Spatial interaction of crime incidents in Japan

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 78 (2008) 276–282

Spatial interaction of crime incidents in Japan Kazuhiko Kakamu a,∗ , Wolfgang Polasek b , Hajime Wago c b c

a Faculty of Law and Economics, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, 263-8522, Japan Department of Economics & Finance, Institute for Advanced Studies, Stumpergasse 56, 1060 Vienna, Austria Department of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8555, Japan

Available online 17 January 2008

Abstract We analyze the development of 18 types of criminal records in Japan for the period 1991–2001 across 47 prefectures with spatial lag and spatio-temporal heteroscedasticity. We explore the hypothesis that crime data are related to socio-economic variables in Japan. We extend the Bayesian approach of LeSage [J.P. LeSage, Bayesian estimation of spatial autoregressive models, Int. Regional Sci. Rev. 20 (1997) 113–129] for spatio-temporal Bayesian models. Additionally we analyze unobserved heteroscedasticity in the panel model by variance inflation factors as in Geweke [J. Geweke, Bayesian treatment of the independent Student-t linear model, J. Appl. Econ. 8 (1993) 19–40]. Positive and significant spatial dependencies can be found for 12 types of crimes and the influence of the socio-economic variables varies over the type of crimes. © 2008 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Crime incidents; Heteroscedasticity; Markov chain Monte Carlo (MCMC); Panel data; Spatial autoregressive models (SAR)

1. Introduction Since the seminal work of Becker [2], a large empirical literature has developed around the estimation and testing of economic models of crime. Recently, spatial interactions of crime incidents have been analyzed in economics in connection with the progress of spatial econometric estimation techniques, e.g. Anselin [1]. But these models of crime are based on aggregated data and use simple cross-sectional econometric techniques, with no attempt do not control for unobserved heterogeneity. Cornwell and Trumbull [4] show that panel data lead to a large model class for criminal data modeling.1 Although new methods have been developed for spatial panel models, e.g. Elhorst [5], homoscedasticity is mainly assumed in these models. Now, with the progress of Bayesian techniques such models with heteroscedastic variances can be estimated for panel data. For example, LeSage [9] proposes the Bayesian estimation of spatial autoregressive models (SAR) with heteroscedasticity. In this paper, we extend the approach by LeSage [9] to Bayesian panel spatial autoregressive models, using a spatio-temporal heteroscedasticity approach. We apply the methods to 18 types of criminal time series in Japan for the period 1991–2001 across 47 prefectures. As explanatory variables we use income, the unemployment rate, the number of registered foreigners and the police force. We also use the (lagged) arrest rate, since the occurrence of crimes might be related to the ∗

Corresponding author. E-mail addresses: [email protected] (K. Kakamu), [email protected] (W. Polasek), [email protected] (H. Wago). 1 This approach differs from our approach in so far as they also focus on simultaneity. But the authors address the fact that the panel data might be modeling loss for criminal models. 0378-4754/$32.00 © 2008 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2008.01.019

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apprehension rate. Additionally, a potential relationship to tourism is modeled by a proxy variable counting hotel nights. The empirical results suggest three major trends: (1) we can confirm the spatial interaction for 12 types of crime incidents. (2) The criminal rate increase when the unemployment rate, the Gross Regional Product (GRP) per capita, the number of foreigners (per capita) and the police force (per capita) are high (across prefectures) and becomes low when the arrest rate is high. And finally: (3) the relative variance of metropolitan areas like Tokyo and Osaka is much larger than in other prefectures and additionally the relative variances in metropolitan areas increase over time. This paper is organized as follows. In the next section, we introduce the Bayesian heteroscedastic panel SAR model to analyse the spatial interactions of crime incidents. Section 3 derives the full conditional distributions for our model. Section 4 presents the empirical results based on 18 types of criminal records in Japan from 1990 to 2001. Section 5 summarizes the results with concluding remarks. 2. The panel spatial autoregressive model with heteroscedasticity Let yit and xit for i = 1, . . . , N, t = 1, . . . , T denote dependent and independent variables, where xit is a 1 × k vector on ith unit and t th period, respectively. Also let wij denote the spatial weight on jth unit with respect to ith unit. Then, the panel spatial autoregressive model with heteroscedasticity conditional on parameters αi , β, ρ, σ 2 and vit is written as follows; yit = αi + xit β +

N 

ρwij yjt + it ,

it ∼ N(0, σ 2 vit ).

