Lott’s Results

К оглавлению
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 
68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 
102 103 104 105 106 107 108 109 110 111 112 113 114 115 

Table 6-1 (first row) displays Lott’s estimates from Model 6.1. Lott

finds that where they have been adopted, right-to-carry laws have reduced

homicide by about 8 percent, rapes by about 5 percent, and aggravated

assaults by about 7 percent (Lott, 2000:51). Lott also finds that adoption of

right-to-carry laws may increase the rates of nonviolent property crimes

(burglary, larceny, auto theft). In theory, this is possible, as criminals substitute

away from crimes that involve contact with victims toward crimes

that do not involve encounters with victims.

Rows 2 and 3 of Table 6-1 report the results of the committee’s replication

of these estimates. In row 2, we use the revised original data set and

Lott’s computer programs. The committee was unable to replicate Lott’s

estimate of the reduction in the murder rate, although the estimates are

close and consistent with the conclusion that right-to-carry laws reduce the

incidence of murder. Through communication with Lott, the committee

learned that the data used to construct Table 4.1 of Lott (2000) were lost

and that the data supplied to the committee are a reconstruction and not

necessarily identical to the original data.

Row 3 displays estimates using the revised new data set restricted to

period 1977-1992. The estimates from these revised data are substantially

different from those originally reported by Lott (2000). In the dummy

variable model, the magnitude of the estimated reduction in the rates of

violent crime and aggravated assault was reduced, the estimated reduction

in the murder rate increased, and the sign of the estimated effects of rightto-

carry laws on robbery reversed. Moreover, the effects of right-to-carry

laws on violent crime are no longer statistically significantly different from

zero at the 5 percent significance level. Finally, the estimated increase in the

rates of all property crimes increased substantially.

Table 6-2 presents estimates of the trend model. The first row displays

Lott’s estimates. Lott finds the passage of right-to-carry laws to be associated

with changes in the crime trend. He finds a 0.9 percent reduction in the

annual rate of growth of violent crime overall, and a 0.6 percent reduction

in the rate of growth of property crimes. Row 2 of Table 6-2 shows the

committee’s attempt to replicate Lott’s results using the revised original

data set. The committee was unable to replicate most of the results in Lott’s

Table 4.8. Through communication with Lott, the committee learned that

TABLE 6-1 Dummy Variable Model with Common Time Pattern,

Original and Revised Dataa

Violent

Sample Years Crime Murder Rape

1. Lott (2000) Original 1992 1992 –4.9% –7.7% –5.3%

2. Committee

replication Revised 1992b 1992 –4.91 –7.30 –5.27

SE (0.98)** (1.57)** (1.22)**

3. Committee

replication Revised 2000c 1992 –1.76 –9.01 –5.38

SE (1.07) (1.70)** (1.33)**

aThe regressions use the covariates and specification from the original Lott and Mustard

(1997) models that do not control for state poverty, unemployment, death penalty execution

rates, or regional time trends. The controls include the arrest rate for the crime category in

question (AOVIOICP), population density in the county, real per capita income variables

(RPCPI RPCUI RPCIM RPCRPO), county population (POPC), and variables for the percentage

of the population that is in each of many race x age x gender categories (e.g., PBM1019 is

the percentage of the population that is black, male, and between ages 10 and 19). The “no

this is because there are many misprints in Table 4.8. Nonetheless, Lott’s

and the committee’s results have the same signs for all crimes except aggravated

assault. Row 3 displays estimates using the revised new data set

restricted to the period 1977-1992. These new results tend to show larger

reductions in the violent crime trends than those found using the revised

original data.

Other Statistical Evaluations of Right-to-Carry Laws

Researchers have estimated the effects of right-to-carry laws using Lott’s

or related data and models. Many of these studies have found that the use

of plausible alternative data, control variables, specifications, or methods

of computing standard errors, weakens or reverses the results. Tables 6-3

and 6-4 display estimates from selected studies that illustrate variability in

the findings about the effects of right-to-carry laws. The committee does

not endorse particular findings or consider them to provide better estimates

of the effects of right-to-carry laws than do Lott’s results. Moreover, the

committee recognizes that several independent investigators have used alternative

models or data to obtain results that are consistent with Lott’s.

