The Spatial Analysis of Terrorism in Pakistan
5.5 Empirical Strategy
For an empirical analysis, the spatial methodology developed by Anselin (1988) is used. The spatial lag model assumes that terrorists in one district may also commit terrorist activities in the
neighborhood, and thus, generate spillover effect from one district to another. Given the research question, terrorism prevention efforts might decrease attacks in a district, but, displace them to the neighborhood. Therefore, over time, the spatial lag model captures the spillover effect of a policy.
Similarly, the spatial error model controls for the spatial dependence in the error term.
Consider the following simple regression equation:
y=xβ+e (5.1)
whereyis the dependent variable (terrorism)k×1 vector;xis the 1×krow vector of the covariates;
β is the corresponding matrix in thek×1 vector of the coefficients; andeis the error term. To account for the spatial dependency, the spatial lag term (the weighted average of response variable in the contiguous districts) is added to equation 5.1, which becomes:
y=λwy+xβ+e (5.2)
where
e=ρwe+u (5.3)
In equation 5.2, λ is the spatial dependence parameter andwy is the weight matrix of the spatially lagged dependent variable. Alternative choices are available to measurewmatrix.14 For this study, the inverse distance and simple contiguity to the 1storder districts are considered.15 The contiguitywmatrix is calculated to measure the displacement/diffusion effects of an intervention.
Likewise,ρis the spatial error dependence parameter in the equation 5.3.
14These include inverse distance matrix; inverse of the euclidean distance between the geographic coordi-nates; the arc distance between the geographic coordicoordi-nates; a matrix based on social, economic and political indicators; the simple contiguity among districts; etc.
15The first order contiguity weight matrix is calculated when a given district is connected to the neigh-boring districts with the shared borders and vertex. The value of the spatial lag is equal to the weighted average (terrorist attacks) in the contiguous districts.
5.5.1 The Standardized Spatial Weight Matrix
Lets define a spatial weight matrixwwith the following properties:
wij=
⎧⎪
⎨
⎪⎩
1 ifiis contiguous toj 0 otherwise
(5.4)
If districtsiandjare neighbors, the weight is 1, otherwise 0. Equation 5.4 can be written inn×n matrix as:
w=
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎣
w11 ... ... w1n
w21 ... ... w2n
... ... ... ...
wn1 ... ... wnn
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎦
(5.5)
In the context of our study, the weight matrix is 115×115 dimensions. Each row is divided by the sum total of that row to get the standardized spatial weight matrix. After the row standardization, the spatial value is calculated as the weighted average:
yi=
j
wijyj (5.6)
wherewijare the row standardized weights andyj is the value ofyin the districtj.16
5.5.2 Identification
Pooled data is used to provide precise estimates of the relationship between terrorism prevention interventions and terrorist incidents. Firstly, it considers thefull sample of districts to study the externalities of terrorism prevention in the form of displacement or diffusion of benefits, that is, to test (H3). Equation 5.7 is designed to estimate the externalities for thefull samplethrough spatial lag and spatial error regression models. Secondly, to be more specific, spatial regression disconti-nuity design (SRDD) is also used to capture the average treatment effect of terrorism prevention
16In equation 5.2,wyis endogenous and must be instrumented. Thus, we estimate equation 5.2 with the Generalized Method of Moments (GMM) estimator, in particular, we use the generalized spatial two stage least squares (GS2SLS) estimator of Kelejian and Prucha (1998). GS2SLS in our estimates become the usual 2SLS estimator, where instrumental variables forwyare chosen among the vector ofx(wx, w2x, ..., wnx).
efforts. The use of SRDD would allow an analysis of displacement/diffusion to the nearest of the neighborhood. It is estimated only for theterritorial control sampleand its neighborhood districts due to the explicit boundary line between treatment and control groups. The SRDD is explained in equations 5.8-5.10, given below:
Consider the following simple regression equation:
T errorismi=α+β WT Pi+γXi+φp+ei (5.7)
whereT errorismiis the pooled average/total terrorist incidents;WT Piis the weighted average of terrorism prevention in the neighborhood around districti, which is calculated to analyze the spillover of terrorism prevention to districti. Theoretically, terrorists can move in any direction in 360◦around a district, when terrorism prevention strategies raise the opportunity cost of terrorist activities in a place. Likewise, it is costly for the government to police the 360◦circle around a district. If there are interventions in more than one district in a 360◦circle around a district, it would be challenging to observe the spillovers of interventions separately from each other. Therefore, the weighted average of interventions in the 360◦circle around a district is constructed. If there is no intervention in the circle, the value is zero. If intervention(s) is observed in the circle, it is calculated as the weighted averages, while the weight is assigned on the basis of the number of districts in the circle. The radii of the circle range between 0-75 kilometers. Xiis a set of control variables;φp are province-fixed effects; and finallyei is the error term. The set of observableXi
is comprised of demographic and socioeconomic determinants of terrorism. Similarly, equations 5.8-5.10 are designed to explore the externalities from terrorism prevention through SRDD:
T errorismi=αt+βt(X−b) +ei (5.8)
T errorismi=αc+βc(X−b) +ei (5.9) whereαtandαc are the intercepts in treatment and control districts on the opposite side of the border; b represents the border line; (X−b) is the distance of a district from the border.
By estimating equations 5.8 and 5.9, the impact of terrorism prevention can be computed as the difference between the two regression intercepts,αtandαc, on the two sides of the boundary.
The pooled version of equations 5.8 and 5.9, as suggested by Lee and Lemieux (2009) are used.
Thus, lettingτ=αt-αc, and usingDto indicate the treatment variable, the estimating equation is:
T errorismi=α+τ∗Di+ (βt−βc)(X−b) + (βt−βc)∗D∗(X−b) +ei (5.10) Equation 5.10 is flexible enough to allow the regression function to differ on both sides of the border by including interaction terms betweenDandX. The parameter of interest isτ, i.e., the average effect of having a treated (compared to a control group) and can be interpreted as the jump between the two regression lines at the border. Importantly, the estimating equation 5.10 is run for samples of increasing widths (w) around the border. The idea is that reallocation of terrorism due to terrorism prevention efforts should be higher with smaller bandwidths, while decreasing with larger ones.
Finally, to test H4, the following equation is used:
(5.11) T errorismi=α+β T Pi+γXi+φp+i
where (T P) is the terrorism prevention intervention in a district and∈ {0,1}.