(1)

j=1

Let α = (α1 , . . . , αN ) and θ = (α , β ) , then the likelihood function of model (1) is written as follows:   −1  eV e 2 2 −NT/2 T −1/2 , L(Y |θ, ρ, σ , V, Z, W) = (2πσ ) |IN − ρW| |V | exp − 2σ 2

(2)

where Y = (Y1 , . . . , YT ) , Yt = (y1t , . . . , yNt ) , Z = (Z1 , . . . , ZT ) , Zt = (IN , Xt ), IN is N × N unit matrix, Xt =  , . . . , x ) , V = diag(V . . . , V ), V = diag(v , . . . , v ), W denotes weight matrix (e.g. Anselin [1]) and e = (x1t 1 T t 1t Nt Nt Y − ρ(IT ⊗ W)Y − Zθ. While this model could be estimated using maximum likelihood method like, e.g. in Elhorst [5], if V is known, the following Bayesian approach allows a numerical analysis of the joint parameter distribution including the estimation of the variance factors vit (i = 1, . . . , N, t = 1, . . . , T ) by simulating the posterior distribution of the model. 3. Posterior analysis 3.1. The joint posterior distribution Given a prior density p(θ, ρ, σ 2 , V ) and the likelihood function given in (2), the joint posterior distribution can be expressed as p(θ, ρ, σ 2 , V |Y, Z, W) = p(θ, ρ, σ 2 , V )L(Y |θ, ρ, σ 2 , V, Z, W).

(3)

Since we adopt a Bayesian approach, we complete the model by specifying the prior distribution for the parameters. all the relevant variance parameters for i = 1, . . . , N, Following Geweke [7], a hierarchical prior p(v−1 it ) is assigned for   −1 t = 1, . . . , T and we assume p(θ, ρ, σ 2 , V ) = p(θ)p(ρ)p(σ 2 ) Tt=1 N i=1 p(vit ). Finally, we give the following prior distributions: p(θ) ∼ N(θ∗ , ∗ ),

p(σ 2 ) ∼ G−1 (ν∗ /2, λ∗ /2),

−1 −1 2 p(ρ) ∼ U(λ−1 min , λmax ), p(vit ) ∼ χ (q∗ )/q∗ ,

for

i = 1, . . . , N, t = 1, . . . , T,

where G−1 (a, b) denotes an inverse gamma distribution with scale and shape parameters a and b. λmin and λmax denote the minimum and maximum eigenvalue of W, respectively. As is shown in Sun et al. [11], we have λ−1 min < 0 and −1 −1 −1 λmax > 0 and the spatial ρ must lie in this interval. Therefore, we restrict the prior space to ρ ∈ (λmin , λmax ).

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3.2. Posterior simulation Since the joint posterior distribution is given by (3) is much simplified, we can now use MCMC methods. The Markov chain sampling scheme can be constructed from the full conditional distributions of θ, ρ, σ 2 and vit for i = 1, . . . , N, t = 1, . . . , T . 3.2.1. Sampling θ and σ 2 Let ρ be given, then Y¯ = Y − ρ(IT ⊗ W)Y becomes a constant, and (1) is reduced to a linear regression model. Therefore, the full conditional distributions for θ and σ 2 can be obtained as θ|ρ, σ 2 , V, Y, Z, W ∼ N(θ∗∗ , ∗∗ ), σ 2 |θ, ρ, V, Y, Z, W ∼ G−1 (ν∗∗ /2, λ∗∗ /2), −2  −1 Z + −1 )−1 , ν = NT + ν and λ = e V −1 e + λ where θ∗∗ = ∗∗ (σ −2 Z V −1 Y¯ + −1 ∗∗ ∗ ∗∗ ∗ ∗ θ∗ ), ∗∗ = (σ Z V ∗ like the Gibbs sampling in Gelfand and Smith [6].

