LITERATURE REVIEW

The literature has affirmed that crime is a determinant social force. Individuals, households, and communities know first-hand the effects of crime on quality of life. ^{Houser et al. (2015)} have highlighted that in areas where crime is prevalent, residents notice direct effects on housing prices, education, and job availability. Then, fear of crime within urban areas has a powerful influence on perceptions of area deprivation and may discourage home-buyers, inhibit local regeneration, and catalyse a downward spiral in neighbourhood status (^{Gibbons, 2004}). The literature has tried to measure the effect of crime on housing prices using two main methodologies: contingent evaluation and hedonic models.

Contingent evaluation tries to estimate the value of goods that are not transacted in a market. The strategy is to ask how much people would be willing to pay for them. For instance, in ^{Cohen et al. (2004)}, respondents were asked if they would be willing to vote for a proposal requiring each household in their community to pay a certain amount to prevent one in ten crimes in their community. Meanwhile, in ^{Atkinson et al. (2005)}, respondents were told the characteristics of a type of crime and the current risk of victimization. Then, they were asked to express their willingness to pay to reduce the chance of being a victim of this offence by 50% over the next 12 months.

In hedonic models (^{Rosen, 1974}), a house may derive its value from the quality of its physical characteristics (e.g. living space, number of bedrooms, garage, amenities) and its location. Furthermore, the level of crime and violence in the surrounding area may be additional attributes of the property, and individuals may be willing to pay more to live in an area with lower levels of crime. Then, an estimate of how much the attribute "lower-level-of-crime" is worth in the housing price provides an estimation of the cost of crime.

Asking individuals how they would react in a certain hypothetical situation is not the same as how they will react in a real decision-making situation. Consequently, economists have long been sceptical of information extracted from stated preferences, rather than revealed ones (^{Carson et al., 2001}; ^{Levitt & List, 2007}). This is the reason why hedonic price models have been mostly used to estimate the relationship between crime and housing prices. Hedonic models rely on the preferences revealed by market behaviour by analysing the actual amount that people pay to avoid living in high crime areas. Some literature that uses this methodology has found a negative and significant relationship between crime and housing prices.

Because there could be a potential simultaneity relationship between crime and housing prices, Table 1 and Table 2 present the literature review divided into two categories.¹ The first is related to studies that have not instrumented for crime, and the second part includes studies that have instrumented for crime. For Rochester, New York, one standard deviation increase in the crime rate caused a 3% reduction in house prices (^{Thaler, 1978}. The measures of crime used by the author are total offences, property crimes, crimes against persons, and property crimes committed in or around homes. ^{Hellman and Naroff (1979)} obtained an elasticity of property value concerning crime equal to -0.63 for Boston. For Jacksonville, Florida, ^{Lynch and Rasmussen (2001)} found an elasticity of -0.05 for violent crimes. Meanwhile, for Atlanta, ^{Bowes and Ihlanfeldt (2001)} argued that crime may be higher in train station areas; moreover, train stations may be the source of more crime in higher-income than in lower-income neighbourhoods. Authors found that an additional crime per acre per year decreases housing prices by around 3%. ^{Besley and Mueller (2012)} presented evidence that supports the assumption that housing prices depend on the level and persistence of historical crime rates. Authors argued that houses are assets whose prices reflect the present and future expected attractiveness of living in an area. They used the information from 11 regions of Northern Ireland to evaluate the increased housing prices in response to a reduction in murders. Using a Markov switching model, the authors predicted that peace in Northern Ireland leads to an increase in housing prices of between 1.3% and 3.5% percent (nevertheless, the result is heterogeneous across regions).

Table 1 Studies That Have Not Instrumented for Crime 

Table 2 Studies That Have Instrumented for Crime 

The literature shown in Table 1 has deemed crime as an exogenous regressor. Relaxing that assumption in Table 2, ^{Rizzo (1979)} found similar qualitative results for Chicago and remarked that in the housing market people reveal the cost of crime as they perceive it. ^{Gibbons (2004)} estimated the impact of recorded domestic property crime on property prices in the London area. This author considered information for five types of crime: home burglaries, burglary in other buildings, criminal damage to a dwelling, criminal damage to other buildings, and theft from shops. As a result, a one-tenth standard deviation increase in the recorded density of incidents of criminal damage had a capitalized cost of just under 1% of property values. These results meant that incoming residents perceived criminal damage as a deterioration of the neighbourhood.

