capital <- read.csv("data/capitalJapan.csv", header=T)
capital <- subset(capital, 1948<=YEAR & YEAR<=2018)

1 Analysis of the Deterrent Effect on the Homicide Rate (Section 3.3 of the Paper)

1.1 Descriptive Statistics of Variables (Section 3.3.1)

1.1.1 Preparing for variables

capital$U_L <- capital$U/capital$L*100
capital$K2_N <- capital$K2/capital$N*100
capital$A2_K2 <- capital$A2/capital$K2*100
capital$C2_A2 <- capital$C2/capital$A2*100
capital$C2_K2 <- capital$C2/capital$K2*100
capital$I2_C2 <- capital$I2/capital$C2*100
capital$S2_C2 <- capital$S2/capital$C2*100
capital$X2_C2 <- capital$X2/capital$C2*100

capital$LU_L <- log(capital$U_L)
capital$LK2_N <- log(capital$K2_N)
capital$LA2_K2 <- log(capital$A2_K2)
capital$LC2_A2 <- log(capital$C2_A2)
capital$LC2_K2 <- log(capital$C2_K2)
capital$LI2_C2 <- log(capital$I2_C2)

capital$S2_C2_1 <- (capital$S2+1)/capital$C2*100
capital$X2_C2_1 <- (capital$X2+1)/capital$C2*100
capital$LS2_C2 <- log(capital$S2_C2_1)
capital$LX2_C2 <- log(capital$X2_C2_1)

capital$Y1955 <- ifelse(capital$YEAR<1955,0,1)
capital$Y1972 <- ifelse(capital$YEAR<1972,0,1)
capital$Y2005 <- ifelse(capital$YEAR<2005,0,1)

1.1.2 Descriptive Statistics of Variables (Table 2)

library(RcmdrMisc)
sum_K2 <- numSummary(capital[,c("U_L","K2_N","A2_K2","C2_A2","C2_K2","I2_C2","S2_C2","X2_C2")],statistics=c("mean", "sd", "quantiles"), quantiles=c(0,.5,1))
sum_K2a <- cbind(sum_K2$table,sum_K2$n)
print(round(sum_K2a,3))
##         mean    sd     0%    50%    100%   
## U_L    2.644 1.229  1.118  2.269   5.367 71
## K2_N   1.709 0.858  0.705  1.465   3.490 71
## A2_K2 96.739 1.556 91.784 96.688 101.279 71
## C2_A2 50.352 9.073 25.161 52.636  64.401 71
## C2_K2 48.671 8.646 25.435 50.792  62.681 71
## I2_C2  2.020 1.096  0.616  1.764   5.714 71
## S2_C2  0.554 0.680  0.000  0.356   5.025 71
## X2_C2  0.349 0.640  0.000  0.157   4.833 71

1.1.3 Correlation Coefficients between Variables (Table 3)

cor_K2 <- cor(capital[,c("U_L","A2_K2","C2_A2","C2_K2","I2_C2","S2_C2","X2_C2")]) 
print(round(cor_K2,3))
##          U_L  A2_K2  C2_A2  C2_K2  I2_C2  S2_C2  X2_C2
## U_L    1.000 -0.054 -0.126 -0.138  0.476  0.154  0.110
## A2_K2 -0.054  1.000 -0.289 -0.217 -0.279 -0.373  0.019
## C2_A2 -0.126 -0.289  1.000  0.997 -0.264 -0.082 -0.426
## C2_K2 -0.138 -0.217  0.997  1.000 -0.294 -0.119 -0.433
## I2_C2  0.476 -0.279 -0.264 -0.294  1.000  0.629  0.215
## S2_C2  0.154 -0.373 -0.082 -0.119  0.629  1.000  0.118
## X2_C2  0.110  0.019 -0.426 -0.433  0.215  0.118  1.000

