Estimating the Causal Effect of Gun Prevalence on Homicide Rates: A Local Average Treatment Effect Approach

Abstract

Objective

This paper uses a “local average treatment effect” (LATE) framework in an attempt to disentangle the separate effects of criminal and noncriminal gun prevalence on violence rates. We first show that a number of previous studies have failed to properly address the problems of endogeneity, proxy validity, and heterogeneity in criminality. We demonstrate that the time series proxy problem is severe; previous panel data studies have used proxies that are essentially uncorrelated in time series with direct measures of gun relevance.

Methods

We adopt instead a cross-section approach: we use US county-level data for 1990, and we proxy gun prevalence levels by the percent of suicides committed with guns, which recent research indicates is the best measure of gun levels for crosssectional research. We instrument gun levels with three plausibly exogenous instruments: subscriptions to outdoor sports magazines, voting preferences in the 1988 Presidential election, and numbers of military veterans. In our LATE framework, the estimated impact of gun prevalence is a weighted average of a possibly negative impact of noncriminal gun prevalence on homicide and a presumed positive impact of criminal gun prevalence.

Results

We find evidence of a significant negative impact, and interpret it as primarily “local to noncriminals”, i.e., primarily determined by a negative deterrent effect of noncriminal gun prevalence. We also demonstrate that an ATE for gun prevalence that is positive, negative, or approximately zero are all entirely plausible and consistent with our estimates of a significant negative impact of noncriminal gun prevalence.

Conclusions

The policy implications of our findings are perhaps best understood in the context of two hypothetical gun ban scenarios, the first more optimistic, the second more pessimistic and realistic. First, gun prohibition might reduce gun ownership equiproportionately among criminals and noncriminals, and the traditional ATE interpretation therefore applies. Our results above suggest that plausible estimates of the causal impact of an average reduction in gun prevalence include positive, nil, and negative effects on gun homicide rates, and hence no strong evidence in favor of or against such a measure. But it is highly unlikely that criminals would comply with gun prohibition to the same extent as noncriminals; indeed, it is virtually a tautology that criminals would violate a gun ban at a higher rate than noncriminals. Thus, under the more likely scenario that gun bans reduced gun levels more among noncriminals than criminals, the LATE interpretation of our results moves the range of possible impacts towards an increase in gun homicide rates because the decline in gun levels would primarily occur among those whose gun possession has predominantly negative effects on homicide.

Appendices

Appendix 1: Data and Sources

We use cross-sectional data for US counties which had a population of 25,000 or greater in 1990, and for which relevant data were available (N = 1,456). Alaska and Washington, DC were excluded from the analysis: the former, because we did not have compatible data for one of our instruments (voting in 1988); the latter, because it is itself a single county and thus drops out of a fixed-effects specification. Data for most county level variables were obtained from the US Bureau of the Census, County and City Data Book, 1994. Other data sources were as follows:

Homicide rates are averages for the 7 years 1987–1993 (bracketing the decennial census year of 1990). 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 NCHS 1997). The data include all intentional homicides in the county with the exception of those due to legal intervention (e.g., killings by police and executions).

Similar to homicide, data for the percent of suicides committed with guns are also 1987–93 averages and were obtained using special Part III Mortality Detail File computer tapes made available by the NCHS. Unlike widely available public use versions, the tapes permit the aggregation of death counts for even the smallest counties (US NCHS 1997).

Subscriptions per 100,000 county population to three of the most popular outdoor/sport magazines (Field and Stream, Outdoor Life, and Sports Afield) in 1993 were obtained from Audit Bureau of Circulations (1993). In the earlier version of this paper, we used a principal components index based on the three separate subscription rates; the measure we use here is more convenient and generates almost identical results.

The percent of the county population voting for the Republican candidate in the 1988 Presidential election is from ICPSR (1995). Rurality measures are from US Bureau of the Census (1990).

The statistical package Stata was used for all estimations. The main IV/GMM estimation programs, ivreg2 and xtivreg2, were co-authored by one of us (Schaffer), and can be freely downloaded via the software database of RePEc.³¹ For further discussion of how the estimators and tests are implemented, see Baum et al. (2003, 2007, 2008), Schaffer (2007), and the references therein.

Appendix 2: The OLS and IV Estimators with Population Heterogeneity

Model Setup

The “true model” is one with population heterogeneity (Eq. 7c in the main text):

$$ h_{i} = \beta_{0} + {{\upbeta}}_{ 1}^{\text{C}} g_{\text{i}}^{\text{C}} + {{\upbeta}}_{ 1}^{\text{NC}} g_{\text{i}}^{\text{NC}} + u_{i} $$

(19)

Criminal and noncriminal gun prevalence are not separately observable. A proxy for aggregate gun prevalence is available (Eq. 14 in the text):

$$ p_{i} = \delta_{0} + \delta_{ 1}^{\text{C}} g_{\text{i}}^{\text{C}} + \delta_{ 1}^{\text{NC}} g_{\text{i}}^{\text{NC}} + v_{i} $$

