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Crime under-reporting in Bogotá: a spatial panel model with fixed effects

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Abstract

We examine spatial spillovers in violent crime and its under-reporting in Bogotá, Colombia, using a cuadrante (quadrant) level data. To model spatial spillovers, we use a spatial panel model with fixed effects, and to address under-reporting, we use the stochastic frontier approach as a tool. The novel statistical approach is combined with a database of police-reported crimes in Bogotá to examine how influential surrounding areas with high criminal offenses are on crime (under)reporting. The results suggest that spatial correlations are highly significant and that under-reporting is mainly related to interactions with other localities, which have important public policy implications.

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Data availability statements

The crime data that support the findings of this study were obtained from La Secretaría Distrital de Seguridad, Convivencia y Justicia de Bogotá, https://scj.gov.co/es/oficina-oaiee/estadisticas-mapas and Datos Abiertos, https://datos.gob.cl/. Luis Sandoval accessed to the data in 2019.

Notes

  1. Crime is a critical issue in most developing countries. For instance, the United Nations Office on Drugs and Crime reported that in 2017 Latin America accounted for about 33% of the homicides in the world, while its population was only about 9%. Pessino et al. (2018) presents a review of the challenges that this issue puts on public spending in the region.

  2. Glaeser et al. (1996) state that this is perhaps one of the most interesting aspects of crime.

  3. To illustrate, Donohue and Levitt (2001, 2019) study whether the legalization of abortion and some other socioeconomic variables explain the reductions in crime in the USA; or, more recently, Higney et al. (2022) review the role of lead pollution.

  4. It is often argued that SAR production models violate axioms of the production function. One can produce more without increasing its use of inputs and/or increasing efficiency—simply because the neighboring producers are producing more, and there is a positive feedback/spillover effect. There is no such problem in the spatial crime model because crimes of a locality can go up (down) if crimes in the neighboring localities go up (down), ceteris paribus.

  5. Although we wrote two equations (viz. (4) and (5)), they are connected through \(\varepsilon \) and are part of the same equation in (3) and for this is why the same \(\xi \) in both. Also two separate spatial parameters in (4) and (5) could not be identified.

  6. A general nesting spatial (GNS) econometric model may be a more flexible specification to model the spatial relations because it allows for spatial dependency in all the different variables (e.g., the dependent variable, the independent variables, and/or the error term). Thus, although our model is flexible, it may still miss spatial dependency in the explanatory variables. However, we do not find any compelling reason for including spatially lagged covariates in the present application. Also, in empirical applications using fine-gridded data, as we do in this paper by analyzing cuadrantes finding the information for spatial dependence in the covariates is challenging. Nonetheless, methodologically speaking, the proposed spatial modeling approach we present for studying spillovers could naturally be extended to also include, for instance, spatially lagged explanatory variables, making the specification more general. For this, one needs to find economic reasons for including spatially lagged explanatory variables and better data sets.

  7. To illustrate, after the transformation, a correlation matrix emerges over various time spans, posing computational complexities for the application of the ML method.

  8. Note that \({\mathbb {E}}(\varepsilon _{it}^{*})=0\), although \(u_{it}\) follows the SMA structure in (4).

  9. Perhaps this discussion is more appropriate for a review in the spatial econometrics literature. Here we emphasize that an advantage of our first step is that one can recover estimates using standard spatial models without having to deal with under-reporting and distributional assumptions associated with it and the noise term.

  10. The classical weighting scheme in wild bootstrap. To illustrate, following Mammen (1993), \(\eta _{it}\) is equal to \(-(\sqrt{5}-1)/2\) with probability \((\sqrt{5}+1)/(2\sqrt{5})\) and \((\sqrt{5}+1)/2\) with probability \((\sqrt{5}-1)/(2\sqrt{5})\).

  11. Kutlu (2018) and Kutlu et al. (2020) state that the approach in Glass et al. (2016) may be highly sensitive to outliers, proposing the use of the share approach as a more intuitive and robust to an outlier.

  12. Note that there are zero values for each crime type. So one cannot use a log transformation of y. To deal with zero values, we use a transformed dependent variable, the inverse hyperbolic sine (IHS) of the number of reported crimes, y. The IHS transformation of y is \(\ln (y+\sqrt{1+y^2})\).

  13. There is no personal information in the data that may reveal a victim’s identity.

  14. Data from La Secretaría Distrital de Seguridad, Convivencia y Justicia de Bogotá, https://scj.gov.co/es/oficina-oaiee/estadisticas-mapas and Datos Abiertos, https://datos.gob.cl/; accessed in 2019.

