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Gun Prevalence, Homicide Rates and

Causality: A GMM Approach to Endogeneity

Bias

In: The SAGE Handbook of Criminological Research Methods

By: Tomislav Kovandzic, Mark E. Schaffer & Gary Kleck

Edited by: David Gadd, Susanne Karstedt & Steven F. Messner

Pub. Date: 2012

Access Date: November 17, 2019

Publishing Company: SAGE Publications Ltd

City: London

Print ISBN: 9781849201759

Online ISBN: 9781446268285

DOI: https://dx.doi.org/10.4135/9781446268285

Print pages: 76-92

This PDF has been generated from SAGE Research Methods. Please note that the pagination of the

online version will vary from the pagination of the print book.

https://dx.doi.org/10.4135/9781446268285

Gun Prevalence, Homicide Rates and Causality: A GMM Approach to

Endogeneity Bias

TomislavKovandzic, Mark E.Schaffer, GaryKleck

Introduction

As is well known, guns are heavily involved in violence in America, especially homicide. In 2008, 66.9 percent

of homicides were committed by criminals armed with guns (US Federal Bureau of Investigation, 2010).

Probably an additional 100,000 to 150,000 individuals were medically treated for non-fatal gunshot wounds

(Kleck, 1997: 5). Further, relative to other industrialized nations, the US has higher rates of violent crime, both

fatal and non-fatal, a larger private civilian gun stock (about 90 guns of all types for every 100 Americans), and

a higher fraction of its violent acts committed with guns (Killias, 1993; Kleck, 1997: 64). These simple facts

have led many to the logical conclusion that America's high rate of gun ownership must be at least partially

responsible for the nation's high rates of violence, or at least its high homicide rate (e.g., Killias, 1993; Zimring

and Hawkins, 1999).

But while gun levels may affect crime rates, higher crime rates may also increase gun levels by stimulating

people to acquire guns, especially handguns, for self-protection. At least ten macro-level studies have found

effects of crime rates on gun levels (Bordua and Lizotte, 1979; Clotfelter, 1981; Duggan, 2001; Kleck, 1979,

1984; Kleck and Patterson, 1993; Magaddino and Medoff, 1984; McDowall and Loftin, 1983; Rice and

Hemley, 2002; Southwick, 1997), and individual-level survey evidence (not afflicted by problems of inferring

the direction of causal influences) indicates that people buy guns in response to higher crime rates (Kleck and

Kovandzic, 2009).

Thus, causality in the guns–crime relationship may run in either or both directions. If such a simultaneous

relationship exists, but analysts fail to take account of it using appropriate methods, their results will be almost

meaningless. What is asserted to be the impact of gun levels on crime rates will in fact also include the impact

of crime rates on gun levels.

The result is that researchers who want to use macro-level data to estimate the relationship between gun

prevalence and crime rates face a classic problem of ‘endogeneity bias’ resulting from reverse causation. In

an econometric estimation in which the dependent variable is a crime measure and the explanatory variable of

interest is gun prevalence or a proxy for it, the estimated coefficient on gun prevalence will be biased. Indeed,

in such an estimation the crime–guns relationship could quantitatively dominate the guns–crime relationship,

in which case the analyst will misinterpret an effect of crime on gun levels as an effect of gun levels on crime.

Nor is reverse causality the only reason to expect endogeneity bias in practice; omitted variable bias and

measurement error are also likely to be encountered by a researcher seeking to use macro-level data to

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Endogeneity Bias

investigate the gun–crime relationship.

This chapter shows how to address the problem of endogeneity bias by applying the modern estimation

and specification testing procedures provided by the generalized method of moments (GMM), using the

relationship of gun prevalence to homicide as an illustration. GMM can be thought of as an extension

and generalization of the older method of instrumental variables (IV). These extensions are very useful in

practice: GMM provides a framework that allows the empirical researcher to address not only endogeneity

bias but also other practical problems with standard errors and inference such as heteroskedasticity and

correlation of errors across observations (clustering). We apply these procedures in a cross-sectional setting

to study the relationship between gun ownership levels and homicide rates using county-level data from

the US in 1990. Gun ownership levels were measured using the percent of suicides committed with guns,

which recent research indicates is the best measure of gun levels for cross-sectional research (Kleck, 2004).

Our application provides a simple and easy-to-follow illustration of how researchers working on empirical

problems in criminology can use GMM. Endogeneity problems are widespread in quantitative criminology,

especially in aggregate studies of crime rates, as many of the explanatory variables (e.g., adoption of crime-

control strategies, police levels, criminal sanctions) are likely to be simultaneously determined with the crime

rate. The goal of this chapter is to set out the methodology of GMM in a way that enables an empirical

researcher without a specialist training in econometrics to understand and to employ these techniques

successfully.

The chapter is organized as follows. The next section discusses the problems of endogeneity bias,

unrobustness of standard errors, and a third problem unrelated to GMM, namely the choice and calibration of

a proxy when an explanatory variable is not directly available. The method of GMM is then described together

with its relationship to other, older estimators such as ordinary least squares (OLS) and instrumental variables

(IV), and the various tests of specification that an empirical researcher should employ and report are set out.

This is followed by a description of the dataset and variables used. The results are then presented, followed

by some conclusions, and finally suggestions for further reading and applications are given.

Endogeneity, Robustness and Mismeasurement

Consider a researcher who wants to estimate the impact of gun availability on the homicide rate. The

researcher has available cross-sectional data on localities (we consider the potential alternative of

longitudinal data in the next section). The researcher estimates the following simple linear model using OLS:

where hi is the homicide rate in locality i, gi is the level of gun ownership, ui is an error term, and for

expositional convenience the constant term is suppressed. The parameter of interest to the researcher in

equation (6.1) is β, the impact of gun levels on the homicide rate. There are three potential pitfalls facing the

researcher that we consider in this section: the variable gi may be ‘endogenous’ (correlated with the error ui),

in which case the OLS estimate of β will be biased; the OLS standard errors needed to test the significance

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of β may not be ‘robust’; and a satisfactory direct measure of gun ownership may simply not be available,

in which case the researcher needs to work with a proxy measure of gun prevalence. In the next section

we outline an estimation strategy built on the GMM that can address the first two pitfalls, as well a practical

solution for the proxy problem.