(20)

A single instrument Z _i is available that is correlated with both criminal and noncriminal gun prevalence, but the strength of the correlation may differ (Eqs. 11 and 12 in the text):

$$ g_{\text{i}}^{\text{C}} = \pi_{ 0}^{\text{C}} + \pi_{ 1}^{\text{C}} Z_{i} + \eta_{\text{i}}^{\text{C}} $$

(21)

$$ g_{\text{i}}^{\text{NC}} = \pi_{ 0}^{\text{NC}} + \pi_{ 1}^{\text{NC}} Z_{i} + \eta_{\text{i}}^{\text{NC}} $$

(22)

We assume that $\pi_{ 1}^{\text{C}} ,\pi_{ 1}^{\text{NC}} \ge 0$ and at least one is strictly greater than zero, and similarly for $\delta_{ 1}^{\text{C}}$ and $\delta_{ 1}^{\text{NC}}$. If gun prevalence is directly observable, the estimating equation is (Eq. 1 in the text):

$$ h_{i} = \beta_{0} + \beta_{1} g_{i} + u_{i} $$

(23)

If only the proxy for gun prevalence is observable, the estimating equation is (Eq. 3 in the text):

$$ h_{i} = {\text{b}}_{0} + {\text{b}}_{ 1} p_{i} + e_{i} $$

(24)

The derivations below follow the format of those in Stock and Watson (2007), Appendix 13.4.

The Average Treatment Effect (ATE) of Gun Prevalence

Rewrite Eq. (19) as a “random coefficient” model:

$$ h_{i} = \beta_{0} + {{\upbeta}}_{{ 1 {\text{i}}}} g_{\text{i}} + u_{i} $$

(25)

where

$$ {{\upbeta}}_{{ 1 {\text{i}}}} \equiv \frac{{g_{\text{i}}^{\text{C}} }}{{g_{\text{i}}^{\text{C}} + g_{\text{i}}^{\text{NC}} }}{{\upbeta}}_{ 1}^{\text{C}} + \frac{{g_{\text{i}}^{\text{NC}} }}{{g_{\text{i}}^{\text{C}} + g_{\text{i}}^{\text{NC}} }}{{\upbeta}}_{ 1}^{\text{NC}} $$

(26)

and by definition $g_{\text{i}}^{{}} \equiv g_{\text{i}}^{\text{C}} + g_{\text{i}}^{\text{NC}}$, i.e., our measures of gun prevalence are in levels. The average treatment effect of gun prevalence β^ATE is:

$$ \begin{aligned} {\text{E(}}\beta_{{1{\text{i}}}} ) & = {\text{E}}\left( {\frac{{g_{\text{i}}^{\text{C}} }}{{g_{\text{i}}^{\text{C}} + g_{\text{i}}^{\text{NC}} }}} \right)\beta_{ 1}^{\text{C}} + {\text{E}}\left( {\frac{{g_{\text{i}}^{\text{NC}} }}{{g_{\text{i}}^{\text{C}} + g_{\text{i}}^{\text{NC}} }}} \right)\beta_{ 1}^{\text{NC}} \\ & = \left( {\frac{{{{\upmu}}^{\text{C}} - \text{cov} \left( {g_{i} ,\frac{{g_{\text{i}}^{\text{C}} }}{{g_{i} }}} \right)}}{{{{\upmu}}^{\text{C}} + {{\upmu}}^{\text{NC}} }}} \right)\beta_{ 1}^{\text{C}} + \left( {\frac{{{{\upmu}}_{{}}^{\text{NC}} + \text{cov} \left( {g_{i} ,\frac{{g_{\text{i}}^{\text{C}} }}{{g_{i} }}} \right)}}{{{{\upmu}}^{\text{C}} + {{\upmu}}^{\text{NC}} }}} \right)\beta_{ 1}^{\text{NC}} \\ \end{aligned} $$

(27)

where μ^C ≡ E($g_{\text{i}}^{\text{C}}$) and μ^NC ≡ E($g_{\text{i}}^{\text{NC}}$) and where we make use of the result in Frishman (1971) for the expectation of a ratio; the covariance terms account for the fact that the expectation of the ratio of two random variables does not, in general, equal the ratio of the expectations.³² In the special case that total gun prevalence is uncorrelated with the criminal/noncriminal share of gun prevalence, the covariance terms in (27) are zero and the ATE takes the following simple form:

$$ {\text{E(}}\beta_{{ 1 {\text{i}}}} ) = \frac{{\mu^{\text{C}} }}{{\mu^{\text{C}} + \mu^{\text{NC}} }}\beta_{ 1}^{\text{C}} + \frac{{\mu^{\text{NC}} }}{{\mu^{\text{C}} + \mu^{\text{NC}} }}\beta_{ 1}^{\text{NC}} $$

(28)

Equations (27) and (28) are Eqs. (8) and (9) in the main text.