These investigators include Bartley and Cohen (1998) and Moody (2001).

We focus on the conflicting results in this section because they illustrate a

variability of the findings that is central to the committee’s evaluation of

their credibility.

Aggravated Property

Assault Robbery Crimes Auto Theft Burglary Larceny

–7.0% –2.2% 2.7% 7.1% 0.05% 3.3%

–7.01 –2.21 2.69 7.14 0.05 3.34

(1.14)** (1.33) (0.72)** (1.14)** (0.76) (0.89)**

–5.60 1.17 5.84 10.28 4.12 6.82

(1.25)** (1.45) (0.76)** (1.24)** (0.83)** (0.82)**

controls” specification” includes county fixed effects, year dummies, and the dummy for

whether the state has a right-to-carry law.

bUsing Lott’s reconstruction of his original 1977-1992 data.

cUsing the revised new data set, which contains observations, 1977-2000, even though the

estimates in this row use data only through 1992.

NOTE: All samples start in 1977. SE = standard error. Standard errors are in parentheses,

where * = significant at 5% and ** = significant at 1%.

TABLE 6-2 Trend Model with Common Time Pattern, 1977-1992a

Violent

Sample Years Crime Murder Rape

1. Lott (2000) Original 1992 1992 –0.9% –3.0% –1.4%

2. Committee

replication Revised 1992b 1992 –0.50 –4.25 –1.37

SE (0.41) (0.65)** (0.51)**

3. Committee

replication Revised 2000c 1992 –2.15 –3.41 –3.37

SE (0.39)** (0.62)** (0.48)**

aThe regressions use the covariates and specification from the original Lott and Mustard

(1997) models that do not control for state poverty, unemployment, death penalty execution

rates, or regional time trends. The controls include the arrest rate for the crime category in

question (AOVIOICP), population density in the county, real per capita income variables

(RPCPI RPCUI RPCIM RPCRPO), county population (POPC), and variables for the percentage

of the population that is in each of many race age gender categories (e.g., PBM1019 is

the percentage of the population that is black, male, and between ages 10 and 19).

Control Variables and Specification

The most common modifications to Lott’s original analyses of right-tocarry

laws has been to assess the sensitivity of the findings to variation in the

control variables or the specification of the model. Lott’s basic model relies

on dozens of controls, but concerns have been raised that some controls may

be missing, others may be unnecessary, and still others may be endogenous

(that is, related to the unobserved determinates of county crime rates).

Duggan (2001), for example, raises concerns that county-level control

variables may not be precisely measured on an annual basis and that the

arrest rate control variable, which includes the crime rate in the denominator,

may bias the estimates. In response to these concerns, Duggan estimated

a simple dummy variable model that controls only for year and

county fixed effects.7 Duggan drops all other covariates from the model.

When estimated on all county-year observations with nonmissing crime

7Duggan also changed the coding of the dates of adoption of right-to-carry laws, although

this had only a minimal effect on the estimates. According to Duggan (2001) and others (see, for

example, Ayres and Donohue, 2003a), there is an inconsistency in the coding used by Lott and

Mustard. Duggan finds that in 8 of the 10 right-to-carry states, the adoption date is defined as

the year the law was passed, but in 2 states, Florida and Georgia, the adoption date is set to the

calendar year after the law was passed. Lott, in personal communications, maintains that the

dates are coded correctly. The committee does not take a stand on which coding is correct.

Aggravated Property

Assault Robbery Crimes Auto Theft Burglary Larceny

–0.5% –2.7% –0.6% –0.1% –0.3% –1.5%

0.46 –2.72 –0.69 –0.31 –1.58 –0.11

(0.48) (0.56)** (0.30)* (0.48) (0.32)** (0.37)

–2.63 –3.02 –1.13 0.25 –1.80 –0.84

(0.45)** (0.53)** (0.27)** (0.45) (0.30)** (0.30)**

bUsing Lott’s reconstruction of his original 1977-1992 data.

cUsing the revised new data set, which contains observations, 1977-2000, even though the

estimates in this row use data only through 1992.

