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Computing Generalized Method of Moments and

Generalized Empirical Likelihood with R

Pierre Chauss´e

Abstract

This paper shows how to estimate models by the generalized method of moments

and the generalized empirical likelihood using the R package gmm. A brief discussion is

oﬀered on the theoretical aspects of both methods and the functionality of the package is

presented through several examples in economics and ﬁnance. It is a modiﬁed version of

Chauss´e (2010) published in the Journal of Statistical Software. It has been adapted to

the version 1.4-0.

Keywords:˜generalized empirical likelihood, generalized method of moments, empirical likeli-

hood, continuous updated estimator, exponential tilting, exponentially tilted empirical likeli-

hood, R.

1. Introduction

The generalized method of moments (GMM) has become an important estimation procedure

in many areas of applied economics and ﬁnance since Hansen (1982) introduced the two step

GMM (2SGMM). It can be seen as a generalization of many other estimation methods like

least squares (LS), instrumental variables (IV) or maximum likelihood (ML). As a result, it is

less likely to be misspeciﬁed. The properties of the estimators of LS depend on the exogeneity

of the regressors and the circularity of the residuals, while those of ML depend on the choice of

the likelihood function. GMM is much more ﬂexible since it only requires some assumptions

about moment conditions. In macroeconomics, for example, it allows to estimate a structural

model equation by equation. In ﬁnance, most data such as stock returns are characterized

by heavy-tailed and skewed distributions. Because it does not impose any restriction on the

distribution of the data, GMM represents a good alternative in this area as well. As a result

of its popularity, most statistical packages like Matlab, Gauss or Stata oﬀer tool boxes to use

the GMM procedure. It is now possible to easily use this method in R with the new gmm

package.

Although GMM has good potential theoretically, several applied studies have shown that the

properties of the 2SGMM may in some cases be poor in small samples. In particular, the

estimators may be strongly biased for certain choices of moment conditions. In response to

this result, Hansen, Heaton, and Yaron (1996) proposed two other ways to compute GMM: the

iterative GMM (ITGMM) and the continuous updated GMM (CUE)

. Furthermore, another

family of estimation procedures inspired by Owen (2001), which also depends only on moment

conditions, was introduced by Smith (1997). It is the generalized empirical likelihood (GEL).

See also Hall (2005) for a detailed presentation of most recent developments regarding GMM.

2 GMM and GEL with R

So far, this method has not reached the popularity of GMM and it was not included in any

statistical package until gmm was developed for R which also includes a GEL procedure.

Asymptotic properties of GMM and generalized empirical likelihood (GEL) are now well

established in the econometric literature. Newey and Smith (2004) and Anatolyev (2005)

have compared their second order asymptotic properties. In particular, they show that the

second order bias of the empirical likelihood (EL) estimator, which is a special case of GEL,

is smaller than the bias of the estimators from the three GMM methods. Furthermore, as

opposed to GMM, the bias does not increase with the number of moment conditions. Since

the eﬃciency improves when the number of conditions goes up, this is a valuable property.

However, these are only asymptotic results which do not necessarily hold in small sample as

shown by Guggenberger (2008). In order to analyze small sample properties, we have to rely

on Monte Carlo simulations. However, Monte Carlo studies on methods such as GMM or

GEL depend on complicated algorithms which are often home made. Because of that, results

from such studies are not easy to reproduce. The solution should be to use a common tool

which can be tested and improved upon by the users. Because it is open source, R oﬀers a

perfect platform for such tool.

The gmm package allows to estimate models using the three GMM methods, the empirical

likelihood and the exponential tilting, which belong to the family of GEL methods, and the

exponentially tilted empirical likelihood which was proposed by Schennach (2007), Also it

oﬀers several options to estimate the covariance matrix of the moment conditions. Users can

also choose between optim, if no restrictions are required on the coeﬃcients of the model to

be estimated, and either nlminb or constrOptim for constrained optimizations. The results

are presented in such a way that R users who are familiar with lm objects, ﬁnd it natural.

In fact, the same methods are available for gmm and gel objects produced by the estimation

procedures.

The paper is organized as follows. Section 2 presents the theoretical aspects of the GMM

method along with several examples in economics and ﬁnance. Through these examples,

the functionality of the gmm packages is presented in details. Section 3 presents the GEL

method with some of the examples used in section 2. Section 4 concludes and Section 5 gives

the computational details of the package.