3.2.2. Sampling ρ From the joint posterior distribution (3), the full conditional distribution of ρ is proportional to   −1  eV e , p(ρ|θ, σ 2 , V, Y, Z, W) ∝ |IN − ρW|T exp − 2σ 2

(4)

which cannot be sampled by standard methods (e.g. LeSage [10]). Therefore, we use the Metropolis algorithm (see, e.g. Tierney [12]). The following Metropolis step is used: sample ρnew from ρnew = ρold + cφ,

φ ∼ N(0, 1),

where the scaler c is called tuning parameter and ρold is the parameter of the previous sampling. Next, we evaluate the acceptance probability   p(ρnew ) , 1 , α(ρold , ρnew ) = min p(ρold ) where p is the full conditional distribution in (4). Finally we set ρ = ρnew with probability α(ρold , ρnew ), otherwise ρ = ρold . The scalar c is tuned to produce an acceptance rate between 40 and 60% as is suggested in Holloway et al. [8]. −1 It should be mentioned that the proposal density of ρt is not truncated to the interval (λ−1 min , λmax ) since the constraint is part of the target density. Thus, if the proposal value of ρ is not within the interval, the conditional posterior is zero, and the proposal value is rejected with probability one (see Chib and Greenberg [3]). 3.2.3. Sampling vit for i = 1, . . . , N, t = 1, . . . , T From the joint posterior distribution (3), we find 2 2 −2 2 v−1 it |θ, ρ, σ , V−it , Y, Z, W ∼ χ (q∗ + 1)/(σ eit + q∗ ),

where V−it denotes V without the it th element. The (positive) hyper-parameter q∗ controls the amount of dispersion in the vit across observations. Alternative values for this parameter produces different families of prior densities for vit : small values of q∗ produce leptokurtic distributions and large values (q∗ → ∞) imply homoscedasticity (see Geweke [7] in detail). 4. Empirical results 4.1. The available data set First we would like to explain the data set used in this paper. The criminal records are obtained from the (annual reports on) Crime Statistics prepared by the National Police Agency of Japan, which reports 18 types of crimes. The

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Table 1 Types of crimes Symbol

Explanation

MURDER ROBBERY

To kill a person intentionally. To make a third party deprive others of valuable things by using assault or threat, to obtain unlawful profits or to obtain property. Crime that sets fire by intention and burns buildings, etc. Raping a woman of 13 years or older. Crimes in which a lethal weapon is used so that two or more people may jointly harm other people’s lives, bodies, and properties or the preparation for this. To exercise physical power on the bodies of others. (The case where it does not become injury.) Injuries to other people’s bodies by intention. Crime that involves the threat to harm other people’s (or their relatives’) bodies, lives, freedom, honor, and properties. To make a third party be afraid in order to obtain unlawful profit or property. To steel other people’s properties. Crime to make the third person’s property delivered or to obtain profit out of unlawfulness. The fraudulent conversion of property (or lost articles, etc.) of another by a person in lawful possession of that property (or the benefits thereof). Currency, documents, securities, and stamps are counterfeited on purpose. Crime that abuses official power and position, and approves of illegal acts by taking bribes. Crime that involves a malfeasance act so that the one to process the clerical work for others may measure the third person’s profit or add damage to the person in question, and causes damage to the third person’s property. Crime to obtain merits and demerits of properties (games of luck). Crime that does indecent (sexual) acts in public. Other crimes.

ARSON RAPE ASSEMBLY VIOLENCE INJURY THREAT1 THREAT2 THEFT FRAUD EMBEZZLE FORGERY GRAFT TRUST GAMBLING OBSCENITY OTHERS

types of crimes and the details are listed in Table 1. In this paper, we explore the hypothesis that crime data are related to socio-economic variables, like, e.g. income, unemployment, foreigners and the size of the police force. For income, we use the Gross Regional Product (GRP) from Vital Statistics prepared by the Cabinet Office of Japan, for the unemployment rate (UNEMP) the Basic Survey on Wage Structure prepared by the Ministry of Health, Labour and Welfare of Japan, for the registered foreigners (FOREIGN) we use Statistics on Immigration Control prepared by the Ministry of Justice of Japan and for the number of policemen (POLICE) the Survey on Wages of Local Government Employees prepared by the Ministry of Internal Affairs and Communications of Japan. Crimes might also depend on the arrest rate (ARREST),which is calculated as the number of arrests by criminal records. Additionally, the relationship to tourism is explored. Since there are only a few statistics available on tourism in Japan, we use the number of hotels (HOTEL) from the Reports of Health Administration prepared by the Ministry of Health, Labour and Welfare of Japan as a proxy variable for tourism. The crime records (dependent variable) are obtained for the period 1991–2001. Avoiding the causality problem, the lagged variables are used for POLICE and ARREST. That is, these (independent) variables are obtained for the period 1990 to 2000 across 47 prefectures and the other independent variables are obtained for the period 1991–2001. Following the previous literature we use standardized (per capita) data: all dependent and independent variables except the UNEMP and ARREST variables are divided by the population of each prefecture (from the Vital Statistics of Japan provided by the Ministry of Health, Labour and Welfare). As the spatial weight matrix W, we use the contiguity dummy variables (see Anselin [1]). All except one (Okinawa) of the Japanese prefectures are situated on the four major islands, Hokkaido, Honshu, Shikoku and Kyushu. But these four islands are connected by train and roads, despite the fact that islands are separate geographical entities. For example, the most northern island Hokkaido is connected by the Seikan railway tunnel to Honshu. And Honshu is connected by the Awaji and Seto bridge to Shikoku, and the southern island of Kyushu is also connected by the Kanmon tunnel and bridge to Honshu. Therefore, considering such connections we have built a contiguity weight matrix where Okinawa is the only prefecture which is independent of all other prefectures. 4.2. Estimation First of all, for the prior distributions, we have fixed the hyper-parameters in the following way: θ∗ = 0,

