Chapter 20 Generalized Method of Moments Estimators and ...

Web Extension 10 

Chapter 20 

Generalized Method of Moments 

Estimators and Identification 

“A foolish consistency is the hobgoblin of little minds, adored by little 

statesmen and philosophers and divines.” 1 

—RALPH WALDO EMERSON (1803–1882) 

Despite Emerson’s contempt for foolish consistency, consistency 

is not a foolish property for an estimator. Indeed, its 

absence would seem troubling. Who would want to admit 

that his or her estimator wouldn’t get the right answer 

with “all the data in the world”? Consequently, a widely 

applicable method for devising consistent estimators is a useful tool. This 

section develops two such tools, the method of moments and the 

generalized method of moments; these are the most recently popularized estimation 

techniques presented in this book. In addition to its usefulness in 

applications, method of moments estimation provides a natural bridge to a 

fundamental concept in econometrics, identification, which we’ll study in 

the next section. 

20.1 Method of Moments Estimators 

The ungeneralized method of moments harkens back to 

Chapter 2, where we sought estimators for a straight line 

through the origin. How were we to construct estimators 

from a sample of data? The method of moments offers a 

straightforward strategy for devising estimators that prove 

consistent under very general assumptions. Just as our intuition 

led to several estimators in Chapter 2, the method of 

moments may also lead to multiple estimators. In such cases, 

we turn to the generalized method of moments to settle on a 

single consistent estimator. 

EXT 10-1

EXT 10-2 Web Extension 10 

Moments 

Statisticians call the expected values of variables or of products of variables moments. 

E(X , and E(X i e 2 i ), E(e i ), E(e 2 i ), E(X i e i ) 

i ) are all examples of moments. We 

commonly make assumptions about moments in our data-generating processes. 

For example, we might assume 

and 

Y i = b 0 + b 1 X i + e i , 

E(e i ) = 0, 

E(X i e i ) = 0, 

as is true under the Gauss–Markov Assumptions for a straight line with unknown 

slope and intercept. (Notice that in this example, the explanators need not be 

fixed across samples. The assumption that E(X i e i ) = 0 is weaker than the assumption 

that the X i are fixed across samples.) 

The Ungeneralized Method of Moments 

Method of moments estimation devises an estimator by insisting that a moment 

expectation that is true in the population holds true exactly for the corresponding 

moment within a given sample. That is, the method of moments insists that something 

true on average of the disturbances in the population be true on average 

about the residuals in any one sample. In our example, the mean residual in the 

sample is forced to be zero, or the covariance of the X’s and the residuals in the 

sample is forced to be zero, analogously to the expected value of the disturbances 

in the population or the covariance of X and the disturbances in the population 

both being zero. The method of moments estimators for b and , b ~ and b ~ 0 b 1 0 1, in 

our example are, therefore, found by solving 

1 

n a e ~ i = 1 n a (Y i - b ~ 0 - b ~ 1X i ) = 0 

20.1 


1 

n a X ie ~ i = 1 n a X i(Y i - b ~ 0 - b ~ 1X i ) = 0, 

20.2 

the within-sample versions of the moment conditions assumed in the DGP. The e~ 

are the residuals obtained using the estimators b ~ and b ~ i 

0 1. In this example, these 

relationships happen to be the same relationships required to minimize the sum of 

squared residuals, as we learned in Chapter 5. Solving these equations for b ~ 0 and

Generalized Method of Moments Estimators and Identification EXT 10-3 

b ~ 1, 

therefore, leads to bN 0 and bN 1 . The method of moments estimators in this example 

coincide with the ordinary least squares (OLS) estimators, bN 0 and bN 1 . 

The method of moments estimators are consistent. An intuition for their consistency 

is that the Law of Large Numbers says that under suitable conditions 

Equations 20.1 and 20.2 are almost surely very close to correct if b ~ and b ~ 0 1 are 

replaced by b 0 and b 1 . Thus, if the DGP satisfies the conditions for the Law of 

Large Numbers, the method of moments estimators tend to coincide with the true 

parameter values as the sample size grows without bound. Appendix 20.1 more 

formally shows the consistency of the method of moments estimator in this 

model. In general, method of moments estimators are consistent whenever the 

Law of Large Numbers ensures that the sample moments in the DGP converge in 

probability to the corresponding population moments. 