OLS Estimation

We consider first estimation of Eq. (23), when total gun prevalence is directly observable. The OLS estimator is

$$ \hat{\beta }_{1}^{\text{OLS}} = \frac{{{\text{s}}_{\text{gh}} }}{{{\text{s}}_{\text{g}}^{ 2} }}\mathop \to \limits^{\text{P}} \frac{{{\text{cov(}}g_{i} ,h_{i} )}}{{{\text{var(}}g_{i} )}} $$

(29)

where s denotes a sample covariance and $\mathop \to \limits^{\text{P}}$ denotes convergence in probability. The numerator is

$$ \begin{aligned} {\text{cov(}}g_{i} ,h{}_{i} )& = {\text{cov}}\left\{ {\left( {g_{i}^{C} + g_{i}^{NC} } \right),\left( {\beta_{0} + \beta_{1}^{C} g_{i}^{C} + \beta_{1}^{NC} g_{i}^{NC} + u_{i} } \right)} \right\} \\ & = \beta_{1}^{C} {\text{var(}}g_{i}^{C} ) + \beta_{1}^{C} {\text{cov(}}g_{i}^{C} ,g_{i}^{NC} ) + \beta_{1}^{NC} {\text{var(}}g_{i}^{NC} ) + \beta_{1}^{NC} {\text{cov(}}g_{i}^{C} ,g_{i}^{NC} ) \\ & \quad + {\text{cov(}}g_{i}^{C} ,u_{i} ) + {\text{cov(}}g_{i}^{NC} ,u_{i} ) \\ & = \beta_{1}^{C} \left\{ {{\text{var(}}g_{i}^{C} ) + {\text{cov(}}g_{i}^{C} ,g_{i}^{NC} )} \right\} + \beta_{1}^{NC} \left\{ {{\text{var(}}g_{i}^{NC} ) + {\text{cov(}}g_{i}^{C} ,g_{i}^{NC} )} \right\} \\ & \quad + {\text{cov(}}g_{i}^{C} ,u_{i} ) + {\text{cov(}}g_{i}^{NC} ,u_{i} ) \\ \end{aligned} $$

(30)

The denominator is simply

$$ \begin{aligned} {\text{var(}}g{}_{i} )& = {\text{var}}\left( {g_{i}^{C} + g_{i}^{NC} } \right) \\ & = {\text{var(}}g_{i}^{C} ) { } + \, \text{var} (g_{i}^{NC} ) + 2\text{cov} (g_{i}^{C} ,g_{i}^{NC} ) \\ \end{aligned} $$

(31)

and therefore

$$ \begin{aligned} \hat{\beta }_{1}^{\text{OLS}} & = \frac{{{\text{s}}_{\text{gh}} }}{{{\text{s}}_{\text{g}}^{ 2} }}\mathop \to \limits^{\text{P}} \frac{{\left\{ {{\text{var(}}g_{i}^{C} ) + {\text{cov(}}g_{i}^{C} ,g_{i}^{NC} )} \right\}}}{{{\text{var(}}g_{i}^{C} ) { } + \, \text{var} (g_{i}^{NC} ) + 2\text{cov} (g_{i}^{C} ,g_{i}^{NC} )}}\beta_{1}^{C} \\ & \quad + \frac{{\left\{ {{\text{var(}}g_{i}^{NC} ) + {\text{cov(}}g_{i}^{C} ,g_{i}^{NC} )} \right\}}}{{{\text{var(}}g_{i}^{C} ) { } + \, \text{var} (g_{i}^{NC} ) + 2\text{cov} (g_{i}^{C} ,g_{i}^{NC} )}}\beta_{1}^{NC} \\ & \quad + \frac{{{\text{cov(}}g_{i}^{C} ,u_{i} ) + {\text{cov(}}g_{i}^{NC} ,u_{i} )}}{{{\text{var(}}g_{i}^{C} ) { } + \, \text{var} (g_{i}^{NC} ) + 2\text{cov} (g_{i}^{C} ,g_{i}^{NC} )}} \\ \end{aligned} $$

(32)

The OLS estimator differs from the ATE Eq. (27) for two reasons. First, if criminal or noncriminal guns are endogenous, then the third term in Eq. (32) is nonzero. Second, even if gun prevalence is exogenous and the third term in Eq. (32) drops out, the resulting OLS estimator is a weighted average of ${{\upbeta}}_{ 1}^{\text{C}}$ and ${{\upbeta}}_{ 1}^{\text{NC}}$, but the weights differ from those for the ATE in Eq. (27); whereas the ATE weights are relative gun prevalence (plus the Frishman correction for the expectation of a ratio), the OLS weights are driven by the variances and covariances of gun prevalence, i.e., by gun variability. The intuition is that the identifying variation in the estimation of Eq. (23) comes from the variation in criminal and noncriminal gun prevalence, and these may differ. To take an extreme example, if criminal and noncriminal gun prevalence are uncorrelated so that $\text{cov} (g_{i}^{C} ,g_{i}^{NC} )$ = 0, and noncriminal gun prevalence varies little or not at all across localities so that $\text{var} (g_{i}^{NC} )$≈0, then the OLS estimator $\hat{\beta }_{1}^{\text{OLS}}$will be approximately equal to the impact of criminal guns ${{\upbeta}}_{ 1}^{\text{C}}$, because the identifying variation in the data is driven solely by variation in criminal gun prevalence.