The Endogeneity Problem

The first property that empirical researchers look for in an estimator is usually that it is unbiased, or at least

‘consistent’. (‘Consistency’ is an ‘asymptotic’ property of an estimator; i.e., a property that emerges as the

sample size increases. Intuitively, an estimator that is consistent is asymptotically unbiased–there might be

a bias, but the bias gets small as the sample size gets larger.) The key requirement for the OLS estimator

OLS to be consistent is that gun levels gi must be ‘exogenous’. This ‘exogeneity’ requirement can be stated

in various ways: gi is uncorrelated with the error term ui, gi and ui are ‘orthogonal’ (i.e., independent), or, in

statistical terms, E (giui) = 0. This last statistical expression is vsometimes called a ‘moment condition’ or

‘orthogonality condition’. All these are different ways of stating the same requirement.

If the exogeneity condition fails, then we say gi is ‘endogenous’. The consequences for the OLS estimator

OLS are serious: it is no longer unbiased or even consistent. The bias of OLS is upwards if gi and ui are

positively correlated, and downwards if gi and ui are negatively correlated. This bias can arise for various

reasons, and the literature sometimes uses different terms for the bias depending on the cause. In this

chapter we will use the general term ‘endogeneity bias’ irrespective of the reason. Unfortunately, there are

three good reasons to think that the exogeneity condition will fail and the OLS estimator OLS will suffer from

endogeneity bias if it is used to estimate the impact of gun prevalence on homicide rates. All three–reverse

causality bias, omitted variables bias and measurement error bias–are forms of endogeneity bias.

Reverse (or ‘simultaneous’) causality arises when there is a second relationship in which gun levels depend

on homicide rates:

As already noted, there is empirical evidence from the US that high crime rates do lead people to acquire guns

for self-protection, so this is a serious practical problem for the researcher. The impact of reverse causality

onv OLS depends on δ given that δ and the correlation between gi and ui will have the same sign. The

evidence cited previously suggests that δ > 0–high crime levels lead to higher gun levels–and hence gi and

ui will be positively correlated. The result is that OLS will be biased upwards. Indeed, if the reverse causality

is strong enough, the researcher could find that OLS > 0 and conclude that more guns means more crime

even if the ‘true’ β–the true impact of guns on crime, all else being equal–is negligible or negative.

The second reason the exogeneity assumption may fail is because of omitted variable bias. Say that the ‘true’

relationship between homicide rates and gun prevalence is actually

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but the researcher does not include the variable Xi in the estimation, and simply estimates equation (6.1)

using OLS. This will result in ‘omitted variable bias’, depending on whether gun prevalence gi is correlated

with the omitted variable Xi. The reason is straightforward: if equation (6.1b) is the ‘true’ model, then the error

term in the equation actually estimated, equation (6.1), can be written ui = γXi + ?i. It is easy to see that if gi

is correlated with Xi, it will also be correlated with ui, and the OLS estimate of equation (6.1) will once again

suffer from endogeneity bias. The sign of the bias in OLS is determined by the sign of γ and the sign of the

correlation of gi and Xi; if both are positive or both are negative, OLS will be biased upwards, otherwise it

will be biased downwards.

The simplest way to address this problem is to include Xi in the estimation, but often this is impossible

because Xi is not available to the researcher, and the guns–homicide application is no exception. For

example, we expect that pro-violence sub-cultural norms in a locality will be a positive determinant of

homicide, and will also be positively correlated with gun prevalence. The result will be a OLS that is

biased upwards, because in effect we are mistakenly attributing the pro-violence effect of local norms to gun

prevalence. Local laws are another example of a potentially important omitted variable that in practice may

not be available to the researcher.

The third reason the exogeneity assumption may fail is because of classical (i.e., random) measurement

error: the true relationship between gun and homicide is

but the observable level of gun prevalence, measured with error ηi, is all we have available:

If we estimate (6.1) using gi instead of the unavailable true gi*, the result is again an endogeneity bias in

OLS, because measurement error creates a correlation between gi and ui. In this case, the endogeneity bias

is a form of attenuation bias, because OLS will be biased away from the true β and towards zero.

The standard solution to endogeneity bias is to estimate equation (6.1) using the method of IV or the more

modern framework of GMM. These methods require the researcher to have a variable Zi that is correlated

with guns (‘instrument relevance’) and that is also uncorrelated with the error term ui in the homicide equation

(‘instrument exogeneity’). We describe the method of GMM in the next section. First, however, we discuss the

issue of robustness of standard errors.

Robustness of Standard Errors

Exogeneity is all that is required for the OLS estimator to be consistent. But we also need an estimate of the

variance of the OLS estimator in order to calculate standard errors, test hypotheses, construct confidence

intervals, and so forth.

The classical textbook formula for OLS standard errors can fail for two reasons that are likely to be important

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in practice. First, ui, the error term in equation (6.1) may be conditionally heteroskedastic; that is, the

variance of the error term ui may be correlated with the variance of the explanatory variable in (1), gi. This

phenomenon–that when ui is highly variable, gi is also highly variable–is very common in cross-sectional and

panel data. If, for example, the localities in our dataset–say, US counties–vary a lot in size, we should expect

that all our variables will bounce around a lot more in very large counties than in very small ones. The effect

on the OLS standard errors is typically to make them ‘too small’, and hence to lead the researcher to conclude

that OLS is a more precise estimate than it actually is.

Second, the classical textbook formula for OLS standard errors assumes ‘independence’. More precisely, it

assumes that giui is uncorrelated across localities; that is, giui and gjuj. This too is likely to be untrue in cross-

sectional and panel data. Here omitted variables make an unwelcome reappearance, because they are a

prime culprit behind the failure of the independence assumption in cross-sectional data. For example, state-

level variables such as laws or economic shocks would affect counties in that state but not in other states; this

can be enough to create a correlation between giui and gjuj, where i and j refer to two counties in the state.

Another example, very common in panel data applications, is serial correlation. If we were using panel data

on counties, we should expect that gituit and git-1uit-1, the observations for county i in periods t and t-1, are

likely to be correlated. The effect of this ‘clustering’ or ‘within-group correlation’ is again typically to make the

OLS standard errors ‘too small’.

The modern approach to this problem is to use standard errors that are ‘robust’ to heteroskedasticity and

clustering; in other words, to use standard errors that are correct (consistent) whether or not there is

heteroskedasticity and/or clustering. These standard errors are known variously as Eicker–Huber–White or

‘sandwich’ standard errors; for conciseness we will simply call them ‘robust’. As we discuss in the next

section, this approach to obtaining consistent standard errors is closely linked to the GMM approach to

obtaining consistent estimates of β.

