Measuring  the  causal  effect  of  air  temperature  on  violent  crime

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Uppsala  University     Department  of  Statistics   Autumn  –  14  

Bachelor  Thesis  

Supervisor:  Ronnie  Pingel   Authors:  Karl  Hammarström  &            Fabian  Söderdahl                    



Measuring  the  causal  effect  of  air  

temperature  on  violent  crime  




This   thesis   aimed   to   apply   the   causal   framework   with   potential   outcomes   to   examine   the   causal   effect   of   air   temperature   on   reported   violent   crimes   in   Swedish  municipalities.  The  Generalized  Estimating  Equations  method  was  used   on  yearly,  monthly  and  also  July  only  data  for  the  time  period  2002-­‐2014.  One   significant  causal  effect  was  established  but  the  majority  of  the  results  pointed  to   there   being   no   causal   effect   between   air   temperature   and   reported   violent   crimes.    


Keywords:   Causal   inference,   propensity   score,   Inverse   Probability   Weighing,   Generalized  Estimating  Equations  



We  would  like  to  thank  our  supervisor  Ronnie  Pingel  for  his  undying  devotion   and  support  in  guiding  us  through  our  work  on  this  thesis.


Table  of  contents


1.  Introduction  ...  4  

2.  Previous  research  on  the  link  between  violent  crime  and  temperature  ...  6  

3.  Method  ...  7  

3.1  Rubin  causal  framework  and  potential  outcomes  ...  7  

3.2  Generalized  Estimating  Equations  ...  10  

3.2.1  Covariance  structures  for  repeated  measurements  ...  12  

3.3  GEE  using  a  reweighted  sample  ...  13  

3.3.1  Estimation  of  the  propensity  score  ...  13  

3.3.2  The  Inverse  Probability  Weighing  estimator  ...  14  

3.3.2  Variable  selection  ...  14  

3.4  Spatial  correlation  ...  15  

4.  Data  ...  16  

5.  Results  ...  20  

5.1  Spatial  correlation  ...  20  

5.2  Generalized  estimating  equations  ...  20  

5.2.1  Monthly  data  ...  20  

5.2.2  Yearly  data  ...  21  

5.3  GEE  using  reweighted  data  ...  22  

5.3.1  Monthly  data  ...  22  

5.3.2  Yearly  data  ...  24  

5.4  July  data  ...  25  

5.4.1  Generalized  estimating  equations  ...  25  

5.4.1  GEE  with  reweighted  data  ...  25  

6.  Discussion  ...  27  

References  ...  29  


1.  Introduction


Causality   is   often   an   important   goal   for   researchers.   Causality   means   that   the   value   of   one   variable   affects   the   value   of   another   variable   (Imbens   and   Wooldridge,  2008).  While  proving  an  association  is  easier  to  do  than  proving  a   causal  relationship,  it  often  leaves  a  lot  to  be  desired  when  drawing  conclusions   from   a   study.   Showing   that   there,   for   instance,   is   an   association   between   university  students  and  higher  future  salary  might  lead  us  to  instinctually  see  a   causal  relationship,  going  to  university  leads  to  higher  future  salary  which  does   not  have  to  be  the  case.  Instead  it  might  be  that  individual  traits  (i.e.  intelligence,   ambition)  make  it  more  likely  for  an  individual  to  acquire  a  higher  degree,  which   is  also  related  to  higher  salaries.    


In   experiments   randomization   is   a   straightforward   method   for   inferring   causality  (Rubin,  1978).  In  randomization  treatments  are  randomly  assigned  to   different   groups,   i.e.   in   the   case   of   two   treatments   a   control   group   and   a   treatment  group.  Since  the  treatments  are  randomly  assigned  the  groups  should   be  similar  with  regard  to  the  background  variables  and  therefore  any  differences   in   the   outcome   of   interest   between   the   treated   group   and   the   control   group   should  only  be  due  to  the  treatment  received.  


However  it  is  not  always  possible  to  conduct  these  experiments  in  the  real  world.   If   examining   smoking’s   effect   on   lung   cancer,   ethics   prevent   researchers   from   randomly  selecting  people  to  smoke.  In  other  situations  it  is  just  not  physically   possible   to   control   the   assigned   treatment.   This   is   the   case   with   the   subject   of   this   thesis,  the   temperatures   effect   on   violent   crimes.   When   looking   at   the   temperatures   causal   effect   on   crime   there   is   obviously   not   any   way   to   control   what  temperature  will  be  at  different  locations  at  different  times.  Luckily,  there   are  methods,  given  certain  assumptions,  for  measuring  the  causal  effect  when  a   randomized   experiment   is   not   possible   (Imbens   and   Wooldridge,   2008;   Rubin,   1974).  



Sorg  and  Taylor,  2011).  However,  the  issue  of  the  real  causal  effect  seems  to  be   in  the  background  of  the  discussion.  Applying  the  causal  inference  framework  on   this  seems  to  be  a  natural  progression  for  the  research  in  this  area.  


The   aim   of   this   thesis   is   to   examine   if   there   is   a   causal   relation   between   air   temperatures  and  violent  crime  in  Swedish  municipalities.  This  thesis  applies  the   causal   inference   framework   to   see   if   higher   temperatures   cause   an   increase   in   violent  crime.  