1.1.4 Plot of time-series data of each variable (Figure 2)

par(mar = c(4, 4, 2, 1))
plot(capital$YEAR,capital$U_L,type="l",xlab="",ylab="",main="Unemployment rate")

plot(capital$YEAR,capital$K2_N,type="l",xlab="",ylab="",main="Homicide rate")

plot(capital$YEAR,capital$A2_K2,type="l",xlab="",ylab="",main="Clearance rate")

plot(capital$YEAR,capital$C2_A2,type="l",xlab="",ylab="",main="Conviction rate 1")

plot(capital$YEAR,capital$C2_K2,type="l",xlab="",ylab="",main="Conviction rate 2")

plot(capital$YEAR,capital$I2_C2,type="l",xlab="",ylab="",main="Life Sentence rate")

plot(capital$YEAR,capital$S2_C2,type="l",xlab="",ylab="",main="Death Sentence rate")

plot(capital$YEAR,capital$X2_C2,type="l",xlab="",ylab="",main="Execution rate")

1.2 Unit Root Test (Section 3.3.2)

1.2.1 ADF Test (Table 4)

1.2.1.1 Level variables

1.2.1.1.1 Trend and Constant
library(CADFtest)
print(CADFtest(capital$LK2_N, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LK2_N
## ADF(1) = -2.078, p-value = 0.548
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.149623
print(CADFtest(capital$LA2_K2, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LA2_K2
## ADF(3) = -1.143, p-value = 0.913
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.253265
print(CADFtest(capital$LC2_A2, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LC2_A2
## ADF(2) = -1.811, p-value = 0.688
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.129821
print(CADFtest(capital$LC2_K2, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LC2_K2
## ADF(2) = -1.99, p-value = 0.596
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.147804
print(CADFtest(capital$LI2_C2, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LI2_C2
## ADF(3) = -2.826, p-value = 0.194
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.431698
print(CADFtest(capital$LS2_C2, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LS2_C2
## ADF(3) = -3.238, p-value = 0.0863
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.623564
print(CADFtest(capital$LX2_C2, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LX2_C2
## ADF(5) = -1.601, p-value = 0.782
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.379475
print(CADFtest(capital$LU_L, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LU_L
## ADF(2) = -1.47, p-value = 0.83
## alternative hypothesis: true delta is less than 0
## sample estimates:
##      delta 
## -0.0707852
1.2.1.1.2 Constant
print(CADFtest(capital$LK2_N, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LK2_N
## ADF(1) = -1.193, p-value = 0.673
## alternative hypothesis: true delta is less than 0
## sample estimates:
##    delta 
## -0.01944
print(CADFtest(capital$LA2_K2, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LA2_K2
## ADF(3) = -1.373, p-value = 0.59
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.293543
print(CADFtest(capital$LC2_A2, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LC2_A2
## ADF(1) = -0.8775, p-value = 0.789
## alternative hypothesis: true delta is less than 0
## sample estimates:
##      delta 
## -0.0621418
print(CADFtest(capital$LC2_K2, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LC2_K2
## ADF(1) = -0.9995, p-value = 0.749
## alternative hypothesis: true delta is less than 0
## sample estimates:
##      delta 
## -0.0732074
print(CADFtest(capital$LI2_C2, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LI2_C2
## ADF(3) = -1.608, p-value = 0.473
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.210157
print(CADFtest(capital$LS2_C2, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LS2_C2
## ADF(3) = -1.577, p-value = 0.489
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.245687
print(CADFtest(capital$LX2_C2, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LX2_C2
## ADF(4) = -0.5082, p-value = 0.882
## alternative hypothesis: true delta is less than 0
## sample estimates:
##      delta 
## -0.0854885
print(CADFtest(capital$LU_L, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LU_L
## ADF(3) = -1.58, p-value = 0.487
## alternative hypothesis: true delta is less than 0
## sample estimates:
##      delta 
## -0.0423042