∗ = 100 · IN+k ,

ν∗ = 0.01,

λ∗ = 0.01,

q∗ = 5.

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Table 2 Empirical results: posterior means and standard deviations (in parentheses)

UNEMP GRP POLICE FOREIGN HOTEL ARREST ρ σ2

MURDER

ROBBERY

ARSON

RAPE

ASSEMBLY

VIOLENCE

−0.021 (0.904) −0.192 (0.551) 2.208∗ (0.459) 7.099∗ (2.511) 4.475 (8.108) −0.130 (0.090) 0.006 (0.052) 0.021 (0.002)

9.876∗ (2.634) −3.651∗ (1.588) 16.919∗ (1.558) 6.322 (6.987) 14.171 (9.615) −0.048 (0.117) 0.057∗ (0.014) 0.176 (0.019)

1.142 (2.099) −1.164 (1.433) 6.320∗ (1.403) 12.724∗ (4.923) 1.577 (9.509) −0.062 (0.131) 0.004 (0.028) 0.141 (0.014)

3.845 (1.992) −0.146 (0.900) 1.563 (1.356) 6.780 (4.722) 3.548 (9.748) −0.202 (0.128) 0.091∗ (0.038) 0.092 (0.008)

−0.150 (0.150) 0.181∗ (0.079) 0.000 (0.049) −1.024∗ (0.451) 0.799 (1.934) 0.004 (0.014) 0.056 (0.036) 0.001 (0.000)

−0.644 (6.037) 3.321 (2.311) 23.759∗ (3.016) 18.167∗ (8.886) 0.053 (10.104) −2.974∗ (0.684) 0.105∗ (0.021) 1.405 (0.158)

R2

0.992 INJURY

0.989 THREAT1

0.957 THREAT2

0.974 THEFT

UNEMP GRP POLICE FOREIGN HOTEL ARREST ρ σ2

16.703∗ (7.774) 25.276∗ (2.620) 11.742∗ (5.027) 20.745∗ (9.187) 2.199 (9.862) −7.001∗ (1.411) 0.206∗ (0.028) 3.912 (0.377)

4.421∗ (1.479) 0.290 (1.164) 3.170∗ (0.618) 11.187∗ (4.139) 9.698 (9.434) −0.018 (0.079) 0.160∗ (0.024) 0.057 (0.005)

12.153 (7.008) 7.304∗ (3.165) 34.680∗ (6.560) 14.826 (9.186) 3.760 (9.699) −1.683∗ (0.814) 0.165∗ (0.030) 2.878 (0.301)

0.372 (9.923) 13.946 (10.057) 4.120 (10.185) 2.964 (9.957) −0.018 (9.772) 3.931 (10.073) 0.363∗ (0.023) 904418.169 (85738.118)

R2

0.992 FORGERY

0.969 GRAFT

UNEMP GRP POLICE FOREIGN HOTEL ARREST ρ σ2

−33.072 (6.909) 7.346∗ (2.719) 24.715∗ (4.842) 23.370∗ (9.319) −4.925 (9.828) 0.631 (0.389) 0.031 (0.026) 2.881 (0.290)

−1.045∗ (0.531) −0.318 (0.232) 0.560∗ (0.184) 0.875 (1.021) 1.990 (6.334) 0.055∗ (0.025) 0.031 (0.037) 0.007 (0.001)

