广义矩估计讲义.doc

广义矩估计 基本知识： 矩方法是一种古老的估计方法。 其基本思想是：在随机抽样中，样本统计量将依概率收敛于某个常数。这个常数又是分布中未知参数的一个函数。 在严格意义上，一个统计量是观察的n维随机向量即子样的一个（波雷尔可测）函数，且要求它不包含任何未知参数。 基本定义： 统计量 为子样的ν阶矩（ν阶原点矩）； 统计量 为子样的ν阶中心矩。 子样矩的均值与方差 约定，我们用到时假定它是存在的。 基本做法： 设：母体的可能分布族为，其中来自参数空间的是待估计的未知参数。假定母体分布的k阶矩存在，则母体分布的ν阶矩 是的函数。 对于子样，其ν阶子样矩是 现在用子样矩作为母体矩的估计，即令： （1）式确定了包含k个未知参数的k个方程式。求解（1）式就可以得到的一组解。因为是随机变量，故解得的也是随机变量。将分别作为的估计称为矩方法的估计，这种求估计量的方法称为矩方法。 定理 若存在2ν阶矩，则对子样的ν阶原点矩，有 。 证明： 。 Idea of GMM Estimation under weak assumptions; based on so-called moment conditions. Moment conditions are statements involving the data and the parameters. Arise naturally in many contexts. For example: (A) In a regression model,, we might think that . This implies the moment condition . (B) Consider the economic relation Under rational expectation, the expectation error, , should be orthogonal to the information set, , and for we have the moment condition . Properties of GMM GMM is a large sample estimator. Desirable properties as . ? Consistent under weak assumptions. No distributional assumptions like in maximum likelihood (ML) estimation. ? Asymptotically efficient in the class of models that uses the same amount of information. ? Many estimators are special cases of GMM. Unifying framework for comparing estimators. ? GMM is a nonlinear procedure. We do not need a regression setup . We can have . Moment Conditions and MM Estimation ? Consider a variable with some (possibly unknown) distribution. Assume that the mean exists. We want to estimate . ? We could state the population moment condition: , or , where . ? The parameter ? is identified by the condition if there is a unique solution, in the sense only if . ? We cannot calculate from an observed sample,. Define the sample moment condition as (1) ? By Law of Large Numbers, sample moments converge to population moments, for . (2) The method of moments estimator, , is the solution to (1), i.e