Greene W.H. Econometric Analysis

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CHAPTER 12

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449

in the class of consistent and asymptotically normally distributed (CAN) estima-

tors. There are two important practical considerations in this setting. First, the

researcher will want to know that he or she has not made demonstrably suboptimal

use of the data. (The literature contains discussions of GMM estimation of fully

speciﬁed parametric probit models—GMM estimation in this context is unambigu-

ously inferior to maximum likelihood.) Thus, when possible, one would want to

avoid obviously inefﬁcient estimators. On the other hand, it will usually be the case

that the researcher is not choosing from a list of available estimators; he or she has

one at hand, and questions of relative efﬁciency are moot.

12.5.2 EXTREMUM ESTIMATORS

An extremum estimator is one that is obtained as the optimizer of a criterion function

q(θ |data). Three that have occupied much of our effort thus far are

•

Least squares:

= Argmax



−(1/n)



i=1

− h(x

, θ

))



•

Maximum likelihood:

= Argmax



(1/n)



i=1

ln f (y

, θ

)



, and

•

GMM:

GMM

= Argmax[−

m(data, θ

GMM

)



m(data, θ

GMM

)].

(We have changed the signs of the ﬁrst and third only for convenience so that all three

may be cast as the same type of optimization problem.) The least squares and max-

imum likelihood estimators are examples of M estimators, which are deﬁned by op-

timizing over a sum of terms. Most of the familiar theoretical results developed here

and in other treatises concern the behavior of extremum estimators. Several of the es-

timators considered in this chapter are extremum estimators, but a few—including the

Bayesian estimators, some of the semiparametric estimators, and all of the nonparamet-

ric estimators—are not. Nonetheless. we are interested in establishing the properties of

estimators in all these cases, whenever possible. The end result for the practitioner will

be the set of statistical properties that will allow him or her to draw with conﬁdence

conclusions about the data generating process(es) that have motivated the analysis in

the ﬁrst place.

Derivations of the behavior of extremum estimators are pursued at various levels

in the literature. (See, for example, any of the sources mentioned in Footnote 1 of this

chapter.) Amemiya (1985) and Davidson and MacKinnon (2004) are very accessible

treatments. Newey and McFadden (1994) is a rigorous analysis that provides a current,

standard source. Our discussion at this point will only suggest the elements of the anal-

ysis. The reader is referred to one of these sources for detailed proofs and derivations.

12.5.3 ASSUMPTIONS FOR ASYMPTOTIC PROPERTIES

OF EXTREMUM ESTIMATORS

Some broad results are needed in order to establish the asymptotic properties of the

classical (not Bayesian) conventional extremum estimators noted above.

1. The parameter space (see Section 12.2) must be convex and the parameter vector

that is the object of estimation must be a point in its interior. The ﬁrst requirement

rules out ill-deﬁned estimation problems such as estimating a parameter which

can only take one of a ﬁnite discrete set of values. Thus, searching for the date of

a structural break in a time-series model as if it were a conventional parameter

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leads to a nonconvexity. Some proofs in this context are simpliﬁed by assuming

that the parameter space is compact. (A compact set is closed and bounded.)

However, assuming compactness is usually restrictive, so we will opt for the weaker

requirement.

2. The criterion function must be concave in the parameters. (See Section A.8.2.)

This assumption implies that with a given data set, the objective function has an

interior optimum and that we can locate it. Criterion functions need not be “glob-

ally concave”; they may have multiple optima. But, if they are not at least “locally

concave,” then we cannot speak meaningfully about optimization. One would nor-

mally only encounter this problem in a badly structured model, but it is possible to

formulate a model in which the estimation criterion is monotonically increasing or

decreasing in a parameter. Such a model would produce a nonconcave criterion

function.

The distinction between compactness and concavity in the preceding

condition is relevant at this point. If the criterion function is strictly continuous in

a compact parameter space, then it has a maximum in that set and assuming con-

cavity is not necessary. The problem for estimation, however, is that this does not

rule out having that maximum occur on the (assumed) boundary of the parameter

space. This case interferes with proofs of consistency and asymptotic normality.

The overall problem is solved by assuming that the criterion function is concave

in the neighborhood of the true parameter vector.

3. Identiﬁability of the parameters. Any statement that begins with “the true param-

eters of the model, θ

are identiﬁed if ...” is problematic because if the parameters

are “not identiﬁed,” then arguably, they are not the parameters of the (any) model.