2.  Previous  research  on  the  link  between  violent  crime  and  


The   association   between   temperature   and   conflict   is   an   area   that   has   been   studied   in   many   different   scenarios.   Hsiang   and   Burke   (2014)   provide   a   thorough   summary   of   previous   research   in   this   area.   Associations   are   found   between  temperature  and  group  conflict  (Burke  et  al.,  2009;  Dell  et  al.,  2012)  as   well  as  temperature  and  violent  crime  (Anderson  et  al.,  2000;  Jacob  et  al.,  2007;   Mares,  2013;  Rotton  and  Cohn,  2000).  


There   are   differing   psychological   theories   for   the   cause   of   the   relationship   between   temperature   and   violent   crimes.   The   general   affect   aggression   model   states  that  higher  temperatures  increases  aggressive  behavior  by  psychological   changes   caused   by   higher   temperatures   (Anderson   et   al.,   2000).   This   theory   implicates  a  linear  relationship  between  temperature  and  violent  crime.  On  the   other   side,   the   negative   affect   escape   model   states   that   violent   behavior   increases  to  a  certain  point  in  a  linear  fashion  but  then  decreases  as  extremely   high   temperatures   causes   individuals   to   instead   try   to   escape   the   discomfort   caused   by   the   temperatures   (Rotton   and   Cohn,   2000).   This   theory   suggests   a   curvelinear   relationship   but   is   not   something   that   is   particularly   considered   in   this   thesis,   as   it   is   uncommon   that   temperatures   in   Sweden   reach   such   high   temperatures.  



3.  Method  

This   thesis   applies   a   causal   framework   to   estimate   the   relationship   between   temperature   and   violent   crime.   Two   different   methods   in   this   framwork   are   utilized   to   see   if   there   is   a   causal   relation   between   temperature   and   violent   crime.  The  first  method  is  the  Generalized  Estimating  Equations  (GEE).  GEE  is  a   commonly   used   method   when   examining   longitudinal   data   that   is   correlated   over  time.    


The   second   part   of   the   method   involves   reweighing   the   sample   using   the   propensity  score  the  Inverse  Probability  Weighing  (IPW)  and  then  applying  the   GEE.   This   section   first   presents   the   causal   framework   which   this   analysis   is   based  upon.  Secondly  the  GEE  method  is  described  and  then  the  theory  behind   the   propensity   score   and   the   IPW   estimator   are   introduced.   Lastly   this   section   presents   the   issue   of   spatial   correlation   and   how   to   test   for   it   as   that   could   possibly  be  a  problem  in  this  study.  

3.1  Rubin  causal  framework  and  potential  outcomes  

Rubin  (1974)  presents  the  framework  for  potential  outcomes  that  is  widely  used   in  causal  inference.  This  framework  is  called  the  Rubin  causal  model  (RCM).  The   aim  of  causal  inference  is  to  estimate  the  effect  of  a  treatment  on  an  individual.   Rubin  (1974)  considers  a  causal  effect  as  the  effect  of  a  treatment  in  relation  to   another  treatment.    


The  fundamental  aspect  of  the  RCM  is  potential  outcomes.  Assuming  a  sample  of  

i=1,   2,   …,   N   individuals.   Each   individual   is   associated   with   a   value   for   the  


potential  outcomes  for  each  individual.  The  observed  response  is  then  for  each   individual  


𝑌! = 𝑇×𝑌! 1 + 1 − 𝑇 ×𝑌! 0                                                                                          (1)    

This   equation   leads   to   what   is   known   as   the   Fundamental   Problem   of   Causal   Inference   (Holland,   1986).   For   each   individual   i   we   can   only   observe   one   treatment   T   at   the   same   time,   meaning   that   we   can   only   observe   one   of   either  

Yi(0)   or   Yi(1).     This   makes   it   impossible   to   estimate   what   the   goal   of   causal   inference  is,  which  is  the  individual  causal  effect  


                                                                                                                             𝑌! 1 − 𝑌! 0                                                                                                                              (2)    

The  solution  to  this  problem  is  to  look  at  the  sample  as  a  whole  instead  of  at  each   individual  and  compare  the  untreated  group    (control  group)  against  the  treated   group  (Rubin,  1974).  For  this  to  be  possible  the  two  groups  need  to  be  similar  on   certain   background   variables   Xi,   called   covariates.   In   a   randomized   experiment   where   the   control   and   treatment   groups   are   randomly   assigned,   it   can   be   assumed   that   the   two   groups   are   similar   on   the   background   variables   (Rubin,   1978).  In  an  observational  study  however,  this  cannot  be  assumed.  While  there   are  several  variations  when  estimating  the  causal  effect  such  as  median  or  ratios,   this  thesis  focuses  on  the  most  used  variant  that  is  the  average  treatment  effect    

                                                                                                                       𝐴𝑇𝐸 = 𝐸 𝑌 1 − 𝑌(0)                                                                                            (3)    

The  average  treatment  effect  can  be  estimated  in  an  observational  study  given  an   assumption  of  unconfoundedness  where  X  is  a  vector  of  covariates  