NOTE: All samples start in 1977. SE = standard error. Standard errors are in parentheses,

where * = significant at 5% and ** = significant at 1%.

data, this reduced the magnitude of the estimated reduction in the rates of

murder and aggravated assault, and it reversed the signs of the estimated

effects of right-to-carry laws on rape, robbery, and all violent crime. That

is, according to Duggan’s estimates, adoption of right-to-carry laws increases

the frequencies of rape, robbery, and violent crime as a whole.

Moreover, Duggan found there is no statistically significant effect of rightto-

carry laws on violent crimes (at the 5 percent significance level).

Other researchers have varied the specification of the model, allowing

for the effects of right-to-carry laws to be more heterogeneous. Black and

Nagin (1998), for example, estimated a dummy variable model in which

the effects of right-to-carry laws are allowed to vary among states (that is,

the coefficient d is allowed to take different values for different states).

Plassmann and Tideman (2001) estimate a nonlinear Poisson regression

model with a restricted set of covariates, but otherwise similar to Model

6.1. Ayres and Donohue (2003a) combined Models 6.1 and 6.2, thereby

obtaining a hybrid model in which adoption of right-to-carry laws can

affect both the level and the trend of crime. The results from these analyses,

which vary the way in which right-to-carry laws can effect crime, are highly

variable, with some suggesting that the laws increase crime, others suggesting

that they decrease crime, and many being statistically insignificant.

In Black and Nagin (1998), for example, only Florida has a statistically

significant decrease in the murder rate following adoption of a right-tocarry

law, and only West Virginia has a statistically significant increase in

TABLE 6-3 Summary of Selected Studies: Dummy Variable Model

(percentage) (shaded cells indicate a positive coefficient)

Violent

Source Modification Crime Murder Rape

Lott (2000) Original specification and data –5* –8* –5*

Moody Unweighted –6* –4* –5*

State-level analysis –11 15 –22*

Duggana County and time effects only –1 –6 3

All counties 0 –1 6

Black and Nagin Large counties –9* –4

Exclude Florida –1 1

Florida –27.7* –17*

Georgia –5.2 –5

Idaho –21 –10

Maine 7.2 4

Mississippi 5.4 32*

Montana –36.7 –97*

Oregon –5.9 4

Pennsylvania –8.9 4

Virginia 3.9 –8

West Virginia 72* –29*

Plassmann and No control for arrest rate –7* –6*

Tideman All counties –2 –5

Count model (Poisson) –11* –4*

Florida –24* –16*

Georgia –8* –16*

Idaho –6 10*

Maine 1 –2

Mississippi 5 11*

Montana –7 –4

Oregon –10* –2

Pennsylvania –5 14*

Virginia 8* –3

West Virginia 5 –1

Ayres and State trends 0 –9* –2

Donohue (2003a) 1977-1997 data 2 0 3

State level analysis

State and time effects only –3 –8 –1

1977-1999 data 9* –2 6*

Plassmann and Regional trend + others

Whitleya,b 1977-2000 data –3 –6* –7*

Ayres and Regional trends + other controls

Donohue (2003b)a,b 1977-2000 corrected data 0 –4 –5

continued

Aggravated Property Auto

Assault Robbery Crimes Theft Burglary Larceny

–7* –2 3* 7* 0 3*

–9* –1 3* 3 1 4*

–18* –10 1 –9 4 3

–6 4 6* 9* 8* 5

–5 10 7* 11* 10* 5

–7* –3

–6* –5

–7 7

–4 8

–31* –64*

–52* –33*

–45* 10

–71* –14

–17* –4

7* –5

–16* –12

–3 9

–1

2

6*

–3*

1

–41*

–22*

25*

–27*

–48*

–14*

–5*

–9*

3 –8 –1* –1* –4* 1

7* 0 –1 4 1 4

–10 –5 7* 9* 9* 7*

4* 16* 16* 23* 14* 16*

–2 –5 4 9* 0 6

1 –3 6* 11* 2 8*

aUses clustered sampling standard errors.

bAdded covariates for state poverty, unemployment, death penalty execution rates, and

regional time trends.