(For example, there is no “true” parameter vector in the unidentiﬁed model of Ex-

ample 2.5.) A useful way to approach this question that avoids the ambiguity of

trying to deﬁne the true parameter vector ﬁrst and then asking if it is identiﬁed

(estimable) is as follows, where we borrow from Davidson and MacKinnon (1993,

p. 591): Consider the parameterized model, M, and the set of allowable data gener-

ating processes for the model, μ. Under a particular parameterization μ, let there

be an assumed “true” parameter vector, θ (μ). Consider any parameter vector θ

in the parameter space, . Deﬁne

(μ, θ) = plim

(θ |data).

This function is the probability limit of the objective function under the assumed

parameterization μ. If this probability limit exists (is a ﬁnite constant) and more-

over, if

[μ, θ(μ)] > q

(μ, θ) if θ = θ(μ),

then, if the parameter space is compact, the parameter vector is identiﬁed by the

criterion function. We have not assumed compactness. For a convex parameter

In their Exercise 23.6, Grifﬁths, Hill, and Judge (1993), based (alas) on the ﬁrst edition of this text, suggest a

probit model for statewide voting outcomes that includes dummy variables for region: Northeast, Southeast,

West, and Mountain. One would normally include three of the four dummy variables in the model, but

Grifﬁths et al. carefully dropped two of them because in addition to the dummy variable trap, the Southeast

variable is always zero when the dependent variable is zero. Inclusion of this variable produces a nonconcave

likelihood function—the parameter on this variable diverges. Analysis of a closely related case appears as a

caveat on page 272 of Amemiya (1985).

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451

space, we would require the additional condition that there exist no sequences

without limit points θ

such that q(μ, θ

) converges to q[μ, θ (μ)].

The approach taken here is to assume ﬁrst that the model has some set of

parameters. The identiﬁability criterion states that assuming this is the case, the

probability limit of the criterion is maximized at these parameters. This result rests

on convergence of the criterion function to a ﬁnite value at any point in the interior

of the parameter space. Because the criterion function is a function of the data, this

convergence requires a statement of the properties of the data—for example, well

behaved in some sense. Leaving that aside for the moment, interestingly, the results

to this point already establish the consistency of the M estimator. In what might

seem to be an extremely terse fashion, Amemiya (1985) deﬁned identiﬁability

simply as “existence of a consistent estimator.” We see that identiﬁcation and the

conditions for consistency of the M estimator are substantively the same.

This form of identiﬁcation is necessary, in theory, to establish the consistency

arguments. In any but the simplest cases, however, it will be extremely difﬁcult to

verify in practice. Fortunately, there are simpler ways to secure identiﬁcation that

will appeal more to the intuition:

•

For the least squares estimator, a sufﬁcient condition for identiﬁcation is that

any two different parameter vectors, θ and θ

, must be able to produce dif-

ferent values of the conditional mean function. This means that for any two

different parameter vectors, there must be an x

that produces different val-

ues of the conditional mean function. You should verify that for the linear

model, this is the full rank assumption A.2. For the model in Example 2.5, we

have a regression in which x

= x

+ x

. In this case, any parameter vec-

tor of the form (β

,β

− a,β

+ a,β

+ a) produces the same conditional

mean as (β

,β

) regardless of x

, so this model is not identiﬁed. The

full rank assumption is needed to preclude this problem. For nonlinear regres-

sions, the problem is much more complicated, and there is no simple generality.

Example 7.2 shows a nonlinear regression model that is not identiﬁed and how

the lack of identiﬁcation is remedied.

•

For the maximum likelihood estimator, a condition similar to that for the re-

gression model is needed. For any two parameter vectors, θ =θ

, it must be pos-

sible to produce different values of the density f (y

, θ ) for some data vector

, x

). Many econometric models that are ﬁt by maximum likelihood are “in-

dex function” models that involve densities of the form f (y

, θ ) = f (y



θ).

When this is the case, the same full rank assumption that applies to the regres-

sion model may be sufﬁcient. (If there are no other parameters in the model,

then it will be sufﬁcient.)