                                                                                                                          𝑌(0), 𝑌(1) ∐  𝑇 𝑋                                                                                                            (4)    

and  an  overlap  assumption  


The   unconfoundedness   assumption   states   that   the   treatment   should   be   independent,  ∐   of   the   outcomes   conditional   on   a   vector   of   covariates   and   the   second  assumption  is  that  each  individual  should  have  a  non-­‐zero  chance  to  be   given  one  or  the  other  treatment.  Given  the  assumptions  the  average  treatment   effect  is  identified  by  


         𝐴𝑇𝐸 = 𝐸 𝐸 𝑌 1 − 𝑌 0 𝑋 = 𝐸 𝐸 𝑌 1 𝑇 = 1, 𝑋 − 𝐸 𝑌 0 𝑇 = 0, 𝑋 =                     𝐸 𝐸 𝑌 𝑇 = 1, 𝑋 − 𝐸 𝑌 𝑇 = 0, 𝑋 = 𝐸 𝑌 1 − 𝑌(0)                                                                          (6)    

Looking   at   this   equation   we   see   that   identification   of   the   average   treatment   effect   is   now   possible,   given   the   assumptions,   to   be   estimated   by   comparing   treated  and  untreated  individuals  conditional  on  the  vector  of  covariates  X.  The   ATE  can  for  example  be  estimated  by  the  difference  on  the  regression  of  treated   group  and  the  control  group.    


Alternatively,   it   can   be   shown   that   (Imbens   and   Wooldridge,   2008)   that   by   weighing  the  sample,  an  estimate  of  the  average  treatment  effect  is  given  by                                      

                                                                                         𝐴𝑇𝐸 = 𝐸 !"

! ! −

!!! !

!!! ! = 𝐸 𝑌 1 − 𝑌(0)                          (7)                                                                                                                                                                                                             were  p(x)  is  the  propensity  score  which  is  described  more  extensively  in  section  

3.3.1.  The  propensity  score  is  often  modeled  using  a  logit  model                                                                      𝑃𝑟𝑜𝑝𝑒𝑛𝑠𝑖𝑡𝑦  𝑠𝑐𝑜𝑟𝑒 = p(x) = !!!!!!!!!"!!!!!!"

!!!!!!!!!!!"!!!!!!"                                                (8)    

From  equation  7  appropriate  estimators  can  then  be  developed.      



In   this   analysis   the   data   used   is   measured   both   over   several   observations   and   over  time  and  this  will  pose  a  problem  when  estimating  ATE.  Consider  that  one   might  estimate  the  ATE  for  each  point  in  time  where  j  indicates  time  and  𝜏!  is  the   ATE  for  each  time  point  k  

  𝐸 𝑌! 1 −𝑌! 0 =𝜏! 𝐸 𝑌! 1 −𝑌! 0 =𝜏! ⋮ 𝐸 𝑌! 1 −𝑌! 0 =𝜏!    

The  estimated  ATE:s  will  due  to  the  correlation  over  time  be  dependent.  To  deal   with  this  issue  the  model  that  is  used  must  account  for  the  correlation  in  the  data   over  time.  A  model  suitable  for  this  is  the  Generalized  Estimating  Equations  that   is  described  in  the  following  section.    

3.2  Generalized  Estimating  Equations  

Generalized   Estimating   Equations   (GEE)   is   an   extension   of   Generalized   Linear   Models   (GLM)   and   was   first   introduced   by   Zeger   et   al.   (1988).   The   GLM     is   suitable  for  independent  observations  and  the  Generalized  Estimating  Equations   extends   these   models   to   work   for   time   dependent   data.   The   GEE   has   a   population-­‐average  approach,  meaning  that  the  focus  is  to  draw  inferences  about   the   population   as   a   whole   and   not   on   individuals.   This   is   of   importance   when   interpretating  the  regression  coefficients  (Zeger  et  al.,  1988).    



variable  Y,  covariates  X  and  the  suffixes  i  for  individuals  and  j  for  time  the  model   looks  as  follows:  


                                                                   𝐸 𝑌!" = 𝛽!+ 𝛽!𝑋!!"+𝛽!𝑋!!"+𝛽!𝑋!!"+ ⋯                                                          (9)    

By   the   link   function   the   mean   of   Yij   is   related   to   the   covariates   as   seen   in   equation  9.  


Given  the  effect  of  the  covariates,  the  variance  of  each  Yij  are  dependent  on  the   mean  response,  𝜇!",    as  


                                                                                                                               𝑉𝑎𝑟 𝑌!" = 𝜙𝜇!"                                                                                                          (10)    

ϕ  is  a    time-­‐invariant  scale  parameter,  meaning  it  is  not  dependent  on  time.  It  is  

required  to  estimate  this  parameter.    

The   GEE   is   dependent   on   both   estimates   of   α   (Equation   14)   and   β.   Because   of   this,  the  estimation  procedure  is  an  iterative  process  in  two  steps  meaning  that  it   repeats  the  calculations  of  the  two  steps  until  the  results  are  satisfactory.  Given   values   of   α   and   ϕ,   the   β   are   estimated   by   solving   the   generalized   estimating   equation  which  is  the  first  step  in  the  iterative  process  

                                                                                             𝑈 𝛽 = !!!! !!∗𝑉!!! 𝛼 𝑌! − 𝜇! = 0 ! !!!                                                              (11)    

The  other  step  in  the  iterative  process  is  to  given  the  values  of  the  β  estimate  the  

α  and  ϕ  based  on  the  standardized  residuals  


                                                                                                                                     𝑒!" =(!!"!!!") !(!!")

