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The social costs of gun ownership: Spurious regression and unfounded public policy advocacy

08 Jul 2013-Social Science Research Network (Marburg: Philipps-University Marburg, Faculty of Business Administration and Economics)-
TL;DR: This article used a panel regression of 200 U.S. counties across 20 years to find a significant elasticity of homicides with respect to firearm ownership in the United States and made the public policy recommendation of taxing gun ownership.
Abstract: In 2006, a study, published in the Journal of Public Economics, employing a panel regression of 200 U.S. counties across 20 years, found a significant elasticity of homicides with respect to firearms ownership. Based on this finding the authors made the public policy recommendation of taxing gun ownership. However that study fell prey to the ratio fallacy, a trap known since 1896. All the "explanatory power" (goodness-of-fit-wise and significance-wise) of the original analysis was due to regional and inter-temporal differences and population being explained by itself. When the ratio fallacy is accounted for, all authors’ results can no longer be found. This is illustrated in this paper using a balanced panel from the data for 1980 to 2004. My findings are robust to (i) alternative specifications not subject to the ratio problem, (ii) using only data from 1980 to 1999 as in the original paper, (iii) using an unbalanced panel for 1980 to either 1999 or 2004, (iv) applying weighting as done by the original authors and (v) using data aggregated at the state level.

Summary (4 min read)

6 Alternative specifications 19

  • The association between guns and crime has been and continues to be a topic of intense debate in society at large and among social scientists.
  • This finding sparked a furious academic debate across disciplines.
  • A), which begins with the bold statement “There is A Proven Correlation between the Availability of Handguns and Incidents of Violence.” and then goes on to draw on findings from Duggan (2001).
  • The original objective was to address specialized econometric problems, such as the noisy proxy used and truncation of the data due to the logarithmic model, and to also possibly confirm the results with five more years of data.

2 AN OVERVIEW 2

  • This statistical property is the only reason C&L arrived at their result, based on which they advocated taxing gun ownership.
  • If these findings are biased or spurious, any public policy based on them may not have its intended effect and in the worst case could actually be harmful.
  • Indeed, only those readers interested in such a replication need read Subsections 3.1 and 3.2.
  • The results from Cook and Ludwig (2006) are repeated in Sections 3.3 and 4.3, with a sharp twist in the results in Section 4.4 nullifying C&L’s original conclusion.
  • The analysis assumes that gun ownership may impose externalities on society (Cook and Ludwig 2006: 379–380), specifically that more guns may result in more homicides.

2 AN OVERVIEW 3

  • That proxy is the fraction of suicides committed with a firearm (Cook and Ludwig 2006: 380), that is, “firearm suicides” divided by “suicides” (FSS or FS/S).
  • (b) is confirmed to function in the way intended by other studies (Azrael, Cook and Miller 2004; Kleck 2004), at least for the cross-section.
  • Statistics on population size and number of homicides and suicides, as well as some sociodemographic controls for each county, are available.
  • The panel of ratios is analyzed by a two-way (individual and time) fixed effects panel model on the logarithms.

3 DATA ACQUISITION 4

  • And Cook and Ludwig (2003) without giving any reasons for doing so.
  • Kleck (2009) has several criticisms, including (a) that C&L’s method of dealing with causal dependence is overly simple, (b) that the FSS proxy may not be valid for measuring trends in gun ownership, and, similar to Moody and Marvell’s argument, (c) that the controls used are arbitrarily chosen and that some possible necessary controls are missing from the model.
  • The aggregated data set used in Cook and Ludwig (2006) is not published and the authors chose not to share it with me.
  • 4The set of selected counties does not change if the 1990 census population from United States Census Bureau (1990) is used instead.
  • 5United States Department of Health and Human Services.

3 DATA ACQUISITION 5

  • The total population is taken from Table P3; the respective number of households from P5.
  • 10For detailed download procedures see Westphal (2013).