The difference with previous papers is that Gibbon paid attention to identification issues and deals with the endogeneity problem using instrumental variables. ^{Ceccato and Wilhelmsson (2011)} analysed the relationship between apartment prices and different measures of crime in Stockholm. The authors found that when total crime increases by 1%, apartment prices are expected to fall by 0.04%. ^{Buonanno et al. (2013)}, for Barcelona, found that one standard deviation increase in perceived security is associated with a 0.57% increase in the valuation of districts.

Although comparing results is somewhat arbitrary because of the differences in the types of crimes, the empirical literature presented evidence of a negative relationship between crime and housing prices. This relationship implies that high crime rates deter new residents and motivate those-who-can to move out to lower-crime rate neighbourhoods (^{Gibbons, 2004}).

DATA AND RESULTS

In this section, we present data, theoretical approximation, quantitative methodologies, and the estimated results of the relationship between homicide rates and housing prices in Cali and Bogotá D. C. (Appendix A presents a map of Colombia that shows the geographic location of both cities). In this context, it is relevant to note that Cadastral information in Colombia is one of the oldest and most extensive in Latin America. Nevertheless, it is limited to formulate politics (^{Departamento Administrativo de Hacienda Municipal, 2012}).

For Cali, we used the cadastral housing prices of 2012 to estimate the relationship between homicide and housing prices, using homicide rates from a neighbourhood level. For Bogotá D.C., we used information registered in the Encuesta Multipropósito (2017), specifically the hypothetical lease value and effective lease value. As we stated in the introduction, the homicide rate is defined as the number of homicides committed in a year per 100,000 inhabitants, excluding homicides committed as a result of armed conflict.

Table 3 summarizes the key variables from the housing price and homicide data for Cali. The mean log of housing prices is 17.16, with a standard deviation of 1.21 (Figure 2 shows the distribution of this variable); the homicide rate that we used is a ten-year average (2000-2010) at neighbourhood level. This average measure was included in the econometric model to avoid simultaneity effects with the housing prices variable, which is from 2012. This rate could be interpreted as the violence environment for each neighbourhood. The variable had a mean of 104 with a standard deviation equal to 155.²

Source: Cadastral housing prices of 2012.

Figure 2 Distribution of Log Housing Prices for Cali 

The urban area is divided into 338 neighbourhoods: the average size of a neighbourhood is 360 square meters; the average distance between the Central Business District (CBD) and the centroid of a neighbourhood is 4.76 kilometres. The CBD is the area within the city where the most important economic activity is concentrated (the CBD could or could not coincide with the historical place where the city was founded). The average distance from the centroid of a neighbourhood to the closest main road is 0.41 kilometres.

The left map in Figure 3 displays the average housing prices per square meter in Colombian pesos while the second map (right) exhibits the homicide rates at neighbourhood level. The map of housing prices shows that neighbourhoods with higher housing prices are distributed in the western part of the city. In the second, it can be observed that there are high homicide rates in central areas and some peripheral areas of the city. We hypothesized that persistent cases of homicide will be capitalized in housing prices.

Source: Cadastral housing prices of 2012 and Observatorio Social de Cali.

Figure 3 Housing Prices per Square Meter (left) and Homicide Rates (right) in Cali 

Table 4 summarises key variables related to housing prices and homicide data for Bogotá D.C. The mean log of hypothetical lease value was 13.31, with a standard deviation of 1.27, and the mean log of effective lease value was 13.13 with a standard deviation of 0.8 (Figure 4 shows the distribution of these variables). The mean for the homicide rate was 18.93 per 100,000 inhabitants with a standard deviation equal to 27.55. It is important to highlight that this homicide rate of 18.93 was slightly different from the homicide rate of 17, which was the official data from the INML (2015). The difference is due to disparate official calculations. One estimation came from the 2005 national census population projections conducted by Departamento Administrativo Nacional de Estadística, DANE; however, the rate that we are reporting in this paper was calculated with projections from the planning office of Bogotá D. C. (Secretaría de Planeación Distrital, SDP).

Figure 4 shows the distribution of the logarithm of housing prices: hypothetical lease value (left) and the effective lease value (right) from the Encuesta Multipropósito report (2017). As we observed in Table 4, the hypothetical lease value was slightly higher than the effective lease value. This characteristic is relevant because in the econometric models, the set of explanatory variables was similar, and the estimated coefficient related to the homicide rate was not that different.