Next we consider OLS estimation of Eq. (24), when only a proxy is available. To simplify the algebra, we assume that the homicide error u _i and the proxy error ν _i are uncorrelated with gun prevalence and with each other. The OLS estimator is

$$ \hat{b}_{1}^{\text{OLS}} = \frac{{{\text{s}}_{\text{ph}} }}{{{\text{s}}_{\text{p}}^{ 2} }}\mathop \to \limits^{\text{P}} \frac{{{\text{cov(}}p_{i} ,h_{i} )}}{{{\text{var(}}p_{i} )}} $$

(33)

The numerator is

$$ \begin{aligned} {\text{cov(}}p_{i} ,h{}_{i} )& = {\text{cov}}\left\{ {\left( {\delta_{0} + \delta_{ 1}^{\text{C}} g_{\text{i}}^{\text{C}} + \delta_{ 1}^{\text{NC}} g_{\text{i}}^{\text{NC}} + \nu_{i} } \right),\left( {\beta_{0} + \beta_{1}^{C} g_{i}^{C} + \beta_{1}^{NC} g_{i}^{NC} + u_{i} } \right)} \right\} \\ & = \delta_{ 1}^{\text{C}} \beta_{1}^{C} \text{var} (g_{\text{i}}^{\text{C}} ) + \delta_{ 1}^{\text{NC}} \beta_{1}^{NC} \text{var} (g_{\text{i}}^{\text{NC}} ) + (\delta_{ 1}^{\text{C}} \beta_{1}^{NC} + \delta_{ 1}^{\text{NC}} \beta_{1}^{C} )\text{cov} (g_{\text{i}}^{\text{C}} ,g_{\text{i}}^{\text{NC}} ) \\ \end{aligned} $$

(34)

since we’ve assumed that the error terms are uncorrelated with gun levels. The denominator is

$$ \begin{aligned} {\text{var(}}p{}_{i} )& = {\text{var}}\left( {\delta_{0} + \delta_{ 1}^{\text{C}} g_{\text{i}}^{\text{C}} + \delta_{ 1}^{\text{NC}} g_{\text{i}}^{\text{NC}} + \nu_{i} } \right) \\ & = (\delta_{ 1}^{\text{C}} )^{ 2} {\text{var(}}g_{i}^{C} ) { } + (\delta_{ 1}^{\text{NC}} )^{ 2} \, \text{var} (g_{i}^{NC} ) + 2\delta_{ 1}^{\text{C}} \delta_{ 1}^{\text{NC}} \text{cov} (g_{i}^{C} ,g_{i}^{NC} ) + \text{var} (\nu_{i} ) \\ \end{aligned} $$

(35)

Thus

$$ \begin{aligned} \hat{b}_{1}^{\text{OLS}} & \mathop \to \limits^{\text{P}} \frac{{\delta_{ 1}^{\text{C}} \text{var} (g_{\text{i}}^{\text{C}} ) + \delta_{ 1}^{\text{NC}} \text{cov} (g_{\text{i}}^{\text{C}} ,g_{\text{i}}^{\text{NC}} )}}{{(\delta_{ 1}^{\text{C}} )^{ 2} {\text{var(}}g_{i}^{C} ) { } + (\delta_{ 1}^{\text{NC}} )^{ 2} \, \text{var} (g_{i}^{NC} ) + 2\delta_{ 1}^{\text{C}} \delta_{ 1}^{\text{NC}} \text{cov} (g_{i}^{C} ,g_{i}^{NC} ) + \text{var} (\nu_{i} )}}\beta_{1}^{C} \\ & \quad + \frac{{\delta_{ 1}^{\text{NC}} \text{var} (g_{\text{i}}^{\text{NC}} ) + \delta_{ 1}^{\text{C}} \text{cov} (g_{\text{i}}^{\text{C}} ,g_{\text{i}}^{\text{NC}} )}}{{(\delta_{ 1}^{\text{C}} )^{ 2} {\text{var(}}g_{i}^{C} ) { } + (\delta_{ 1}^{\text{NC}} )^{ 2} \, \text{var} (g_{i}^{NC} ) + 2\delta_{ 1}^{\text{C}} \delta_{ 1}^{\text{NC}} \text{cov} (g_{i}^{C} ,g_{i}^{NC} ) + \text{var} (\nu_{i} )}}\beta_{1}^{NC} \\ \end{aligned} $$

(36)

