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.

The method requires that a certain number of moment conditions be specified for the model. These moment conditions are functions of the model parameters and the data, such that their expectation is zero at the parameters' true values. The GMM method then minimizes a certain norm of the sample averages of the moment conditions, and can therefore be thought of as a special case of minimum-distance estimation.^[1]

The GMM estimators are known to be consistent, asymptotically normal, and most efficient in the class of all estimators that do not use any extra information aside from that contained in the moment conditions. GMM were advocated by Lars Peter Hansen in 1982 as a generalization of the method of moments,^[2] introduced by Karl Pearson in 1894. However, these estimators are mathematically equivalent to those based on "orthogonality conditions" (Sargan, 1958, 1959) or "unbiased estimating equations" (Huber, 1967; Wang et al., 1997).

Description

Suppose the available data consists of T observations {Y_t }_{t = 1,...,T}, where each observation Y_t is an n-dimensional multivariate random variable. We assume that the data come from a certain statistical model, defined up to an unknown parameter θ ∈ Θ. The goal of the estimation problem is to find the “true” value of this parameter, θ₀, or at least a reasonably close estimate.

A general assumption of GMM is that the data Y_t be generated by a weakly stationary ergodic stochastic process. (The case of independent and identically distributed (iid) variables Y_t is a special case of this condition.)

In order to apply GMM, we need to have "moment conditions", that is, we need to know a vector-valued function g(Y,θ) such that

${\displaystyle m(\theta _{0})\equiv \operatorname {E} =0,}$

where E denotes expectation, and Y_t is a generic observation. Moreover, the function m(θ) must differ from zero for θ ≠ θ₀, otherwise the parameter θ will not be point-identified.

The basic idea behind GMM is to replace the theoretical expected value E with its empirical analog—sample average:

${\displaystyle {\hat {m}}(\theta )\equiv {\frac {1}{T}}\sum _{t=1}^{T}g(Y_{t},\theta )}$

and then to minimize the norm of this expression with respect to θ. The minimizing value of θ is our estimate for θ₀.

By the law of large numbers, ${\displaystyle \scriptstyle {\hat {m}}(\theta )\,\approx \;\operatorname {E} \,=\,m(\theta )}$ for large values of T, and thus we expect that ${\displaystyle \scriptstyle {\hat {m}}(\theta _{0})\;\approx \;m(\theta _{0})\;=\;0}$. The generalized method of moments looks for a number ${\displaystyle \scriptstyle {\hat {\theta }}}$ which would make ${\displaystyle \scriptstyle {\hat {m}}(\;\!{\hat {\theta }}\;\!)}$ as close to zero as possible. Mathematically, this is equivalent to minimizing a certain norm of ${\displaystyle \scriptstyle {\hat {m}}(\theta )}$ (norm of m, denoted as ||m||, measures the distance between m and zero). The properties of the resulting estimator will depend on the particular choice of the norm function, and therefore the theory of GMM considers an entire family of norms, defined as

${\displaystyle \|{\hat {m}}(\theta )\|_{W}^{2}={\hat {m}}(\theta )^{\mathsf {T}}\,W{\hat {m}}(\theta ),}$

where W is a positive-definite weighting matrix, and ${\displaystyle m^{\mathsf {T}}}$ denotes transposition. In practice, the weighting matrix W is computed based on the available data set, which will be denoted as ${\displaystyle \scriptstyle {\hat {W}}}$. Thus, the GMM estimator can be written as

${\displaystyle {\hat {\theta }}=\operatorname {arg} \min _{\theta \in \Theta }{\bigg (}{\frac {1}{T}}\sum _{t=1}^{T}g(Y_{t},\theta ){\bigg )}^{\mathsf {T}}{\hat {W}}{\bigg (}{\frac {1}{T}}\sum _{t=1}^{T}g(Y_{t},\theta ){\bigg )}}$

Under suitable conditions this estimator is consistent, asymptotically normal, and with right choice of weighting matrix ${\displaystyle \scriptstyle {\hat {W}}}$ also asymptotically efficient.

Properties

Consistency

Consistency is a statistical property of an estimator stating that, having a sufficient number of observations, the estimator will converge in probability to the true value of parameter:

${\displaystyle {\hat {\theta }}{\xrightarrow {p}}\theta _{0}\ {\text{as}}\ T\to \infty .}$

Sufficient conditions for a GMM estimator to be consistent are as follows:

1. ${\displaystyle {\hat {W}}_{T}{\xrightarrow {p}}W,}$ where W is a positive semi-definite matrix,
2. ${\displaystyle \,W\operatorname {E} =0}$  only for ${\displaystyle \,\theta =\theta _{0},}$
3. The space of possible parameters ${\displaystyle \Theta \subset \mathbb {R} ^{k}}$ is compact,
4. ${\displaystyle \,g(Y,\theta )}$  is continuous at each θ with probability one,
5. ${\displaystyle \operatorname {E} <\infty .}$

The second condition here (so-called Global identification condition) is often particularly hard to verify. There exist simpler necessary but not sufficient conditions, which may be used to detect non-identification problem:

• Order condition. The dimension of moment function m(θ) should be at least as large as the dimension of parameter vector θ.
• Local identification. If g(Y,θ) is continuously differentiable in a neighborhood of ${\displaystyle \theta _{0}}$, then matrix ${\displaystyle W\operatorname {E} }$ must have full column rank.

In practice applied econometricians often simply assume that global identification holds, without actually proving it.^[3]: 2127

Asymptotic normalityedit

Asymptotic normality is a useful property, as it allows us to construct confidence bands for the estimator, and conduct different tests. Before we can make a statement about the asymptotic distribution of the GMM estimator, we need to define two auxiliary matrices:

${\displaystyle G=\mathrm{E}<!--\; \; -->{\mathrm{\nabla <!--\; \nabla \; -->}}_{}}$Zdroj:https://en.wikipedia.org?pojem=Generalized_method_of_moments
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