•

For the GMM estimator, not much simplicity can be gained. A sufﬁcient con-

dition for identiﬁcation is that E[

m(data, θ )] = 0 if θ = θ

4. Behavior of the data has been discussed at various points in the preceding text.

The estimators are based on means of functions of observations. (You can see

this in all three of the preceding deﬁnitions. Derivatives of these criterion func-

tions will likewise be means of functions of observations.) Analysis of their large

sample behaviors will turn on determining conditions under which certain sample

means of functions of observations will be subject to laws of large numbers such as

the Khinchine (D.5) or Chebychev (D.6) theorems, and what must be assumed in

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order to assert that “root-n” times sample means of functions will obey central

limit theorems such as the Lindeberg–Feller (D.19) or Lyapounov (D.20) theo-

rems for cross sections or the Martingale Difference Central Limit theorem for

dependent observations (Theorem 20.3). Ultimately, this is the issue in establish-

ing the statistical properties. The convergence property claimed above must occur

in the context of the data. These conditions have been discussed in Sections 4.4.1

and 4.4.2 under the heading of “well-behaved data.” At this point, we will assume

that the data are well behaved.

12.5.4 ASYMPTOTIC PROPERTIES OF ESTIMATORS

With all this apparatus in place, the following are the standard results on asymptotic

properties of M estimators:

THEOREM 12.1

Consistency of M Estimators

If (a) the parameter space is convex and the true parameter vector is a point in

its interior, (b) the criterion function is concave, (c) the parameters are identiﬁed

by the criterion function, and (d) the data are well behaved, then the M estimator

converges in probability to the true parameter vector.

Proofs of consistency of M estimators rely on a fundamental convergence result

that, itself, rests on assumptions (a) through (d) in Theorem 12.1. We have assumed

identiﬁcation. The fundamental device is the following: Because of its dependence on

the data, q(θ |data) is a random variable. We assumed in (c) that plim q(θ |data) = q

(θ)

for any point in the parameter space. Assumption (c) states that the maximum of q

(θ)

occurs at q

(θ

),soθ

is the maximizer of the probability limit. By its deﬁnition, the

estimator

θ, is the maximizer of q(θ |data). Therefore, consistency requires the limit of

the maximizer,

θ be equal to the maximizer of the limit, θ

. Our identiﬁcation condition

establishes this. We will use this approach in somewhat greater detail in Section 14.4.5.a

where we establish consistency of the maximum likelihood estimator.

THEOREM 12.2

Asymptotic Normality of M Estimators

(i)

θ is a consistent estimator of θ

where θ

is a point in the interior of the

parameter space;

(ii) q(θ |data) is concave and twice continuously differentiable in θ in a neigh-

borhood of θ

;

(iii)

√

n[∂q(θ

|data)/∂θ

]

−→ N[0, ];

(iv) for any θ in , lim

n→∞

Pr[|(∂

q(θ |data)/∂θ

∂θ

) − h

(θ)| >ε] = 0 ∀ε>0

where h

(θ) is a continuous ﬁnite valued function of θ;

(v) the matrix of elements H(θ ) is nonsingular at θ

, then

√

θ − θ

)

−→ N



0, [H

−1

(θ

)H

−1

(θ

)]



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453

The proof of asymptotic normality is based on the mean value theorem from calculus

and a Taylor series expansion of the derivatives of the maximized criterion function

around the true parameter vector;

√

∂q(

θ |data)

∂

= 0 =

√

∂q(θ

|data)

∂θ

∂

θ |data)

∂

θ∂



√

θ − θ

The second derivative is evaluated at a point

θ that is between

θ and θ

, that is,

θ =

θ +(1 −w)θ

for some 0 < w < 1. Because we have assumed plim

θ = θ

, we see that

the matrix in the second term on the right must be converging to H(θ

). The assumptions

in the theorem can be combined to produce the claimed normal distribution. Formal

proof of this set of results appears in Newey and McFadden (1994). A somewhat more

detailed analysis based on this theorem appears in Section 14.4.5.b, where we establish

the asymptotic normality of the maximum likelihood estimator.

The preceding was restricted to M estimators, so it remains to establish counterparts

for the important GMM estimator. Consistency follows along the same lines used earlier,

but asymptotic normality is a bit more difﬁcult to establish. We will return to this issue

in Chapter 13, where, once again, we will sketch the formal results and refer the reader

to a source such as Newey and McFadden (1994) for rigorous derivation.