Figure 4 Distribution of Log Hypothetical Lease Value (left) and the Effective Lease Value (right) for Bogotá D.C. 

The maps in Figure 5 display the hypothetical lease value (left) and the effective lease value (right) from the information in the Encuesta Multipropósito (2017). Both maps indicate that the higher lease values at the level of Unidades de Planeación Zonal (UPZ)³ were in the north-eastern part of the city. The lower lease values were distributed in the periphery: predominantly the south and west.

Source: Encuesta Multipropósito (2017) and own calculation.

Figure 5 Hypothetical Lease Value (left), Effective Lease Value (right) 

Finally, Figure 6 shows the homicide rates at the UPZ level for 2015. In the case of Cali, we used the average homicide rate for one decade to avoid simultaneity effects with housing prices. Meanwhile, for Bogotá, we used homicides for 2015 and housing prices for 2017. The urban area is divided into 112 UPZs according to the Encuesta Multipropósito (2017). The map shows that higher homicide rates were in UPZs in the geographical centre of the city and the south area. As Table 4 demonstrates, the average population in a UPZ was 70,024; the average of the log accessibility indicator was 9.88. The log accessibility indicator was calculated as the weighted average of the number of jobs for each UPZ. In the next section, which presents the regression model, we will explain the construction of this variable.

This section exhibited the geographical distribution of housing prices and homicide rates for Cali and Bogotá D.C. The sources of data used were different as were the years and the spatial units of reference. Nevertheless, in both cases, it was possible to use a theoretical model of reference and econometric estimations methodologies.

Source: INLM (2015) and own calculations.

Figure 6 Homicide Rates in Bogotá D.C. 2015 

THEORETICAL MODEL

In this section, we present a theoretical model to identify the relationship between both variables of interest: homicide rates and housing prices in the urban structures of Cali and Bogotá D.C., and the model's main equation, which we estimated with econometric methodologies. It is worth highlighting that in the literature review ^{Hellman and Naroff (1979)} provided a theoretical alternative to consider a crime indicator within the residential location model.

The theoretical problem for the typical household was to maximize a utility function, U = U[z,Q Hr( j )], subject to Y = z + P(j)Q + T(j). In this model, the variable z represents a composite good (numeraire), Q represents the quantity of housing attributes per unit of land, Hr(j) is the homicide rate in a neighbourhood j, Y represents the income, P(j) is the price per unit of housing, and T(j) the transportation cost.

The maximization problem, for which the utility function of an individual agent or household includes the residential location and for which the budget constraint includes the commuting cost, was replicated in more current literature, including ^{Brueckner (2011}, chapter 2), ^{Fujita and Thisse (2013}, chapter 3), and others. The difference was that ^{Hellman and Naroff (1979)} assumed that the homicide rate only enters the utility function for the household in the location with a negative relationship: ∂U / ∂Hr < 0.

Using a Cobb-Douglas utility function U = z ^α Q ^β Hr ^-ү and the budget constraint, the indirect utility function can be obtained, which can be solved for P to get the bid-rent function (remember that P is the price per unit of housing in the budget constraint):

where equals the constant and equal utility level for each household in the city. The bid-rent in the equation above can be interpreted as the willingness to pay for housing. This is positively related to income and negatively related to the transportation cost. The equation demonstrates that when the homicide rate increases beyond a threshold level (Hr > l),the bid-rent shifts down:⁴

It is possible to demonstrate that the function decreases at an increasing rate, which means that the second derivative is positive. ^{Hellman and Naroff (1979)} argued that the impact on residential property values at any location j is the combined effect of a decrease in bid rents and a resulting decline in density. Then, the total residential property value at any location j is given by:

Taking the log of both sides of the equation resulted in an equation where housing price was a linear function of income, distance from the Central Business District (CBD), and the homicide rate. In the literature, the value of the coefficient of the homicide rate variable represents the elasticity of housing prices regarding the homicide rate. Considering the model developed by ^{Hellman and Naroff (1979)} and following ^{Thaler (1978)}, the econometric model can be estimated as follows:

where V _ij . is the logarithm of housing price i in a neighbourhood j; μ is the constant term; Hr _j is the homicide rate in the neighbourhood j; Q _ik is the attribute k of a house i; β _k is the estimated price associated with each attribute k; and u _ij .. is an error term. Indeed, ^{Thaler (1978)} argued that the willingness to pay for housing depends not only on the structure and land, but also on the surrounding neighbourhood. ^{Ridker and Henning (1967)} estimated the effect of air pollution on property values with a similar econometric specification, and ^{Gibbons (2004)} and ^{Tita et al. (2006)} evaluated the cost of urban property crime in London and Columbus, Ohio, respectively. With these references in mind, we estimated the relationship between homicide rates and housing prices for the Colombian cities.