R2

0.969

0.548

0.988 TRUST −0.001 (0.207) −0.516∗ (0.164) −0.367∗ (0.112) 1.888∗ (0.583) 6.055∗ (3.062) −0.005 (0.009) 0.071 (0.043) 0.001 (0.000) 0.791

0.885 FRAUD −7.451 (9.324) 33.430∗ (3.487) 12.962 (8.041) 27.452∗ (9.609) −0.049 (9.911) 8.663∗ (1.529) 0.108∗ (0.023) 36.413 (3.489)

0.988 EMBEZZLE −5.896∗ (1.959) −4.079∗ (1.635) 4.398∗ (0.877) 24.352∗ (6.716) −6.508 (9.526) −0.091 (0.124) 0.121∗ (0.035) 0.112 (0.011)

0.456 GAMBLING

0.968 OBSCENITY

0.970 OTHERS

−6.408∗ (1.499) 0.303 (0.818) 0.717 (0.690) 0.682 (3.757) −21.314∗ (9.523) −0.005 (0.084) 0.153∗ (0.028) 0.055 (0.006)

17.166∗ (5.426) 4.650∗ (1.946) 17.314∗ (3.120) 2.823 (8.376) 5.741 (10.100) −1.809∗ (0.434) 0.096∗ (0.033) 1.063 (0.104)

7.065 (9.856) 143.180∗ (5.990) 75.175∗ (8.895) 31.424∗ (9.829) 0.976 (9.903) −20.518∗ (3.128) 0.117∗ (0.010) 661.210 (90.295)

0.915

0.981

0.959

Note: Asterisk (∗ ) means that the 95% credible interval does not include zero.

Next, we ran the MCMC algorithm for 2000 iterations following a burn-in phase of 1000 iterations. Table 2 shows the regression estimates for the 18 types of crimes. We have marked the significance of the estimated parameters, by indicating if zero is included in the 95% (posterior) credible interval or not. First of all, we notice that the spatial correlation coefficients are significant in 12 types of crimes (2/3 of the crime types). In addition, all the correlations in the 12 types of crimes are positive; this is similar to the results in Anselin [1]. Furthermore, high crime rates in the neighboring prefectures tend to moderate spill-overs into neighboring prefectures for 2/3 of the crimes. This shows a rather small spatial neighborhood dependence of crime incidents in Japan. 4.3. The socio-economic profiles of crimes In this subsection, we will discuss general aspects of the estimated models by looking at the explanatory variables. Recall that POLICE and ARREST are lagged by 1 year. The UNEMP variable: this variable is significant in 7 types of crimes. In all but 3 cases (EMBEZZLE, GRAFT and GAMBLING) of these 7 types of crimes, the coefficient is positive, implying that, negative job market conditions lead to more crimes in Japan. The GRP variable: this variable is significant in 10 types of crimes. In all but 3 cases (ROBBERY, EMBEZZLE and TRUST) of these 10 types of crimes, the coefficient is positive, implying that in contrast to UNEMP, richer regional economic conditions lead to more crimes in Japan.

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Fig. 1. Heteroscedastic variance factor of ROBBERY.

The POLICE variable: this variable is significant in 12 types of crimes. In all but 1 (TRUST) cases of the 11 types of crimes, the coefficient is positive, implying that high crime rates are associated with large police force. TRUST seems to be difficult to deterred by a large policy force. The FOREIGN: this variable is significant in 11 types of crimes. These are the same variables as for the GRP which indicates that the presence of foreigners in a prefecture in strongly correlated with the macroeconomic conditions. In all but 1 (ASSEMBLY) of the 11 types of crimes, the coefficient is positive, implying that – in the present model – a higher foreigner rate is connected with a higher crime rate. Clearly, such a correlation can be spurious, i.e. due to third effects influencing the crime and the foreigner rate at the same time. The HOTEL variable: this variable is significant in 2 types of crimes (TRUST and GAMBLING). However, the sign is quite different. This shows that the crime rate may not be connected to tourism. The ARREST variable: this variable is significant in 7 types of crimes. In all but 2 cases of the 7 types of crimes, the coefficient is negative. In these 5 types of crimes, the (lagged) arrest rate keeps the crime rate down. Surprisingly, positive and significant coefficients can be found for FRAUD and GRAFT which could indicate that these crimes are more resistant to deterrent effects of a high (general) arrest rate. Summarizing, we found the following tendencies: in general, negative job market and better macroeconomic conditions lead to more crimes in Japan. High crime rates are also associated with a (lagged) large police force. A higher foreigner rate is connected with a higher crime rate. In 6 types of crimes, the arrest rate helps to drive down the crime rate, but for FRAUD and GRAFT, the association is positive. Also it is important to mention that each type of crime has different features since the related socio-economic variables are different in each types of crimes.