The preceding results are not straightforward in all estimation problems. For exam-

ple, the least absolute deviations (LAD) is not among the estimators noted earlier,

but it is an M estimator and it shares the results given here. The analysis is com-

plicated because the criterion function is not continuously differentiable. Nonethe-

less, consistency and asymptotic normality have been established. [See Koenker and

Bassett (1982) and Amemiya (1985, pp. 152–154).] Some of the semiparametric and

all of the nonparametric estimators noted require somewhat more intricate treatments.

For example, Pagan and Ullah (Sections 2.5 and 2.6) are able to establish the familiar

desirable properties for the kernel density estimator

f (x

∗

), but it requires a somewhat

more involved analysis of the function and the data than is necessary, say, for the lin-

ear regression or binomial logit model. The interested reader can ﬁnd many lengthy

and detailed analyses of asymptotic properties of estimators in, for example, Amemiya

(1985), Newey and McFadden (1994), Davidson and MacKinnon (2004), and Hayashi

(2000). In practical terms, it is rarely possible to verify the conditions for an estima-

tion problem at hand, and they are usually simply assumed. However, ﬁnding viola-

tions of the conditions is sometimes more straightforward, and this is worth pursuing.

For example, lack of parametric identiﬁcation can often be detected by analyzing the

model itself.

12.5.5 TESTING HYPOTHESES

The preceding describes a set of results that (more or less) uniﬁes the theoretical un-

derpinnings of three of the major classes of estimators in econometrics, least squares,

maximum likelihood, and GMM. A similar body of theory has been produced for the

familiar test statistics, Wald, likelihood ratio (LR), and Lagrange multiplier (LM). [See

Newey and McFadden (1994).] All of these have been laid out in practical terms else-

where in this text, so in the interest of brevity, we will refer the interested reader to the

background sources listed for the technical details.

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12.6 SUMMARY AND CONCLUSIONS

This chapter has presented a short overview of estimation in econometrics. There are

various ways to approach such a survey. The current literature can be broadly grouped

by three major types of estimators—parametric, semiparametric, and nonparametric.

It has been suggested that the overall drift in the literature is from the ﬁrst toward the

third of these, but on a closer look, we see that this is probably not the case. Maximum

likelihood is still the estimator of choice in many settings. New applications have been

found for the GMM estimator, but at the same time, new Bayesian and simulation

estimators, all fully parametric, are emerging at a rapid pace. Certainly, the range of

tools that can be applied in any setting is growing steadily.

Key Terms and Concepts

•

Bandwidth

•

Bayesian estimation

•

Bootstrap

•

Conditional density

•

Copula function

•

Criterion function

•

Data generating

mechanism

•

Density

•

Empirical likelihood

function

•

Entropy

•

Estimation criterion

•

Extremum estimator

•

Fundamental probability

transform

•

Generalized method of

moments

•

Histogram

•

Identiﬁability

•

Kernel density estimator

•

Least absolute deviations

(LAD)

•

Likelihood function

•

M estimator

•

Maximum empirical

likelihood estimator

•

Maximum entropy

•

Maximum likelihood

estimator

•

Method of moments

•

Nearest neighbor

•

Nonparametric estimators

•

Parameter space

•

Parametric estimation

•

Partially linear model

•

Quantile regression

•

Semiparametric estimation

•

Simulation-based estimation

•

Sklar’s theorem

•

Smoothing function

•

Stochastic frontier model

Exercise and Question

1. Compare the fully parametric and semiparametric approaches to estimation of a

discrete choice model such as the multinomial logit model discussed in Chapter 17.

What are the beneﬁts and costs of the semiparametric approach?

MINIMUM DISTANCE

ESTIMATION AND THE

GENERALIZED METHOD

OF MOMENTS

13.1 INTRODUCTION

The maximum likelihood estimator presented in Chapter 14 is fully efﬁcient among con-

sistent and asymptotically normally distributed estimators, in the context of the speciﬁed

parametric model. The possible shortcoming in this result is that to attain that efﬁciency,

it is necessary to make possibly strong, restrictive assumptions about the distribution,

or data generating process. The generalized method of moments (GMM) estimators

discussed in this chapter move away from parametric assumptions, toward estimators

that are robust to some variations in the underlying data generating process.

This chapter will present a number of fairly general results on parameter estimation.