1.2.1.2 First-differenced variables

1.2.1.2.1 Trend and Constant
print(CADFtest(diff(capital$LK2_N), type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LK2_N)
## ADF(0) = -9.634, p-value = 9.13e-09
## alternative hypothesis: true delta is less than 0
## sample estimates:
##   delta 
## -1.1976
print(CADFtest(diff(capital$LA2_K2), type="trend", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LA2_K2)
## ADF(0) = -17.729, p-value = 1e-04
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.712858
print(CADFtest(diff(capital$LC2_A2), type="trend", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LC2_A2)
## ADF(0) = -11.552, p-value = 3.491e-07
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.414344
print(CADFtest(diff(capital$LC2_K2), type="trend", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LC2_K2)
## ADF(0) = -11.562, p-value = 3.614e-07
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.398386
print(CADFtest(diff(capital$LI2_C2), type="trend", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LI2_C2)
## ADF(0) = -11.477, p-value = 2.691e-07
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.365994
print(CADFtest(diff(capital$LS2_C2), type="trend", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LS2_C2)
## ADF(0) = -15.901, p-value = 1e-04
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.613699
print(CADFtest(diff(capital$LX2_C2), type="trend", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LX2_C2)
## ADF(0) = -12.406, p-value = 1.153e-05
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.443024
print(CADFtest(diff(capital$LU_L), type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LU_L)
## ADF(2) = -3.243, p-value = 0.0855
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.535715
1.2.1.2.2 Constant
print(CADFtest(diff(capital$LK2_N), type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LK2_N)
## ADF(3) = -3.893, p-value = 0.00357
## alternative hypothesis: true delta is less than 0
## sample estimates:
##    delta 
## -1.16436
print(CADFtest(diff(capital$LA2_K2), type="drift", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LA2_K2)
## ADF(0) = -17.821, p-value = 2.056e-09
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.709708
print(CADFtest(diff(capital$LC2_A2), type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LC2_A2)
## ADF(3) = -3.118, p-value = 0.0302
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.967469
print(CADFtest(diff(capital$LC2_K2), type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LC2_K2)
## ADF(3) = -3.234, p-value = 0.0225
## alternative hypothesis: true delta is less than 0
## sample estimates:
##    delta 
## -1.02566
print(CADFtest(diff(capital$LI2_C2), type="drift", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LI2_C2)
## ADF(0) = -11.557, p-value = 6.107e-13
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.364579
print(CADFtest(diff(capital$LS2_C2), type="drift", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LS2_C2)
## ADF(0) = -16.005, p-value = 1.184e-11
## alternative hypothesis: true delta is less than 0
## sample estimates:
##    delta 
## -1.61214
print(CADFtest(diff(capital$LX2_C2), type="drift", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LX2_C2)
## ADF(0) = -12.465, p-value = 3.127e-13
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.440928
print(CADFtest(diff(capital$LU_L), type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LU_L)
## ADF(2) = -3.27, p-value = 0.0205
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.535222

1.2.2 KPSS Test (Table 4)

1.2.2.1 Level variables

1.2.2.1.1 Trend and Constant
library(urca)
summary(ur.kpss(capital$LK2_N,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.2387 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LA2_K2,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.1132 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LC2_A2,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.273 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LC2_K2,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.2779 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LI2_C2,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.3239 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LS2_C2,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.2432 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LX2_C2,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.1953 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LU_L,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.1685 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
1.2.2.1.2 Constant
summary(ur.kpss(capital$LK2_N,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 1.808 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LA2_K2,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.1444 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LC2_A2,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.4069 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LC2_K2,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.4098 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LI2_C2,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.5876 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LS2_C2,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.4806 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LX2_C2,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.8529 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LU_L,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 1.3555 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739