The Proxy Problem

Ideally, researchers investigating the relationship between gun levels and homicide would have well-defined

measures of both variables at the level of the locality. Unfortunately, direct measures of gun prevalence have

severe drawbacks (which we discuss in the next section), and researchers have to make do with a proxy for

gun prevalence. The relationship between the proxy pi and the (unobserved) level of gun prevalence gi is

and instead of estimating (6.1), the researcher estimates

The task facing the researcher now is similar to, but more challenging than, the case of classical

measurement error discussed previously. In addition to the other problems already discussed, the researcher

now has the problem that the parameter δ relating gun prevalence to its proxy is unobservable and

can't be directly estimated–if it could, the proxy would of course be unnecessary. This has two practical

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consequences. First, it is incumbent on the researcher to validate the proxy; that is, to assemble evidence that

supports its use. If pi is uncorrelated with actual gun levels gi, it is, of course, useless. Second, the researcher

should ideally also have available some estimate of the parameter δ (e.g., from a validation exercise). The

coefficient b is not the quantitative impact of gun levels on crime rates; this is given by β = bδ, and δ is

not observed directly. A test of the estimated b may enable the researcher to say if there is a statistically

significant non-zero impact of guns on crime; but without an estimate of δ, the researcher will be unable to

say anything about the actual size of the impact.

Instrumental Variables and GMM

GMM is a modern and increasingly popular approach to the problem of estimation with endogenous

regressors. GMM provides a unified framework for estimation and testing that is naturally suited to empirical

situations where endogeneity is a central problem. The literature on GMM is now vast and many good

expositions are available. All modern graduate-level econometrics textbooks cover it, in varying degrees of

detail. Hayashi (2000) is an advanced text that sets out many of the tests and results used and cited in this

paper. Baum et al. (2003, 2007) set out the basics of IV and GMM estimation and specification testing, and

describe the set of extended estimation and testing routines implemented for the Stata statistical package

used here.

In this section we provide a brief non-specialist introduction to GMM. We show how GMM is related to older,

standard estimators, how the GMM framework relates to the ‘robust’ approach to obtaining standard errors,

and how GMM can be used in our application to obtain consistent estimates of the impact of gun prevalence

on homicide rates. We then provide a checklist of specification tests that a researcher using GMM should

employ. Throughout we use our example of the guns–crime relationship, but the recommended procedures

and the specification testing checklist are generally applicable.

Box 6.1 A Checklist

We have described previously all the tools needed for GMM-based estimation and testing

of the guns-homicide relationship. The GMM procedure that we follow is:

Estimate equation (6.1) using efficient GMM or inefficient OLS and all the

instruments Z1i, Z2i, etc., assuming that the measure of gun prevalence gi

is exogenous. Obtain the J statistic for the efficient GMM estimator. Allow for

possible heteroskedasticity or clustering. If the J statistic is large, take this as

evidence that one or more of Z1i, Z2i, …, and gi is endogenous, and proceed to

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Step 2. If the J statistic is small, take this as evidence that all the aforementioned

variables are exogenous. However, prior research suggests that the assumption

that gun prevalence is exogenous is questionable, so we proceed to Step 2

anyway.

Estimate equation (6.1) using efficient GMM or inefficient IV, this time treating

gun prevalence as endogenous. Obtain the J statistic for the efficient GMM

estimator. Allow for possible heteroskedasticity or clustering. If the J statistic is

small, take this as evidence that the instruments Z1i, Z2i, … are exogenous and

proceed to Step 3. If the J statistic is large, take this as evidence that one or

more of the instruments is endogenous. Unless there are a priori good reasons

to suspect one instrument in particular is endogenous (and hence this suspicion

can be tested using a C test), stop here–consistent estimation is not possible.

Test the relevance of the instruments Z1i, Z2i in the specification estimated in

Step 2. If the first-stage F statistic is large (greater than the Staiger–Stock rule

of thumb value of 10 or greater than the Stock–Yogo critical values), proceed

to Step 4. If the first-stage F statistic is small, conclude that the instruments are

‘weak’ and consider using alternative estimation methods (see ‘Recommended

Reading’ section).

Test whether gun prevalence is endogenous using a GMM distance test using

J–J 2, where J is the J statistic using all the instruments plus treating gun levels

as exogenous (Step 1) and J2 is the J statistic that does not assume exogeneity

of gi. If the C statistic is small, take this as evidence that gun prevalence

is exogenous, and proceed to Step 5. If the C statistic is large, take this as

evidence that gun prevalence is endogenous and go to Step 6.

(Where gi is exogenous): Consider using the Step 1 estimates–efficient GMM

or OLS–as final estimates. The former is consistent and efficient; the latter

is consistent, but inefficient if errors are not homoskedastic and independent.

Alternatively, because prior evidence and research suggests that gun levels

may be subject to endogeneity bias from various sources, treat gun prevalence

as endogenous anyway and go to Step 6.

(Where gi is endogenous): Use the Step 2 estimates–efficient GMM or IV–as

final estimates. The former is consistent and efficient, but may be more prone to

problems in small samples; the latter is consistent, but inefficient if errors are not

homoskedastic and independent. In the case IV is used, be sure to use standard

errors that are robust to heteroskedasticity and clustering (this is automatic in

the case of efficient GMM).

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The Classical Method of Moments (MM)

We begin by repeating the equation to be estimated:

We cannot use the OLS estimator because gi is endogenous–it is correlated with the error term ui. If, however,

we have available a different variable or ‘instrument’ Zi, we can obtain a consistent estimate of β, provided

two conditions are met:

Assumption EX: Zi is exogenous; i.e., uncorrelated with ui

Assumption R: Zi is correlated with gi.

What we have done is replaced the failed requirement that gi is exogenous with the requirement that the

instrument Zi is exogenous, and added the requirement that Zi is ‘relevant’; that is, correlated with gi.

If these two assumptions are satisfied, we can use the classical ‘method of moments’ (MM) introduced by Karl

Pearson in 1894 to obtain a consistent estimate of β. The intuition behind the MM is simple. Formally stated,

Assumption EX means that E(ziui) = 0. This is an ‘orthogonality’ or ‘moment’ condition. It is an assumption,

because we cannot observe the true error term ui. But what we can do is calculate the residual, defined in

the usual way as ûi hi– gi. The crucial but simple point is that the residual depends on our estimate of β.

