CHAPTER III. Methods 62
Table III-3 Number of complete observations for calculating the effects of terrorism, major violence and control variables on homicide rates
Models with control variables
only Terrorism models MEPV models
Region Obs. Years Countries Obs. Years Countries Obs. Years Countries
Africa 73 17
(1991-2012) 26 55 16
(1991-2012) 20 57 17
(1991-2012) 20 Americas 356 25
(1990-2014) 23 306 22
(1990-2013) 23 326 23
(1990-2012) 23
Asia 274 25
(1990-2014) 33 189 22
(1990-2013) 28 254 23
(1990-2012) 32
Europe 557 25
(1990-2014) 35 522 22
(1990-2013) 35 517 23
(1990-2012) 33
Oceania 17 13
(1990-2010) 3 16 12
(1990-2010) 3 17 13
(1990-2010) 3 Global 1278 25
(1990-2014) 121 1088 22
(1990-2013) 109 1171 23
(1990-2012) 111
III.4. Analytical strategy 63
Given the uncertainties as to whether victims of terror attacks might already have been included in the given homicide rates, adjusted homicide rates were calculated in order as to avoid measuring a spurious relationship. This was achieved by detracting the terrorism mortality rate from the homicide rate. The adjusted homicide rate served as the dependent variable throughout all of the analyses presented in this study, except for the models that tested the relationship of the homicide rate with the control variables without adding the independent variables of interest (terrorism, warfare, major violence).
Both the homicide rate and the terrorism mortality rate exhibited a significant positive skew in their distributions, with particularly long tails to the right. A natural logarithmic transformation (Phillips and Greenberg 2008, 57) was performed to correct for the skew and to pull and smoothen the data. In the case of the terrorism mortality rate, a constant of 1.0 was added before applying the transformation method. This was necessary because the terrorism mortality rate contained frequent zero-values which cannot be transformed by a natural logarithm. It has to be noted that log-transformed variables imply a specific interpretation of the coefficients. In the case that both the dependent and independent variables had been transformed, the relationship takes the form of a log-log model. In econometrics such a relationship is referred to as “elastic” (Benoit 2011, 4). Unlike untransformed variables, the coefficients indicate changes, not unit-changes. The value of the regression coefficients indicates percent-changes in the dependent variable (untransformed) while the independent variable (untransformed) consistently increases by 1 percent. If only the dependent variable is transformed, the coefficients of the untransformed independent variable need to be read as a 100 * coefficient percent changes in the dependent variable while the independent variable changes by 1 unit.
III.4.2. Regression analyses
For the purpose of this study, most regression analyses were performed using fixed effects models. Fixed effects models are one of two major types of regression models used for panel analyses, the other one being the random effects model (Phillips and Greenberg 2008). The appropriateness of one of the other can be shown by conducting a Hausman test (Hausman and Taylor 1981) which was performed on all models an confirmed the appropriateness of a fixed-effects approach.
Panel models, i.e. fixed and random effects models, allow to account for unobserved heterogeneity in the observed units (countries). This is not possible when conducting general
CHAPTER III. Methods 64
multiple regression models, also referred to as pooled OLS (ordinary least squares) in a panel context. In the case of pooled OLS, individual differences in the units of observation are not accounted for. Given that such differences almost certainly occur in the data, the estimation becomes inconsistent as far as the error terms are correlated with the predictor variables, and become inefficient when terror terms are heteroscedastic and serially correlated. These problems can be accounted for by using models for panel data analysis. The formula that applies to both random and fixed effects models is displayed below. The placeholder i thereby represents the observed entities and t the period of observation. Y is the dependent variable (homicide rate), X stands for the k predictor variables (independent variable of interest plus control variables), β the corresponding regressor coefficients, α the unobserved fixed effects specific to each country and ϵ the error term.