                                                                                                               (12)   where  then  ϕ  can  be  estimated  by  


                                                                                                                                     𝜙 = !!" ! !! !!! ! !!! !! ! !!!                                                                                                          (13)  

and  α  when  using  for  example  the  autoregressive  model,  AR(1),  presented  below   as  the  model  for  the  within-­‐subject  association  can  be  estimated  by  


                                                                                                                                   𝛼 = 𝜌𝛼!,!!!+ 𝑒!"                                                                                                    (14)  

3.2.1  Covariance  structures  for  repeated  measurements  

When  working  with  data  that  is  both  measured  cross-­‐sectionally  as  well  as  over   time   there   are   certain   aspects   that   need   to   be   taken   into   account.   One   issue   is   that   observations   can   be   correlated   over   time.   To   correct   for   this,   the   proper   covariance  structure  needs  to  be  estimated.  We  show  some  examples  for  three   time  points.     Σ = 𝑉𝑎𝑟(𝑌!) 𝐶𝑜𝑣(𝑌!, 𝑌!) 𝐶𝑜𝑣(𝑌!, 𝑌!) 𝐶𝑜𝑣(𝑌!, 𝑌!) 𝑉𝑎𝑟(𝑌!) 𝐶𝑜𝑣(𝑌!, 𝑌!) 𝐶𝑜𝑣(𝑌!, 𝑌!) 𝐶𝑜𝑣(𝑌!, 𝑌!) 𝑉𝑎𝑟(𝑌!)    

Clearly,  there  are  several  different  covariance  structures  available  depending  on   the  data.  The  compound  symmetry  structure  is  the  structure  that  is  the  simplest   among  the  covariance  structures.  It  assumes  that  the  each  covariance  is  constant   over  time  


Σ = 𝜎! 1 𝜌 𝜌𝜌 1 𝜌

𝜌 𝜌 1  


The  most  commonly  used  covariance  structure  when  observations  closer  to  each   other  in  time  are  more  highly  correlated  than  observations  further  apart  is  the   first  order  autoregressive  model  or  AR(1)  for  short  


The   last   covariance   structure   considered   in   this   analysis   is   the   Toeplitz   model.   While   the   Toeplitz   model   is   similar   to   the   AR(1),   it   does   require   more   parameters  to  be  estimated  

  Σ = 𝜎!! 𝜌1 𝜌! 𝜌! ! 1 𝜌! 𝜌! 𝜌! 1 = 𝜎!! 𝜎 !" 𝜎!" 𝜎!" 𝜎!! 𝜎!" 𝜎!" 𝜎!" 𝜎!!    

The   Toeplitz   model   and   the   AR(1)   model   are   often   suitable   when   the   observations  are  equally  spaced  over  time  which  is  the  case  in  this  analysis  and   could  speak  in  their  favor.    

3.3  GEE  using  a  reweighted  sample  

The   IPW   estimator   have   been   used   to   account   for   missing   data   in   longitudinal   studies   (Little   and   Rubin,   2002).   Liang   and   Zeger   (1986)   extended   the   GEE   approach  by  using  GEE  with  reweighted  data.  Seeing  as  the  IPW  estimator  can   also   be   used   to   make   the   reweighted   sample   independent   of   background   variables,  this  study  will  also  test  the  data  by  applying  the  GEE  to  the  reweighted   sample  to  see  if  that  will  give  differing  results  from  the  normal  use  with  GEE.  

3.3.1  Estimation  of  the  propensity  score  

The  propensity  score  is  the  conditional  probability  that  an  individual  is  assigned   treatment  given  a  vector  of  observed  covariates  (Rosenbaum  and  Rubin,  1983a).   The  propensity  allows  for  easily  adjusting  for  many  covariates  by  reducing  the   covariates  from  a  multidimensional  issue  to  be  one-­‐dimensional  instead.    


One   issue   with   the   estimation   of   the   propensity   score   is   that   the   temperature   observations  in  consecutive  months  could  be  correlated.  For  the  yearly  and  July   data  this  does  not  pose  a  problem.    Due  to  time  constraints  we  do  not  take  this   into  consideration  but  it  should  be  noted  as  a  possible  source  of  weakness  in  the   estimated  propensity  score.  



assumption   of   unconfoundedness   is   fulfilled,   meaning   treatment   is   unconfounded  given  X,  then  it  is  also  unconfounded  given  the  propensity  score.   Given  this  assumption,  at  any  value  of  the  propensity  score,  unbiased  estimates   of  the  ATE  are  given  by  the  difference  between  the  treated  and  the  control  group   individuals.  One  way  to  check  for  the  suitability  of  the  propensity  score  is  to  see   if   the   propensity   score   is   balanced   between   the   treated   and   control   group   (Caliendo  and  Kopeinig,  2008).    