By choosing a specific estimate , we are also choosing a set of residuals. But what estimate should we

choose? Pearson's suggestion was to choose so that the residuals ûi ‘behave like’ the true error term ui.

The true error term is uncorrelated with Zi–it satisfies the ‘moment condition’ E(ZiUi) = 0.

Pearson's suggestion is to choose MM so that the residuals ûi from the sample data we have at hand

behave the same way; in other words, so that the sample mean of (Ziûi) is zero; that is, . We

refer to this as sample moment condition.

This is just one equation in one unknown, namely MM. To solve the equation for MM we substitute for the

residual ûi to obtain , and after simplifying and rearranging, we have the MM estimate

of the impact of guns on homicide: . The MM estimator in this simple case is none other

than the standard IV estimator. Provided assumptions EX and R are satisfied, MM is a consistent estimate

of the impact of guns levels on homicide rate.

In fact, OLS is also a special case of a MM estimator. It is a straightforward exercise to replace assumption

EX above with our original (but flawed) assumption that gi is exogenous, and then derive the OLS estimator

as the MM estimator.

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GMM

A natural question is, what if we have more than one Zi? Say we want to estimate (6.1), and we have two

variables, Z1i and Z2i, that we believe are also exogenous. We now have two exogeneity assumptions; that is,

two moment conditions: E(Z1iUi) = 0 (assumption EX1) and E(Z2iUi) = 0 (assumption EX2). We can't use the

classical method of moments, for the following reason. MM says to choose MM so that vthe sample moment

conditions mimic assumptions EX1 and EX2; that is, so that and . However, we

can't, because there is a basic problem: we have two equations but only one unknown, MM. The parameter

β is overidentified: there are more exogenous variables than there are parameters to be estimated.

The GMM estimator GMM addresses this problem by selecting a GMM that gets the two sample moment

conditions as ‘close’ to zero as possible. More precisely, GMM proceeds by defining an objective function

J(.) that is a function of the data, the parameters and a set of weights in a weighting matrix W. The GMM

objective function is a quadratic form in the sample moment conditions; that is, the sample moment conditions

are weighted using the weights in W and summed to produce a scalar that is minimized: J(.) can be thought

of as the ‘GMM distance’–the distance from zero, which is the value the objective function would take if all

the sample moment conditions were satisfied–and the definitions of the GMM estimator GMM is that it is the

value of that minimizes J given W and the data.

There are as many different GMM estimators as there are different possible Ws to use in J. Where GMM

comes into its own is when an optimal weighting matrix is chosen. Using an optimal weighting matrix

guarantees that the GMM estimator is ‘efficient’–roughly speaking, it is the most precise estimator possible (it

has the smallest possible asymptotic variance) given all the assumptions made by the researcher. Hansen,

who introduced GMM in a seminal paper in 1982, showed that the optimal GMM weighting matrix is the

inverse of the covariance matrix of moment conditions, which we will denote by Ω. In our example, the

elements of Ω are the variances and covariances of (Z1iUi) and (Z2iUi).

The matrix Ω is unknown, but it can be readily estimated. This is where the link with robust standard errors

comes in. The matrix that is used for calculating robust standard errors is none other than an estimate of

this same Ω used to calculate the efficient GMM estimator. In GMM estimation, efficiency and robustness go

hand-in-hand. Thus:

If we use an that is obtained under the assumption of classical homoskedastic and independent

errors as in traditional OLS, the GMM estimator will be efficient under the assumption of

homoskedasticity and independence.

If we use an that allows for arbitrary heteroskedasticity, the GMM estimator will be efficient

for arbitrary heteroskedasticity, and the GMM standard errors will be robust in the presence of

heteroskedasticity.

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If we use an that allows for arbitrary heteroskedasticity and within-group correlation

(‘clustering’), the GMM estimator will be efficient for arbitrary heteroskedasticity and clustering,

and the GMM standard errors will be robust in the presence of heteroskedasticity and clustering.

It is helpful to contrast the use of the efficient GMM estimators in (a), (b) and (c) with the use of the IV

estimator combined with robust standard errors, a practice commonplace in empirical work. In cases (b) and

(c), both the efficient GMM estimator and the IV estimator will be consistent, and the standard errors in both

cases will also be consistent. The difference is that the efficient GMM estimator is more precise. In case (a),

where we assume that the errors are classical, homoskedastic and independent errors, the efficient GMM

estimator is the IV estimator–they are one and the same. The IV estimator is simply a special case of a

GMM estimator, one that is consistent and efficient under homoskedasticity and independence, but inefficient

(although still consistent) if the errors are heteroskedastic or clustered.

We can make a similar contrast between efficient GMM and OLS in (a), (b) and (c) when gi is considered

exogenous. Assumption R about the instruments is no longer needed in this case, but Assumption EX is still

useful because it means that efficient GMM can deliver more precise estimates than OLS. The efficient GMM

estimator in this special case is known as ‘heteroskedastic OLS’ or HOLS, and was introduced by Cragg

(1983). When gi is exogenous, OLS is a special case of a GMM estimator, consistent and efficient if errors

are classical, and consistent but inefficient if the errors are heteroskedastic or clustered, just like IV.

Specification Testing: Exogeneity

An attractive feature of GMM for the empirical researcher is that is makes specification testing very

straightforward. A ‘specification test’ is a test of one or more of the assumptions required for the model. In

our application, these are tests of the assumptions of exogeneity and relevance. Reporting such tests is now

essential in any piece of empirical work using IV or GMM methods.

GMM provides a straightforward framework for testing exogeneity when the equation is overidentified. Under

the null hypothesis that the all the exogeneity assumptions are valid–all the variables that were assumed

to be exogenous are indeed exogenous–the minimized value of J is distributed as χ2 with degrees of

freedom equal to the degree of overidentification. This ‘overidentification test’ is known in the literature as the

Hansen or Sargan–Hansen J statistic and is, conveniently, an automatic by-product of GMM estimation. The

consequence of a failure of exogeneity of instruments is the same as in the case of OLS: the estimated

GMM will be inconsistent.