TABLE 6-3 Continued

Violent

Source Modification Crime Murder Rape

Standard errors

Lott (2000) Unadjusted standard errors 0.98 1.57 1.22

Duggan State clustered standard errors 2.31 2.95 2.32

Helland and Placebo standard errors 4.9 6.4 5.6

Tabarrok

its murder rate. The estimated changes in the murder rates of other states

that adopted right-to-carry laws are sometimes positive (three cases) and

sometimes negative (five cases) and are not statistically significantly different

from zero. Black and Nagin also report variations in the directions and

statistical significance of changes in the rates of rape and aggravated assault.

They report no statistically significant increases in robberies, but only

2 of the 10 states that adopted right-to-carry laws had statistically signifi-

TABLE 6-4 Summary of Selected Studies: Trend and Hybrid Variable

Model (shaded cells indicate a positive coefficient)

Violent

Source Modification Crime Murder Rape

Lott (2000) Original specification and data 2* –3* –1*

Lott (2000)a 1977-1996 –2* –2* –3*

Ayres and Hybrid model: Level 7* 3 7*

Donohue (2003a) Trend –2* –5* –3*

1977-1997 data: Level 0 7* 6*

Trend –2* –4* –3*

Plassmann and Regional trend + others

Whitleya,b 1977-2000 data –1 –2 –3*

Ayres and Regional trends + other controls

Donohue (2003b)a,b 1977-2000 corrected data 0 –2 –2

aAdded covariates for state poverty, unemployment, death penalty execution rates, and

regional time trends.

bStandard errors adjusted for state clustering.

NOTES: Shaded cells indicate a positive coefficient estimate and * indicates the estimate is

statistically significant at the 5% significance level. Unless otherwise noted, the standard

errors are not adjusted for state-level clustering. Exceptions: Duggan, Plassmann and Tideman,

Ayres and Donohue.

Aggravated Property Auto

Assault Robbery Crimes Theft Burglary Larceny

1.14 1.33 0.72 1.14 0.76 0.89

2.77 3.34 1.89 2.59 2.29 2.27

6.6 7.5 5.1 6.5 5.7 5.7

Aggravated Property Auto

Assault Robbery Crimes Theft Burglary Larceny

–1* –3* –1* 0* –2* 0

–3* –3* –2* –3* –1* –2*

10* –3 0 0 –3 0

–2 –1 0 0 0 1

6* 4 –1 9* 4* 5*

–3* –4* 0 –2* –3* –2*

–2 –3* 0 0 –2 –1

–1 –2 0 0 –1 0

NOTES: Shaded cells indicate a positive coefficient estimate and * indicates the estimate is

statistically significant at the 5% significance level. Unless otherwise noted, the standard

errors are not adjusted for state-level clustering. Exceptions: Duggan, Plassmann and Tideman,

Ayres and Donohue.

cant decreases. In summary, according to Black and Nagin, adoption of a

right-to-carry law may increase, decrease, or have no discernible effect on

the crime rate depending on the crime and the state that are involved.8

8To avoid selection problems associated with using counties with positive crime rates,

Black and Nagin also restricted their analysis to counties with populations of 100,000 or

more. This was done to mitigate a possible bias arising from Lott’s use of the arrest rate as an

explanatory variable. The arrest rate is the number of arrests divided by the number of crimes