1.2.2.2 First-differenced variables

1.2.2.2.1 Trend and Constant
summary(ur.kpss(diff(capital$LK2_N),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.069 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LA2_K2),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.0814 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LC2_A2),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.0586 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LC2_K2),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.0529 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LI2_C2),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.061 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LS2_C2),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.087 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LX2_C2),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.0432 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LU_L),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.115 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
1.2.2.2.2 Constant
summary(ur.kpss(diff(capital$LK2_N),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.067 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LA2_K2),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.1483 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LC2_A2),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.1741 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LC2_K2),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.1982 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LI2_C2),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.2022 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LS2_C2),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.2755 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LX2_C2),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.1993 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LU_L),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.1628 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739

1.3 Cointegration Test (Section 3.3.3)

1.3.1 Preparing for variables

dat_K2_2 <- capital[,c("LU_L","LK2_N","LC2_K2","LI2_C2","LS2_C2","LX2_C2")]
dum <- capital[,c("Y1955","Y1972","Y2005")]

1.3.2 Determining the lag length

library(vars)
VARselect(dat_K2_2, lag.max=5, type="const", exogen=dum)
## $selection
## AIC(n)  HQ(n)  SC(n) FPE(n) 
##      5      1      1      5 
## 
## $criteria
##                    1             2             3             4             5
## AIC(n) -1.835699e+01 -1.826033e+01 -1.814427e+01 -1.820428e+01 -1.943218e+01
## HQ(n)  -1.757041e+01 -1.700180e+01 -1.641380e+01 -1.600185e+01 -1.675781e+01
## SC(n)  -1.636639e+01 -1.507538e+01 -1.376496e+01 -1.263061e+01 -1.266416e+01
## FPE(n)  1.080907e-08  1.245378e-08  1.545413e-08  1.750231e-08  6.999735e-09

1.3.3 Johansen Cointegration Test (Table 5)

1.3.3.1 Constant

rank_K2_const_2 <- ca.jo(dat_K2_2, ecdet=c("const"),type="trace",K=2,spec="longrun",dumvar=dum)
summary(rank_K2_const_2)
## 
## ###################### 
## # Johansen-Procedure # 
## ###################### 
## 
## Test type: trace statistic , without linear trend and constant in cointegration 
## 
## Eigenvalues (lambda):
## [1] 4.323190e-01 3.803469e-01 3.345632e-01 1.841003e-01 1.032335e-01
## [6] 6.242613e-02 6.661338e-16
## 
## Values of teststatistic and critical values of test:
## 
##            test 10pct   5pct   1pct
## r <= 5 |   4.45  7.52   9.24  12.97
## r <= 4 |  11.97 17.85  19.96  24.60
## r <= 3 |  26.00 32.00  34.91  41.07
## r <= 2 |  54.11 49.65  53.12  60.16
## r <= 1 |  87.13 71.86  76.07  84.45
## r = 0  | 126.20 97.18 102.14 111.01
## 
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
## 
##               LU_L.l2  LK2_N.l2   LC2_K2.l2   LI2_C2.l2   LS2_C2.l2   LX2_C2.l2
## LU_L.l2    1.00000000  1.000000  1.00000000    1.000000  1.00000000   1.0000000
## LK2_N.l2  -0.08563392  1.101482  1.48568634  -44.881139 -3.53141500   1.2914279
## LC2_K2.l2 -0.86329780 -7.257757 -0.36696700   98.515022 -0.20822702   2.3029327
## LI2_C2.l2 -2.11812176  1.639968 -0.09638776  -27.193273 -0.07045611   0.2685567
## LS2_C2.l2  0.87016505 -1.433542 -0.45424965  -25.822654 -0.05179951   0.2891275
## LX2_C2.l2 -0.19493510 -1.761629  0.61569218    2.193684  0.22517002   0.1594403
## constant   4.37350289 18.341556  0.14810258 -298.880639  4.43662536 -12.0198141
##             constant
## LU_L.l2    1.0000000
## LK2_N.l2  -0.2646032
## LC2_K2.l2  1.5357304
## LI2_C2.l2 -0.2399128
## LS2_C2.l2 -0.4670023
## LX2_C2.l2 -0.2280600
## constant  -4.8498921
## 
## Weights W:
## (This is the loading matrix)
## 
##              LU_L.l2      LK2_N.l2   LC2_K2.l2     LI2_C2.l2    LS2_C2.l2
## LU_L.d   -0.02047251 -0.0064457569 -0.00234825  4.640915e-04 -0.010814601
## LK2_N.d  -0.02690932  0.0007915992  0.00858428  7.224453e-05  0.015474179
## LC2_K2.d  0.04711451  0.0161822875  0.07485182 -3.047768e-04 -0.005775784
## LI2_C2.d  0.35368910 -0.0134935828 -0.01761382  6.701195e-03  0.014486979
## LS2_C2.d -0.18513229  0.1413606673  0.31208257  1.183133e-02 -0.003847439
## LX2_C2.d  0.02877281  0.2474760389 -0.64766868 -2.964487e-04  0.012056018
##             LX2_C2.l2      constant
## LU_L.d   -0.030786984  2.651460e-16
## LK2_N.d  -0.006144532 -2.353504e-15
## LC2_K2.d -0.009416477 -2.271823e-14
## LI2_C2.d -0.021938491  3.646985e-14
## LS2_C2.d -0.008900883  1.882460e-14
## LX2_C2.d -0.043552887 -2.275182e-13