  15. To check the quality of our dataset, we compute some aggregated statistics using larger geographical areas, such as Localidades When figures in our dataset do not match other governmental records (e.g., reports by the Secretaría de Seguridad, Convivencia y Justicia de Bogotá, retrieved from: scj.gov.co/en/oficina-oaiee/boletines; accessed: January 2021), we report a missing value in our data. Therefore, in the case of personal injuries, we dropped the observations for 2014. For robbery and residential burglary the period was reduced to 2013–2018.

  16. There may be permanent under-reporting which will be captured by the fixed effects. It is not possible to identify time-invariant (permanent) under-reporting from time-invariant fixed effects. See Greene (2005) and Chen et al. (2014).

  17. We used three software: ArcGIS, Stata, and R. The first two software, ArcGIS and Stata, were used to create the spatial matrices and manage the (raw) crime reports. We then, taking the spatial matrices and the dataset as inputs, code and estimate the econometric model using R. Some R commands involved, used for instance to compare results with other standard models in the literature, are the spgm that is part of the splm package (version 1.6-2) (Millo and Piras 2012), the stsls that is part of the spdep package (version 1.2-8), and the spatialreg package (version 1.2-9) (Pebesma and Bivand 2023). To facilitate replication and empirical implementation of our proposed model, we have made available an online Jupyter notebook with the central parts of the R code we developed. It also includes the main results.

  18. The figure provides an illustration rather than a precise scaled map.

  19. The statistics on the table are based on analytical solutions. In this case, the use of a Monte Carlo simulation approach generates similar results (very small p values of about 0.001) for all crimes.

  20. Note that we are modeling under-reporting as a MA process. In such a case, there are no requirements for the coefficient to be below one in absolute value.

  21. Under-reporting for each crime is measured relative to a benchmark (frontier), which is estimated separately for each crime and therefore varies among crimes. As a result, comparing under-reporting across different crimes is not recommended. In this context, comparisons should be made among localidades considering each crime separately. The same principle applies to making cross-country comparisons of, for example, banking efficiency when frontier estimation is conducted separately for each country.

  22. This model combines the similarity of under-reporting with locational similarity and endogenously determines the number of regions. Because of space constraint, we decided not to delve into max-p further.

  23. In the case of sexual assault, we notice that the coefficient is somehow larger than one. GMM estimates may produce this kind of result. However, in this case, the standard errors are large enough to consider that the estimate is still in a reasonable range.

  24. The most recent surveys are also representative at the UPZ level, a smaller geographical area, but creating panel data with this unit is not feasible yet.

  25. We use those localidades that are well-defined across the different surveys.

  26. The transformation approach of using the eigenvector matrix implies losing one year’s observations. Reducing the sample size in this way and including time effects in the model raises multicollinearity issues due to the small sample size. Nonetheless, we checked for the effect of using the within transformation in the full dataset of cuadrantes and results in Table 2 were unaffected.

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Acknowledgements

We thank Tomás Berríos and Jorge Lobos for excellent research assistance. We thank seminar participants at the Society for Economic Measurement (SEM) 2022 conference and the Latin American and Caribbean Economic Association (LACEA) 2022 Annual Meeting for helpful comments. We thank the Associate Editor and two anonymous referee for their constructive comments.

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Authors and Affiliations

  1. Escuela de Ingeniería Comercial, Facultad de Economía y Negocios, Universidad Santo Tomás, Santiago, Chile

    Luis Chanci

  2. Binghamton University, Binghamton, NY, USA

    Subal C. Kumbhakar

  3. Czech University of Life Sciences Prague, Prague, Czech Republic

    Subal C. Kumbhakar

  4. Universidad Militar Nueva Granada, Bogotá, Colombia

    Luis Sandoval

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  1. Luis Chanci
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  2. Subal C. Kumbhakar
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  3. Luis Sandoval
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Contributions

Luis Chanci contributed to conceptualization, methodology, writing original draft preparation, writing review and editing, and software. Subal C. Kumbhakar contributed to conceptualization, methodology, writing original draft preparation, and writing review and editing. Luis Sandoval performed data curation.

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Correspondence to Luis Chanci.

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Competing interests

Luis Chanci acknowledges the financial support from Universidad Santo Tomás, Chile, proyecto interno de investigación “Spillovers and Efficiency: A Spatial Autoregressive Stochastic Frontier Panel Data Model with Fixed Effects” in 2022. Luis Sandoval acknowledges the financial support from Universidad Militar Nueva Granada, project INV ECO 3170 “Temporal and spatial changes in the measurement of crime in Bogotá during 2010-2020.”

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Chanci, L., Kumbhakar, S.C. & Sandoval, L. Crime under-reporting in Bogotá: a spatial panel model with fixed effects. Empir Econ 66, 2105–2136 (2024). https://doi.org/10.1007/s00181-023-02517-4

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