Math 2.0

Statistics Definition

Generalized method of moments

Builds on Method of momentsEstimating equationsPositive semidefinite matrixIndependent and identically distributed random variablesExpected valuePoisson distributionEstimator

Definition

Let X1,…,XnX_1, \dots, X_n be i.i.d. observations with values in a set 𝒳\mathcal{X}, and let Θ⊆ℝk\Theta \subseteq \R^k be a parameter set. Let g ⁣:𝒳×Θ→ℝmg \colon \mathcal{X} \times \Theta \to \R^m with m≥km \ge k be a moment function such that the true parameter θ0\theta_0 satisfies E[g(X1,θ0)]=0E[g(X_1, \theta_0)] = 0, where EE is the expectation. Let g^n(θ)=1n∑i=1ng(Xi,θ)\hat{g}_n(\theta) = \frac{1}{n}\sum_{i=1}^n g(X_i, \theta), and let WnW_n be an m×mm \times m positive semidefinite weight matrix computed from the data. The generalized method of moments estimator is an estimator θ^n∈arg min⁡θ∈Θ  g^n(θ)T Wn g^n(θ).\hat{\theta}_n \in \operatorname*{arg\,min}_{\theta \in \Theta} \; \hat{g}_n(\theta)^T \, W_n \, \hat{g}_n(\theta).

Remarks

If m=km = k and the equations g^n(θ)=0\hat{g}_n(\theta) = 0 have a solution, then that solution is the estimator for every positive definite WnW_n. This is the method of moments, a case of estimating equations. When m>km > k the model is over-identified, and the choice of WnW_n matters. Under regularity conditions, the choice Wn→S−1W_n \to S^{-1} in probability, with S=E[g(X1,θ0)g(X1,θ0)T]S = E[g(X_1, \theta_0) g(X_1, \theta_0)^T] nonsingular, gives the smallest asymptotic variance, (GTS−1G)−1(G^T S^{-1} G)^{-1} with G=E[∇θg(X1,θ0)]G = E[\nabla_\theta g(X_1, \theta_0)].

Example: for Poisson(λ\lambda) data, take g(x,λ)=(x−λ,  x2−λ−λ2)Tg(x, \lambda) = (x - \lambda, \; x^2 - \lambda - \lambda^2)^T, with m=2>k=1m = 2 > k = 1. Both components have mean zero at the true λ\lambda, since EX=λE X = \lambda and EX2=λ+λ2E X^2 = \lambda + \lambda^2.

Non-example: in a normal model with the two parameters (μ,σ2)(\mu, \sigma^2), take the single condition E[X−μ]=0E[X - \mu] = 0, so m=1<k=2m = 1 < k = 2. The clause m≥km \ge k fails, and the objective does not depend on σ2\sigma^2, so it cannot determine σ2\sigma^2.

Reading path

To understand this page from the ground up, read these 84 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Estimating equations concept
  4. Factorial definition
  5. Integers object
  6. Matrix definition
  7. Ordered field definition
  8. Proposition concept
  9. Random experiment concept
  10. Rational numbers object
  11. Real numbers object
  12. Set definition
  13. Ring of integers statement
  14. Square matrix definition
  15. Transpose definition
  16. Disjunction definition
  17. Implication definition
  18. Predicate definition
  19. Ordered pair definition
  20. Union definition
  21. Conjugate transpose definition
  22. Sample space definition
  23. The rationals form a field statement
  24. Existential quantifier definition
  25. Universal quantifier definition
  26. Cartesian product definition
  27. Hermitian matrix definition
  28. Subset definition
  29. The rationals form an ordered field statement
  30. Uniqueness quantifier definition
  31. Binary relation definition
  32. Positive semidefinite matrix definition
  33. Universal set definition
  34. Order on the reals definition
  35. Power set definition
  36. Complement definition
  37. Function definition
  38. Interval definition
  39. Injective function definition
  40. Sigma-algebra definition
  41. Preimage of a set definition
  42. Countable set definition
  43. Event definition
  44. Borel sigma-algebra definition
  45. Probability axioms concept
  46. Probability measure definition
  47. Probability distribution definition
  48. Probability space definition
  49. Conditional probability definition
  50. Random variable definition
  51. Cumulative distribution function definition
  52. Discrete random variable definition
  53. Law of a random variable definition
  54. Independence of two events definition
  55. Convergence in probability definition
  56. Random variable with a density definition
  57. Joint probability distribution definition
  58. Probability mass function definition
  59. Mutual independence of events definition
  60. Expected value definition
  61. Identically distributed random variables definition
  62. Independence of two random variables definition
  63. Probability density function definition
  64. Marginal distribution definition
  65. Independent sigma-algebras definition
  66. Poisson distribution definition
  67. Variance definition
  68. Law of the unconscious statistician statement
  69. Linearity of expectation statement
  70. Independence of random variables definition
  71. Independence means the joint law is the product of the marginals statement
  72. Covariance definition
  73. Moment definition
  74. Independent and identically distributed random variables definition
  75. Expectation of a product of independent random variables statement
  76. Variance of a sum statement
  77. Statistic definition
  78. Independent random variables are uncorrelated statement
  79. Estimator definition
  80. Variance of a sum of independent random variables statement
  81. Sample mean definition
  82. Consistent estimator definition
  83. Method of moments concept
  84. Generalized method of moments definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Generalized_method_of_moments
About this page
TagMRCBJ0WDR0
URIm2://MRCBJ0WDR0
Kinddefinition
WritingW0 stub
MSC62F10, 62P20
DefinesGeneralized method of moments
UsesMethod of moments, Estimating equations, Positive semidefinite matrix, Independent and identically distributed random variables, Expected value, Poisson distribution, Estimator

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