The Spatial Context of Homicide

The inclusion of a spatially lagged dependent variable is consistent with prior literature that explores the diffusion process of homicide (Baller et al. 2001). Would-be offenders in county B may observe the high (or low) number of homicides in neighboring county A and determine that homicide is (is not) an acceptable resolution to interpersonal conflicts. The number of homicides in a particular county might also be influenced, through a retaliatory process, by the number of homicides in neighboring counties, if there exists a contentious relationship between factions in the neighboring counties. There is often little demarcation in the urban form between adjacent counties, particularly on the denser East Coast, and the daily life of many residents may be carried out on both sides of a county border. The arbitrariness of these borders may have implications for the homicide totals in each county, as an attack that occurred in one county could have conceivably been carried out in the adjacent county.

The inclusion of spatially lagged independent variables accounts for any potential relationship between the structural factors associated with homicide in a target county and the homicide rates in neighboring counties. As an example, consider a county which houses a large number of establishments that sell guns. While the ready availability of weapons might be expected to affect homicide rates in the host county, it may also influence homicide rates in neighboring counties, as individuals in those counties have easy access to these same businesses. The exclusion of the spatially lagged independent variables could also result in biased parameter estimates for the remaining model

variables, if there are unmeasured spatial processes that operate between the neighboring counties that are correlated with the included control variables (LeSage and Pace 2009; Elhorst and Fréret 2009). In the above example, the presence of a large number of gun

stores may be correlated with the existence of an underlying violent subculture, yet this latter variable is unobserved or unmeasured. To the extent that the violent subculture diffuses to neighboring counties, its exclusion from the regression equation may produce biased coefficient estimates unless the number of gun stores in neighboring counties is controlled.

Incorporating spatially lagged endogenous and spatially lagged exogenous variables into the model presents a challenge in the interpretation of the coefficient estimates. The raw coefficient estimates (β in equation (3) above) include feedback, as the effect of each exogenous variable on the county homicide rate is reintroduced into the regression equation through the spatially lagged dependent variable. An illustration of the relationship between neighboring spatial units a and b in the SDM is shown in Figure 2.1. Each exogenous covariate (Xa) in unit a will encompass both a direct effect, the impact of the variable on the outcome (Ya) within its own spatial unit, and an indirect effect, the impact of the variable on the outcome (Yb) in neighboring spatial units. The inclusion of the outcome variable in the neighboring spatial unit as an additional

explanatory variable is the source of the feedback loop, as Xa affects Ya both directly and through the path Xa → Yb → Ya. Methods to estimate both the direct and indirect effects from models which incorporate spatially lagged independent and dependent variables are described in LeSage and Pace (2009). Notationally, the impact of a change in an

explanatory variable k on the outcome y for a sample of n spatial units is given by: 𝜕𝑦

𝜕𝑥𝑘 = �𝐼𝑛 − 𝛿̂𝑊� −1

In this equation, 𝛿̂ is the estimated coefficient of spatial dependence, 𝛽̂_𝑘 and 𝜃�_𝑘 are the estimated coefficients for explanatory variable k and the neighbor weighted variable k,

respectively, W is the spatial weights matrix, and In is the identity matrix. The right hand

side of this equation is characterized by a symmetrical n x n matrix, with the diagonal

elements representing the direct effect of a change in k in a spatial unit on the y in that

spatial unit. The off-diagonal elements represent indirect effects of changes in k in all

other spatial units on y in each spatial unit. LeSage and Pace propose summarizing the

information contained in this matrix by averaging the diagonal elements of the matrix to determine the average direct impact of a change in k and averaging either the summed

column or row elements of the matrix to determine the average indirect impact of a change in k.

It is worth noting that in a non-spatial regression, in which there is neither a spatially lagged dependent variable nor spatially lagged covariates, both the 𝛿̂ and 𝜃�_𝑘 terms are assumed to be equal to 0. In this case the off-diagonal elements of the n x n

matrix on the right hand side of the equation will all be 0 and each diagonal element will be 𝛽̂_𝑘. Thus, the non-spatial regression has no indirect impacts and the average direct impact is equal to 𝛽̂_𝑘, identical to the usual interpretation of coefficient estimates.