We begin with perhaps the oldest formalized theory of estimation, the classical theory

of the method of moments. This body of results dates to the pioneering work of Fisher

(1925). The use of sample moments as the building blocks of estimating equations is

fundamental in econometrics. GMM is an extension of this technique that, as will be

clear shortly, encompasses nearly all the familiar estimators discussed in this book.

Section 13.2 will introduce the estimation framework with the method of moments. The

technique of minimum distance estimation is developed in Section 13.3. Formalities of

the GMM estimator are presented in Section 13.4. Section 13.5 discusses hypothesis

testing based on moment equations. Major applications, including dynamic panel data

models, are described in Section 13.6.

Example 13.1 Euler Equations and Life Cycle Consumption

One of the most often cited applications of the GMM principle for estimating econometric

models is Hall’s (1978) permanent income model of consumption. The original form of the

model (with some small changes in notation) posits a hypothesis about the optimizing be-

havior of a consumer over the life cycle. Consumers are hypothesized to act according to

the model:

Maximize E



T −t



τ =0



1 + δ



U( c

t+τ

)|



subject to

T −t



τ =0



1 + r



t+τ

− w

t+τ

) = A

The information available at time t is denoted 

so that E

denotes the expectation formed

at time t based on the information set 

. The maximand is the expected discounted stream

of future utility from consumption from time t until the end of life at time T. The individual’s

subjective rate of time preference is β =1/(1+δ). The real rate of interest, r ≥δ is assumed to

be constant. The utility function U(c

) is assumed to be strictly concave and time separable

(as shown in the model). One period’s consumption is c

. The intertemporal budget constraint

states that the present discounted excess of c

over earnings, w

, over the lifetime equals

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total assets A

not including human capital. In this model, it is claimed that the only source of

uncertainty is w

. No assumption is made about the stochastic properties of w

except that

there exists an expected future earnings, E

t+τ

|

]. Successive values are not assumed

to be independent and w

is not assumed to be stationary.

Hall’s major “theorem” in the paper is the solution to the optimization problem, which

states



t+1

)|

] =

1 + δ

1 + r



For our purposes, the major conclusion of the paper is “Corollary 1” which states “No in-

formation available in time t apart from the level of consumption, c

, helps predict future

consumption, c

t+1

, in the sense of affecting the expected value of marginal utility. In particu-

lar, income or wealth in periods t or earlier are irrelevant once c

is known.” We can use this as

the basis of a model that can be placed in the GMM framework. To proceed, it is necessary

to assume a form of the utility function. A common (convenient) form of the utility function

is U (c

) =c

1−α

/(1 − α) , which is monotonic, U



−α

> 0 and concave, U





=−α/c

< 0.

Inserting this form into the solution, rearranging the terms, and reparameterizing it for con-

venience, we have



(1+ r )



1 + δ



t+1



−α

− 1|



= E



β(1+ r ) R

t+1

− 1|



= 0,

where R

t+1

= c

t+1

and λ =−α.

Hall assumed that r was constant over time. Other applications of this modeling frame-

work [for example, Hansen and Singleton (1982)] have modiﬁed the framework so as to

involve a forecasted interest rate, r

t+1

. How one proceeds from here depends on what is

in the information set. The unconditional mean does not identify the two parameters. The

corollary states that the only relevant information in the information set is c

. Given the form

of the model, the more natural instrument might be R

. This assumption exactly identiﬁes the

two parameters in the model:





β(1+ r

t+1

) R

t+1

− 1











As stated, the model has no testable implications. These two moment equations would

exactly identify the two unknown parameters. Hall hypothesized several models involving

income and consumption which would overidentify and thus place restrictions on the model.

13.2 CONSISTENT ESTIMATION: THE METHOD

OF MOMENTS

Sample statistics such as the mean and variance can be treated as simple descriptive

measures. In our discussion of estimation in Appendix C, however, we argue that, in

general, sample statistics each have a counterpart in the population, for example, the

correspondence between the sample mean and the population expected value. The

natural (perhaps obvious) next step in the analysis is to use this analogy to justify using

the sample “moments” as estimators of these population parameters. What remains to

establish is whether this approach is the best, or even a good way to use the sample data

to infer the characteristics of the population.

The basis of the method of moments is as follows: In random sampling, under

generally benign assumptions, a sample statistic will converge in probability to some

constant. For example, with i.i.d. random sampling, ¯m



= (1/n)



i=1

will converge

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457

in mean square to the variance plus the square of the mean of the random variable, y

This constant will, in turn, be a function of the unknown parameters of the distribution.