In section 3.2, for Cali, we use a cross section of cadastral housing prices from 2012 and the average homicide rates from 2000-2010 at the neighbourhood level. Due to data availability, it was possible to estimate two model specifications, one was similar to those from ^{Thaler (1978)}, ^{Gibbons (2004)} and ^{Tita et al. (2006)}, and the second was an econometric strategy developed following ^{Buonanno et al. (2013)}. In section 3.3, for Bogotá D.C., we used information about lease values and housing characteristics registered in the Encuesta Multipropósito (2017), and information of crime from 2015 registered in the INML.

HOMICIDE RATES AND HOUSING PRICES IN CALI

The literature mentioned that it is possible to estimate standard errors of the coefficients using an Ordinary Least Square (OLS) equation, assuming that all observations in the database are unrelated. However, the correct standard error estimation procedure was given by the underlying structure of the data. Indeed, some economic phenomena did not affect observations individually, but they affected groups of observations within each cluster. In our case, it was more appropriate to use clusters at neighbourhood level. This resulted in a data structure where unob-servable elements of housing prices within a cluster were correlated, while they were uncorrelated across clusters. This kind of correlation occurs because some regressors take the same value for all observations within clusters. In fact, Hr _j is perfectly correlated within neighbourhoods (^{Cameron & Miller, 2015}).

Table 5 shows the results for different estimation strategies using data for Cali: OLS model; the heteroscedasticity correction suggested by ^{White (1980)}, which is known as having robust standard errors; cluster-robust standard errors;⁵ and a Random Effects (RE) model, where j represents clusters and i represents individual housing prices as follows:

Table 5 Dependent Variable: Log Housing Prices for Cali 

Note: t-statistics in parenthesis. Significant level: *** 1%; ** 5%; * 10%.

In the RE model, the estimation of standard errors is valid because it considers correlations of residuals within each cluster. Furthermore, if residuals α _j . are asymptotically correlated with explanatory variables, coefficients of the RE model tend to coefficients of a fixed-effects model. Consequently, the correlation between random effects and explanatory variables is corrected. Therefore, a consistent estimate with N observations was obtained.⁶

We found a significant negative relationship between housing prices and homicide rates. The elasticity is around -0.24, which means that a 10% increase in the homicide rate is related to a 2.4% decrease in housing prices. The reported elasticity was calculated using the estimated coefficient of the model as follows:

where E represents the elasticity for the housing price; _Hr is the estimated coefficient; and Hr is the homicide rate variable.

The coefficient of the area variable was significant and positive. In contrast, the coefficient of the squared area was significantly negative (this variable was included to control for nonlinear effects). As an additional control variable, we included a geological variable called liquefaction risk. Geological variables such as soil composition, rock depth, water capacity, soil erodibility, and seismic and landslide hazard have been used in the literature about urban structures (^{Combes et al., 2010}; Combes & Gobillon, 2015; ^{Rosenthal & Strange, 2008}).

The soil characteristics were important to be able to identify original settlements, and they have been determinant in developing agglomeration processes in those areas. Liquefaction risk refers to the strength and hardness of the soil when it is affected by earthquakes. The risk is present in areas where the soil is saturated with water and then acts like a liquid when shaken by an earthquake. Earthquake waves cause water pressure to increase in the sediment, so sand grains lose contact with each other, and the soil loses its ability to support high buildings. Nobody would be willing to pay the same price for two equivalent houses when one is in an at-risk area and the other is not. The coefficient of the liquefaction risk variable was significantly negative. We included the distance to main roads, and we also found a significantly negative coefficient.

The results reported in Table 5 reveal that in the presence of correlation within each cluster (neighbourhood), the OLS standard errors can overvalue the estimator precision. The literature mentioned that when there is a large number of clusters, statistical inference after OLS should be based on cluster-robust standard errors (^{Cameron & Miller, 2015}).⁷ An alternative to this methodology is to estimate a RE model because it considers the within dependency and, in this way, provides efficient estimates. Furthermore, the RE model is consistent in N even in when there is correlation between the random effects and the explanatory variables.