3 DATA ACQUISITION 6

  • These contain reported crime numbers aggregated at the county level.
  • Study dataset 0654512 for the 1993 Uniform Crime Report data was not available for download at the time of writing.
  • Different geographical coding schemes are found in the data: NCHS17 coding and FIPS18 coding.
  • These changes may lead to mismatched assignment of values if ignored during data extraction; thus each data source and each year had to be individually checked for such changes.

3 DATA ACQUISITION 8

  • Index, and nchs and fips, and the tuple of state and county as interchangeable individual identifiers for the counties.
  • I then used these numbers to calculate the percentages with the appropriate denominator.
  • Switching the denominator to either total or UCRpop changes the results only marginally from those reported below; correlation between pop and total is > 0.999 and correlation between pop and UCRpop is > 0.989.3.3 Comparison of Des riptives.

4 REGRESSION ANALYSIS 9

  • This column allows comparing my data to those of C&L while at the same time giving the correct descriptives.
  • Differences may be due to slightly different data sources,20 a slightly different set of observations used for computation,21 or, possibly, revised data.
  • They include the proxy FSS = E955/E95 lagged by one year to circumvent possible reverse causation, i.e., people buying guns because of a higher homicide rate.

4 REGRESSION ANALYSIS 11

  • Dependent variable Y is the homicide rate E96/pop.
  • There are several ways of excluding observations containing a ratio of zero: unbalance the panel or remove counties or years (whichever is less costly) in order to keep the panel balanced.
  • Results for this and those presented in the following sections are qualitatively the same and numerically close when using (various subsets of) the unbalanced panel.
  • Descriptive statistics do not differ much from those set out in Table 4.
  • The results are only slightly different from the results in Cook and Ludwig (2006: Table 2, final column).

4 REGRESSION ANALYSIS 12

  • Table 3, model 3), who needed weighting to achieve significance on β1, significance on the balanced panel is achieved without weighting, also known as and Ludwig (2006.
  • The within R2 reported in Table 5 is magnitudes smaller than the R2 of around 0.9 reported by Cook and Ludwig (2006: Table 2) for all their models.
  • The coefficient on the female household heads changes sign between the original study and my estimation, but this does not affect the arguments in Sections 4.4, 5 or 6.
  • To understand what happens when the authors estimate the first difference model, the estimating equation (2) needs to be written out in full.

5 DISCUSSION 14

  • Values do not exhibit too much orthogonal variation to population itself, it will be able to explain itself.
  • This is a variant of the ratio fallacy, first discovered by Pearson (1896) and discussed in detail by Kronmal (1993),29 which here appears disguised in a logarithmic model.
  • 31 Together with Table 6 this shows that all other values vary much more strongly over time than does the value based on population.
  • 30Computed by analysis of variance decomposition of variance: within-county variance is variance over time, between-county variance is variance between counties.

5 DISCUSSION 15

  • Of interest β1(∆ ln E955k,t−1−∆ ln E95k,t−1), the term from the numerator is double the mean squared distance from zero and double the variance than the term from the denominator.
  • This means that in this specific data set, taking the first differences at least partially removes the numbers causing spurious correlations between ratios.
  • Here, it basically removes population from all the terms and only the (growth rates) of the numerators remain in the model after taking first differences.
  • One could now argue E95k,t−1 is not population and therefore the results from Cook and Ludwig (2006) are not due to the ratio fallacy.
  • When the authors look at the correlation matrix (Table 7) they immediately see that the correlation between suicides and population is far superior to any other correlation between the lefthand side and the right-hand side, at least in regard to the four variables shown in Table 7.

5 DISCUSSION 16

  • It seems unlikely that none of these situations occurred in the original analysis, and thus there may be more sources for spurious results than just the ratio problem.
  • Therefore, not all trends will be accounted for in C&L’s original model.
  • This is further evidence that the differentiated model has in this case taken the ratio problem out of the data.
  • 32Using the original denominators and testing all four linear hypotheses leads to an even more significant rejection of the null hypothesis.
  • The estimation result from this model is set out in the column labeled “Eq. (17)” in Table 9.

