The Traditional Definition

Due to rather limited numerber of literatures applying either the SAR or the SMA model on crime rate analysis, only two articles have been found for each case.

can simply be found by maximizing the concentrated likelihood function within the acceptable

interval and . (5.10)

In the concentrated likelihood function given by equation (5.10), is the residual sum of squares from equation (5.9): regressing the spatially filtered dependent variable on the spatially filtered

Martin (2002) applies the SAR model to analyse the pattern of residential burglary in the city of Detroit based on the social disorganization theory. The spatial unit of analysis is the census tract and three hundred and twenty of them have entered the sample as proxies for neighbourhoods in Detroit. The dependent variable is the average burglary rate over the years 1995-1997. The independent variables have been selected according to the social disorganization theory. Initially, 11 variables have been chosen to represent the factors that influence the degree of social disorganization.

However, as some of the variables are highly correlated, a principle components factors analysis has been conducted looking for a small number of linear combinations of these variables. Four factors have been produced as a result, namely, concentrated poverty, social capital, age composition and residential stability.

An OLS estimation has been performed as the first step. The results suggest that age composition is the strongest predictor for residential burglary rate, as it is positively correlated with the proportion of young people. The percentage population living in poverty also has positive effect on burglary rate as expected. The residential stability, though, has surprisingly obtained positive coefficient. The positive effect of residential stability does not necessarily mean that stable and familiar neighbourhood cannot improve the guardian effect that neighbours have for each other. It is possibly because more affluent and stable areas could be more appealing to potential criminals.

For further investigation, the SAR model is applied to detect the potential spatial dependence in burglary rate. The results have shown that the overall fit of the model has been substantially improved by introducing the spatial lag of the dependent variable. It has demonstrated that approximately 60 percent of the variation in

burglary rate can now be explained comparing to 46 percent in the OLS estimation.

Nevertheless, including the spatial lag of the dependent variable has reduced the magnitude of each coefficient for the independent variables. In addition, the residential stability becomes no insignificant as a result of including the spatial lag of burglary rate.

The application of SMA model has been given an example in Andresen (2006). The aim of this article is to analyse the spatial dependence in crime rates of different types using the data of Vancouver in 1996. The dependent variables under investigation are the crime rates of automotive theft, break and entering and violent crime. After separately mapping the crime rates over space with Geographic Information System software (GIS), the distributions of different types of crime rates have all shown very uneven patterns which have been taken as the evidence for applying the SMA model on the analysis. The census tract has been chosen to be the spatial analysis unit and there are totally 87 of them in Vancouver.

Initially, 13 independent variables have been selected to represent the factors influencing the degree of social disorganization such as ethnic heterogeneity, economic status, population composition and so on. Due to the potentially high correlation between these independent variables, Andresen has applied the general-to-specific method to reduce the influence of such correlation: for each type of crime rate, the model begins with including all independent variables; then the variable with the most insignificant coefficient will be dropped and the equation will be re-estimated.

The same process will be repeated until all the remaining variables have significant coefficients.

The main findings of this paper can be summarized by the following points. Firstly, the variations in all three types of crime have been largely explained by the independent variables: 53, 65 and 78 percent respectively for automotive theft, break and entering and violent crime. Secondly, unemployment rate, young people percentage and the standard deviation of average family income have all exhibited positive correlations with the three crime rates in question. In particular, the unemployment rate has the strongest effect on these three crime rates. Thirdly, the proportion of single-parent households has positive impact on break and entering but shows no significant impact on automotive theft and violent crime. Finally, the ethnic heterogeneity shows significant and negative relationship with automotive theft and violent crime but insignificant correlation with break and entering. The negative effect of ethnic heterogeneity is opposite to the expectation that increasing diversity in ethnic composition will increase crime rate since it is supposed to measure the communal stability. One possible explanation could be the definition of this variable:

ethnic heterogeneity has been defined as the percentage of recent (1981-1996) immigrants within the total census tract population. This is a different way of defining ethnic composition contrary to traditionally measurement on the proportions of different ethnic groups. Since the immigrants in Vancouver mainly containing economic-class, entrepreneurial-class and investor class, the proportion of them is not expected to directly affect crime rates such as automotive theft.

Martin (2002) and Andresen (2006) both try to explain the observed uneven and non-random distribution of crime rates with the spatial dependence model, either the SAR or the SMA. The unmentioned assumption behind both papers is that the non-random distribution of crime rates is caused by the spatial dependent effect. A major

difference between these papers is that Martin (2002) applies the SAR model as it assumes the spatial dependence exists in the dependent variable. In other words, the crime rate in one region is affected not only by local explanatory variables, but also by the crime rates of neighbouring regions. This assumption is true when crimes can actually spill over into neighbouring regions due to the change in relative conditions of neighbouring regions. On the other hand, Andresen (2006) attributes the non-random distribution of crime to the feature of the error term. It assumes that the error terms are spatially dependent rather than normally distributed. Such assumption is reasonable when there are independent variables omitted from the equation and the omitted variables are spatially dependent.

Cahill and Mulligan (2007) and Malczewski and Poetz (2005) have also observed the non-random distribution of crime rates. Both papers attempt to explain this pattern with spatial heterogeneity. With visualization mapping equipment, it is found that the dependent variables in both papers, violent crime in Portland and residential burglary in London (Ontario) respectively, are highly concentrated at city centre areas. This clustering pattern reduces with the distance from city centre. Instead of assuming crime rates are spatially dependent, the two papers employ models allowing the coefficients of independent variables to vary over space. In other words, the crime rates of different regions are allowed to response differently to their local predictors.

In order to estimate the coefficients varying across regions, both papers have employed the Geographically Weighted Regression (GWR) and compared the results to those generated by the conventional OLS regression. One of the most important propositions for this is that the GWR regression has substantially improved the

explanatory power of the independent variables and some of the coefficients do display significant variations across neighbourhoods.