The same framework can be used to test the validity of a subset of exogeneity conditions; that is, to test

whether or not selected variables are exogenous. For example, say we have doubts about whether Z1i is

actually exogenous. Consider the J statistic resulting from two different efficient GMM estimations: J is the

J statistic from an efficient GMM estimation that uses both Z1i and Z2i as instruments, and J2 is the J

statistic from an efficient GMM estimation that uses only Z2i. Under the null hypothesis, both Z1i and Z2i

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are exogenous; under the alternative hypothesis, only Z2i is exogenous. A ‘GMM distance’ or ‘C’ test of the

exogeneity of Z1i is given by the quantity C = J–J2. Under the null hypothesis, C should be distributed as χ 2

with one degree of freedom. A large-test statistic suggests that in fact Z1i is not a valid instrument, but rather

is endogenous.

To take another example that is important for our application: say we want to see if we have evidence that

indeed gi is endogenous. The GMM distance approach suggests that we compare the J statistic from an

efficient GMM estimation that uses Z1i and Z2i as instruments to a J statistic from an efficient GMM estimation

that uses Z1i, Z2i and gi. (This means treating gi as exogenous, which in the GMM framework is either similar

to or identical to OLS–but we can still get a J statistic from the estimation.) The difference between these

two J statistics will distributed as χ2 with one degree of freedom if gi is actually exogenous. For the reasons

discussed earlier, a priori we suspect this is unlikely, and the GMM distance statistic will probably be large.

The robustness features of GMM estimation carry over to J and C statistics. If the used in the GMM

estimation is robust under arbitrary heteroskedasticity, then not only will the efficient GMM estimator be

efficient for arbitrary heteroskedasticity, the J and C statistics will also be consistent for tests of exogeneity.

This is one of the most important practical reasons for using the GMM framework. Older specification tests

such as Sargan's (1958) NR2 overidentification statistic and the Durbin–Wu–Hausman endogeneity test are

not robust–they require the assumptions of homoskedasticity and independence. In fact, these older statistics

can be derived as special cases of the more general GMM test statistics. In practice there is rarely any reason

for empirical researchers to use these older, un-robust tests given that robust GMM-based statistics can be

obtained just as easily.

The intuition behind the J statistic is actually quite straightforward. The power of these tests to detect

specification failures–that is, that instruments are endogenous–comes from a ‘vector of contrasts’. In our

simple one-endogenous-regressor example, we could, if we wanted to, use our two instruments separately,

and thereby obtain two different estimates, Z1 and Z2. If our two instruments Z1i and Z2i are both

exogenous, then these two estimates should be fairly similar, since both are consistent estimates of the true

β. But what if Z1 and Z2 are very different? The only way these two estimators can be different is if at

least one of them is wrong: either Z1i is actually endogenous and Z1 suffers from endogeneity bias; and/or

Z2i is actually endogenous and Z2 suffers from endogeneity bias. The J statistic in this case is essentially a

measure of the difference between Z1 and Z2; the bigger the difference, the more likely it is that either Z1i

and/or Z2i is endogenous. The C statistic has a similar intuitive interpretation for the subset of instruments

being tested for endogeneity.

This intuitive interpretation of the J statistic also makes clear when the J test will not have the power to

detect endogeneity problems. Say that both Z1i and Z2i are endogenous, and hence both Z1 and Z2 suffer

from endogeneity bias. What if the size of the endogeneity bias is similar in both cases? Unfortunately for

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us, there will be little difference between Z1 and Z2, and the J statistic will be small. In other words, the

two estimates will be similar–similarly ‘wrong’, but still similar–and our ‘vector of contrasts’ test will show a

small contrast. One lesson from this discussion is that the J test will have more power to detect problems if

the instruments used derive from different sources, since variables from different sources are less likely to

share endogeneity biases that are very similar. (For further discussion of this point, see Stock and Watson,

2007, and Wooldridge, 2008, for simple expositions, and Kovandzic et al., 2008, for an application to the

guns—homicide model.)

Specification Testing: Relevance

The second requirement for a GMM estimator is assumption R: an instrument Zi must be correlated with gi.

This assumption can and should also be subjected to specification testing.

In the case of a single endogenous regressor, as in our application, the standard specification test is

straightforward and very easy to implement: the statistic for instrument relevance is the F test of the excluded

instruments in the ‘first-stage’ regression. The ‘first-stage’ regression is identical to the first stage of ‘two-

stage least squares’ (2SLS), an alternative method of calculating the traditional IV estimator. In the first-stage

regression, the endogenous variables are regressed on the instruments plus any exogenous covariates used

in the main equation. In our simple example of two instruments and no exogenous covariates, for example,

the researcher should estimate

and, after examining θ1 and θ2, for their consistency with theory and prior evidence, calculate a standard F

statistic for their joint significance; a large value indicates that the model is identified. If heteroskedasticity or

clustering is suspected, a heteroskedastic- or cluster-robust test statistic can be used. If the main equation

(6.1) being estimated includes exogenous covariates, these should also be included in (6.5).

It is not enough for this F statistic to be significant at, say, the 5 percent or 1 percent levels. If the instruments

are correlated with the endogenous regressor but only weakly, the F statistic will be significant at conventional

levels, but the IV and GMM may still suffer from serious problems. Recent research (e.g., Bound et al., 1995;

Staiger and Stock, 1997; Stock et al., 2002, which provides a survey) has shown that when instruments

are weak, IV/GMM estimates of parameters will be badly biased (in the same direction as OLS), estimated

standard errors will be too small and the null hypothesis will be rejected too often. Staiger and Stock (1997)

recommend an F statistic of at least 10 as a rule of thumb for the standard IV estimator. Stock and Yogo

(2005) provide more detailed advice based on Monte Carlo studies, and show inter alia that the Staiger–Stock

rule of thumb is a reasonable guideline to follow for the single-endogenous-regressor case when the number

of instruments is small.

Caveats

Useful as it is, GMM is not a panacea, and we note here several caveats to its use besides those already

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mentioned. First, the use of large numbers of instruments can generate bias problems, and the general

advice here is to be parsimonious with the number of instruments employed. Second, there is some evidence

that the standard errors for some efficient GMM estimators may be biased downwards in small samples.

A conservative estimation strategy adopted by some researchers is consequently to use the inefficient IV

estimator, rely on robust standard errors for inference, and use the robust GMM J and C statistics for

specification testing.