3.3.2  The  Inverse  Probability  Weighing  estimator  

The  Inverse  Probability  Weighting  (IPW)  creates  a  synthetic  sample  where  the   baseline   covariates   are   independent   of   treatment   assignment.   This   is   done   by   utilizing  the  propensity  score  (Austin,  2011).    The  new,  weighted  municipalities   can  be  defined  as      


𝑌!" = 𝑌!"  ! !!! + 𝑌!"!!! !(!!!!)  .  

The  weight  of  a  municipality  is  equal  to  the  inverse  probability  of  receiving  a   positive  deviation  in  average  temperature  that  the  municipality  actually   received.  GEE  is  then  performed  on  the  new  synthetic  sample.  

𝐸 𝑌!" = 𝛽!+ 𝛽!𝑇!"  

Seeing  as  the  background  variables  are  already  accounted  for,  only  treatment  is   included  in  the  regression.  

3.3.2  Variable  selection  

Brookhart  et  al.  (2006)  examines  which  variables  to  include  in  the  model.  They   test  three  different  variables;  X1  which  is  only  related  to  the  treatment,  X2  which  

is  a  true  confounder  and  is  related  to  both  the  treatment  and  the  outcome  and  X3  

which  is  related  only  to  the  outcome.  The  results  show  that  the  true  confounder   X2  should  always  be  included  in  the  model.  The  variable  related  to  the  outcome,  

X3,  improves  the  model  by  decreasing  the  variance.  X1,  the  variable  related  only  

to  the  treatment  should  however  not  be  included  in  the  model.  These  results  are   also  supported  by  earlier  findings  by  Rubin  and  Thomas  (1996).  


  Figure  1:  Illustration  of  variable  selection  


In   this   thesis,   precipitation   is   expected   to   be   a   true   confounder   affecting   both   temperature   and   violent   crime.   Other   background   variables   are   variables   concerning  socioeconomic  factors  and  demographic  factors  that  are  believed  to   be  related  only  to  the  response  variable.  Seeing  as  according  to  theory  there  are   no   other   variables   other   than   precipitation   that   can   have   an   influence   on   both   temperature  and  the  variable  violent  crimes,  the  unconfoundedness  assumption   in  formula  4  should  hold.  

3.4  Spatial  correlation  


4.  Data  

The   data   is   both   monthly   and   yearly   municipality   data   for   the   time-­‐period   January  2002  up  until  October  2014  with  all  the  data  collected  from  government   sources.  We  also  research  the  month  of  July  on  its  own  to  test  if  the  effect  only   exists  for  higher  temperatures.  Due  to  limitations  in  the  accessibility  of  weather   data   our   dataset   is   narrowed   down   to   seventy   municipalities   spread   evenly   across  the  counties  in  Sweden.    


The   treatment   in   this   paper   is   defined   in   two   different   ways.   Firstly   for   the   monthly  and  yearly  analyses  it  is  defined  as    

• Deviation   in   air   temperature:  This  variable  is  a  binary  variable  with  1   being   air   temperatures   more   than   1   degree   Celsius   over   the   historical   mean   temperature   for   each   month   and   year   respectively   and   0   for   observations  with  air  temperatures  below  that.  The  observations  are  for   every   weather   station   located   in   the   different   municipalities,   collected   from  The  Swedish  institute  for  meteorology  and  hydrology.  

Secondly  for  the  July  data  

• Deviation   in   air   temperature:  This  variable  is  a  binary  variable  with  1   being   air   temperatures   more   than   18   degree   Celsius   and   0   for   observations  with  air  temperature  with  18  and  below  for  each  month  and   year   respectively   for   every   weather   station   located   in   the   different   municipalities,  collected  from  The  Swedish  institute  for  meteorology  and   hydrology.  

The  effect  or  response  in  this  paper  is    


The  following  variables  are  the  covariates  or  background  variables  used.  All   variables  are  available  for  yearly  data  but  only  the  unemployment  variable  is   available  for  the  monthly  and  July  data  

• Disposable   income:   The   individual   average   disposable   income   in   Swedish  crowns.  

• Age:  The  mean  age  for  the  municipalities.  

• Population:   The   population   density,   recorded   as   individuals   per   square   kilometer.  

• Foreigners:  The  percentage  of  foreign  born  citizens.  

• Precipitation:   The   mean   amount   of   downfall   measured   in   millimeters,   also  collected  from  SMHI.  

• Unemployment:   The   total   number   of   unemployed   persons   in   the   age   group  16-­‐64  years  of  age.  

Variables  measured  in  total  numbers  as  unemployment  and  violent  crimes  were   controlled   for   population   by   entering   them   as   the   total   number   per   every   100  000  inhabitants.  Table  1  displays  descriptive  statistics  for  the  variables.    

Table  1:  Descriptive  statistics  

Variable   Mean   Standard  Deviation  

Crime     65.71   167.71   Unemployment   1639.69   2971.29   Precipitation   55.56   40.96   Temperature   5.41   8.29   Population  density   142.62   553.90   Foreign   12.05   8.01   Income   4.79   0.57   Age   42.79   2.85    


loud  and  the  other  one  proofreading  to  catch  eventual  errors  made  by  the  one   entering  the  data.  A  visual  inspection  of  the  dataset  up  on  completion  was  also   made  and  found  errors  were  corrected.    