2. Generalized method of moments

This section presents an overview of the GMM method. It is intended to help the users

understand the options that the gmm package oﬀers. For those who are not familiar with the

method and require more details, see Hansen (1982) and Hansen et˜al. (1996) for the method

itself, Newey and West (1994) and Andrews (1991) for the choice of the covariance matrix or

Hamilton (1994).

We want to estimate a vector of parameters θ

∈ R

from a model based on the following

q × 1 vector of unconditional moment conditions:

E[g(θ

, x

)] = 0, (1)

where x

is a vector of cross-sectional data, time series or both. In order for GMM to pro-

duce consistent estimates from the above conditions, θ

has to be the unique solution to

E[g(θ, x

)] = 0 and be an element of a compact space. Some boundary assumptions on higher

Pierre Chauss´e 3

moments of g(θ, x

) are also required. However, it does not impose any condition on the

distribution of x

, except for the degree of dependence of the observations when it is a vector

of time series.

Several estimation methods such as least squares (LS), maximum likelihood (ML) or instru-

mental variables (IV) can also be seen as being based on such moment conditions, which make

them special cases of GMM. For example, the following linear model:

Y = Xβ + u,

where Y and X are respectively n × 1 and n × k matrices, can be estimated by LS. The

estimate

β is obtained by solving min

kuk

and is therefore the solution to the following ﬁrst

order condition:

u(β) = 0,

which is the estimate of the moment condition E(X

(β)) = 0. The same model can be

estimated by ML in which case the moment condition becomes:



(β)

dβ



= 0,

where l

(β) is the density of u

. In presence of endogeneity of the explanatory variable X,

which implies that E(X

) 6= 0, the IV method is often used. It solves the endogeneity

problem by substituting X by a matrix of instruments H, which is required to be correlated

with X and uncorrelated with u. These properties allow the model to be estimated by the

conditional moment condition E(u

) = 0 or its implied unconditional moment condition

E(u

) = 0. In general we say that u

is orthogonal to an information set I

or that

E(u

) = 0 in which case H

is a vector containing functions of any element of I

. The model

can therefore be estimated by solving

u(β) = 0.

When there is no assumption on the covariance matrix of u, the IV corresponds to GMM. If

E(X

) = 0 holds, generalized LS with no assumption on the covariance matrix of u other

than boundary ones is also a GMM method. For the ML procedure to be viewed as GMM,

the assumption on the distribution of u must be satisﬁed. If it is not, but E(dl

(θ

)/dθ) = 0

holds, as it is the case for linear models with non normal error terms, the pseudo-ML which

uses a robust covariance matrix can be seen as being a GMM method.

Because GMM depends only on moment conditions, it is a reliable estimation procedure for

many models in economics and ﬁnance. For example, general equilibrium models suﬀer from

endogeneity problems because these are misspeciﬁed and they represent only a fragment of

the economy. GMM with the right moment conditions is therefore more appropriate than

ML. In ﬁnance, there is no satisfying parametric distribution which reproduces the properties

of stock returns. The family of stable distributions is a good candidate but only the densities

of the normal, Cauchy and L´evy distributions, which belong to this family, have a closed form

expression. The distribution-free feature of GMM is therefore appealing in that case.

Although GMM estimators are easily consistent, eﬃciency and bias depend on the choice of

moment conditions. Bad instruments implies bad information and therefore low eﬃciency.

4 GMM and GEL with R

The eﬀects on ﬁnite sample properties are even more severe and are well documented in the

literature on weak instruments. Newey and Smith (2004) show that the bias increases with

the number of instruments but eﬃciency decreases. Therefore, users need to be careful when

selecting the instruments. Carrasco (2009) gives a good review of recent developments on

how to choose instruments in her introduction.

In general, the moment conditions E(g(θ

, x

)) = 0 is a vector of nonlinear functions of θ

and

the number of conditions is not limited by the dimension of θ

. Since eﬃciency increases with

the number of instruments q is often greater than p, which implies that there is no solution

¯g(θ) ≡

i=1

g(θ, x

) = 0.