1.3.3.2 Trend

rank_K2_trend_2 <- ca.jo(dat_K2_2, ecdet=c("trend"),type="trace",K=2,spec="longrun",dumvar=dum)
summary(rank_K2_trend_2)
## 
## ###################### 
## # Johansen-Procedure # 
## ###################### 
## 
## Test type: trace statistic , with linear trend in cointegration 
## 
## Eigenvalues (lambda):
## [1] 4.671144e-01 4.052823e-01 3.345194e-01 1.953452e-01 1.034380e-01
## [6] 3.224415e-02 4.163336e-17
## 
## Values of teststatistic and critical values of test:
## 
##            test  10pct   5pct   1pct
## r <= 5 |   2.26  10.49  12.25  16.26
## r <= 4 |   9.80  22.76  25.32  30.45
## r <= 3 |  24.79  39.06  42.44  48.45
## r <= 2 |  52.89  59.14  62.99  70.05
## r <= 1 |  88.75  83.20  87.31  96.58
## r = 0  | 132.18 110.42 114.90 124.75
## 
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
## 
##              LU_L.l2   LK2_N.l2    LC2_K2.l2   LI2_C2.l2   LS2_C2.l2
## LU_L.l2    1.0000000  1.0000000  1.000000000  1.00000000  1.00000000
## LK2_N.l2  -3.9958474  7.7211266  1.566231143 -2.72477665 -4.66048757
## LC2_K2.l2  1.2186532 -6.6862864  0.011715527 -2.79088822  0.12240172
## LI2_C2.l2 -1.5715624 -1.8498059 -0.170257166  0.83174535 -0.40294384
## LS2_C2.l2  1.0940887 -0.2662846 -0.443675894  0.83816160 -0.40930898
## LX2_C2.l2  0.3166758 -1.4746860  0.738273147  0.01449794  0.05205570
## trend.l2  -0.1178612  0.2154370  0.001968889 -0.12663513 -0.04265582
##             LX2_C2.l2    trend.l2
## LU_L.l2    1.00000000  1.00000000
## LK2_N.l2   0.48163912 -4.51183275
## LC2_K2.l2  1.97362190  0.44688165
## LI2_C2.l2 -0.08008088 -0.29947027
## LS2_C2.l2 -0.24785478 -0.03653680
## LX2_C2.l2 -0.08614736 -0.05042698
## trend.l2   0.01165801 -0.14772562
## 
## Weights W:
## (This is the loading matrix)
## 
##               LU_L.l2     LK2_N.l2     LC2_K2.l2    LI2_C2.l2    LS2_C2.l2
## LU_L.d   -0.023109708 -0.005948199 -0.0005290804 -0.033352646 -0.003763235
## LK2_N.d  -0.025654003 -0.008706464  0.0079626810  0.005467172  0.020250000
## LC2_K2.d  0.000655055  0.031425404  0.0619898277  0.006484110 -0.004436507
## LI2_C2.d  0.341330472  0.103489132 -0.0035258417 -0.231036448  0.048013924
## LS2_C2.d -0.614430000  0.055015214  0.2632540525 -0.317866514  0.053631910
## LX2_C2.d -0.481819240  0.171271667 -0.6867236399  0.045202650  0.040954197
##             LX2_C2.l2      trend.l2
## LU_L.d   -0.037896502  1.180439e-14
## LK2_N.d  -0.004615909 -3.367341e-16
## LC2_K2.d -0.011862136 -2.815699e-14
## LI2_C2.d  0.004367677  5.909522e-14
## LS2_C2.d  0.044769843  1.077742e-13
## LX2_C2.d -0.028983354 -1.680801e-13