The Generalized Method of Moments 

Our method of moments estimation of a line with unknown intercept and slope 

exploited two moment conditions embedded in the Gauss–Markov Assumptions. 

But what would we do to estimate the slope of a line through the origin? The two 

moment restrictions are still E(e i ) = 0 and E(X i e i ) = 0, but because there is only 

one parameter to estimate, the slope, b, the method of moments provides a surfeit 

of riches. It tells us to devise an estimate of b from 

1 

n a e ~ i = 1 n a (Y i - b ~ X i ) = 0 


1 

n a X i e ~ i = 1 n a X i(Y i - b ~ X i ) = 0. 

But with two equations and only one unknown, solving both of these equations 

for b ~ b ~ , 

yields two estimators, not one. The first equation yields 

b ~ = 

1 

n a Y i 

1 

n a X i 

= aY i 

a X i 

= b g2 , 

whereas the second equation yields 

b ~ = 

1 

n a X iY i 

1 

n a X i 2 

= aX iY i 

a X i 2 = b g4 .


b g2 

b g4 

Both and are method of moments estimators of the slope of a line 

through the origin under the Gauss–Markov Assumptions (or in any DGP in 

which E(e i ) = 0 and E(X i e i ) = 0, including some in which the explanators are 

not fixed). Notice that in satisfying one of two moment restrictions, neither b g2 

nor b g4 satisfies both moment restrictions. When b g2 is the estimator, the mean 

1 

residual is zero, so n g e~ i matches its corresponding population moment. But when 

1 

b is the estimator, n gX ~ g2 i e i does not equal zero (except by occasional accident), 

so this second sample moment does not equal its population counterpart. When 

1 

1 

b is the estimator, n gX ie ~ g4 

i equals zero, but n g e~ i does not (except by occasional 

accident). Except in the accidental case in which b g2 and b g4 are equal, neither 

method of moments estimator makes both sample moments equal to their population 

counterparts. 

The generalized method of moments provides an estimation strategy when 

the number of restricted moments in the DGP exceeds the number of parameters 

to be estimated. Rather than satisfy one moment condition and violate another, 

the generalized method of moments (GMM) strategy chooses an estimator that 

balances each population moment condition against the others, seeking residuals 

that trade off violations of one moment restriction against violations of the other 

moment restrictions. A GMM estimator may satisfy no one moment condition, 

but it may come close to satisfying them all. 

In the example of a line through the origin, the BLUE property of b g4 makes 

it clear which is the most preferred method of moments estimator. We should 

choose our estimator so that 

and ignore the restriction that 

1 

n a X ie ~ i = 1 n a X i(Y i - b ~ X i ) = 0 

1 ~ n a e i = 1 n a (Y i - b ~ X i ) = 0. 

This strategy yields b g4 , which we know to be BLUE. With more complex DGPs, 

the optimal choice is not so obvious. When the choice among method of 

moments estimators is not clear, GMM offers a strategy for devising a single estimator. 

GMM does not require that the sample moments equal the population moments. 

For example, in the simple case of a line through the origin with two moment 

restrictions, the GMM estimator does not insist that 

1 

n a e~ i = 1 n a (Y i - b ~ X i ) = 0



1 

n a X ie ~ i = 1 n a X i(Y i - b ~ X i ) = 0. 

Such insistence would be futile—generally, no estimator can satisfy both restrictions. 

Nor does GMM insist that one sample moment or the other equal zero. Instead, 

GMM looks at how much an estimator makes the sample moments differ 

from their population counterparts, as in 

1 

n a e * i = 1 n a (Y i - b * X i ) = n 1 


1 

n a X i e i * = n 1 a X i(Y i - b * X i ) = n 2 , 

where n 1 and n 2 are the amounts by which the first and second moment restrictions 

are violated by the residuals implied by b * . One estimation strategy that 

would yield consistent estimators would be to minimize (n 2 1 + n 2 ). This strategy 

would aim to make as small as possible the squared deviations of the sample moments 

from the their population analogs. This strategy shares an intuitive foundation 

with ordinary least squares, but here the goal is to make small not the 

squared residuals, but the squared deviations from the population moment restrictions. 