Plassmann and Tideman (2001) document similar variability in the

estimates. To account for the fact that county-level crime data include a

large number of observations for which the outcome variable equals zero,

Plassmann and Tideman estimate a nonlinear count data model. Using data

from all counties with reported crime figures, the resulting estimates on

murder and rape are consistent with Lott’s findings, but the sign of the

estimated effect of right-to carry laws on robbery is reversed. Furthermore,

when the effects of right-to-carry laws are allowed to vary among states,

Plassmann and Tideman found that adoption of a right-to-carry law may

increase, decrease, or have no effect on the crime rate depending on the

crime and state that are involved. Consider, for example, murder. Right-tocarry

laws are estimated to have a statistically significant decrease in the

murder rate in Florida, Georgia, and Oregon following adoption of a rightto-

carry law. Virginia has a statistically significant increase in its murder

rate. The changes in the murder rates of other states that adopted right-tocarry

laws are not statistically significantly different from zero. Plassmann

and Tideman conclude by noting the fragility in the estimated effects of

right-to-carry laws: “While this ambiguous result is somewhat discouraging,

it is not very surprising. Whenever the theoretically possible and in

practice plausible effects of public policy are ambiguous, it can be expected

that the effects of such a policy will differ across localities that are clearly

different from each other” (p. 797).

Finally, the added flexibility of the hybrid model estimated by Ayres

and Donohue (2003a) produces estimation results that are different from

Lott’s.9 The results found when using the revised original data (1977-

and is undefined in counties that report no crimes of the types analyzed. Therefore, these

counties are not included in Lott’s analysis. Because the denominator of the arrest rate variable

contains the dependent variable in Lott’s models, it is possible that dropping no-crime

counties biases the results of his analysis. Nearly all of the low-crime counties have populations

below 100,000. Therefore, use of only counties with larger populations largely overcomes

the problem of missing arrest rate data without creating a bias.

Lott (1999:8-9; 2000:142-143), however, has argued that Black’s and Nagin’s results are

unreliable because they eliminated 85 percent of the counties in the nation (all the counties with

populations of less than 100,000). In particular, they used only one county in West Virginia.

Lott (2000: Table 4.9) presents his own estimation results according to which his findings are

largely unaffected by disaggregating the right-to-carry effect by state. However, Lott does not

report the details of his analysis or the statistical significance levels of his estimates. Moreover,

his response does not explain why Black and Nagin found statistically significant increases in

some crime rates for some states following passage of right-to-carry laws.

9The committee takes no position on whether the hybrid model provides a correct description

of crime levels or the effects of right-to-carry laws. The important feature of the hybrid

model is that it nests Models 6.1 and 6.2.

1992) are illustrated in Figure 6-1, which shows the “relative trend” in

the logarithm of the violent crime rate obtained from the Ayres and

Donohue model for a hypothetical county in which a right-to-carry law is

adopted in year 8. The relative trend is the difference between the crime

trend in the adopting county and the trend in a nonadopting county with

the same values of the explanatory variables X. According to the figure,

adoption of the law increased the level of violent crime but accelerated a

decreasing (relative) trend. Ayres and Donohue obtained similar results

for rape and aggravated assault. For murder, the shift in the level is not

statistically significant, but there is a statistically significant downward

shift in the trend. There is no statistically significant effect on either the

level or the trend for robbery and property crimes. Ayres and Donohue

also report estimates from an expanded data set that includes the years

1977-1999. The results found using these data, which are reported in

Table 6-4, are similar.

Updated Sample Endpoint

Several researchers, including Lott, have assessed whether the basic

findings from Models 6.1 and 6.2 continue to hold when using more recent

data. In the epilogue to the second edition of his book, Lott (2000: Table

9.1) analyzes data covering the period 1993-1996. Plassmann and Whitley

(2003) use data through 2000. In addition to updating the data, these Trend of Log(Crime Rate)

Year

0 5 10 15

-0.2

-0.15

-0.1

-0.05

0

FIGURE 6-1 Trend in the logarithm of the violent crime rate.

researchers also change the model specification. In particular, these analyses

include additional covariates (i.e., state poverty, unemployment and

death penalty execution rates) and allow for region-interacted time patterns,

as opposed to a common time trend used in the original Lott models

(Lott 2000:170).