1.4 Constructing VECM (Section 3.3.4)

1.4.1 Estimating the VECM equation (Equation (2))

1.4.1.1 Coefficients of the equation

vecm_K2_2 <- cajorls(rank_K2_trend_2, r=2)
print(vecm_K2_2, digits=3)
## $rlm
## 
## Call:
## lm(formula = substitute(form1), data = data.mat)
## 
## Coefficients:
##             LU_L.d    LK2_N.d   LC2_K2.d  LI2_C2.d  LS2_C2.d  LX2_C2.d
## ect1        -0.02906  -0.03436   0.03208   0.44482  -0.55941  -0.31055
## ect2         0.04642   0.03529   0.24002  -0.56485   2.87995   3.24769
## constant    -0.07977  -0.17288   0.41226   1.96416  -1.22845   0.76847
## Y1955       -0.08151  -0.05222   0.06618   0.27922   0.43067   0.45861
## Y1972        0.07208   0.03394  -0.04894  -0.11489   0.53924   0.03891
## Y2005       -0.08152  -0.02014  -0.09687   0.10154  -0.79298  -0.48622
## LU_L.dl1     0.19997   0.08007   0.16021   0.07902  -1.10774   0.32949
## LK2_N.dl1    0.26242  -0.24151   0.62968   0.77928   3.91210  -1.49922
## LC2_K2.dl1  -0.00325   0.01613  -0.52989   0.09074  -1.03692  -3.34693
## LI2_C2.dl1   0.01117  -0.00533   0.02921  -0.69887   0.55675   0.30755
## LS2_C2.dl1  -0.00755  -0.00203  -0.01067   0.14281  -0.97881  -0.18809
## LX2_C2.dl1   0.02253   0.00450  -0.02632   0.02136  -0.14422  -0.64247
## 
## 
## $beta
##              ect1      ect2
## LU_L.l2    1.0000  2.78e-17
## LK2_N.l2   0.0000  1.00e+00
## LC2_K2.l2 -1.4772 -6.75e-01
## LI2_C2.l2 -1.6665 -2.37e-02
## LS2_C2.l2  0.6302 -1.16e-01
## LX2_C2.l2 -0.2942 -1.53e-01
## trend.l2  -0.0042  2.84e-02

