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In econometrics and statistics, the generalized method of moments (GMM) is a generic method for estimating parameters in statistical models.Usually it is applied in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the data's distribution function may not be known, and therefore maximum likelihood estimation is not applicable.
An example application of the method of moments is to estimate polynomial probability density distributions. In this case, an approximating polynomial of order is defined on an interval [,]. The method of moments then yields a system of equations, whose solution involves the inversion of a Hankel matrix. [2]
In statistics, the method of estimating equations is a way of specifying how the parameters of a statistical model should be estimated.This can be thought of as a generalisation of many classical methods—the method of moments, least squares, and maximum likelihood—as well as some recent methods like M-estimators.
The EM algorithm consists of two steps: the E-step and the M-step. Firstly, the model parameters and the () can be randomly initialized. In the E-step, the algorithm tries to guess the value of () based on the parameters, while in the M-step, the algorithm updates the value of the model parameters based on the guess of () of the E-step.
To estimate parameters of a conditional moment model, the statistician can derive an expectation function (defining "moment conditions") and use the generalized method of moments (GMM). However, there are infinitely many moment conditions that can be generated from a single model; optimal instruments provide the most efficient moment conditions.
GEEs belong to a class of regression techniques that are referred to as semiparametric because they rely on specification of only the first two moments. They are a popular alternative to the likelihood-based generalized linear mixed model which is more at risk for consistency loss at variance structure specification. [5]
An empirical likelihood ratio function is defined and used to obtain confidence intervals parameter of interest θ similar to parametric likelihood ratio confidence intervals. [7] [8] Let L(F) be the empirical likelihood of function , then the ELR would be: = / (). Consider sets of the form
This method is known as system GMM. Note that the consistency and efficiency of the estimator depends on validity of the assumption that the errors can be decomposed as in equation (1). This assumption can be tested in empirical applications and likelihood ratio test often reject the simple random effects decomposition. [2]