In
econometrics
Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics", '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
and
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the generalized method of moments (GMM) is a generic method for estimating
parameters in
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repre ...
s. Usually it is applied in the context of
semiparametric model In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components.
A statistical model is a parameterized family of distributions: \ indexed by a parameter \theta.
* A parametric model is a model i ...
s, 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
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
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
Norm, the Norm or NORM may refer to:
In academic disciplines
* Normativity, phenomenon of designating things as good or bad
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy), a standard in normative e ...
of the sample averages of the moment conditions, and can therefore be thought of as a
special case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of .Brown, James Robert.� ...
of
minimum-distance estimation
Minimum-distance estimation (MDE) is a conceptual method for fitting a statistical model to data, usually the Empirical distribution function, empirical distribution. Often-used estimators such as ordinary least squares can be thought of as special ...
.
The GMM
estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s are known to be
consistent
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
,
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
Lars Peter Hansen (born 26 October 1952 in Urbana, Illinois) is an Americans, American economy, economist. He is the David Rockefeller Distinguished Service Professor in Economics, Statistics, and the Booth School of Business, at the Universi ...
in 1982 as a generalization of the
method of moments, introduced by
Karl Pearson
Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English biostatistician and mathematician. He has been credited with establishing the discipline of mathematical statistics. He founded the world's first university ...
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 , where each observation ''Y
t'' is an ''n''-dimensional
multivariate random variable
In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge ...
. We assume that the data come from a certain
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repre ...
, defined up to an unknown
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
. The goal of the estimation problem is to find the “true” value of this parameter, ''θ''
0, 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
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
. (The case of
independent and identically distributed
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in Pennsylvania, United States
* Independentes (English: Independents), a Portuguese artist ...
(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
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
''g''(''Y'',''θ'') such that
:
where E denotes
expectation, and ''Y
t'' is a generic observation. Moreover, the function ''m''(''θ'') must differ from zero for , 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:
:
and then to minimize the norm of this expression with respect to ''θ''. The minimizing value of ''θ'' is our estimate for ''θ''
0.
By the
law of large numbers
In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law o ...
,
for large values of ''T'', and thus we expect that
. The generalized method of moments looks for a number
which would make
as close to zero as possible. Mathematically, this is equivalent to minimizing a certain norm of
(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
:
where ''W'' is a
positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite ...
weighting matrix, and
denotes
transposition. In practice, the weighting matrix ''W'' is computed based on the available data set, which will be denoted as
. Thus, the GMM estimator can be written as
:
Under suitable conditions this estimator is
consistent
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
,
asymptotically normal, and with right choice of weighting matrix
also
asymptotically efficient.
Properties
Consistency
Consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
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:
:
Sufficient conditions for a GMM estimator to be consistent are as follows:
#
where ''W'' is a
positive semi-definite matrix,
#
only for
# The
space
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
of possible parameters
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
,
#
is continuous at each ''θ'' with probability one,
#
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
, then matrix