1.4.1.2 The t values of coefficients

print(coef(summary(vecm_K2_2$rlm)),digits=3)
## Response LU_L.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1       -0.02906     0.0287 -1.0112   0.3162
## ect2        0.04642     0.1306  0.3554   0.7236
## constant   -0.07977     0.1500 -0.5319   0.5968
## Y1955      -0.08151     0.0552 -1.4770   0.1452
## Y1972       0.07208     0.0345  2.0921   0.0409
## Y2005      -0.08152     0.0439 -1.8570   0.0685
## LU_L.dl1    0.19997     0.1299  1.5399   0.1291
## LK2_N.dl1   0.26242     0.2427  1.0811   0.2842
## LC2_K2.dl1 -0.00325     0.1440 -0.0226   0.9821
## LI2_C2.dl1  0.01117     0.0359  0.3112   0.7568
## LS2_C2.dl1 -0.00755     0.0223 -0.3390   0.7359
## LX2_C2.dl1  0.02253     0.0165  1.3674   0.1769
## 
## Response LK2_N.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1       -0.03436    0.01562  -2.200   0.0319
## ect2        0.03529    0.07098   0.497   0.6210
## constant   -0.17288    0.08151  -2.121   0.0383
## Y1955      -0.05222    0.02999  -1.741   0.0870
## Y1972       0.03394    0.01873   1.812   0.0752
## Y2005      -0.02014    0.02386  -0.844   0.4020
## LU_L.dl1    0.08007    0.07058   1.134   0.2613
## LK2_N.dl1  -0.24151    0.13193  -1.831   0.0724
## LC2_K2.dl1  0.01613    0.07829   0.206   0.8375
## LI2_C2.dl1 -0.00533    0.01950  -0.274   0.7854
## LS2_C2.dl1 -0.00203    0.01211  -0.167   0.8678
## LX2_C2.dl1  0.00450    0.00896   0.502   0.6176
## 
## Response LC2_K2.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1         0.0321     0.0234   1.373 1.75e-01
## ect2         0.2400     0.1062   2.261 2.76e-02
## constant     0.4123     0.1219   3.381 1.31e-03
## Y1955        0.0662     0.0449   1.475 1.46e-01
## Y1972       -0.0489     0.0280  -1.747 8.60e-02
## Y2005       -0.0969     0.0357  -2.714 8.78e-03
## LU_L.dl1     0.1602     0.1056   1.517 1.35e-01
## LK2_N.dl1    0.6297     0.1974   3.191 2.31e-03
## LC2_K2.dl1  -0.5299     0.1171  -4.525 3.12e-05
## LI2_C2.dl1   0.0292     0.0292   1.001 3.21e-01
## LS2_C2.dl1  -0.0107     0.0181  -0.589 5.58e-01
## LX2_C2.dl1  -0.0263     0.0134  -1.964 5.44e-02
## 
## Response LI2_C2.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1         0.4448      0.119   3.749 4.17e-04
## ect2        -0.5649      0.539  -1.048 2.99e-01
## constant     1.9642      0.619   3.172 2.44e-03
## Y1955        0.2792      0.228   1.225 2.25e-01
## Y1972       -0.1149      0.142  -0.808 4.23e-01
## Y2005        0.1015      0.181   0.560 5.78e-01
## LU_L.dl1     0.0790      0.536   0.147 8.83e-01
## LK2_N.dl1    0.7793      1.002   0.778 4.40e-01
## LC2_K2.dl1   0.0907      0.595   0.153 8.79e-01
## LI2_C2.dl1  -0.6989      0.148  -4.717 1.59e-05
## LS2_C2.dl1   0.1428      0.092   1.552 1.26e-01
## LX2_C2.dl1   0.0214      0.068   0.314 7.55e-01
## 
## Response LS2_C2.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1         -0.559     0.1743   -3.21 2.18e-03
## ect2          2.880     0.7921    3.64 5.96e-04
## constant     -1.228     0.9095   -1.35 1.82e-01
## Y1955         0.431     0.3347    1.29 2.03e-01
## Y1972         0.539     0.2090    2.58 1.25e-02
## Y2005        -0.793     0.2662   -2.98 4.25e-03
## LU_L.dl1     -1.108     0.7876   -1.41 1.65e-01
## LK2_N.dl1     3.912     1.4722    2.66 1.02e-02
## LC2_K2.dl1   -1.037     0.8736   -1.19 2.40e-01
## LI2_C2.dl1    0.557     0.2176    2.56 1.32e-02
## LS2_C2.dl1   -0.979     0.1352   -7.24 1.26e-09
## LX2_C2.dl1   -0.144     0.0999   -1.44 1.54e-01
## 
## Response LX2_C2.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1        -0.3105      0.205  -1.514 1.36e-01
## ect2         3.2477      0.932   3.483 9.59e-04
## constant     0.7685      1.071   0.718 4.76e-01
## Y1955        0.4586      0.394   1.164 2.49e-01
## Y1972        0.0389      0.246   0.158 8.75e-01
## Y2005       -0.4862      0.313  -1.551 1.26e-01
## LU_L.dl1     0.3295      0.927   0.355 7.24e-01
## LK2_N.dl1   -1.4992      1.733  -0.865 3.91e-01
## LC2_K2.dl1  -3.3469      1.028  -3.255 1.91e-03
## LI2_C2.dl1   0.3075      0.256   1.201 2.35e-01
## LS2_C2.dl1  -0.1881      0.159  -1.182 2.42e-01
## LX2_C2.dl1  -0.6425      0.118  -5.461 1.08e-06

