ELEMENTOS QUE HACEN PARTE DE LOS MECANISMOS DE CONTROL IMPLEMENTADOS POR EL DEPARTAMENTO DEL META COMO

The first step in the analysis is to test whether the inclusion of county fixed effects and/or time fixed effects is warranted, based on a comparison of model fit statistics. To this end, the data is pooled and non-spatial OLS models are run without fixed effects, with county fixed effects, with time fixed effects, and with both county and time fixed effects. The likelihood ratio (LR) statistics for the specification with county fixed effects (χ2 = 6674.4, p<.001) and the specification with time fixed effects (χ2 = 246.7 , p<.001)

are both significant with respect to the base model with no fixed effects. In addition the LR tests of the specification with both county and time fixed effects indicates that this model is significantly improved over the model with only county fixed effects (χ2 = 203.9, p<.001) or the model with only time fixed effects (χ2 = 6631.7, p<.001). As the interest in this analysis is on isolating the effect of changes in within-county foreign born populations on within-county homicide rates, the use of county fixed effects is warranted, and the spatial panel analysis which follows will incorporate both spatial and time period effects.27

To highlight how the focus on within-county change alters the interpretation of the model parameters, a between-county fixed effects model was estimated and compared to the specification with within-county fixed effects. The coefficients from these two models are shown in Table 2.2. The between-county effects model estimates the effect of each exogenous variable using the county mean of the variable for all time points, thus leveraging differences in structural covariates between counties but not differences in structural covariates over time. In the period 1970-2000 the mean effect of a county’s foreign born population on its homicide rate is insignificant, while the mean effects of most of the other covariates on the homicide rate, excepting the proportion young and

27 In cases where the unobserved heterogeneity between counties is the result of the clustering of unobserved or unmeasured variables at the regional level, researchers will often include regional indicator variables as crude controls to reduce bias in the remaining covariates. A fifth pooled model including time fixed effects and region indicators is also estimated, to assess whether the improvement in model fit achieved through the addition of county fixed effects compensates for the large number of degrees of freedom lost through the inclusion of these additional parameters. The specification which includes regional indicators and time fixedeffects is nested within thespecification which includes county and time fixed effects, and the LR test of the two models indicates that the spatial fixed effects model is preferred. A comparison of the Information Criterion of these two models reveals that the model with county fixed effects is likewise preferred based on the AIC, while the model with regional indicator variables only is preferred based on the BIC. This is unsurprising, as the BIC penalizes additional parameters more heavily than does the AIC. Overall, these test statistics are inconclusive in highlighting a preferred model fit, so we proceed with the model with county fixed effects, which more closely aligns with the original aim of the research.

male and the proportion of housing owner occupied, are positive. Focusing on the mean value of an explanatory variable over the whole period, however, ignores any trend in the variable over time, an oversight that may be especially salient in the case of a foreign born population which was substantially increasing between 1970 and 2000. A

comparison of the between-effects and the fixed-effects specifications reveals that many of the structural factors used in homicide studies have explanatory power in predicting differences in homicide rates between counties, but not necessarily within counties. Given that this paper is interested in the dynamics of population change via immigration, a focus on the within-county effects is thus warranted.

2.4.4 Spatial Models

Having established the preferred fixed effect specification, the most suitable spatial model can be identified using the three step procedure suggested by Elhorst (2010) and explained above. The panel LM test of the spatial lag and spatial error specifications indicates limited support for a lag specification over an error specification, although this choice is somewhat ambiguous. In the simple version of these tests, both the LM lag value (91.5, p<.001) and the LM error value (80.5, p<.001) are highly

significant; the same is true for the robust version of the tests (LM lag=40.8, p<.001; LM error=29.9, p<.001). Although a preference for a spatial lag specification may thus be based on the assumption that coefficient estimates derived from this model are unlikely to be biased, further testing reveals that a spatial Durbin model is preferred over either a spatial lag model or a spatial error model. A Wald test (χ2 = 42.8, p<.001) of the restricting assumption that θ = 0 (from equation (3)) suggests the rejection of this

hypothesis, indicating that the SDM does not simplify to a SAM and justifying the choice of the SDM. Likewise, a Wald test (χ2 = 49.0, p<.001) of the restricting assumption that

θ + δβ = 0 (from equation (3)) points to rejection of the SEM model in favor of the SDM. Thus, there is some assurance that the SDM is the preferable model with which to

proceed.

While these objective test statistics indicate a preference for an SDM specification, it is worth noting that the underlying structure of the SDM is also

conceptually appealing. The inclusion of spatially lagged independent variables allows social processes to cross borders, and the impacts of structural features of the population are therefore not limited to a single spatial unit. For example, a county with a very low rate of poverty that is surrounded by counties with high rates of poverty may still suffer some of the social effects of increased poverty due to its close proximity to the high poverty counties. This spillover effect is absent from both the SAR and SEM models, although the SAR model would include some feedback effects through the spatially lagged dependent variable.

The results from the estimation of the panel SDM described in Equation 3, in which the county homicide rate is a function of the structural covariates in the county, the lagged homicide rate in neighboring counties, and the lagged values of structural

covariates in neighboring counties, are shown in Table 2.3. As detailed above, the coefficient estimates of the SDM, reported in the first column, are not directly interpretable, owing to the feedback effects present between neighboring counties. Feedback exists due to the introduction of the spatially lagged homicide rate, which itself is determined in part through the values of the variables in the target county, as well as

the introduction of the spatially lagged covariates. The direct effect is calculated as the average, over all spatial units, partial derivative of the homicide rate with respect to changes in the covariate value in that county, while the indirect effect is the average, over all spatial units, partial derivative of the homicide rate with respect to changes in the covariate values in all other counties (Lesage and Pace 2009). The total effect is the sum

of these direct and indirect effects.

The direct effects shown in column 2 indicate that increases in a county’s black population and increases in the gun ownership rate (proportion of suicides in which a firearm are used) are associated with higher homicide rates within that county. The differences between the coefficient estimates in column 1 and the direct effect estimates in column 2 are small in this model, suggesting that the feedback effects are minimal. While there is no significant direct effect of a county’s foreign born population on its homicide rate, a sizable negative indirect effect is present between the two variables, suggesting homicide reductions in those counties which neighbor counties experiencing increases in foreign born concentration. A similar negative effect is seen in the

proportion of the population that is residentially stable, while positive homicide spillover is associated with an increasing black population, growth in the proportion of the

population without a high school diploma and increasing rates of gun ownership. The total impact of growth in the foreign born population on homicide rates is negative, as shown in the 4th column of Table 2.3. This total includes the direct effect of the foreign born population on rates in a county, as well as the indirect effect from growth in the foreign born population in neighboring counties. Residential stability is likewise correlated with decreased rates of homicide, while variables commonly used as proxies

for economic disadvantage, the proportion of the population that is black and the proportion of the population without a high school diploma, are associated with higher rates of homicide. None of the other economic disadvantage variables (poverty rate, mean family income, proportion of household female-headed) demonstrate a significant impact on homicide rates, either direct or indirect. Because a panel analysis focuses on within-unit change, the lack of a significant relationship between the homicide rate and some of the measures of economic disadvantage is likely the result of these measures exhibiting little change over time, or exhibiting non-uniform change over time.28