15.2 The generalised method of moments
When \(\ell > k\) the equations \(\bar g_n(\theta ) = 0\) have no solution in general, because a sample will not satisfy more conditions than there are parameters. GMM makes \(\bar g_n\) as close to zero as possible in a chosen metric.
Definition 15.2.1. Let \(W_n\) be an \(\ell \times \ell \) positive definite weighting matrix with \(W_n \xrightarrow {p} W\). The GMM estimator minimises \[ Q_n(\theta ) = \bar g_n(\theta )'W_n\,\bar g_n(\theta ) . \]
The estimator is consistent under the identification condition that \(E\,g(w, \theta ) = 0\) only at \(\theta = \theta _0\), together with regularity conditions (a compact parameter space and a uniform law of large numbers for \(\bar g_n\)). The argument is that \(Q_n(\theta )\) converges uniformly to \(Q(\theta ) = E\,g(\theta )'W\,E\,g(\theta )\), which is zero at \(\theta _0\) and positive elsewhere, so the minimiser of \(Q_n\) converges to the minimiser of \(Q\). We take consistency as given and derive the distribution.
Theorem 15.2.2. Under the conditions of Theorem 15.1.3, but with \(\ell \geq k\) and \(G\) of rank \(k\), the GMM estimator satisfies \[ \sqrt n(\hat \theta - \theta _0) \xrightarrow {d} N\bigl (0,\ (G'WG)^{-1}G'WSWG(G'WG)^{-1}\bigr ) . \] The variance is smallest, in the sense of positive semi-definite matrices, when \(W = S^{-1}\), and is then \[ V_{\text {eff}} = (G'S^{-1}G)^{-1} . \]
Proof. The first-order condition is \(\bar G_n(\hat \theta )'W_n\bar g_n(\hat \theta ) = 0\). Expanding \(\bar g_n(\hat \theta ) = \bar g_n(\theta _0) + \bar G_n(\tilde \theta )(\hat \theta - \theta _0)\) as before and substituting, \[ \sqrt n(\hat \theta - \theta _0) = -\bigl [\bar G_n(\hat \theta )'W_n\bar G_n(\tilde \theta )\bigr ]^{-1}\bar G_n(\hat \theta )'W_n\sqrt n\,\bar g_n(\theta _0) \xrightarrow {d} -(G'WG)^{-1}G'W\,N(0, S), \] which has the stated variance.
For optimality, write \(A = (G'WG)^{-1}G'W\) and \(A_0 = (G'S^{-1}G)^{-1}G'S^{-1}\). Both satisfy \(AG = A_0G = I\). The variance for weight \(W\) is \(ASA'\), and \[ ASA' - A_0SA_0' = (A - A_0)S(A - A_0)' + A_0S(A - A_0)' + (A - A_0)SA_0' . \] Now \(A_0S = (G'S^{-1}G)^{-1}G'\), so \(A_0S(A - A_0)' = (G'S^{-1}G)^{-1}(AG - A_0G)' = 0\), and likewise for the last term. Hence \(ASA' - A_0SA_0' = (A - A_0)S(A - A_0)'\), which is positive semi-definite. Finally \(A_0SA_0' = (G'S^{-1}G)^{-1}G'S^{-1}SS^{-1}G(G'S^{-1}G)^{-1} = (G'S^{-1}G)^{-1}\). □
The optimal weight gives more weight to the moments that are estimated precisely and takes account of their correlation, exactly as GLS does for observations. \(S\) is unknown, but it can be estimated consistently from a preliminary consistent estimate, which leads to the standard procedure.
- 1.
- First step. Minimise \(Q_n\) with a convenient weight, such as \(W_n = I\) or one that scales the moments, to obtain a consistent \(\hat \theta _1\).
- 2.
- Second step. Estimate \(\hat S = \frac 1n\sum g(w_i, \hat \theta _1)g(w_i, \hat \theta _1)'\) and minimise \(Q_n\) with \(W_n = \hat S^{-1}\).
The two-step estimator has the efficient asymptotic variance \(V_{\text {eff}}\), estimated by \((\hat G'\hat S^{-1}\hat G)^{-1}/n\). The two steps can be repeated until convergence (iterated GMM), or \(S\) can be made a function of \(\theta \) inside the objective (the continuously updated estimator of Hansen, Heaton and Yaron 1996); all three are asymptotically equivalent. With time-series data the \(g(w_t, \theta _0)\) are usually serially correlated, and \(\hat S\) must be a HAC estimator such as Newey and West’s of Theorem 11.4.3, with \(g_t\) in place of \(e_tx_t\).
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