Equation (36) shows the OLS estimator using a proxy for gun levels is a weighted average of the criminal and noncriminal effects. The weights sum to less than one because of the var(ν _i ) term; this is the attenuation bias attributable to the measurement error in the proxy. The weights on ${{\upbeta}}_{ 1}^{\text{C}}$ and ${{\upbeta}}_{ 1}^{\text{NC}}$ now depend not only on gun variability, but also on the relative strength of the correlations between the proxy and criminal/noncriminal gun levels: if $\delta_{ 1}^{\text{NC}}$ ≫ $\delta_{ 1}^{\text{C}}$, then the OLS estimator will put a high weight on the noncriminal impact gun prevalence, and vice versa if $\delta_{ 1}^{\text{NC}}$ ≪ $\delta_{ 1}^{\text{C}}$. Note that even if $\delta_{ 1}^{\text{C}}$ = 0, the weight on ${{\upbeta}}_{ 1}^{\text{C}}$ may be positive if criminal and noncriminal gun prevalence are correlated. Note also that sign{$\hat{b}_{1}^{\text{OLS}}$} is not in general a consistent estimator of sign{$\hat{\beta }_{1}^{\text{OLS}}$}.

IV Estimation

Again we start with the case where gun levels are observable. The IV estimator can be written

$$ \hat{\beta }_{1}^{\text{IV}} = \frac{{{\text{s}}_{\text{Zh}} }}{{{\text{s}}_{\text{Zg}} }}\mathop \to \limits^{\text{P}} \frac{{{\text{cov(}}Z_{i} ,h_{i} )}}{{{\text{cov(}}Z_{i} ,g_{i} )}} $$

(37)

Taking the numerator first,

$$ \begin{aligned} {\text{cov(}}Z_{i} ,h{}_{i} )& = {\text{cov}}\left\{ {Z_{i} ,\left( {\beta_{0} + \beta_{1}^{C} g_{i}^{C} + \beta_{1}^{NC} g_{i}^{NC} + u_{i} } \right)} \right\} \\ & = {\text{cov(}}Z_{i} ,\beta_{1}^{C} g_{i}^{C} ) + {\text{cov(}}Z_{i} ,\beta_{1}^{NC} g_{i}^{NC} ) + {\text{cov(}}Z_{i} ,u_{i} ) \\ & = \beta_{1}^{C} {\text{cov(}}Z_{i} ,g_{i}^{C} ) + \beta_{1}^{NC} {\text{cov(}}Z_{i} ,g_{i}^{NC} ) \\ \end{aligned} $$

(38)

since Z is exogenous and orthogonal to the error term. Using Eqs. (21) and (22), we have

$$ \begin{aligned} {\text{cov(}}Z_{i} ,g_{i}^{C} ) & = {\text{cov(}}Z_{i} ,\pi_{ 0}^{\text{C}} + \pi_{ 1}^{\text{C}} Z_{i} + \eta_{i} ) = \pi_{ 1}^{\text{C}} \text{var} (Z_{i} ) \\ {\text{cov(}}Z_{i} ,g_{i}^{NC} ) & = {\text{cov(}}Z_{i} ,\pi_{ 0}^{\text{NC}} + \pi_{ 1}^{\text{NC}} Z_{i} + \eta_{i} ) = \pi_{ 1}^{\text{NC}} \text{var} (Z_{i} ) \\ \end{aligned} $$

(39)

since Z is also uncorrelated with η. Substituting (39) into (38), we have

$$ {\text{cov(}}Z_{i} ,h{}_{i} )= \beta_{1}^{C} \pi_{ 1}^{\text{C}} \text{var} (Z_{i} ) + \beta_{1}^{NC} \pi_{ 1}^{\text{NC}} \text{var} (Z_{i} ) = \text{var} (Z_{i} )\left\{ {\beta_{1}^{C} \pi_{ 1}^{\text{C}} + \beta_{1}^{NC} \pi_{ 1}^{\text{NC}} } \right\} $$

(40)

Now taking the denominator of (37),

$$ \begin{aligned} {\text{cov(}}Z_{i} ,g{}_{i} )& = {\text{cov}}\left\{ {Z_{i} ,\left( {g_{i}^{C} + g_{i}^{NC} } \right)} \right\} = {\text{cov(}}Z_{i} ,g_{i}^{C} )+ {\text{cov(}}Z_{i} ,g_{i}^{NC} )\\ & = \pi_{ 1}^{\text{C}} \text{var} (Z_{i} ) + \pi_{ 1}^{\text{NC}} \text{var} (Z_{i} ) = \text{var} (Z_{i} )\left\{ {\pi_{ 1}^{\text{C}} + \pi_{ 1}^{\text{NC}} } \right\} \\ \end{aligned} $$

