14.2 Identification
The reduced form contains everything the data can tell us. The question is whether the structural parameters can be recovered from it. They cannot in general: if \(F\) is any non-singular \(G \times G\) matrix, the model \(FBy_t + F\Gamma x_t = Fu_t\) has the same reduced form, since \((FB)^{-1}F\Gamma = B^{-1}\Gamma \). The two structures are observationally equivalent. An equation is identified if the prior restrictions on it, usually that certain variables are excluded, rule out every such \(F\) except those that leave it unchanged.
Theorem 14.2.1 (Order and rank conditions). Write \(A = [B\ \ \Gamma ]\), of order \(G \times (G + K)\), and suppose the restrictions on the first equation are that certain variables are excluded, so that \(\alpha _1'\Phi = 0\), where \(\alpha _1'\) is the first row of \(A\) and the columns of \(\Phi \) select the excluded variables. Then the first equation is identified if and only if \[ \rank (A\Phi ) = G - 1 \qquad \text {(rank condition)}. \] A necessary condition is that at least \(G - 1\) variables be excluded from the equation; equivalently, the number of exogenous variables excluded from the equation must be at least the number of endogenous variables included on its right-hand side (order condition).
Proof. An admissible transformation replaces the first equation by a linear combination \(f'A\) of all the equations, and it must satisfy the same restrictions, \(f'A\Phi = 0\). The true equation corresponds to \(f = e_1\), and \(e_1'A\Phi = \alpha _1'\Phi = 0\), so the first row of \(A\Phi \) is zero. The equation is identified if the only solutions are \(f = ce_1\), that is, if the solution space of \(f'(A\Phi ) = 0\) has dimension one. That dimension is \(G - \rank (A\Phi )\), so identification holds if and only if \(\rank (A\Phi ) = G - 1\). Since \(A\Phi \) has as many columns as there are exclusions, the rank cannot exceed their number, which gives the order condition. If the equation includes \(G_1\) endogenous variables and \(K_1\) exogenous ones, it excludes \((G - G_1) + (K - K_1)\) variables, and the order condition \((G - G_1) + (K - K_1) \geq G - 1\) is \(K - K_1 \geq G_1 - 1\). □
The order condition counts; the rank condition checks that the excluded variables actually appear elsewhere in the system, in a way that distinguishes the equation from combinations of the others. An equation is exactly identified if the order condition holds with equality and the rank condition holds, overidentified if more variables are excluded than needed, and underidentified otherwise. In the market model each equation excludes one exogenous variable and includes one endogenous regressor, so both are exactly identified. If the supply equation had no \(w\), the demand equation would be underidentified, as the right panel of Figure 15 would then collapse to a single point.
The excluded exogenous variables are exactly the instruments of Section 13, and identification is the rank condition of the instrumental variables estimator. For an exactly identified equation, indirect least squares and 2SLS coincide, by Theorem 13.3.1(i), since both are the IV estimator with the exogenous variables as instruments.
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