15.3 Linear GMM and its relation to 2SLS
For the linear model \(y = x'\beta + u\) with instruments \(z\), the moment conditions are \(E[z(y - x'\beta )] = 0\), so \(\bar g_n(\beta ) = \frac 1nZ'(Y - X\beta )\) is linear in \(\beta \) and the GMM problem has a closed-form solution.
- (i)
- For weight \(W_n\), the linear GMM estimator is \[ b(W_n) = (X'ZW_nZ'X)^{-1}X'ZW_nZ'Y . \]
- (ii)
- With \(W_n = (Z'Z/n)^{-1}\) it is 2SLS.
- (iii)
- The efficient weight is \(S^{-1}\) with \(S = E(u^2zz')\), estimated by \(\hat S = \frac 1n\sum \hat u_i^2z_iz_i'\) from 2SLS residuals. Under homoscedasticity, \(S = \sigma ^2E(zz')\), and 2SLS is the efficient GMM estimator; under heteroscedasticity, two-step GMM is more efficient than 2SLS.
Proof. (i) \(Q_n(\beta ) = \frac {1}{n^2}(Y - X\beta )'ZW_nZ'(Y - X\beta )\) is a quadratic in \(\beta \), and setting its derivative \(-\frac {2}{n^2}X'ZW_nZ'(Y - X\beta )\) to zero gives the formula.
(ii) Substituting \(W_n = n(Z'Z)^{-1}\), the factor \(n\) cancels, and \(b = (X'P_ZX)^{-1}X'P_ZY = b_{\text {2SLS}}\) by Theorem 13.3.1.
(iii) Here \(g = z(y - x'\beta )\) and \(S = E(u^2zz')\), and the efficient weight is \(S^{-1}\) by Theorem 15.2.2. If \(E(u^2 \mid z) = \sigma ^2\), then \(S = \sigma ^2E(zz')\), and a scalar factor in the weight does not change the minimiser, so \(W = E(zz')^{-1}\), which is the probability limit of \((Z'Z/n)^{-1}\), is efficient: 2SLS is efficient GMM. Otherwise the efficient weight differs, and by Theorem 15.2.2 the efficient GMM estimator has a smaller asymptotic variance. □
When \(\ell = k\), \(Z'X\) is square and \(b(W) = (Z'X)^{-1}Z'Y = b_{\text {IV}}\) for every \(W\): the weighting matrix matters only when there are surplus moments. The variance of the efficient estimator, \(\bigl (X'Z\hat S^{-1}Z'X\bigr )^{-1}n\) in the notation of (iii), is valid under heteroscedasticity of unknown form, so GMM delivers heteroscedasticity-robust inference automatically.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.