This reasonable approach is not the one followed by GMM. 

GMM modifies the strategy of minimizing (n 2 1 + n 2 ) much as generalized 

least squares modifies ordinary least squares. Instead of minimizing the unweighted 

sum of squared deviations, (n 2 1 + n 2 ), GMM minimizes a weighted sum 

of the squared deviations, in which the weights reflect the variances and covariances 

of the n i . GMM does not guarantee an efficient estimator, but it does provide 

a consistent estimator, and its weighting scheme is more efficient than the 

simpler unweighted scheme. GMM provides a powerful tool for finding consistent 

estimators in models that are otherwise mathematically quite cumbersome. 

20.2 Identification 

WHAT IS THE DGP? 

We have just seen that when the DGP has more moment conditions than parameters 

to be estimated, we have a surfeit of riches. The problem isn’t finding a consistent 

estimator, but choosing among several method of moments estimators. But 

what if there are fewer moment restrictions than parameters to be estimated? 

When there are too few restrictions in the DGP to allow consistent estimation of


some or all parameters, we say the parameters are underidentified. When there 

are more restrictions than necessary to estimate the parameters consistently, we 

say the parameters are overidentified. Underidentified parameters cannot be estimated 

consistently. 

Underidentified Parameters 

The following example helps us understand underidentification. Suppose that in 

the DGP for a straight line with an unknown intercept term, the X-values are not 

fixed across samples, but rather X is a random variable. If X is a random variable, 

it might be correlated with the disturbance term. For example, in a wage equation 

in which education and experience are the only explanators, the disturbance contains 

all other influences on wages, such as punctuality, diligence, and native intelligence. 

If the education one attains is correlated with those same traits, then our 

explanator education is correlated with the disturbances. In general, if the explanators 

are correlated with the disturbances, we cannot say that E(X i e i ) = 0. 

In 

such a case, we have only one moment condition, E(e i ) = 0, but two parameters 

to estimate. 

With only one moment condition and two parameters to estimate, we can 

still choose estimates of b 0 and b 1 that make the population moment condition 

true in our sample: 

1 

n a (e i) = 1 n a (Y i - b ~ 0 - b ~ 1X i ) = 0. 

But with one equation and two unknowns, we can choose any value for and 

then compute the value for b ~ b ~ 0 

1 that makes the moment condition true. Conversely, 

we could choose any value for b ~ and then compute the value for b ~ 1 

0 that makes 

the moment condition true. Such arbitrary parameter estimates do not have the 

property of consistency. 2 When the DGP offers too few restrictions to pin down 

the parameters of interest, we say the relationship’s parameters are underidentified. 

Exactly Identified Parameters 

In contrast to the DGP with unknown slope and intercept, in the DGP for a 

straight line through the origin, abandoning the population restriction that 

E(X i e i ) = 0 does not pose a problem for consistent estimation. With only one parameter 

to be estimated, the one moment restriction, E(e i ) = 0, provides a basis 

for consistent estimation. Indeed, in this DGP, abandoning the assumption that 

E(X i e i ) = 0 relieves the surfeit of riches that plagued us earlier. With one restriction 

and one parameter, there will be only one method of moments estimator for


the slope of the line through the origin, . In a classic application, presented in 

Chapter 2, Milton Friedman estimated the marginal propensity to consume from 

permanent income using because he believed that E(e i ) = 0 was true in his 

model and that E(X i e i = 0) was not. When the restrictions in the DGP imply a 

single GMM estimator for each parameter, we say the parameters of the relationship 

are exactly identified, or just identified. 

b g2 

b g2 

(Text continues on page EXT 10-10) 

Overidentified Parameters 

Exact identification permits consistent estimation. Underidentification makes consistent 

estimation impossible. What about overidentification? When the restrictions 

in the DGP yield a surfeit of riches, with more restrictions than parameters to be estimated, 

we say the parameters of the relationship are overidentified. At first look, 

overidentification and underidentification seem similar in their consequences. In 

neither case do the moment restrictions of the DGP yield unique estimators of the 

relationship’s parameters. However, more deeply, the two phenomena are dramatically 

different. The several (or many) estimators offered by overidentification will 

all converge in probability to the same result; all the estimators are consistent. The 

multitude of estimates possible with underidentification, however, remain arbitrary 

and in conflict with one another, even in infinite samples. 