With the estimated coefficients, the increase in housing prices when the homicide rate diminishes can be simulated. In Figure 7, the horizontal axis measures the homicide rate, while the vertical axis indicates the average (percentage) increase of a housing prices index. In this graph, 100 is the observed value of the housing prices for the actual homicide rate of 64.45. As the graph shows, when the homicide rate is "zero", the estimated real estate value (property value in the form of land and buildings) reaches 122.64- a 22.64% increase-which can be interpreted as an increase in people's welfare.

Figure 7 Estimated Housing Prices for Different Values of the Homicide Rate 

AN ALTERNATIVE ESTIMATION: HEDONIC PRICES

A standard method to estimate the effect of crime on housing prices is to use a two-stage model. In the first stage, we estimated the shadow price of the location of houses and in the second, we tested if crime could explain some of the variability of the estimated locational valuation (Buonanno, et al., 2012; ^{Gibbons, 2004}; ^{Thaler, 1978}; and others). Table 6 displays the estimation of the hedonic housing price regarding the size of the property (area in square meters and the square of the area to include nonlinear effects) to obtain the hedonic housing price at the neighbourhood level (see Figure 8).

Table 6 Hedonic First Stage: Dependent Variable Log Housing Prices 

Note: Robust t-statistics in parenthesis. Significant level: *** 1%; ** 5%; * 10%

Figure 8 Hedonic Housing Prices at the Neighbourhood Level 

Figure 9 Homicide Rates at Neighbourhood Level 

In the second stage, we tested if the homicide rate explains some of the variability of the hedonic price at neighbourhood level. It was difficult to find exogenous variables to explain housing prices. We used liquefaction risk as a control variable. As we argued in the previous section, nobody would be willing to pay the same price for two equivalent residences when one is located in an at-risk area and the other is in an area with no risk, so we expected a negative coefficient.

Figure 9 shows the distribution of homicide rates at the neighbourhood level up to 400. However, there are several neighbourhoods with homicide rates of 500, 1000, and 1628. Table 7 exhibits the second stage estimation. The estimated regression excluded neighbourhoods with more than 400, 300, 200, and 100 homicides per 100,000 inhabitants to check whether the coefficient was stable across the distribution of the variable. The number of neighbourhoods was reduced in each econometric estimation. We controlled for liquefaction risk, and the coefficient associated with this variable was negative as expected.

In Table 8, we report the "price-homicide" elasticity derived from the hedonic second stage estimation:

Table 8 Housing Prices, Homicide Rate, and Elasticity by Strata 

Source: Cadastral housing prices of 2012 and Observatorio Social de Cali, own calculations.

where E _h represents the elasticity for the housing price for a neighbourhood in strata; h; _Hr is the estimated coefficient; and H _r is the homicide rate.

We found that a 10% increase in the homicide rate was related to a 2% decrease in housing prices, excluding neighbourhoods with homicide rates higher than 400. The decrease in housing prices is 2.1%, 2.5%, and 2.1%, excluding neighbourhoods with homicide rates of more than 300, 200, and 100, respectively. However, as mentioned by some of the literature, the homicide rate has higher consequences in lower socioeconomic strata neighbourhoods. For that reason, we differentiated results by socioeconomic strata.

In Colombia, residential areas have been divided into six socioeconomic status (strata) since 1988. The stratification system, where 1 is the lowest strata, and 6 is the highest, divides the city into areas of wealth and poverty. With this socioeconomic stratification system, the public administration guarantees that the upper strata pay a higher rate for public utilities (electricity, water, and sewage) to subsidize the cost of those utilities for the lower strata. Thus, the stratification makes low-income families settle in areas where they can afford to pay housing and basic services. Income-based class division explicitly categorized via public policy is strange. However, in our case, we used it to differentiate the negative effect of the homicide rate on housing prices according to different wealth levels because the stratification classifies neighbourhoods with similar characteristics. The second column in Table 8 reveals that around 65% of dwellings in Cali are in strata 1, 2, and 3.

The third column in Table 8 presents the average price of the square meter by strata; column 4 shows the homicide rate per 100,000 inhabitants (it is clear that the rate is higher in lower strata). The last four columns indicate that the price-homicide rate elasticity is more negative in lower strata. In any case, the fact that the elasticity decreases with strata may be the result of the definition of elasticity: (percentage change price) / (percentage change homicide rate). This situation implies that if the homicide rate goes from 1 to 2 in a wealthy neighbourhood, the percentage change is 100%. If the homicide rate goes from 100 to 150 in an impoverished neighbourhood, the percentage of variation is 50%. Lastly, if the homicide rate goes from 0 to 1 in a wealthy neighbourhood, the percentage of variation is infinite, so that elasticity must be zero.

