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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] observations with values in a set $\\mathcal{X}$, and let $\\Theta \\subseteq \\R^k$ be a parameter set.\nLet $g \\colon \\mathcal{X} \\times \\Theta \\to \\R^m$ with $m \\ge k$ be a moment function such that the true parameter $\\theta_0$ satisfies $E[g(X_1, \\theta_0)] = 0$, where $E$ is the [[expectation|0DCT0QZVQ1]].\nLet $\\hat{g}_n(\\theta) = \\frac{1}{n}\\sum_{i=1}^n g(X_i, \\theta)$, and let $W_n$ be an $m \\times m$ [[positive semidefinite|0880F9EQCP]] weight matrix computed from the data.\nThe *generalized method of moments estimator* is an [[estimator|0TTNRP5PFQ]]\n$$\\hat{\\theta}_n \\in \\operatorname*{arg\\,min}_{\\theta \\in \\Theta} \\; \\hat{g}_n(\\theta)^T \\, W_n \\, \\hat{g}_n(\\theta).$$\n\n## Remarks\n\nIf $m = k$ and the equations $\\hat{g}_n(\\theta) = 0$ have a solution, then that solution is the estimator for every positive definite $W_n$.\nThis is the [[method of moments|09J1H07VAH]], a case of [[estimating equations|WTXFYA093D]].\nWhen $m \u003e k$ the model is over-identified, and the choice of $W_n$ matters.\nUnder regularity conditions, the choice $W_n \\to S^{-1}$ in probability, with $S = E[g(X_1, \\theta_0) g(X_1, \\theta_0)^T]$ nonsingular, gives the smallest asymptotic variance, $(G^T S^{-1} G)^{-1}$ with $G = E[\\nabla_\\theta g(X_1, \\theta_0)]$.\n\nExample: for [[Poisson|0VEFSSKX18]]($\\lambda$) data, take $g(x, \\lambda) = (x - \\lambda, \\; x^2 - \\lambda - \\lambda^2)^T$, with $m = 2 \u003e k = 1$.\nBoth components have mean zero at the true $\\lambda$, since $E X = \\lambda$ and $E X^2 = \\lambda + \\lambda^2$.\n\nNon-example: in a normal model with the two parameters $(\\mu, \\sigma^2)$, take the single condition $E[X - \\mu] = 0$, so $m = 1 \u003c k = 2$.\nThe clause $m \\ge k$ fails, and the objective does not depend on $\\sigma^2$, so it cannot determine $\\sigma^2$.\n",
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