E 𝑌𝑖𝑡 = 𝛽1𝑋1,𝑖𝑡 + ⋯ + 𝛽𝑘𝑋𝑘,𝑖𝑡+ 𝛼𝑖+ 𝜖𝑖𝑡
Besides accounting for random or fixed effects specific to each individual country, also omitted effects specific to each year can be included in panel models. This makes for two-ways or time-fixed effects models that can be run in conjunction with both random or time-fixed effects models and read as displayed below. As becomes apparent, the placeholder 𝜆 has been added to the formula as a dummy that accounts for each year (t) under observation:
E 𝑌𝑖𝑡 = 𝛽1𝑋1,𝑖𝑡+ ⋯ + 𝛽𝑘𝑋𝑘,𝑖𝑡+ 𝛼𝑖+ 𝜆𝑡+ 𝜖𝑖𝑡
In the case of the fixed effects model, the calculation is most commonly based on the use of the within estimator which makes also for the most commonly applied model in panel analysis.
For each variable (𝑋, 𝑌𝑘) subject to the analyses, the corresponding mean value (𝑥̅, 𝑦̅) is thereby detracted from each longitudinal observation (t) pertaining to the same cross-section (i). The corresponding formula is shown below. Time-invariant differences between the individuals are factored out before the calculation is subjected to a pooled OLS. This is also why fixed effects models do not provide an interpretable intercept term.
E (𝑌𝑖𝑡− 𝑦̅𝑖) = (𝑋𝑘,𝑖𝑡− 𝑥̅𝑘,𝑖)𝛽𝑘+ (𝜖𝑖𝑡− 𝜖̅𝑖)
In the case of the within estimator, the model is fitted to the effects that variables exert over time within each individual. In order to estimate the effects that the variables exert in determining the time-invariant differences between the individuals, the between estimator has to be used. For that purpose, the mean values of all longitudinal observations pertaining to the
III.4. Analytical strategy 65
same cross-section are calculated before subjecting the estimation to a pooled OLS. This reads as follows:
E 𝑦̅𝑖 = 𝛼 + 𝑥̅𝑖𝛽 + 𝜖̅𝑖
Models were run in various configurations, mainly with fixed effects (within estimator) with and without fixed time effects. Besides running the models with regular standard errors, all models have also been computed with heteroscedasticity-consistent standard errors following the White method (White 1980). Where appropriate, also models with random effects, the between estimator and pooled models are presented for comparative purposes.
All analyses were conducted using the open source programming language R and the adjoined integrated development environment RStudio. Panel models were run using the plm function from the plm package (Croissant, Millo, and others 2008). Robust standard errors were calculated using the coeftest (lmtest package) and vcovHC (sandwich package) functions.
III.4.3. Summaries of trends in homicide, terrorism, warfare and major violence
Apart from running regression analyses, trends in homicide, terrorism, warfare, and other episodes of major violence have been summarized by means of descriptive statistics, visual representations and discussion of relevant context. These analyses were conducted in order as to understand the structure of the datasets incl. its limitations, gain a disaggregated overview of relevant trends in violence, and hence provide the ground for an informed interpretation and discussion of findings from the statistical analyses. Figures were produced using the ggplot2 package for data visualization in R (Wickham 2016). The regional clustering was performed following the regional scheme developed by the UN Department of Economic and Social Affairs Statistics Division.18
18 UN DESA – Standard country or area codes for statistical use (M49)/ Geographic regions, https://unstats.un.org/unsd/methodology/m49/, accessed 09/08/2016
CHAPTER IV. Homicide trends and their association with economic and population-based control variables 66
Homicide trends and their association with economic and population-based control variables
This chapter is divided into three sections. Firstly, the three different sources of homicide data that were consulted for the purpose of this study will be discussed, and the association between them will be examined. Secondly, an overview of global and regional homicide trends between 1950 and 2014 will be provided. Thirdly, associations between the homicide rate and the previously presented set of economic and population-based indicators will be tested in a series of regression models. These are the very indicators that serve as control variables for the models presented in the subsequent chapters on the effects of terrorism (CHAPTER V) as well as warfare and major violence on homicide (CHAPTER VI).