6 ALTERNATIVE SPECIFICATIONS 20

  • Any spurious correlation between the left-hand side and the right-hand side due to population appearing on both sides is no longer possible.
  • Time series problems are not accounted for.
  • 3 Risk Model Duggan (2003: 48–50) proposes a model38 for explaining individual i’s suicide decision39 Pr = α+ X iθ + γGuni + λi + ǫi (20) with X i being individual observable controls, Guni a dummy for gun ownership, andλi individual i’s unobserved individual propensity to commit suicide.
  • The following argument holds for other monotonous link functions as well.

6 ALTERNATIVE SPECIFICATIONS 22

  • Under the simplifying assumption that only gun owners are able to commit suicide by firearm.
  • Now attribute an additional risk to each gun owner,42 then a relation of β0 + β1,1 β0 ∼ RRgunowner, (25) exists, given the number of firearm suicides is somehow linked to the number of gun owners.
  • For burglaries, it might just mean there are around 170 times as many burglaries as homicides.
  • This model is susceptible to criticism for obvious heteroscedasticity across counties with different levels of population.

6 ALTERNATIVE SPECIFICATIONS 23

  • Where Xk,t contains the log growth rates for the controls.
  • Due to nearly perfect correlation with pop−1 k,t , I removed pop5plus−1 k,t and households−1 k,t from the model.
  • Including pop−1 k,t on the right-hand side is obviously ridiculous.
  • This is the only setting showing this result.

7 CONCLUSION 24

  • Thus, from a goodness-of-fit point of view, the only thing C&L’s full model does, is add a lot of noise to Equation (28).
  • “The Latest Misfires in Support of the ’More Guns, Less Crime’ Hypothesis.” Stanford Law Review, 55: 1371.
  • Cook, Philip J, Jens Ludwig, Sudhir Venkatesh, and Anthony A Braga.

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Westphal, Christian
Working Paper
The social costs of gun ownership: Spurious
regression and unfounded public policy advocacy
MAGKS Joint Discussion Paper Series in Economics, No. 32-2013
Provided in Cooperation with:
Faculty of Business Administration and Economics, University of Marburg
Suggested Citation: Westphal, Christian (2013) : The social costs of gun ownership: Spurious
regression and unfounded public policy advocacy, MAGKS Joint Discussion Paper Series in
Economics, No. 32-2013, Philipps-University Marburg, Faculty of Business Administration and
Economics, Marburg
This Version is available at:
http://hdl.handle.net/10419/93516
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Joint Discussion Paper
Series in Economics
by the Universities of
Aachen Gießen Göttingen
Kassel Marburg Siegen
ISSN 1867-3678
No. 32-2013
Christian Westphal
The Social Costs of Gun Ownership: Spurious Regression
and Unfounded Public Policy Advocacy
This paper can be downloaded from
http://www.uni-marburg.de/fb02/makro/forschung/magkspapers/index_html%28magks%29
Coordination: Bernd Hayo Philipps-University Marburg
Faculty of Business Administration and Economics Universitsstraße 24, D-35032 Marburg
Tel: +49-6421-2823091, Fax: +49-6421-2823088, e-mail:
hayo@wiwi.uni-marburg.de

The Social Costs of Gun Ownership: Spurious
Regression and Unfounded Public Policy
Advocacy
a
Christian Westphal
b
This version: July 8, 2013
Abstract
In 2006, a study, published in the Journal of Public Eco nomics, employ-
ing a panel regression of 200 U.S. counties across 20 years, found a sig-
nicant elasticity of homicides with respect to rearms ownership. Based
on this nding the authors m ade the public policy recommendation of tax-
ing gun ownership. However that study fell prey to the ratio fallacy, a trap
known since 1896. All the explanatory power (goodness-of-fit-wise and
signicance-wise) of the original analysis was due to re gional and intertem-
poral differences and popul ation being e xplained by itself. When the ratio
fallacy is accounted for, all authors results c an no longer be found. This is
illustrated in this paper using a balanced panel from the data for 1980 to
2004. My ndings are robust to (i) al ternative specications not subject to
the ratio problem, (ii) using only data from 1980 to 1999 as in the origi-
nal paper, (iii) using an unbalanced panel for 1980 to either 1999 or 2004,
(iv) applying weighting as done by the original authors and (v) using data
aggregated at the state le v el.
JEL Classications: C51; H21; I18; K42
Keywords: Gun O wnership; Social Co s ts; Ratio Fallacy; Spurious Regression
a
Thanks to participants of a Brown Bag Seminar held at Philipps-University Marb ur g and to
participants of a Seminar on replication studies held at University of Göttingen both in which
some good thoughts on the problem were had. Thanks to Bernd Hayo for co ntributing the mov-
ing averages solution for the noisy pr oxy mentioned in section 7 and to Florian Neumeier for
contributing the growth model used in section 6.4.
b
University of Marburg, Faculty of Business Administration and Economics, Department of
Statistics,
hristian.westphalwestphal.de,westphalstaff.uni-marburg.de
i