In the face of overidentification, we are pressed to choose among the multitude 

of consistent estimators. GMM is one common strategy for settling on one 

consistent estimator from among many. GMM is not generally asymptotically efficient, 

however. Later in this extension, we encounter an efficient estimator, 

called the maximum likelihood estimator, which, when it is applicable, is preferable 

to GMM when estimating overidentified relationships. The problem posed 

by overidentification is modest. We can proceed in the face of overidentification 

confident that all consistent estimators give similar results in very large samples. 

In contrast, in the face of underidentification, we cannot consistently estimate the 

parameters of interest at all. 

Exclusion and Covariance Restrictions 

The examples of a straight line through the origin and a straight line with unknown 

intercept and slope illustrate two fundamental ways in which econometricians 

achieve identification of parameters. In the DGP 

Y i = b 0 + b 1 X i + e i i = 1, Á , n 


E(e i ) = 0 i = 1, Á , n,


An Econometric Top 40—A Pop Tune 

Competing in the New Economy 

“How do managerial decisions such as whether 

or not to adopt a Total Quality Management 

(TQM) system or to expand an employee involvement 

program affect labor productivity? 

Does the implementation of ‘high performance’ 

workplace practices ensure better firm performance? 

Does the presence of a union hinder 

or enhance the probability of success associated 

with implementing these practices? Do 

computers really help workers be more productive?” 

3 Economists Sandra Black of the Federal 

Reserve Bank of New York and Lisa Lynch of 

Tufts University posed these pressing contemporary 

questions in 2001. Their informative 

econometric study provides answers of interest 

to many corporate managers. 

Black and Lynch study a sample of more 

than 600 U.S. manufacturing establishments, 

each observed once per year by the Census Bureau 

between 1987 and 1994. In 1994 the data 

on workplace practices were gathered from the 

establishments, along with data on inputs and 

outputs; in the earlier years, only data on inputs 

and outputs were gathered. 

The authors embedded their study in the 

context of a Cobb–Douglas production function 

for manufacturing firms: 

ln(Y>L) i = b 0i + b 1 ln(K>L) i 

+ b 2 ln(M>L) i + e i , 

where Y is output, L is labor, K is capital, and 

M is materials. Notice that this function differs 

from most we have seen in that the intercept 

varies from firm to firm. The authors augmented 

the Cobb–Douglas specification by assuming 

that worker productivity (measured by 

Y/L, output per worker) depends on establish- 

ment-specific workplace practices and worker 

characteristics: 

T 

b 0i = a 0 + a a j Z ji , 

j = 1 

where Z ji is the i-th firm’s value for the j-th trait 

in a list of workplace practices and worker 

characteristics; there are T items in the list in 

all. 

Breaking the specification into two steps 

highlights the underlying Cobb–Douglas assumption. 

The specification actually reduces to 

a single garden variety linear regression with 

inputs per worker (K/L and M/L), workplace 

practices, and worker characteristics as explanators. 

What is not garden variety is the rest 

of the DGP. The Gauss–Markov Assumptions 

surely do not apply here. Each firm is observed 

several times; systematic differences among 

firms (some firms being fundamentally more or 

less productive than others) make it likely that 

the disturbances for each firm are correlated 

over time, even if the disturbances are independent 

across firms. Moreover, were the authors 

to gather a different sample of firms, they 

would surely obtain different values for the explanatory 

variables. The X’s are not fixed in repeated 

samples. And with random X-values 

comes the worry that those firms wise enough 

to choose beneficial X-values might also 

choose beneficial unmeasured practices—the 

inputs per worker and the workplace practices 

might be correlated with the disturbances. Little 

seems left of the Gauss–Markov Assumptions 

on which to build good estimators. 

The wholesale failure of the Gauss–Markov 

Assumptions poses estimation problems for


Black and Lynch. Is consistent estimation possible 

at all? Are the parameters in their model 

identified? And if the parameters are identified, 

how are they estimated? Black and Lynch argue 

that they know enough about the variances and 

covariances of the explanators and disturbances 

to identify the parameters in their 

model through several covariance and exclusion 

restrictions. Consistent estimation is at 

least possible. When the Gauss–Markov Assumptions 

fail thoroughly enough to make 

least squares unattractive, but we do have sufficient 

information about the variances and covariances 

of the explanators and disturbances 

to restrict numerous moments in the population, 

the generalized method of moments 

(GMM) offers an appealing estimation procedure. 