To estimate K parameters, θ

,...,θ

, we can compute K such statistics, ¯m

,..., ¯m

whose probability limits are known functions of the parameters. These K moments are

equated to the K functions, and the functions are inverted to express the parameters

as functions of the moments. The moments will be consistent by virtue of a law of

large numbers (Theorems D.4–D.9). They will be asymptotically normally distributed

by virtue of the Lindeberg–Levy Central Limit theorem (D.18). The derived para-

meter estimators will inherit consistency by virtue of the Slutsky theorem (D.12) and

asymptotic normality by virtue of the delta method (Theorem D.21).

This section will develop this technique in some detail, partly to present it in its own

right and partly as a prelude to the discussion of the generalized method of moments,

or GMM, estimation technique, which is treated in Section 13.4.

13.2.1 RANDOM SAMPLING AND ESTIMATING THE PARAMETERS

OF DISTRIBUTIONS

Consider independent, identically distributed random sampling from a distribution

f (y |θ

,...,θ

) with ﬁnite moments up to E [y

]. The random sample consists of

n observations, y

,...,y

.Thekth “raw” or uncentered moment is

¯m





i=1

By Theorem D.4,

E [¯m



] = μ



= E





and

Var[ ¯m



] =

Var









− μ

2



By convention, μ



= E [y

] = μ. By the Khinchine theorem, D.5,

plim ¯m



= μ



= E





Finally, by the Lindeberg–Levy central limit theorem,

√

n( ¯m



− μ



)

−→ N



0,μ



− μ

2



In general, μ



will be a function of the underlying parameters. By computing K

raw moments and equating them to these functions, we obtain K equations that can (in

principle) be solved to provide estimates of the K unknown parameters.

Example 13.2 Method of Moments Estimator for N

[μ, σ

]

In random sampling from N[μ, σ

plim



i =1

= plim ¯m



= E [ y

] = μ,

and

plim



i =1

= plim ¯m



= Var [ y

] + μ

= σ

+ μ

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Estimation Methodology

Equating the right- and left-hand sides of the probability limits gives moment estimators

ˆμ = ¯m



= ¯y,

and

ˆσ

= ¯m



− ¯m

2





i =1



−





i =1





i =1

( y

− ¯y)

Note that ˆσ

is biased, although both estimators are consistent.

Although the moments based on powers of y provide a natural source of information

about the parameters, other functions of the data may also be useful. Let m

(·) be a

continuous and differentiable function not involving the sample size n, and let

¯m



i=1

), k = 1, 2,...,K.

These are also “moments” of the data. It follows from Theorem D.4 and the corollary,

(D-5), that

plim ¯m

= E [m

)] = μ

(θ

,...,θ

We assume that μ

(·) involves some of or all the parameters of the distribution. With

K parameters to be estimated, the K moment equations,

¯m

− μ

(θ

,...,θ

) = 0,

¯m

− μ

(θ

,...,θ

) = 0,

···

¯m

− μ

(θ

,...,θ

) = 0,

provide K equations in K unknowns, θ

,...,θ

. If the equations are continuous and

functionally independent, then method of moments estimators can be obtained by solv-

ing the system of equations for

[¯m

,..., ¯m

As suggested, there may be more than one set of moments that one can use for estimating

the parameters, or there may be more moment equations available than are necessary.

Example 13.3 Inverse Gaussian (Wald) Distribution

The inverse Gaussian distribution is used to model survival times, or elapsed times from some

beginning time until some kind of transition takes place. The standard form of the density for

this random variable is

f ( y) =

2π y

exp



−

λ( y − μ)

2μ



, y > 0, λ>0, μ>0.

The mean is μ while the variance is μ

/λ. The efﬁcient maximum likelihood estimators of the

two parameters are based on ( 1/n)



i =1

and ( 1/n)



i =1

(1/y

). Because the mean and

variance are simple functions of the underlying parameters, we can also use the sample mean

and sample variance as moment estimators of these functions. Thus, an alternative pair of

method of moments estimators for the parameters of the Wald distribution can be based on

(1/n)



i =1

and ( 1/n)



i =1

. The precise formulas for these two pairs of estimators is

left as an exercise.