(41)

where we have made use of (39). Substituting (40) and (41) into (37), we obtain

$$ \hat{\beta }_{1}^{\text{IV}} = \frac{{{\text{s}}_{\text{Zh}} }}{{{\text{s}}_{\text{Zg}} }}\mathop \to \limits^{\text{P}} \frac{{\text{var} (Z_{i} )\left\{ {\beta_{1}^{C} \pi_{ 1}^{\text{C}} + \beta_{1}^{NC} \pi_{ 1}^{\text{NC}} } \right\}}}{{\text{var} (Z_{i} )\left\{ {\pi_{ 1}^{\text{C}} + \pi_{ 1}^{\text{NC}} } \right\}}} = \frac{{\pi_{ 1}^{\text{C}} }}{{\pi_{ 1}^{\text{C}} + \pi_{ 1}^{\text{NC}} }}{{\upbeta}}_{ 1}^{\text{C}} + \frac{{\pi_{ 1}^{\text{NC}} }}{{\pi_{ 1}^{\text{C}} + \pi_{ 1}^{\text{NC}} }}{{\upbeta}}_{ 1}^{\text{NC}} $$

(42)

which is Eq. (13) in the text, the expression for the LATE estimator when gun levels are observable. The LATE estimator is a weighted average of ${{\upbeta}}_{ 1}^{\text{C}}$ and ${{\upbeta}}_{ 1}^{\text{NC}}$, but now the weights are the relative strengths of the correlations between the instrument Z and criminal/noncriminal gun prevalence. Note that, unlike the OLS estimator, the variation in gun prevalence does not affect the weights.

Now consider the case where gun levels are not observable and the IV estimator uses the proxy p:

$$ \hat{b}_{1}^{\text{IV}} = \frac{{{\text{s}}_{\text{Zh}} }}{{{\text{s}}_{\text{Zp}} }}\mathop \to \limits^{\text{P}} \frac{{{\text{cov(}}Z_{i} ,h_{i} )}}{{{\text{cov(}}Z_{i} ,p_{i} )}} $$

(43)

The numerator is the same as in (37) above. The denominator is

$$ \begin{aligned} {\text{cov(}}Z_{i} ,p{}_{i} )& = {\text{cov}}\left\{ {Z_{i} ,\left( {\delta_{0} + \delta_{ 1}^{\text{C}} g_{\text{i}}^{\text{C}} + \delta_{ 1}^{\text{NC}} g_{\text{i}}^{\text{NC}} + \nu_{i} } \right)} \right\} \\ & = \delta_{ 1}^{\text{C}} {\text{cov(}}Z_{i} ,g_{i}^{C} )+ \delta_{ 1}^{\text{NC}} {\text{cov(}}Z_{i} ,g_{i}^{NC} )\\ & = \delta_{ 1}^{\text{C}} \pi_{ 1}^{\text{C}} \text{var} (Z_{i} ) + \delta_{ 1}^{\text{NC}} \pi_{ 1}^{\text{NC}} \text{var} (Z_{i} ) = \text{var} (Z_{i} )\left\{ {\delta_{ 1}^{\text{C}} \pi_{ 1}^{\text{C}} + \delta_{ 1}^{\text{NC}} \pi_{ 1}^{\text{NC}} } \right\} \\ \end{aligned} $$

(44)

Substituting (40) and (44) into (43) yields

$$ \begin{aligned} \hat{b}_{1}^{\text{IV}} & = \frac{{{\text{s}}_{\text{Zh}} }}{{{\text{s}}_{\text{Zp}} }}\mathop \to \limits^{\text{P}} \frac{{{\text{cov(}}Z_{i} ,h_{i} )}}{{{\text{cov(}}Z_{i} ,p_{i} )}} = \frac{{\text{var} (Z_{i} )\left\{ {\beta_{1}^{C} \pi_{ 1}^{\text{C}} + \beta_{1}^{NC} \pi_{ 1}^{\text{NC}} } \right\}}}{{\text{var} (Z_{i} )\left\{ {\delta_{ 1}^{\text{C}} \pi_{ 1}^{\text{C}} + \delta_{ 1}^{\text{NC}} \pi_{ 1}^{\text{NC}} } \right\}}} \\ & = \frac{{{{\uppi}}_{ 1}^{\text{C}} }}{{{{\updelta}}_{ 1}^{\text{C}} {{\uppi}}_{ 1}^{\text{C}} + {{\updelta}}_{ 1}^{\text{NC}} {{\uppi}}_{ 1}^{\text{NC}} }}{{\upbeta}}_{ 1}^{\text{C}} + \frac{{{{\uppi}}_{ 1}^{\text{NC}} }}{{{{\updelta}}^{\text{C}} {{\uppi}}_{ 1}^{\text{C}} + {{\updelta}}_{ 1}^{\text{NC}} {{\uppi}}_{ 1}^{\text{NC}} }}{{\upbeta}}_{ 1}^{\text{NC}} \\ & = \frac{{{{\uppi}}_{ 1}^{\text{C}} + {{\uppi}}_{ 1}^{\text{NC}} }}{{{{\updelta}}_{ 1}^{\text{C}} {{\uppi}}_{ 1}^{\text{C}} + {{\updelta}}_{ 1}^{\text{NC}} {{\uppi}}_{ 1}^{\text{NC}} }}\left\{ {\frac{{{{\uppi}}_{ 1}^{\text{C}} }}{{{{\uppi}}_{ 1}^{\text{C}} + {{\uppi}}_{ 1}^{\text{NC}} }}{{\upbeta}}_{ 1}^{\text{C}} + \frac{{{{\uppi}}_{ 1}^{\text{NC}} }}{{{{\uppi}}_{ 1}^{\text{C}} + {{\uppi}}_{ 1}^{\text{NC}} }}{{\upbeta}}_{ 1}^{\text{NC}} } \right\} \\ \end{aligned} $$

