Generalized method of moments: 두 판 사이의 차이

2026년 6월 9일 (화) 15:06 기준 최신판

Generalized method of moments (GMM) is an estimation framework that generalizes the classical method of moments. A moment condition is an expectation that equals zero at the true parameter values. Under certain theoretical moment conditions, parameters can be estimated by forcing their sample analogues as close to zero as possible. While GMM yields consistent, asymptotically normal estimates, it is flexible as it does not rely on a specific probability distribution for the data, contrary to other estimation methods such as maximum likelihood.

Relation to OLS and IV

Ordinary least squares (OLS) and instrumental variable (IV) estimators can be seen as special cases of GMM. In an OLS regression $y_{i} = β x_{i} + ϵ_{i}$ , the assumption $𝔼 (x_{i} ϵ_{i}) = 0$ is a moment condition. The OLS estimator can be derived by setting the sample analogue:

$\frac{1}{N} \sum_{i} x_{i} (y_{i} - \hat{β} x_{i}) = 0$

References

Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50(4): 1029–1054.

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-'''Generalized method of moments''' (GMM) is an estimation framework that generalizes the classical method of moments. A '''moment condition''' is an expectation that equals zero at the true parameter values. Under certain theoretical moment conditions, parameters can be estimated by forcing their sample analogues as close to zero as possible.<ref>Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. ''Econometrica'' 50(4): 1029-1054.</ref> While GMM yields consistent, asymptotically normal estimates, it is flexible as it does not rely on a specific probability distribution for the data, contrary to other estimation methods such as maximum likelihood.
+'''Generalized method of moments''' (GMM) is an estimation framework that generalizes the classical method of moments. A '''moment condition''' is an expectation that equals zero at the true parameter values. Under certain theoretical moment conditions, parameters can be estimated by forcing their sample analogues as close to zero as possible. While GMM yields consistent, asymptotically normal estimates, it is flexible as it does not rely on a specific probability distribution for the data, contrary to other estimation methods such as maximum likelihood.
 == Relation to OLS and IV ==
-[[Ordinary least square (OLS)]] and [[instrumental variable (IV)]] estimators can be seen as special cases of GMM. In an OLS regression <math>y_i = \beta x_i + \epsilon_i</math>, the assumption <math>\mathbb{E}(x_i \epsilon_i) = 0</math> is a moment condition.
+[[Ordinary least squares (OLS)]] and [[instrumental variable (IV)]] estimators can be seen as special cases of GMM. In an OLS regression <math>y_i = \beta x_i + \epsilon_i</math>, the assumption <math>\mathbb{E}(x_i \epsilon_i) = 0</math> is a moment condition. The OLS estimator can be derived by setting the sample analogue:
+<div style="text-align: center;">
+<math>\frac{1}{N}\sum_i x_i (y_i - \hat\beta x_i) = 0</math>
+</div>
 == References ==
-<references />
+* Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. ''Econometrica'' 50(4): 1029–1054.
+[[분류:금융수학과 방법론]]