Data, Model and Proxy Validation

Data and Model

To estimate the impact of gun availability on homicide rates, we use cross-sectional data for all US counties

from 49 states for which relevant data were available (N = 3,058). The missing state is Alaska, which had

to be excluded because one of our key variables pertains to election results, and election districts in this

state cannot be confidently linked with particular counties (boroughs). The use of 1990 data is dictated by

two factors. First, most of the control variables included in the homicide equations to mitigate omitted variable

bias are available at the county level only during census years. A second reason for choosing 1990 is the fact

that the firearm crime rate (homicide, robbery and assault) had reached its highest level in nearly 30 years by

1990. It is reasonably argued that if gun availability is responsible for higher homicide rates, the high levels

of firearm crime in 1990 should provide one of the best opportunities to date for testing the gun availability/

homicide relationship.

County-level data were chosen for several reasons. First, the use of counties provides for a diverse

sample of ecological units, including urban, suburban and rural areas. Second, counties are more internally

homogenous than nations, states or metropolitan areas, thereby reducing potential aggregation bias. Third,

counties exhibit great between-unit variability in both gun availability and homicide rates, which is precisely

what gun availability and homicide research is trying to explain. Fourth, county data provide a much larger

sample than previous gun level studies, which have focused mainly on nations, states, or large cities.

Our estimation equation includes state fixed effects; that is, state dummy variables. This enables us to

control for any unobserved county characteristics that vary at the state level and that could be expected to

influence both gun levels and homicide rates. Examples of such confounders would be state laws and judicial

practice relating directly or indirectly to homicide and gun ownership, state-level resources devoted to law

enforcement, and incarceration rates in state prisons. The disadvantage of this approach is that only variables

available at the county level can be used in the estimations, because state-level measures would be perfectly

collinear with the fixed effects.

Although the used of state fixed effects can address some issues relating to unobservables that vary at

the state level, remaining within-state correlation would mean that classical OLS-type standard errors would

be inconsistent. Heteroskedasticity is also likely to be present. The standard approach to addressing these

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problems using disaggregated US locality data is the heteroskedastic and cluster-robust approach, clustering

on states. This allows for the county-level errors to be arbitrarily correlated within states, though we still need

to assume that they are uncorrelated across states.

We follow the convention for crime policy studies and use a linear model in which most variables are specified

in logs. The dependent variable in our model is the log of the gun homicide rate per 100,000 county population

(log CRG); if higher gun levels increase the homicide rate at all, they must do so by, at minimum, increasing

the gun homicide rate. Homicide data for each county were obtained using special Mortality Detail File

computer tapes (not the public use tapes) made available by the National Center for Health Statistics (US

National Center for Health Statistics, 1997). The data include all intentional homicides in the county with

the exception of those due to legal intervention (e.g., police shootings and executions). Homicide rates are

averages for the seven years 1987 to 1993, thus bracketing the census year of 1990 for which data on many

of the control variables were available.

The use of logs poses some minor problems, because even though we are using seven-year averages, some

variables are zeros prior to logging. The loss of observations because the log of zero is undefined gives us a

final sample size of 2,588 counties. Most of the loss of observations is due to counties which had no non-gun

murders in 1987–1993. These counties are mostly fairly small (an average population of about 8,000 versus

93,000 for the counties in the estimation sample) and together accounted for less than 2 percent of the total

population in 1990.

Surveys asking people directly whether they own guns are usually limited to a single large area, such as a

nation or state. Instead, we use the best proxy of gun prevalence for cross-sectional research that is available

at the county level, the percentage of suicides committed with guns (PSG). Our specification uses the log of

the gun suicide percent (log PSG). As was done with homicide rates, PSG was computed for the seven-year

period 1987–1993. Similar to homicide, data for the percent of suicides committed with guns were obtained

using special Part III Mortality Detail File computer tapes made available by the National Center for Health

Statistics. Unlike widely available public-use versions, the tapes permit the aggregation of death counts for

even the smallest counties (US National Center for Health Statistics, 1997).

In addition to the gun prevalence measure, we included a set of county-level control variables as regressors.

Decisions as to which variables to include in the homicide equations were based on a review of previous

macro-level studies linking homicide rates to structural characteristics of ecological units (see Kleck, 1997,

chapter 3; Kovandzic et al., 1998; Land et al., 1990; Sampson, 1986; Vieraitis, 2000, and the studies reviewed

therein).

We were particularly concerned to control for variables that had opposite-sign associations with gun levels

and homicide rates because such variables could suppress evidence of any positive effect of gun levels on

homicide rates. Thus, we controlled for the percent of the population that is rural because rural people are

more likely to own guns, but less likely to commit homicide. Likewise, we controlled for the poverty rate, the

share of the population in the high-homicide ages of 18–24 and 25–34, and the African-American share of the

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population because people in these groups are less likely to own guns, but more likely to commit homicide,

than other people (Cook and Ludwig, 1997; Kleck, 1997; US Federal Bureau of Investigation, 2010). The

other controls used were percentage Hispanic, population density, average education level, unemployment

rate, transient population (born out-of-state), vacant housing units, female-headed households with children,

median household income, households earning less than $15,000, and inequality (ratio of households earning

more than $75,000 to households earning less than $15,000).

The sets of controls for rurality and age structure are, exceptionally, used in percentage rather than log

form. Because the raw percentages sum to 100, including all categories would generate a perfect collinearity

problem, and so one category must be omitted. Using raw percentages has the appealing feature that the

results are invariant to whichever percentage is the omitted category. We omit the percentage rural and the

percentage aged 65+.

The excluded instrumental variables used are (1) log OMAG, the log of the combined subscription rates of

three of the most popular outdoor/sport magazines (Field and Stream, Outdoor Life and Sports Afield) in

1993, per 100,000 county population (Audit Bureau of Circulations, 1993), and (2) log PCTREP88, the log of

the percentage of the county population voting for the Republican candidate in the 1988 presidential election.

Both excluded instruments are theoretically important correlates of gun ownership that are plausibly otherwise

unrelated to homicide. Log OMAG serves as a measure of interest in outdoor sports such as hunting and

fishing, or perhaps as a measure of a firearms-related ‘sporting/outdoor culture’ (Bordua and Lizotte, 1979).