Black and Lynch argue that they know 

enough about the variances and covariances of 

the explanators and disturbances to rely on 

GMM to construct consistent, asymptotically 

normally distributed estimators of the a’s and 

b’s. Given the large number of establishments 

in Black and Lynch’s sample, good asymptotic 

properties are an attractive basis for inference. 

The t-statistics and F-tests we are accustomed 

to analyzing are asymptotically valid here, so 

we can discuss the empirical results of Black 

and Lynch much as we would if they had used 

OLS under the Gauss–Markov Assumptions. 

What do they find? The authors report that 

TQM systems do not raise productivity by 

themselves, but that allowing employees a 

greater voice in decisions does improve productivity 

(and to the extent that TQM includes 

such increased voice, TQM improves productivity). 

Moreover, report the authors, institut- 

ing profit sharing can have a positive effect on 

productivity, but only when the plan includes 

profit sharing for nonmanagerial employees. 

The authors further find that unionized establishments 

that increase worker voice and add 

profit sharing that includes nonmanagerial employees 

get a particularly large boost in productivity 

from such workplace policies. In contrast, 

productivity in unionized establishments 

that do not introduce such new workplace policies 

lags behind that in similarly un-innovative 

nonunionized establishments. 

Black and Lynch also find that increasing 

use of computers can enhance productivity. 

Firms with higher levels of computer usage by 

nonmanagerial workers have higher productivity 

than those with less computer use. 

Final Notes 

Black and Lynch’s paper illustrates how everyday 

economic questions are sometimes best 

tackled with highly sophisticated empirical 

techniques. Relatively simple OLS would not 

reliably answer the questions Black and Lynch 

address. Instead, Black and Lynch combined 

covariance restrictions and exclusion restrictions 

grounded in their understanding of the 

data’s origins to provide both identification for 

their model and the moment restrictions necessary 

for conducting GMM estimation. Simple 

economics can require complex statistics. 

When the DGPs suitable for modeling realworld 

data depart far from the Gauss–Markov 

Assumptions, particularly complicated estimation 

strategies may be needed, rather than OLS 

or GLS. 

■


b 1 

neither the slope of the line, , nor the intercept, , is identified. We do not have 

enough prior information about where the data come from to allow us to consistently 

estimate these parameters. What additional information would identify the 

slope? Our earlier discussion exposes two possibilities. First, we might learn that 

there is, in fact, no intercept term in the model; excluding the intercept from the 

model would identify the slope. Second, we might learn that E(X i e i ) = 0, in which 

case the slope (and the intercept) would be identified; restricting the covariance 

between the explanators and disturbances to be zero would identify the slope of 

the line. Exclusion restrictions and covariance restrictions are not the only ways a 

model’s parameters become identified, but they are common strategies. 

Instrumental variables (IV) estimation relies on exclusion and covariance restrictions 

for its consistency. When the covariance restriction that E(X i e i ) = 0 

fails, IV can nonetheless consistently estimate the parameters of an equation, but 

only if a combination of exclusion restrictions and covariance restrictions hold. IV 

estimation requires that we know that E(Z i e i ) = 0 and E(Z i X i ) Z 0, which are covariance 

restrictions, and that the potential instrument, Z, isnot itself a relevant 

explanator of the dependent variable, which is an exclusion restriction. 

It is important to note that we ought not impose identifying restrictions arbitrarily. 

False restrictions misspecify the DGP and undermine both consistent estimation 

and valid statistical inference. Econometricians look to make plausible 

identifying assumptions and, when possible, to test whether the data support the 

identifying assumptions. 

20.3 An Application: Military Service and Wages 

Underidentification makes econometric analysis futile. Consistent estimators of 

an underidentified parameter do not exist. Essential to econometric success is 

identifying the parameters of interest, and to identify parameters requires that our 

DGP contain sufficient assumptions. Here we see how underidentification can 

threaten a specific empirical project and how an econometrician can use common 

sense and economic reasoning to impose assumptions on a DGP sufficient to 

identify parameters of interest. 