The best we can do is to make it as close as possible to zero by minimizing the quadratic

function ¯g(θ)

W ¯g(θ), where W is a positive deﬁnite and symmetric q × q matrix of weights.

The optimal matrix W which produces eﬃcient estimators is deﬁned as:

∗

lim

n→∞

V ar(

√

n¯g(θ

)) ≡ Ω(θ

)

−1

. (2)

This optimal matrix can be estimated by an heteroskedasticity and auto-correlation consistent

(HAC) matrix like the one proposed by Newey and West (1987). The general form is:

Ω =

n−1

s=−(n−1)

(s)

(θ

∗

), (3)

where k

(s) is a kernel, h is the bandwidth which can be chosen using the procedures proposed

by Newey and West (1987) and Andrews (1991),

(θ

∗

) =

g(θ

∗

, x

)g(θ

∗

, x

i+s

)

and θ

∗

is a convergent estimate of θ

. There are many choices for the HAC matrix. They

depend on the kernel and bandwidth selection. Although the choice does not aﬀect the

asymptotic properties of GMM, very little is known about the impacts in ﬁnite samples. The

GMM estimator

θ is therefore deﬁned as:

θ = arg min

¯g(θ)

Ω(θ

∗

)

−1

¯g(θ) (4)

The original version of GMM proposed by Hansen (1982) is called two-step GMM (2SGMM).

It computes θ

∗

by minimizing ¯g(θ)

¯g(θ). The algorithm is therefore:

1- Compute θ

∗

= arg min

¯g(θ)

2- Compute the HAC matrix

Ω(θ

∗

)

3- Compute the 2SGMM

θ = arg min

¯g(θ)



Ω(θ

∗

)



−1

¯g(θ)

In order to improve the properties of 2SGMM, Hansen et˜al. (1996) suggest two other meth-

ods. The ﬁrst one is the iterative version of 2SGMM (ITGMM) and can be computed as

follows:

Pierre Chauss´e 5

1- Compute θ

(0)

= arg min

¯g(θ)

2- Compute the HAC matrix

Ω(θ

(0)

)

3- Compute the θ

(1)

= arg min

¯g(θ)



Ω(θ

(0)

)



−1

¯g(θ)

4- If kθ

(0)

− θ

(1)

k < tol stops, else θ

(0)

= θ

(1)

and go to 2-

5- Deﬁne the ITGMM estimator

θ as θ

(1)

where tol can be set as small as we want to increase the precision. In the other method, no

preliminary estimate is used to obtain the HAC matrix. The latter is treated as a function of θ

and is allowed to change when the optimization algorithm computes the numerical derivatives.

It is therefore continuously updated as we move toward the minimum. For that, it is called

the continuous updated estimator (CUE). This method is highly nonlinear. It is therefore

crucial to choose a starting value that is not too far from the minimum. A good choice is the

estimate from 2SGMM which is known to be root-n convergent. The algorithm is:

1- Compute θ

∗

using 2SGMM

2- Compute the CUE estimator deﬁned as

θ = arg min

¯g(θ)



Ω(θ)



−1

¯g(θ)

using θ

∗

as starting value.

According to Newey and Smith (2004) and Anatolyev (2005), 2SGMM and ITGMM are

second order asymptotically equivalent. On the other hand, they show that the second order

asymptotic bias of CUE is smaller. The diﬀerence in the bias comes from the randomness

of θ

∗

in Ω(θ

∗

). Iterating only makes θ

∗

more eﬃcient. These are second order asymptotic

properties. They are informative but may not apply in ﬁnite samples. In most cases, we have

to rely on numerical simulations to analyze the properties in small samples.

Given some regularity conditions, the GMM estimator converges as n goes to inﬁnity to the

following distribution:

√

θ − θ

)

→ N(0, V ),

where

V = E



∂g(θ

, x

)

∂θ



Ω(θ

)

−1



∂g(θ

, x

)

∂θ



Inference can therefore be performed on

θ using the assumption that it is approximately

distributed as N(θ

V /n).

If q > p, we can perform a J-test to verify if the moment conditions hold. The null hypothesis

and the statistics are respectively H0 : E[g(θ, x

)] = 0 and:

n¯g(

θ)

[

Ω(θ

∗

)]

−1

¯g(

θ)

→ χ

q−p

3. GMM with R

The gmm package can be loaded the usual way.

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