(45)

which is Eq. (15) in the text, the expression for the LATE estimator when gun prevalence is proxied by p. Note that this is a scaling parameter (assumed positive) times the probability limit of $\hat{\beta }_{1}^{\text{IV}}$ given in Eq. (42). Thus sign {$\widehat{\text{b}}_{1}^{\text{IV}}$} is a consistent estimator of sign {$\hat{\beta }_{1}^{\text{IV}}$}, irrespective of the strength of the correlation between the proxy and criminal/noncriminal gun prevalence.

Appendix 3: Robustness Checks

Checks Using 1990 Data

We tested the robustness of our 1990 results by varying the main specification in a number of ways:

• 1. Weights. We estimated the main specification but weighted by county population in 1990. Because the results using population weights were sometimes sensitive to the inclusion/exclusion of a small number of large counties, we estimated using both the sample of counties with populations in excess of 25,000 as in the results discussed in the main text, and using a subset of this sample that excludes the roughly 100 counties with populations greater than 500,000.

• 2. Functional form. We varied the functional form of the estimating equation by using homicide rates in logs and in levels, and by using PSG in logs and in levels.

• 3. Lagged dependent variable (LDV). One of the anonymous referees suggested we include a lagged measure of the gun homicide rate in the equations to mitigate possible problems of unobserved heterogeneity, i.e., historical factors besides heterogeneity in the criminal population. We note, however, that although this is a useful robustness check, the results of such an estimation are not easily interpreted. In our preferred LATE model, the IV estimator has a very clear interpretation in the presence of heterogeneity in criminality: it is a weighted average of the effects of criminal and non-criminal gun prevalence. Including lagged homicide as a regressor eliminates this clear interpretation offered by our LATE model.

• 4. Criminal justice (CJ) controls. Another referee suggested our specifications may suffer from a specific form of omitted variable bias, namely failing to include controls for formal deterrence measures such as police manpower, incarceration rates, or arrest rates. We examine this possibility by re-estimating our model including as controls two of the most widely used measures in the macro-level deterrence literature and for which data for all US counties are readily available: ICPSR county-level data on sworn police officers per capita in 1992³³ (as a measure of police manpower levels) and a measure of the rate of solving crimes constructed as the ratio of arrests for violent crimes 1989–91 to reports of violent crimes 1989–91.³⁴ We use both controls in log form. Unfortunately, incarceration data are not available for all US counties We should note, however, that to the extent that incarceration levels and other omitted criminal justice measures operate at the state-level to reduce county-level rates of homicide, these effects would be captured by our inclusion of state fixed effects.

We re-estimated the main equation using various combinations of the above specifications. The results for gun homicide are reported below in Table 10; the results for nongun homicide are in Table 11. The first row in each table corresponds to the specification discussed in the main text.

Table 10 1990 gun homicide estimates

Table 11 1990 nongun homicide estimates

GMUR is the gun murder rate; NGMUR is the non-gun murder rate; PSG is defined as in the paper. “LDV” and “CJ” indicate whether a lagged dependent variable or criminal justice measures are included as regressors. In the LDV specifications, the table reports the long-run coefficient on PSG, equal to the coefficient on PSG * 1/(1 − α) where α is the coefficient on lagged homicide. The magnitude of the long-run coefficient can therefore be compared directly to the coefficient on PSG when the LDV is omitted. The “Wt” column indicates whether or not the results weight by 1990 population. The total sample includes counties with a population of at least 25,000 persons in 1990; the “U lim” column indicates whether a subset of counties with a population upper limit of 500,000 persons is used. The HOLS column reports the coefficient on the gun prevalence proxy when it is treated as exogenous; the GMM2S column is the coefficient when treated as endogenous; 2-step efficient GMM is used in both cases. The “F 1st St” column reports the first-stage F statistic; J is the J overidentification statistic; p(J) is the corresponding p value; and N is the sample size. Stars are as in the paper (1, 5, 10 %). Tests are robust to heteroskedasticity and clustering.

The various specifications show that the main results reported in the paper are indeed robust. In the gun homicide estimations, when gun prevalence is treated as exogenous, the estimated impact on gun homicide is generally positive and statistically significant; when it is treated as endogenous, the impact on gun homicide is significantly negative or null. For nongun homicide, the impact of gun prevalence is generally null, whether or not gun prevalence is treated as exogenous or endogenous. The instrument relevance tests are generally satisfactory (a first-stage F statistic in excess of 10), as are the tests of instrument orthogonality (insignificant J statistics).