Let’s begin with the question “Does military service enhance an individual’s 

future earning power?” Many people have thought it does. Unfortunately, the hypothesis 

long proved difficult to test. It might seem simple to specify: 

W i = b 0 + b 1 M i + e i , 

in which W is the person’s wage and M is a dummy variable indicating past military 

service, and to use OLS to estimate b 1 , the effect of past military service on 

wages. Unfortunately, OLS is not a consistent estimator of b 1 because E(M i e i ) 

usually does not equal zero; b 1 is underidentified. Why doesn’t E(M i e i ) equal 

b 0


zero? Because individuals who join the military often differ from other people in 

traits that influence wages, but that the econometrician is unlikely to observe, 

such as self-discipline and self-confidence. Indeed, people often join the military 

with the specific intention of acquiring such traits. We can control for traits such 

as education, work experience, and gender, but some very personal characteristics 

that influence decisions to join the military and also influence wage prospects will 

not be measured and included in our data sets. These unmeasured traits are part 

of the disturbance term, and they may be different for people who serve and people 

who do not, so E(M i e i ) may not equal zero. Military service may give illdisciplined 

enlistees more discipline, but they may still be undisciplined enough 

that they suffer lower wages than otherwise similar workers. If enlistees have unmeasured 

traits that detract from their labor-market prospects, and if military 

service does not fully overcome those disadvantages, the estimated coefficient on 

past military service will be negative, despite what positive effect that service may 

have on those individuals’ earnings potential. Without E(M i e i ) = 0, we have only 

E(e i ) = 0 with which to estimate b 0 and b 1 . The coefficients are not identified. 

Economist Josh Angrist of MIT explored how we might identify the effect of 

military service on wages. 4 He decided that data from a draft lottery that took 

place during the Vietnam conflict would allow identification of the effect of military 

service on earnings. After carefully pondering the draft lottery, Angrist posed 

a DGP for the wages of workers who had been subject to that lottery. He posited 

two covariance restrictions and an exclusion restriction to identify b 1 . In the draft 

lottery, inductees were selected according to their birth dates. A random draw of 

birth dates determined which birth dates would be drafted. It was a lottery few 

wanted to win. Because birth dates were unlikely to be correlated with individual’s 

productivity characteristics, such as self-discipline or self-confidence, entering 

the military through the draft was unlikely to be correlated with those traits. 

Angrist defined a dummy variable L to indicate that a person’s birthday was a 

lottery date. Angrist plausibly argued that E(L i e i ) = 0—individuals’ labor-market 

traits are unlikely to be correlated with their being born on a randomly selected 

date. Angrist also argued that E(L i M i ) does not equal zero—people who won the 

lottery were more likely than others to have military experience. E(L i e i ) = 0 and 

E(L i M i ) Z 0 were Angrist’s two covariance restrictions. Finally, Angrist argued 

that one’s lottery status should not itself directly influence wages, so he also excluded 

L from the wage equation. The method of moments estimator for b based 

on E(e i ) = 0 and E(L i e i ) = 0 is: 

b ~ 1 = al iw i 

a l im i 

, 

where l, w, and m are L, W, and M measured as deviations from their own 

means. Angrist’s second covariance restriction, that E(L i M i ) does not equal zero,


ensures that the denominator in the estimator is highly unlikely to be zero in 

large samples. 

Angrist implemented this estimator and concluded that military service does 

not improve future wage prospects. In the 1980s, years after their military service, 

white lottery veterans’ earnings were 15% less than comparable workers who had 

not served; black lottery veterans’ earnings were statistically indistinguishable 

from those of otherwise comparable nonveterans. Had Angrist not been able to 

argue for the plausibility of his covariance and exclusion restrictions, the effect of 

military service on earnings would not be identifiable from the earnings and draft 

lottery data. Because many economists were persuaded by Angrist’s assumptions, 

his estimates of the effect of military service gained widespread acceptance. Those 

economists who did not accept Angrist’s identifying assumptions remained unpersuaded 

by his parameter estimates. 

An Organizational Structure 

for the Study of Econometrics 

1. What is the DGP? 

GMM requires restrictions on moments. 

