentangle the effects of economies of scale from those of technological change. In some treatments, the solution is to assume that there are constant returns to scale, thereby identifying the effects due to technological change.
5. Restrictions on the disturbance covariance matrix. In the identification of a model, these are similar to restrictions on the slope parameters. For example, if the previous market model were to apply to a microeconomic setting, then it would probably be reasonable to assume that the structural disturbances in these supply and demand equations are uncorrelated. Section 15.3.3 shows a case in which a covariance restriction identifies an otherwise unidentified model.
To formalize the identification criteria, we require a notation for a single equation. The coefficients of the yth equation are contained in the y'th columns of T and B. The /th equation is yl) + x'By = (15-6)
(For convenience, we have dropped the observation subscript.) In this equation, we know that (1) one of the elements in T, is one and (2) some variables that appear elsewhere in the model are excluded from this equation. Table 15.1 defines the notation used to incorporate these restrictions in (15-6). Equation j may be written
^Components of Equation / Endogenous Variables apendent Variable - y) Exogenous Variables
Included Excluded y;- = Mj variables y* = M* variables
The number of equations is Mj + M* + 1 = M. The number of exogenous variables is Kj + K* = K. The coefficient on yj in equation j is 1. *s will always be associated with excluded variables.
The exclusions imply that y* — 0 and /?* = 0. Thus, r; = [l - y) o'] and b;■ = [-/»;■ 0'].
(Note the sign convention.) For this equation, we partition the reduced-form coefficient matrix in the same fashion:
The reduced-form coefficient matrix is n = -Br which implies that nr = -b.
The yth column of this matrix equation applies to the ;'th equation, nr, = -b7.
Inserting the parts from Table 15.1 yields
Now extract the two subequations,
The solution for b in terms of T that we observed at the beginning of this discussion is in (15-8). Equation (15-9) may be written n)Yj - It*. (15-10)
This system is K* equations in M, unknowns. If they can be solved for yj, then (15-8) gives the solution for and the equation is identified. For there to be a solution, there must be at least as many equations as unknowns, which leads to the following condition.
DEFINITION 15.1 Order Condition for Identification of Equation j
The number of exogenous variables excluded from equation j must be at least as large as the number of endogenous variables included in equation j.
The order condition is only a counting rule. It is a necessary but not sufficient condition for identification. It ensures that (15-10) has at least one solution, but it does not ensure that it has only one solution. The sufficient condition for uniqueness follows.
DEFINITION 15.2 Rank Condition for Identification rank[jr*, II*] = rank[n*] = Mj. This condition imposes a restriction on a submatrix of the reduced-form coefficient matrix.
The rank condition ensures that there is exactly one solution for the structural parameters given the reduced-form parameters. Our alternative approach to the identification problem was to use the prior restrictions on [T, B] to eliminate all false structures. An equivalent condition based on this approach is simpler to apply and has more intuitive appeal. We first rearrange the structural coefficients in the matrix
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