With these new models and the updated sample endpoints, Lott found

that the basic conclusions from the trend model are robust to the additional

years of data covering the periods 1977-1996. Likewise, Plassmann and

Whitley (2003) found that when the data are updated to cover the period

1977-2000, the trend model estimates of the effects of right-to-carry laws

on crime continue to be negative, but only the estimates for rape and

robbery are statistically significant. In the dummy variable model, Plassmann

and Whitley found negative coefficient estimates for the right-tocarry

coefficient for each violent crime category and positive coefficients for

each of the property categories.

Ayres and Donohue (2003b), however, document a number of errors in

the data used by Plassmann and Whitley, and Lott’s revised new data

correct these errors. Plassmann, in communications with the committee,

has agreed that the changes to these data are appropriate. Using the revised

new data, the committee exactly replicated the results reported by Ayres

and Donohue (2003b).

In particular, Ayres and Donohue (2003b) found that rerunning the

dummy variable model regressions using the corrected data reduced the

magnitude of the estimated reduction in the rates of violent crime, murder,

rape, and robbery, and it reversed the sign of the estimated effects of rightto-

carry laws on aggravated assault. Moreover, none of the negative estimates

is statistically significant, while effects for larceny, auto theft, and

property crime overall are positive and significant. Likewise, the changes in

the crime trends are generally small in absolute value, and none of the

changes is significantly different from zero (see Table 6-4).10

Maltz and Targonski (2002) do not update the data but instead assess

the quality of the county crime data used in the empirical research on rightto-

carry laws. In particular, they note that not all police jurisdictions report

their crime levels to the FBI and argue that there is systematic underreporting

in the UCR. Maltz and Targonski (2002:298) conclude that “county-level

crime data, as they are currently constituted, should not be used, especially

in policy studies.” However, Maltz and Targonski do not estimate the

magnitude of the effects of underreporting on the results obtained by Lott

and others. Thus, it is not known whether correcting for underreporting, if

it were possible, would change any of the results.

10Both Ayres and Donohue (2003b) and Plassmann and Whitley (2003) use standard errors

that account for state clustering.

Lott and Whitley (2002: Figure 5) report estimates of the effects of

right-to-carry laws that are obtained by dropping from the data counties

with large fractions of missing UCR reports. Lott’s and Whitley’s figure

shows estimated trends in crime levels before and after adoption of right-tocarry

laws, and they claim that these trends support the conclusion that

adoption of right-to-carry laws reduces crime. The committee disagrees.

According to Figure 5b of Lott and Whitley (2002), the murder rate peaks

and begins to decrease at an accelerating rate approximately 5 years before

the adoption of right-to-carry laws. Aggravated assault decreases prior to

adoption and then increases for approximately 3 years following adoption

before starting to decrease again (Figure 5e). Adoption has no effect on rape

(Figure 5c). The rate of violent crimes as a whole decreases up to the time of

adoption and then remains unchanged until approximately 3 years after

adoption before beginning a steeper decline (Figure 5a). Among violent

crimes, only robbery displays a decrease immediately following adoption

(Figure 5d). However, this followed a period during which the robbery rate

first increased and then remained constant for approximately 5 years. In

summary, the committee concludes that it is at least possible that errors in

the UCR data may account for some of Lott’s results.

Standard Errors

A final point that has been argued in the literature is that conventional

standard errors reported by Lott and others are not appropriate. The statistical

analyses of dummy variable and trend models are conducted using a

county-year pair as the unit of analysis. Right-to-carry laws, however, almost

always vary only at the state level. Consequently, some investigators

believe that treating the county-level observations as if they are statistically

independent may lead to estimates of the standard errors that underestimate

their true magnitude. These investigators make adjustments for statelevel

clustering that inflate their standard errors. For example, the standard

error for the dummy variable model estimate of the effect of right-to-carry

laws on violent crime increases from 0.98 when reporting the unadjusted

standard error, to 2.31 when estimating clustered sampling standard errors

(Duggan, 2001), to 4.9 when using the methods advocated by Helland and

Tabarrok (2004) (see Table 6-3). The fact that the adjustments in most

cases greatly increase the standard errors is a reason for concern. Once the

standard errors have been adjusted for clustering, very few of the point

estimates, in any of the models, using any of the data sets, are statistically

different from zero.