2. What Makes a Good Estimator? 

Consistency 

3. How Do We Create an Estimator? 

Method of moments and generalized method of moments 

4. What Are an Estimator’s Properties? 

Identification is a minimal requirement for consistency. 

GMM usually yields consistency. 

5. How Do We Test Hypotheses? 

Summary 

The extension began by introducing a strategy, the method of moments, and its 

generalization, the generalized method of moments (GMM), for constructing 

asymptotically normally distributed consistent estimators in a vast array of 

DGPs. The assumptions about means, variances, and covariances that underpin 

GMM prove to be minimal requirements for consistent estimation of the parame-


ters of a DGP. We learn from this that a DGP may contain too little information 

to support consistent estimation of some or all of its parameters. When a DGP 

suffers such a shortage of information, we say some or all of its parameters are 

underidentified. 

The Law of Large Numbers and the Central Limit Theorem that imply the 

consistency and asymptotic normality of GMM estimators (including OLS estimators) 

rest on bounded variances. When explanators or dependent variables in 

our DGPs have unbounded variances, the normality and even the consistency of 

estimators becomes questionable. 

Concepts for Review 

Exactly identified 

Generalized method of moments (GMM) 

Just identified 

Method of moments 

Moments 

Overidentified 

Underidentified 

Questions for Discussion 

1. “This identification business is nonsense. I can always run a regression, barring multicollinearity. 

That gives me estimates of the parameters of interest. I can always use 

those estimates.” Agree or disagree, and discuss. 

Problems for Analysis 

1. For the DGP 

Y i = bX i + e i 

E(Z i X i ) Z 0 

E(Z i e i ) = 0, 

show that the method of moments estimator of the slope is 

b Z = a Z i Y i > a Z i X i . 

2. For the DGP in problem 1, show that the method of moments estimator is consistent 

if 

plimQ 1 n a Z iX i R Z 0 and plimQ 1 n a Z ie i R = 0. 

3. Show that a valid instrumental variable estimator for a DGP with a straight line 

through the origin is also a method of moments estimator. 

b Z


Endnotes 

1. Ralph Waldo Emerson, “Self-Reliance,” in Essays, 1841. 

2. Notice a special case here. If our DGP provides an increasing number of observations 

for which X = 0, we can consistently estimate b 0 by restricting attention to only 

those observations. 

3. Sandra E. Black and Lisa M. Lynch, “How to Compete: The Impact of Workplace 

Practices and Information Technology on Productivity,” Review of Economics and 

Statistics 83, no. 3 (August 2001): 434–445. 

4. Joshua Angrist, “Lifetime Earnings and the Vietnam Era Draft Lottery: Evidence 

from Social Security Administrative Records,” The American Economic Review 80, 

no. 3 (June 1990): 313–336. 

Appendix 20.A 

The Consistency of Method of Moments 

Estimators 

Method of moments estimators are very often consistent. In particular, they are 

consistent whenever the Law of Large Numbers ensures that the sample moments 

converge in probability to the corresponding population moments. For example, 

in estimating a straight line with unknown slope and intercept, if plim A 1 , 

and plim A 1 , then the method of moments estimators b ~ n ge iB 

and b ~ = 0 

n gX ie i B = 0 

0 1 are 

consistent. Because unbounded variances can undermine the Law of Large Numbers, 

the consistency of b ~ and b ~ 0 1 requires that the joint distribution of the X i and 

the e be bounded appropriately. 1 

i 

20.A.1 

WHAT ARE AN ESTI- 

MATOR’S PROPERTIES? 

Proving Consistency 

This section applies the rules for manipulating probability limits from Table 12.1 

to prove the consistency of the method of moments estimators of the slope and intercept 

of a straight line, assuming that plim A 1 n ge iB = 0 and plim A 1 na X ie iB = 0. 

Assuming instead that E(e i ) = 0 and that both the (homoskedastic) disturbances 

and (fixed) explanators have finite, nonzero variances would suffice to establish, 

by the Law of Large Numbers, that these two probability limits are zero.