Similarly  to  Sorg  and  Taylor  (2011),  the  temperature  is  measured  as  deviations   from  a  historical  mean.  By  doing  this  we  remove  the  effects  of  seasonality  in  the   analysis.  The  historical  mean  is  taken  from  SMHI’s  calculations  from  the  period   1960-­‐1990  for  each  municipality.    


Figure   2   shows   the   distribution   of   the   variable   crime.   The   distribution   looks   roughly   Poisson   distributed.   As   previously   mentioned   the   GEE   is   robust   assumptions   of   distribution   and   a   Gaussian   variance   function   is   used   in   this   thesis.  



  Figure  3:  Distribution  of  temperature  deviation  


Figure  3  displays  the  distribution  of  the  deviation  of  temperature.  It  is  a  lot   pointier  than  a  standard  normal  distribution  but  it  can  be  assumed  to  be   approximately  normally  distributed.  In  appendix  A  there  are  also  time  series   plots  of  crime  and  temperature.  




5.  Results  

This  section  presents  the  results  of  the  computations  made  in  SAS  version  9.4.   The  significance  level  used  is  the  standard  five  percent  level.  

5.1  Spatial  correlation  

Since   the   observations   in   the   data   are   geographically   distributed   there   may   be   spatial   correlation   between   nearby  municipalities   concerning   crime.   Moran’s   I-­‐ test  for  spatial  correlation   shown  in  table  2  yields  a  non-­‐significant  result.  The   null  hypothesis  for  Moran’s  I-­‐test  is  that  there  is  no  spatial  correlation  between   the   observations.   This   means   that   there   is   no   significant   spatial   correlation   on   violent  crime  present  in  the  data.    

Table  2:  Moran’s  I  test  for  violent  crime  

Coefficient   Observed   Standard  

deviation   Z-­‐value   P-­‐value  

Moran's  I   -­‐0.0207   0.0105   -­‐0,649   0,52  

Geary's  c   1,01   0,030   0,336   0,73  


5.2  Generalized  estimating  equations  

This   section   presents   the   estimation   results   for   the   GEE   model   without   a   weighted  sample  with  both  monthly  and  yearly  data.  

5.2.1  Monthly  data  

The  results  when  using  the  GEE  with  monthly  data  are  displayed  in  table  3.  By   comparing   the   estimated   covariance   matrices   with   the   original,   the   Toeplitz   covariance   structure   was   the   best   fit   and   is   therefore   used.   The   covariance   matrices  are  however  excluded  here  as  they  are  large  in  size  and  due  to  that  hard   to   interpret.   The   temperature   deviation   from   the   historical   mean   has   a   coefficient   estimate   of   -­‐0.02,   which   is   the   average   treatment   effect,   indicating   that   when   the   temperature   is   equal   to   one   degree   over   the   historical   temperature   mean   or   colder   there   are   on   average   0.02   fewer   reported   violent   crimes.  This  is  not  in  line  with  the  theories  discussed  earlier  in  this  thesis.    




significant  causal  effect  of  temperature  deviation  on  reported  violent  crimes.  The   residual   plot   can   be   found   in   figure   6   in   appendix   A,   indicating   that   our   covariance   structure   is   correctly   specified.   Even   though   unemployment   has   a   significant  p-­‐value,  it  should  still  not  be  interpreted  as  a  causal  effect  seeing  as   the   assumptions   required   are   only   verified   for   the   temperature   deviation   and   not  for  the  unemployment  variable.    

5.2.2  Yearly  data  

Table   4   shows   the   estimation   using   GEE   on   yearly   data.   Again   the   Toeplitz   covariance   structure   was   the   most   suitable.   The   average   treatment   effect   is   -­‐ 0.018   but   similarly   to   before   the   confidence   interval,   ranging   from   -­‐0.038   to   0.002,   covers   zero   and   therefore   the   causal   effect   is   not   significant.   The   only   variables   that   have   a   significant   impact   on   the   mean   of   violent   crime   are   percentage   of   foreign   born,   income   and   age.   However   it   is   important   to   stress   once  again  that  even  though  they  are  significant,  a  causal  interpretation  cannot   be  concluded  seeing  as  the  assumptions  needed  are  not  examined.  The  residuals   are  shown  in  figure  7  in  appendix  A  and  show  that  the  covariance  structure  used   is  appropriate.  

Table  3:  GEE  with  monthly  data  

Variable   Coefficient   SE   95  %  CI     Z   P-­‐value  



5.3  GEE  using  reweighted  data  

This   section   presents   the   estimation   results   using   the   propensity   score   to   reweigh  the  data  on  both  monthly  and  yearly  data.  

5.3.1  Monthly  data  

Table  5  displays  the  output  of  the  estimated  propensity  score  using  a  logit  model.   The  only  background  variables  used  are  unemployment  and  precipitation.  From   the   p-­‐values   we   see   that   unemployment   has   an   non   significant   effect   while   precipitation   does   have   a   significant   effect,   both   variables   are   however   kept   in   the   calculations   of   the   propensity   score   since   unemployment   is   related   to   the   outcome.   The   values   for   the   dummy   variables   showing   which   municipality   are   not   included   here   seeing   as   there   are   70   dummy   variables,   one   for   each   municipality.  