In Colombia, crime is constant across the income distribution. For that reason, the rich are more likely to adopt costly protective behaviour and neighbourhood watch programmes, install anti-theft devices at home, hire private security personnel or migrate (^{Gaviria & Vélez, 2002}). In fact, in Brazil, homicide victimization is also more common in lower socioeconomic strata (^{Soares, 2006}). Also, in Argentina, most of the increase in burglary rates were shouldered by the poor, since the rich were able to adopt effective protective strategies (^{Di Tella et al., 2010}).

HOMICIDE RATES AND HOUSING PRICES IN BOGOTÁ D. C.

As we previously mentioned, for Bogotá D.C., we used the housing characteristics information and two dependent variables: the hypothetical lease value and the effective lease value. The information comes from the Encuesta Multipropósito (2017), and it is possible to control for different characteristics of each house. The information about the number of homicides corresponds to the data registered in the INML (2015). The estimations reported in Table 9 allow it to be concluded that a 10% increase in the homicide rate is related with a 1.8% decrease in the hypothetical lease value and a decrease of 1.7% in the effective lease value.

Table 9 Models for Bogotá D.C.: Dependent Variable Hypothetical Lease Value (HLV) and Effective Lease Value (ELV) 

Note: Robust t-statistics in parenthesis. Significant level: *** 1%; ** 5%; * 10%

Source: Encuesta Multipropósito (2017), own calculations.

In these regression models, we included an accessibility indicator, z _s , to consider the distribution of employment within the city. This indicator is calculated as a weighted mean of the number of jobs at locations h, with weights that are a decreasing function of distance between s and h (^{Koster & Rouwendal, 2013}; ^{Lucas & Rossi-Hansberg, 2002}). To be specific, z _s is defined as follows:

where J _h denotes the number of jobs at location h, d _sh denotes the distance between locations s and h, and δ > 0 is a given decay parameter. The external effect is more localized for higher values of δ, which implies that for a company the value of establishing itself close to other producers is also higher. As we mentioned, it is important to include the employment gradient of the urban structure in this variable.

The regression models include control variables, for instance, if the housing is an apartment, control variables would be the number of people living in the house, the strata, track quality, and variables of the house quality. The complete regressions are presented in Tables 4A and 4B (Appendix D). Using, quantitative methodologies, we revealed a negative relationship between homicide rates and housing prices in Cali and Bogotá D.C. As we previously mentioned, studying this relationship is essential to quantify crime in terms of its representation in property values, property tax revenues, and family heritage.

A COMMENT ABOUT ENDOGENEITY

The literature mentioned that problems of endogeneity arise through the correlation between the regressors and the random disturbances. Consequently, treating the crime rate as an exogenous variable may result in a biased estimated elasticity in a regression to explain housing prices. This bias appears because crime occurs disproportionately in more impoverished neighbourhoods, with low housing prices, or conversely if criminals target areas with high housing prices. In both cases, the neighbours' behaviour will depend on their individual characteristics, and these may well be systematically related to unobserved housing prices determinants (^{Buonanno et al., 2013}; ^{Gibbons, 2004}). We may infer a causal relationship between local characteristics and housing prices, when, in fact, the unobserved component is the one that drives neighbourhood characteristics.

Estimating the impact of crime on housing prices is empirically challenging because of omitted variables. Furthermore, it is difficult to find valid instruments for crime. To mention several examples, ^{Gibbons (2004)} used instruments such as crimes on non-residential properties, spatial lags of crime, or the distance to the nearest public house or wine bar. Buonnano et al. (2012) used a 20-year lag in the victimization rate and the percentage of youth aged between 15 and 24 as instruments of crime rates. Nevertheless, it is hard to instrument murders.

In the present paper, the explanatory variable referred to homicides, but not to theft or crime in general. Homicide and theft could be correlated, but motives for homicide may differ from the motivations of theft. Indeed, the propensity to report a theft varies with the severity of the incidence. Faced with the difficulty of finding valid instruments, we tried to address endogeneity problems by including homicide rates and all the variables that may provoke the correlation between regressors and random disturbances on the right-hand side of the equation lags.