Log PCTREP88 serves as a measure of political conservatism and hence should be positively correlated

with gun ownership. The 1988 election results were chosen in preference to the 1992 results because the

choice between the two main candidates in 1988 maps more closely to attitudes towards gun ownership. In

the 1992 election, unlike the 1988 election, the politically less conservative candidate (negatively correlated

with gun ownership) was also a southerner (positively correlated with gun ownership). The 1992 results are

also less easily interpreted because of the significant share of the vote that went to the third-party candidate,

Ross Perot. Prior research suggests that both variables are important predictors of gun ownership (Cook and

Ludwig, 1997: 35; Kleck, 1997: 70–72).

Table 6.1 lists and provides a brief description of each variable used along with their means and standard

deviations. Data for the control variables were obtained from the US Bureau of the Census, County and City

Data Book, 1994, except for log PCTREP88, which is from ICPSR (1995), and rurality, which is from US

Census Bureau (1990).

Proxy Validation

We need to validate and calibrate our proxy to available survey-based measures of gun levels. Because

of its availability in the widely used General Social Surveys (GSS), the most convenient calibration is to

the mean percentage of households with guns (HHG). National gun survey prevalence figures have been

available since 1959, though not for every year. Using state-level measures from surveys conducted by the

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Centers for Disease Control (CDC) in 2002 (Okoro et al., 2005) and PSG data for 1995–2002 taken from

CDC's WONDER service, a simple OLS regression based on 50 states of log PSG on log HHG is as follows

(standard errors are in parentheses):

log PSG = 2.31 + 0.481 log HHG + e (0.12) (0.035)

The coefficient on log HHG can be interpreted as an estimate of the log–log calibration δ of the proxy pi to the

level of gun prevalence gi in equation (6.4). The figure suggests that in our log–log estimations, the coefficient

on our proxy log PSG should be approximately halved (δ = 0.481 ≈ 0.5) in order to be interpreted as elasticity

of gun homicide with respect to actual gun levels. This is, however, only an approximation based on limited

data and a simple linear calibration, and should be used with caution.

Results

In this section we discuss the estimation results, following the checklist presented earlier. The main estimation

results for the estimation of the gun homicide/gun prevalence relationship using efficient GMM, IV and OLS

are presented in Table 6.2. Columns 1 and 2 report the efficient GMM and OLS estimations that treat our

proxy for gun prevalence, log PSG, as exogenous. Columns 3 and 4 report the efficient GMM estimation and

IV estimations in which log PSG is assumed endogenous. The bottom half of Table 6.2 reports the details

of the ‘first-stage’ regression corresponding to the estimations in column 3 and 4. All statistics and standard

errors reported are

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Table 6.1 Descriptive statistics, estimation sample (n = 2,588)

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Table 6.2 Log gun homicide equation

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robust to arbitrary heteroskedasticity and within-state correlation (clustering); the efficient GMM estimator is

efficient for arbitrary heteroskedasticity and clustering. The efficient GMM estimator used is two-step feasible

efficient GMM; see Hayashi (2000) or Baum et al. (2003) for a description.

Step 1 in our checklist is to estimate the equation, treating log PSG as exogenous, and examine the J statistic.

The J test in this case is a test of the exogeneity of all the variables assumed to be exogenous: log PSG and

all the other regressors, log OMAG and log PCTREP88. There are no endogenous variables in the equation,

so the test statistic is distributed as χ2 with two degrees of freedom, because there are two (excluded)

instruments–0 endogenous regressors = 2. The table reports only the J test statistic for the efficient GMM

estimator, because the corresponding test statistic for OLS, the Sargan NR2 statistic, would not be robust

to heteroskedasticity and clustering. The J statistic is 10.98, which is very large for a χ2(2) variable (p =

0.004). The null hypothesis that all our exogenous variables are actually exogenous is emphatically rejected.

We conclude that there is evidence that one or more variables is actually endogenous, or perhaps that the

equation is misspecified for other reasons. The immediate implication of this finding is that OLS estimation of

the impact of gun prevalence on homicide levels is likely to be biased, resulting in faulty interpretations.

This is not surprising; a priori, we expected that log PSG is endogenous, and the coefficient in the estimation

is likely to be biased upwards if log PSG is treated as exogenous. The estimated coefficient on log PSG in

columns 1 and 2 is positive and statistically significant: 0.305 and significant at the 1 percent level if estimated

using efficient GMM, or 0.185 with a somewhat larger standard error, but still significant at the 5 percent level,

if estimated using (inefficient) OLS. Our calibration exercise (δ ≈ 0.5) suggests a positive elasticity of gun

homicide with respect to gun prevalence of about 10–15 percent: a doubling (100 percent) increase in gun

prevalence in a county would raise gun homicide rates by 10–15 percent. We will be able to say something

of how much of an upward bias is present in this estimate shortly.

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Step 2 in our checklist is to re-estimate the equation, this time treating log PSG as endogenous, and again

examine the J statistic. The J test is again a test of the exogeneity of all the variables assumed to be

exogenous: the difference is that log PSG is no longer in the list of exogenous variables. There is now one

endogenous variable in the equation, log PSG, so the test statistic is distributed as χ2 with two (excluded)

instruments–1 endogenous regressors = 1 degree of freedom. The table again reports only the J test statistic

for the efficient GMM estimator, because the corresponding test statistic for I V, the Sargan NR2 statistic is

again not robust to heteroskedasticity and clustering. The J statistic is now much smaller: 0.012 (p = 0.913).

We cannot now reject the null hypothesis that our exogenous variables are indeed exogenous. So far, the

evidence suggests that the estimations in columns 3 and 4 of Table 6.2 are well specified.

The explanation for what is happening is apparent from the estimated coefficient on log PSG in columns 3

and 4: efficient GMM and (inefficient) IV give us almost identical estimates of about −2.4, significant at the 1

percent level. Our calibration exercise (δ ≈ 0.5) suggests a negative elasticity of gun homicide with respect to

gun prevalence of about 1.2: a doubling (100 percent) increase in gun prevalence in a county would reduce

gun homicide rates by about 120 percent; that is, gun homicide rates would be halved. But before we proceed

any further, we have to make sure that we have completed the specification testing of the estimations in

columns 3 and 4, namely we have to inspect the instruments log OMAG and log PCTREP88 and confirm that

they are ‘relevant’. This is Step 3 in our checklist.