Table  4:  GEE  with  yearly  data  

Variable   Coeff.   SE   95  %  CI   Z   P-­‐value  



Table  5:  Logit  with  monthly  data  

Variable   Coefficient   Standard  Error   Wald  Chi  

square   P-­‐value   Intercept   -­‐0,0083   0,17   0,0023   0,96   Unemployment   -­‐0,00003   0,000033   0,77   0,38   Precipitation   0,0031   0,00058   28,61   <0,0001    

After   estimating   the   propensity   score,   the   weights   are   calculated   using   the   inverse  probability.  This  creates  a  new  synthetic  sample  that  is  independent  of   the   background   variables.   The   boxplot   of   the   weights   are   shown   in   figure   8   in   appendix  A  and  the  weights  appear  to  be  balanced.  


After  the  sample  is  weighted  a  GEE  regression  of  the  temperature  deviation  on   violent   crimes   is   performed.   Here   the   AR(1)   covariance   structure   was   the   best   fit.   The   results   are   shown   in   table   6.   This   result   is   similar   to   the   two   previous   results,   i.e.   we   find   no   effect   of   temperature   on   crime.   However,   the   point   estimate  of  the  average  treatment  effect  of  temperature  on  crime  is  0.02  and  it   must   be   noted   that   using   the   reweighted   data   we   don’t   adjust   the   standard   errors  due  to  the  estimation  of  the  propensity  score.  That  is  we  assume  that  the   propensity  score  is  true.  Studies  have  shown  (e.g.  Lunceford  and  Davidian,  2004)   that  this  results  in  conservative  estimates  of  ATE  with  standard  errors  that  are   too  wide.  In  theory  this  could  be  adjusted  for  by  using  for  instance  bootstrap  or   using   a   Sandwich   estimator,   however,   this   is   not   feasible   in   this   study.   Once   again  the  95  %  confidence  interval  stretches  over  zero  so  there  is  no  significant   causal  effect.  


Table  6:  GEE  with  IPW  and  monthly  data  

Variable   Coeff.   SE   95  %  CI   Z   P-­‐value  

Intercept   4.31   0.04   4.24   4.38   119.58   <.0001  

Temperature deviation  


5.3.2  Yearly  data    

The  propensity  score  estimate  for  yearly  data  can  be  seen  in  table  7.  All  variables   are  significant  except  for  one  variable  which  is  unemployment.  All  variables  are   still  kept  to  ensure  that  all  background  variables  are  accounted  for,  to  increase   precision.  The  weights  are  then  calculated  the  same  as  earlier.  The  boxplot  of  the   weights   are   shown   in   Appendix   A   in   figure   9   and   unfortunately   they   are   not   perfectly  balanced  but  still  good  enough  for  the  analysis  to  be  performed.  


Table   8   shows   the   final   estimate   of   the   causal   relation   between   temperature   deviation   and   violent   crime.   The   covariance   structure   used   is   the   Toeplitz   structure.  The  average  treatment  effect  is  -­‐0.04  with  a  95  %  confidence  interval   from   -­‐0.08   to   -­‐0.001   and   therefore   the   causal   effect   is   significant.   On   average   municipalities  that  experience  temperatures  equal  to  1  degree  over  the  historical   mean  or  below  have  0.04  fewer  reported  crimes  compared  to  municipalities  with   higher  temperatures.  This  is  not  in  line  with  the  pervious  result  for  monthly  data   using  IPW-­‐estimators.  This  could  be  because  for  the  yearly  data  there  are  more   background   variables   available   but   it   could   also   be   because   there   are   fewer   observations  for  yearly  data.    


Table  8:  GEE  with  weighted  data  yearly  

Variable   Coefficient   SE   95  %  CI   Z   P-­‐value  

Intercept   6.6021   0.0562   6.4920   6.7123   117.47   <.0001  

Temperature   deviation  

 -­‐0.0439   0.0173   -­‐0.0779   -­‐0.0099   -­‐2.53   0.0113  

Table  7:  Logit  model  with  yearly  data  

Variable   Coefficient   Standard  Error   Wald  Chi  



5.4  July  data  

This  section  presents  estimates  of  GEE  and  propensity  score  with  data  for  only   July  months.  This  is  performed  to  see  if  the  relationship  between  violent  crime   and  temperature  is  stronger  at  time  periods  with  higher  temperatures.  

5.4.1  Generalized  estimating  equations  

Table  9  shows  the  results  on  July  data  using  GEE.  Again  the  Toeplitz  covariance   structure   most   resembled   the   original   covariance   structure.   The   average   treatment  effect  is  -­‐0.07.  The  95  %  confidence  interval  is  -­‐0.025  to  0.11  and  as   previously   when   using   the   non-­‐weighted   sample   there   is   no   significant   causal   relationship  found  between  temperature  deviations  and  reported  violent  crimes.   The  variable  unemployment  has  a  p-­‐value  0.006  meaning  that  it  has  a  significant   effect   but   seeing   as   the   assumptions   are   not   accounted   for   it   cannot   be   interpreted   as   a   causal   relationship.   Precipitation   does   not   have   a   significant   effect   on   violent   crime.   The   residuals   are   in   figure   10   in   Appendix   A   and   are   evenly   distributed   around   zero   and   therefore   the   covariance   structure   used   is   appropriate.  