CONTENTS
ii
Contents
1 Introduction 1
2 An overview 2
2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Data Acquisition 4
3.1 Data Sources and Extraction . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Resulting Datase t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Comparison of Descriptives . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Regression Analysis 9
4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Data for Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Conrming results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.4 Estimation on First Differences . . . . . . . . . . . . . . . . . . . . . . 13
5 Discussion 13
5.1 Ratio Fallacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 First Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.3 But E95 is Not Population . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.4 Nons ense Regression Between Time Series . . . . . . . . . . . . . . 16
5.5 Misspecication of the Original Model . . . . . . . . . . . . . . . . . 16
5.5.1 Testing for Misspecication . . . . . . . . . . . . . . . . . . . 16
5.5.2 Theoretical Bias . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Alternative specications 19
6.1 Controlling for Popu lation . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.2 Algebraic Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.3 Risk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.4 Growth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.5 Numerato rs and Denomi nators . . . . . . . . . . . . . . . . . . . . . . 23
7 Conclusion 24

1 INTRODUCTION
1
1 Intro dution
The association between guns and crime has been and continues to be a topic
of intense debate in society at large and amo ng social scientists. The debate
intensied after Lott and Mustard (1997) published results showing that crime
declined follow i ng the passing of shall-issue laws
1
for concealed carry handgun
licenses. This nding sparked a furious academic debate across disciplines. Po-
litical scientists, legal scholars, criminologists, economists and scientists working
in medical elds all added their voi ces to the discussion. From my perspective,
the most noteworthy ( for data and methods used, as well as results) economet-
ric studies on guns and crime appearing after Lott and Mustard (1997) include
Ludwig (1998); Duggan (2001); Le e naars and Lester (2001); Cook and Ludwig
(2003, 2006); Cook et al. (2007) and Leigh and Neill (2010). Some works in
this area bear strongly worded titles that rather clearly reect their authors per-
spectives: Shooting Down More Guns, Less Crime (Ayres and Donohue 2003b),
The Final Bullet in the Body of the More Gun s Less Crime Hypothesis (Dono-
hue 2003) an d The Latest Misres in Support of the More Guns, Less Crime
Hypothesis (Ayres and Donohue 2003a). These titles illustrate the intensity of
the deba te, which is also evident in Lott (2010: Chapter 7). Academic research
results became increasingly important in the public and legal arenas. This can be
seen in Fox and McDowall (2008: III.1.A) , which begins with the bold statement
There is A Proven Correlation between the Availability of Handguns
and Incidents of Violence.
and then goes on to draw on ndings from Duggan (2001). In Fox and McDowall
(III.1.C 2008), a result from (Cook et al. 2007: Section 4) is used to bolster
their argument. Eventually, more rened econometric methods were a p p lie d to
the issue. For example, Cook and Ludwig (2006), following the lead of Duggan
(2001), apply advanced methods to very detailed data, taking i nto consideration
many empirical problems.
I chose to revisi t Cook an d Ludwig (2006) (C&L hereafter) due to its rigor
and the detailed description of the data sources from a preceding working pa-
per (Coo k and Ludwig 2004). The original objective wa s to address specialized
econometric problems, s uch as the noisy proxy used and truncation of the data
due to the logarithmic model, and to also possibly conrm the results with ve
more years of data. In this attempt, I made a surprising discovery: C&L ignored a
statistical property of their data (ratios ) leading to spurious results in regression
analysis. Even more surprising is that this pitfall has been known about for more
1
A shall-issue law forces a state to issue concealed carry licenses to any applicant. No
reasons need to be given by th e applicant; as long as he does not have any convictions or mental
disorders the license must be issued.