The key ‘first-stage’ regression results are reported at the bottom of Table 6.2 for the IV/GMM estimations in

columns 3 and 4. The ‘first-stage’ regression is a regression of the endogenous variable–here, log PSG–on

the two instruments log OMAG and log PCTREP88, plus all the exogenous regressors used in the main

equation (including the state fixed effects). The first-stage coefficients on log OMAG and log PCTREP88 are

positive and significant at the 1 percent level, as expected: greater local interest in outdoor sports and a

politically conservative orientation are both associated with higher levels of gun prevalence as proxied by log

PSG. However, the heteroskedastic and cluster-robust first-stage F statistic is 12.9, only slightly higher than

the Staiger–Stock rule of thumb that the first-stage F should be >10 if we are to conclude that the instruments

are not ‘weak’. The Stock–Yogo test critical values (not reported) are similarly marginal. We conclude that

our instruments are not so weak that our results likely to be are invalid and that we can proceed to Step 4

in our checklist, but some caution in our conclusions will be warranted on this account. We note here that

when we use an alternative (non-GMM-based) estimation method that is robust to weak instruments, our

main conclusions continue to be supported. See the ‘Recommended Reading’ section for details.

Step 4 is to formally test what is already informally apparent, namely that log PSG is endogenous. We do this

using a GMM distance test, defined as C = J from Step 1 (gun levels are exogenous, columns 1–2) minus

J from Step 2 (gun levels are endogenous, columns 3–4). In fact, the C statistic in Table 6.2, columns 1–2,

is very slightly different from this because we report the exogeneity test statistic provided automatically by

the Stata estimation command xtivreg2. This GMM test statistic uses a minor adjustment that guarantees it is

non-negative, and hence differs slightly from what we would obtain if we calculated C ‘by hand’ using the two

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J statistics in columns 1–2 and 3–4. Since we are testing the exogeneity of one variable, the C test is χ2 with

one degree of freedom. The C test statistic is 10.96, which is very large (p = 0.001). We therefore reject the

null hypothesis that log PSG is exogenous, conclude that gun levels are endogenous, and proceed to Step 6.

Step 6 is to take our Step 2 estimates–using efficient GMM and IV, and treating log PSG as endogenous–our

final estimates. We already noted that the estimated coefficient on log PSG, after using our proxy calibration,

implies a roughly unitary gun-levels/gun-homicide elasticity; for example, a ceteris paribus 10 percent

increase in gun prevalence in a locality would lead to a 10 percent decrease in gun homicide. The size of

the endogeneity bias if log PSG is treated as exogenous is substantial: the biased estimates from columns

1–2 imply that a 10 percent increase in gun prevalence would be associated with a 1.0–1.5 percent increase

in gun homicide, rather than the 10 percent decrease from the IV/GMM estimations in which log PSG is

treated as endogenous. The rest of the estimated coefficients appear reasonable. About half of the 18 control

variables are significant, and the significant coefficients have the expected sign. High gun murder rates

are associated poverty measures, the percentage of the population that is black and the percentage that

is Hispanic. Suburban areas have lower levels of gun homicide compared to urban or rural (the omitted

category) areas.

To summarize: our main conclusion is the positive correlation between gun levels and gun homicide rates is

driven by endogeneity bias, and when the endogeneity of gun levels is properly addressed in the estimation,

any positive correlation vanishes and indeed reverses; our results suggest that greater gun prevalence is,

if anything, associated with lower rates of gun homicide. However, because our instruments are marginally

weak, and because the calibration exercise is only approximate, our results, and in particular the size of this

estimated negative impact, should be treated with some caution.

We offer an additional caveat regarding the interpretation of our results and what our simple model does

not do. We have argued in other work (Kovandzic et al., 2008) that heterogeneity in criminality is a hitherto

largely ignored problem in empirical work using ecological data. Heterogeneity in criminality means, in our

guns–homicide application, that the impact of an increase in gun prevalence will depend on the composition

of the county population: an increase in gun prevalence in a county with a large criminal population should

have an impact that is different from an increase in gun prevalence that is the same size but takes place in a

county with a small criminal population. More formally, equation (6.1) is replaced with

and now the impact βi of gun prevalence on homicide is allowed to vary from county to county, rather than

assuming, as we did in this chapter, that β is the same for all counties. It turns out (see Kovandzic et al.,

2008, for details) that if equation (6.1e) is an accurate description of ‘reality’, the estimated obtained using

IV/GMM methods cannot be interpreted as an estimate of the average impact of gun prevalence; that is, as

the impact of gun prevalence in the ‘average’ county. The correct interpretation depends on the correlations

between the instruments used and gun prevalence amongst criminals and non-criminals. This is discussion

further in the ‘Recommended Reading’ section.

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Conclusion

We have shown in this chapter how an empirical researcher can use the modern framework of GMM to

address the problem of endogeneity bias resulting from reverse causality, omitted variable bias, and/or

measurement error, and how the GMM framework relates to another modern empirical technique, namely the

calculation of ‘robust’ standard errors. We applied these procedures to US county level data, and found strong

evidence of the existence of endogeneity problems. When the problem is ignored, gun levels are associated

with higher rates of gun homicide; when the problem is addressed, this association disappears or reverses.

Our presentation has been structured in a way that, by following our examples, empirical researchers should

be able to employ the same techniques to other problems.

As noted earlier, endogeneity problems stemming from simultaneity bias, omitted variable bias, and

measurement error are commonplace in criminology. Take, for example, one of the most often studied

topics in criminological research–the link between increases in police levels and crime rates. Simultaneity

bias is likely to be present in OLS estimates of the police–crime relationship as increases in crime could

encourage policy makers into hiring more police (Levitt, 1997, 2002; Marvell and Moody, 1996; McCrary,

2002). Similarly, studies examining the link between arrest rates and crime rates may also suffer from

endogeneity bias if short-term increases in crime reduce the ability of police to affect arrests (Glaser

and Sacerdote, 1999; Levitt, 1998). Of course, studies examining individual-level correlates of criminal

behaviour are also susceptible to endogeneity bias. A classic example is the plausible two-way relationship

between unemployment and criminal behaviour. While the effects of unemployment on crime are obvious,

unemployment may be an endogenous regressor if income generated as a result of criminal activities leads

to a greater reluctance among criminals to seek out lawful employment (Raphael and Winter-Ebmer, 2001).

In all, there are numerous and varied criminological topics where endogeneity bias is a concern. We hope

criminologists can use the tools and framework provided in this chapter to overcome endogeneity bias

problems.