5.4.1  GEE  with  reweighted  data  

In  table  10  the  results  of  the  logit  estimation  are  shown.  All  variables  are   significant  and  based  on  the  results  the  weights  are  calculated  as  previously.    

Table  10:  Logit  with  July  data  

Variable   Coefficient   Standard  Error   Wald  Chi  

square   P-­‐value  

Intercept   -­‐0,96   0,16   37,45   <0,0001  

Unemployment   -­‐0,00007   0,000025   6,63   0,010  

Precipitation   0,0096   0,0016   35,58   <0,0001  

Table  9:  GEE  with  July  data  and  temperature  

Variable   Coeff.   SE   95  %  CI   Z   P-­‐value  

Intercept   4.3498   0.0549   4.2422   4.4574   79.24   <.0001  

Temperature   -0.0699   0.0905   -0.2472   0.1074   -0.77   0.4397  

Unemployment   0.0000   0.0000   0.0000   0.0000   2.77   0.0056  



The  boxplot  is  shown  in  appendix  A  figure  11  and  shows  that  the  propensity   scores  are  fairly  evenly  distributed.  


Table  11  shows  the  regression  on  the  reweighted  sample.  The  estimated  average   treatment  effect  is  5,06  but  similar  to  most  of  the  previous  results  there  is  not  a   significant   causal   relationship   between   temperature   and   violent   crimes   for   the   July   data   seeing   as   the   confidence   interval   stretches   over   zero.   However,   the   coefficient  is  positive,  and  the  p-­‐value  is  less  than  0.1.  It  would  be  interesting  to   carry   out   bootstrap   to   see   whether   we   actually   establish   a   significant   effect   if   using  correct  standard  errors.  


Table  11:  GEE  with  weighted  July  data    

Variable   Coefficient   Standard  Error   Z-­‐value   P-­‐value  

Intercept   80,85   2,42   33,47   <0,0001  



6.  Discussion  

In  this  study  our  goal  was  to  research  the  causal  effect  that  temperature  has  on   the  total  number  of  reported  violent  crimes  in  Sweden.  When  using  generalized   estimating   equations   our   results   show   that   temperature   does   not   have   a   significant  causal  effect  on  the  number  of  reported  violent  crimes,  neither  on  the   yearly   nor   monthly   data   material   except   for   yearly   data   with   a   reweighted   sample.  Even  though  we  included  several  background  variables  in  the  model  for   yearly  data,  the  result  was  still  inconclusive.  


In  our  testing  with  the  weighted  data  material  temperature  was  insignificant  on   the  monthly  data.  We  got  significant  results  for  our  yearly  data  but  they  showed   that  temperature  had  a  negative  effect  on  the  mean  of  reported  violent  crimes.   So  while  we  found  a  significant  causal  effect,  the  effect  was  not  in  line  with  our   stated  theory.  


The  only  difference  in  results  between  the  two  above  stated  methods  are  that  the   variable  temperature  was  significant  when  using  reweighted  yearly  data.  This  is   perhaps   due   to   an   imbalance   in   the   weight   that   existed   for   the   reweighted   sample  with  yearly  data.  When  using  data  that  spans  over  the  whole  year  as  for   the  monthly  data  and  also  when  using  the  yearly  data,  the  lower  temperatures   during   the   winter   months   make   drawing   conclusions   about   the   general   affect   aggression  theory  not  appropriate.  However  the  routine  activity  theory,  stating   that   people   are   more   outside   when   temperatures   are   higher   could   possibly   be   connected  to  these  results,  which  mostly  disprove  that  theory.  


The   results   for   the   month   of   July   also   yielded   an   insignificant   result   for   the   variable  temperature’s  effect  on  the  mean  of  reported  violent  crimes.  We  were  a   bit  surprised  by  this  result  since  this  goes  against  the  general  affect  aggression   model,  which  states  that  higher  temperatures  increases  aggressive  behavior  by   psychological  changes.    


The   July   results   also   goes   against   the   routine   activity   theory   which   states   that   individuals   tend   to   be   more   out   and   about   when   temperatures   are   higher   and   this  leads  to  an  increased  number  of  violent  confrontations.    


A   possible   explanation   for   our   lack   of   results   is   perhaps   that   many   of   the   municipalities   in   our   data   material   are   scarcely   populated.   Another   critique   of   this  study  is  that  it  doesn’t  take  the  possibly  large  number  of  unreported  violent   crimes   in   to   account.   While   most   of   the   results   yielded   a   negative   average   treatment  effect,  the  result  for  July  data  with  reweighted  sample  gave  a  relatively   higher  positive  average  treatment  effect  that  is  consistent  with  the  theory.  This   could  be  because  the  winter  months  are  too  cold  for  the  underlying  theory  to  be   relevant.  Future  research  could  implement  the  causal  framework  in  regions  with   more  largely  populated  areas  and  where  the  temperatures  reach  higher  levels.   In   conclusion   this   study   does   not   find   any   compelling   evidence   that   higher   temperatures  causes  more  violent  crimes.  



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Appendix  A  


  Figure  4:  TIme  serie  of  crime  

  Figure  5:  Time  serie  of  temperature  


  Figure  6:  Residual  plot  monthly  data  



  Figure  8:  Boxplot  for  monthly  data  



  Figure  10:  Residuals  for  July  data  




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