Because rule (iii) in Table 12.1 states that the plim of a sum is the sum of the 

plims, the convergence of the sample moments can be written 

plimSQ 1 n R a e iT = plimSQ 1 n R a (Y i - b 0 - b 1 X i )T 

= plimQ 1 n a Y iR - b 0 - b 1 plimQ 1 n a X iR = 0 



plimSQ 1 n R a X ie i T = plimSQ 1 n R a X i(Y i - b 0 - b 1 X i )T 

= plimQ 1 n a X iY i R - b 0 plimQ 1 n a X iR - b 1 plimQ 1 n a X 1 2 R = 0. 


When the sample moments converge in probability to their population values, 

what the method of moments insists be true about the residuals in any sample 

proves to be true about the disturbances (in probability limit) as the sample size 

grows large. This buys consistency for the method of moments estimators. More 

1 

1 

formally, because the method of moments always sets and n gX ie ~ n g e~ i 

i equal to 

zero, their probability limits are also zero (according to the first rule in Table 

12.1, that the plim of a constant is the constant): 


plimSQ 1 n R a e~ iT = plimSQ 1 n R a (Y i - b ~ 0 - b ~ 1X i )T = plim(0) = 0 

plimSQ 1 n R a X ie ~ iT = plimSQ 1 n R a X i(Y i - b ~ 0 - b ~ 1X i )T = plim(0) = 0. 

Applying rule (vi) from Table 12.1 (that the plim of a continuous function is the 

function of the plims) then yields 

plimSQ 1 n R a e~ iT = plimSQ 1 n R a (Y i - b ~ 0 - b ~ 1X i )T 

= plimQ 1 n a Y iR - plim(b ~ 0) - plim(b ~ 1) plimQ 1 n a X iR = 0. 



plimSQ 1 n R a X ie ~ iT = plimSQ 1 n R a X i(Y i - b ~ 0 - b ~ 1X i )T 

= plimQ 1 n a X iY i R - plim(b ~ 0) plimQ 1 n a X iR 

- plim(b ~ 1) plimQ 1 n a X i 2 R = 0. 

20.A.4


Comparing Equations 20.A.3 and 20.A.4 with Equations 20.A.1 and 20.A.2 

shows that plim(b ~ 0) and plim(b ~ 1) appear in Equations 20.A.3 and 20.A.4 just 

where b 0 and b 1 do in Equations 20.A.1 and 20.A.2. If Equations 20.A.3 and 

20.A.4 have a unique solution for plim and plim , then plim and 

plim (b ~ (b ~ 0) (b ~ 1) (b ~ 0) = b 0 

1) = b 1 . 

There is one case in which the solution of Equations 20.A.3 and 20.A.4 for 

plim(b ~ 0) and plim(b ~ 1) is not unique. If X takes on only one value, and is therefore 

perfectly collinear with the intercept term, Equation 20.A.4 is equivalent to Equation 

20.A.3: Equation 20.A.4 reduces to Equation 20.A.3 if we divide both sides 

of Equation 20.A.4 by the constant value of X. In this case, we really have only 

one equation in two unknowns. And, in this special case of perfect multicollinearity, 

the two equations do not yield a unique solution for plim and plim . 

Otherwise, the solution is unique, so plim(b ~ (b ~ 

and plim (b ~ 0) (b ~ 1) 

0) = b 0 1) = b 1 . 

Thus, barring perfect multicollinearity, when the sample moments converge 

in probability to their population expectations, b ~ and b ~ 0 1 are consistent estimators. 

Barring perfect collinearity–like problems, method of moments estimators 

are generally consistent when the sample moments converge to their population 

values. Because it is the Law of Large Numbers that ensures that sample means 

converge in probability to their population analogs, the Law of Large Numbers is 

key to the consistency of method of moments estimators. As noted earlier, infinite 

variances in a DGP endanger the applicability of the Law of Large Numbers. 

Endnotes 

1 

1. n gX ie i will converge in probability to zero if its expectation is zero and its variance 

goes to zero as n grows. By assumption, E(X i e i ) = 0, so the first criterion is met. The 

second criterion will also be met if we assume var(X (equal to E(X 2 is a finite, 

nonzero constant, t 4 , so that varA 1 i e 2 i e i ) 

i )) 

n gX ie iB = t 4 >n.

Chapter 20 Generalized Method of Moments Estimators and ...

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