Citations
More filters
Journal ArticleDOI
TL;DR: In this paper, the problem of missing administrative data arises when attempting to measure gun ownership in the United States, making it necessary to find a valid proxy, such as the fraction of suicides by firearm owners.
Abstract: When attempting to measure gun ownership in the United States, the problem of missing administrative data arises, making it necessary to find a valid proxy. Several such proxies are employed in economic studies, one of which is the fraction of “suicides by firearm” of “all suicides” (FSS). My work validates this proxy from out-of-sample data, namely, Austrian administrative data on firearm licenses. I also reevaluate, with appropriate statistical methods, a result on firearms and suicide from the medical that is often used for public policy advocacy. This result is, unfortunately, heavily biased due to ignoring a well-known fallacy and thus can be only partially confirmed.
References
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Abstract: Copyright (©) 1999–2012 R Foundation for Statistical Computing. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions, except that this permission notice may be stated in a translation approved by the R Core Team.

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"The social costs of gun ownership: ..." refers background in this paper

  • ...Regression between time series is known to produce spurious results in the following settings: trending or auto correlated time series (Granger and Newbold 1974), I(1) processes without drift (Phillips 1986), I(1) processes with further stationary regressors (Hassler 1996), stationary AR processes…...

    [...]

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"The social costs of gun ownership: ..." refers methods in this paper

  • ...The lack of efficiency can be dealt with after estimation by applying Driscoll and Kraay (1998) via Croissant and Millo’s (2008) v ovSCC function.25 Contrary to Cook 22An example would be ∑ k dk = 0....

    [...]

Frequently Asked Questions (8)
Q1. What contributions have the authors mentioned in the paper "The social costs of gun ownership: spurious regression and unfounded public policy advocacy" ?

In 2006, a study, published in the Journal of Public Economics, employing a panel regression of 200 U. S. counties across 20 years, found a significant elasticity of homicides with respect to firearms ownership. Based on this finding the authors made the public policy recommendation of taxing gun ownership. When the ratio fallacy is accounted for, all authors ’ results can no longer be found. This is illustrated in this paper using a balanced panel from the data for 1980 to 2004. My findings are robust to ( i ) alternative specifications not subject to the ratio problem, ( ii ) using only data from 1980 to 1999 as in the original paper, ( iii ) using an unbalanced panel for 1980 to either 1999 or 2004, ( iv ) applying weighting as done by the original authors and ( v ) using data aggregated at the state level. 

The first method for removing the spuriousness from C&L’s model is given by Kronmal (1993: 390): include the inverse of the deflating variable as an explanatory variable on the right-hand side. 

The individual fixed effects disappear from the model, the time fixed effects are transformed to the differences between the time fixed effects, and the errors are transformed. 

All the “explanatory power” (goodness-of-fit-wise and significance-wise) of the original analysis was due to regional and intertemporal differences and population being explained by itself. 

The analysis assumes that gun ownership may impose externalities on society (Cook and Ludwig 2006: 379–380), specifically that more guns may result in more homicides. 

25By applying weighting to account for heteroscedasticity (Cook and Ludwig 2006: 382) and calculating standard errors that are robust to heteroscedasticity (Cook and Ludwig 2006: 382), C&L basically “double correct” for heteroscedasticity. 

The authors use the advanced method of panel analysis and analyze a comprehensive data set covering 200 U.S. counties and 20 years. 

Also multicollinearity might be an issue, as all numbers used are part of the population and therefore are in popk,t .6.4 Growth Model A way to standardize without using ratios is to use growth rates.