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4.4. The variance factors Next, we have estimated the relative variances to analyze the heteroscedasticity effects. Fig. 1 shows the relative variance inflation factors for ROBBERY, which shows the typical pattern for the 18 types of crimes. The prefectures where the relative variances are large are the metropolitan areas of Tokyo and Osaka, together with their neighbors. Note that there exists the possibility that unobserved factors –not covered by our set of variables– could influence the estimation results. We also see from Fig. 1 that there exists something like a variance inflation trend over time in the prefectures with large variances, i.e. Tokyo and Osaka. These results indicate that we might need to consider eventually dynamic effects of the socio-economic variables for explaining crime trends in Japanese prefectures. 5. Concluding remarks We analyzed 18 types of crimes by a Bayesian spatial autoregressive panel model with heteroscedasticity, and found some interesting socio-economic profiles for crime incidents in Japan. The results indicate that (1) positive and significant spatial correlation is present in 12 types of crimes; (2) the crime rates of many crime types correlates positively with UNEMP, GRP, POLICE and FOREIGN and negatively with ARREST; (3) the relative variance of the crime rate in metropolitan areas like Tokyo and Osaka is much larger than in other prefectures and become larger over time. Finally, some open issues remain. In this paper, we concentrated on the estimation of spatio-temporal models with heteroscedasticity by an univariate approach. In future applications, we also want to consider correlations among crimes, because some kinds of crimes are recorded simultaneously. Thus, a more elaborate version of these spatio-temporal panel models could be estimated by a seemingly unrelated regression (SUR) model. Potentially, outliers could also be responsible for some of the unexpected results. Nevertheless, the results for the 18 type of crime rates present a first step for more elaborate econometric models to understand the socio-economic background of crimes. Acknowledgements We are indebted to Alan E. Gelfand, Hideo Kozumi and participants at the Workshop on recent advances in modeling spatio-temporal data at University of Southampton, Joint Statistical Meetings 2005 in Minneapolis and MODSIM05 in Melbourne. The research for this work was supported by Grant-in-aid for Scientific Research (A)(1) 15200022. This work was done while Kazuhiko Kakamu was staying at Institute for Advanced Studies. He gratefully acknowledges the hospitality of the school. References [1] L. Anselin, Spatial Econometrics: Methods and Models, Kluwer, Dordrecht, 1988. [2] G.S. Becker, Crime and punishment: an economic approach, J. Polit. Econ. 76 (1968) 169–217. [3] S. Chib, E. Greenberg, Hierarchical analysis of SUR models with extensions to correlated serial errors and time-varying parameter models, J. Econ. 68 (1995) 339–360. [4] C. Cornwell, W.N. Trumbull, Estimating the economic model of crime with panel data, Rev. Econ. Stat. 76 (1994) 360–366. [5] P.J. Elhorst, Specification and estimation of spatial panel data models, Int. Regional Sci. Rev. 26 (2003) 244–268. [6] A.E. Gelfand, A.F.M. Smith, Sampling-based approaches to calculating marginal densities, J. Am. Stat. Assoc. 85 (1990) 398–409. [7] J. Geweke, Bayesian treatment of the independent Student-t linear model, J. Appl. Econ. 8 (1993) 19–40. [8] G. Holloway, B. Shankar, S. Rahman, Bayesian spatial probit estimation: a primer and an application to HYV rice adoption, Agric. Econ. 27 (2002) 384–402. [9] J.P. LeSage, Bayesian estimation of spatial autoregressive models, Int. Regional Sci. Rev. 20 (1997) 113–129. [10] J.P. LeSage, Bayesian estimation of limited dependent variable spatial autoregressive models, Geogr. Anal. 32 (2000) 19–35. [11] D. Sun, R.K. Tsutakawa, P.L. Speckman, Posterior distribution of hierarchical models using CAR(1) distributions, Biometrika 86 (1999) 341–350. [12] L. Tierney, Markov chains for exploring posterior distributions (with discussion), Ann. Stat. 22 (1994) 1701–1762.