However, investigators reporting cluster-adjusted standard errors do

not formally explain the need for these adjustments. These adjustments, in

fact, are not supported in the basic models specified in Equations 6.1 and

6.2. Instead, those who argue for presenting clustered standard errors often

cite Moulton (1990) as the source of their belief that adjustments are needed.

Moulton considered a model in which there is an additive source of variation

(or additive effect) that is the same for all observations in the same

cluster. He showed that ignoring this source of variation leads to standard

errors that are too low. Investigators who make clustering corrections usually

consider the counties in a state to constitute one of Moulton’s clusters

and appear to believe that the absence of state-level additive effects in their

models causes standard errors to be too low. The models estimated in this

literature, including those of Lott and his critics, typically contain countylevel

fixed effects (the constants gi in equations 6.1 and 6.2). Every county

is always in the same state, so, any state-level additive effect simply adds a

constant to the gi’s of the counties in that state. The constant may vary

among states but is the same for all counties in the same state. The combined

county- and state-level effects are indistinguishable from what would

happen if there were no state-level effects but each gi for the counties in the

same state were shifted by the same amount. Therefore, state-level effects

are indistinguishable from county-level effects. Any state-level effects are

automatically included in the gi’s. There is no need for adjustments for

state-level clustering.

Other observationally equivalent but different models can support the

use of adjusted standard errors. If, for example, the effects of right-to-carry

laws (or other explanatory variables) vary across states, then the assumption

of independence across counties would be incorrect. Adjustments to

the standard errors can allow for uncertainty arising from the possibility

that the coefficients of variables in the model that are not allowed to vary

across states, in fact, vary randomly across states. The adjustments made by

Duggan and Plassmann and Whitley, for example, can be used to correct

estimated standard errors for this possibility (see Wooldridge, 2003).

These alternative models have not been discussed in the literature or by

the committee. Thus, it is not clear whether the models that would support

using clustered-sampling-adjusted standard errors are appropriate to evaluate

the effects of right-to-carry laws. At the most basic level, researchers

need to assess whether models that support clustering are of interest.11 If,

for example, coefficients can vary randomly among states, Models 6.1 and

6.2 reveal the mean coefficients. In other words, if different states have

different coefficients, then researchers estimate an average over states. It is

11There are also important technical issues to consider. For example, a commonly used

method for making these corrections is reliable only when the number of “clusters” (here

states) is large, and there is reason to think that the 50 states do not constitute a large enough

set of clusters to make these methods reliable.

not clear why anyone should care about this average, which is not related in

any obvious way to (for example) nationwide benefits of right-to-carry

laws. If coefficients vary among states, then it may be much more useful to

estimate the coefficients for each state. It is entirely possible that the effects

of right-to-carry laws vary among states, even after controlling everything

else that is in the model. If they do, it may be much more useful to know

which states have which coefficients, to see the magnitude of the variation,

and to have a chance of finding out whether it is related to anything else

that is observable. Of course, a number of the studies summarized above

have varied Lott’s model by allowing the effect of right-to-carry laws to

differ by states (see, for example, Black and Nagin, 1998, and Plassmann

and Tideman, 2001). A model in which coefficients are estimated separately

for each state does not require adjustment of standard errors.

In summary, whether adjustment of standard errors is needed depends

on the details of the effects that are being estimated and the model that is

used to estimate them. These issues have not been investigated in studies of

right-to-carry laws to date. Adjusted standard errors are not needed for

Models 6.1 and 6.2. The precision of estimates from these models should be

evaluated using unadjusted standard errors.