The different functional forms generate broadly similar qualitative results, but the specifications in which the homicide rate is in logs, as in the results discussed in the main text, tend to generate smaller quantitative impacts than those in which the homicide rate is used in levels. For the reasons discussed in the text, we regard the log specification, and hence the corresponding quantitative results, as preferable. We discuss here, for illustration, the calibrations for the main specifications for gun homicide (full sample, unweighted, no criminal justice controls or LDV) corresponding to a one percentage point increase in PSG. (1) The specification in line 1 of Table 10 is the specification discussed in the main text; the coefficient of −2.41 on PSG/100 implies that a one percentage point increase in PSG would reduce the gun homicide rate by 0.01 times 2.41 or about 2.4 %. (2) When both the gun homicide rate and PSG are in logs, the estimated coefficient on log(PSG) in the main specification is −1.16 (line 5). A one percentage point increase in PSG evaluated at the sample mean of PSG (67 %; see Table 5) is equivalent to a 1/67th or 1.5 % increase; 0.015 times 1.16 means a fall in the gun homicide rate of 1.7 %. (3) When both the gun homicide rate and PSG are in levels, the coefficient on PSG/100 in the main specification is −28.09 (line 9). A one percentage point increase in PSG would therefore reduce gun homicide by 0.28 persons per 100,000 population. At the sample average of 4.11 gun homicides per 100,000 persons (Table 5), this is equivalent to 6.8 % fall in the gun homicide rate. (4) When the gun homicide rate is in levels and PSG is in logs, the estimated coefficient on log(PSG) in the main specification is −12.35 (line 13). A one percentage point increase in PSG, equivalent to a 1.5 % increase, implies that gun homicide would fall by 0.015 times 12.35 or 0.19 persons per 100,000, equivalent to a 4.5 % fall in the gun homicide rate evaluated at the sample average of 4.11.

Checks Using 1970 and 1980 Data

We constructed datasets for 1970 and 1980 using Census variables, CDC homicide and suicide data, and ICPSR election data. The only variable used in the main specification we did not have available for these earlier years is outdoor sports magazine subscriptions (OMAG), and hence only the voting and veterans variables were available as instruments.

The gun and nongun homicide results for 1970 are reported in Tables 12 and 13, respectively. Because of data availability constraints, some of the control variables are defined slightly differently from the 1990 sample. Age structure categories are 0–14, 15–19, 20–24, 25–44, 45–64, and 65 + . The cutoff for low-income households is $6,000. Inequality is based on household income cutoffs of $6,000 and $25,000. “Hispanic” is based on use of Spanish language. Homicide and PSG are based on 5-year averages, 1968–72. The voting instrument is the percentage voting Republican in the 1968 presidential election. Because 1960 CDC homicide data were not available to us, we do not report the robustness check using a lagged dependent variable. For comparability with the 1990 county coverage, we also report the results of limiting the sample of counties based on 1990 as well as on 1970 population.

Table 12 1970 gun homicide estimates

Table 13 1970 nongun homicide estimates

The 1970 results for gun homicide are similar to those for 1990: when PSG is treated as exogenous, it has a positive and significant coefficient, but when it is treated as endogenous, the significance disappears or (in a few cases) the coefficient becomes negative and significant. The nongun homicide results are also similar to our 1990 results: PSG has a null or (in a few cases) a negative impact on nongun homicide whether treated as exogenous or endogenous. The overidentification statistics are satisfactory. However, the instruments are weak to very weak, and hence the results should be treated with some caution.

Robustness checks using 1980 data are reported below in Tables 14 and 15. Data definitions are the same as for the 1990 sample, except that the cutoff for low-income households is $8,000, and inequality is based on household income cutoffs of $8,000 and $50,000. Homicide and PSG are based on 5-year averages, 1978–82. The voting instrument is the percentage voting Republican in the 1980 presidential election. Lagged homicide is based on the 1968–72 5-year average. For comparability with the 1990 county coverage, we report the results of limiting the sample of counties based on 1990 as well as on 1980 population. The LDV specifications report the long-run impact of gun prevalence, calculated as noted above.

Table 14 1980 gun homicide estimates

Table 15 1980 nongun homicide estimates

The results for both gun and nongun homicide in 1980 are similar to those for 1990 and 1970. As with the 1990 data, including a lagged dependent variable does not noticeably change the results. The main difference with the results in term of the specification tests are that J statistic is sometimes high enough to reject at the 5 % level, and that the instruments are weak less often. We note that when the first-stage F statistic is satisfactorily high (near or above 10), the J statistic is also satisfactorily low, and these particular estimations are consistent with our 1990 results (i.e., PSG typically has a negative and significant coefficient in the gun homicide estimations). Again, however, because of the weakness of the instruments, the results should be treated with some caution.