1.4.2 Diagnostic tests

var_K2_2 <- vec2var(rank_K2_trend_2,r=2)
normality.test(var_K2_2)
## $JB
## 
##  JB-Test (multivariate)
## 
## data:  Residuals of VAR object var_K2_2
## Chi-squared = 18.968, df = 12, p-value = 0.08931
## 
## 
## $Skewness
## 
##  Skewness only (multivariate)
## 
## data:  Residuals of VAR object var_K2_2
## Chi-squared = 11.742, df = 6, p-value = 0.06797
## 
## 
## $Kurtosis
## 
##  Kurtosis only (multivariate)
## 
## data:  Residuals of VAR object var_K2_2
## Chi-squared = 7.2254, df = 6, p-value = 0.3005
serial.test(var_K2_2)
## 
##  Portmanteau Test (asymptotic)
## 
## data:  Residuals of VAR object var_K2_2
## Chi-squared = 507.54, df = 510, p-value = 0.5225
arch.test(var_K2_2)
## 
##  ARCH (multivariate)
## 
## data:  Residuals of VAR object var_K2_2
## Chi-squared = 1344, df = 2205, p-value = 1

1.5 Innovation Accounting (Section 3.3.5)

1.5.1 Cumulative impulse responses of the homicide rate (Figure 3)

cirf_LU_L <- irf(var_K2_2, impulse=c("LU_L"), response=c("LK2_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LK2_N <- irf(var_K2_2, impulse=c("LK2_N"), response=c("LK2_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LC2_K2 <- irf(var_K2_2, impulse=c("LC2_K2"), response=c("LK2_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LI2_C2 <- irf(var_K2_2, impulse=c("LI2_C2"), response=c("LK2_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LS2_C2 <- irf(var_K2_2, impulse=c("LS2_C2"), response=c("LK2_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LX2_C2 <- irf(var_K2_2, impulse=c("LX2_C2"), response=c("LK2_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
plot(cirf_LU_L)

plot(cirf_LK2_N)

plot(cirf_LC2_K2)

plot(cirf_LI2_C2)

plot(cirf_LS2_C2)

plot(cirf_LX2_C2)

### Forecast error variance decomposition of the homicide rate (Figure 4)

fevd_K2 <- fevd(var_K2_2,n.ahead=10)
par(mar = c(4, 4, 1, 1))
plot(fevd_K2$LK2_N[,1]*100,type="o",pch=1,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
axis(1)
axis(2)
par(new=T)
plot(fevd_K2$LK2_N[,3]*100,type="o",pch=6,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
par(new=T)
plot(fevd_K2$LK2_N[,4]*100,type="o",pch=5,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
par(new=T)
plot(fevd_K2$LK2_N[,5]*100,type="o",pch=15,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
par(new=T)
plot(fevd_K2$LK2_N[,6]*100,type="o",pch=4,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
box()
legend("topright", legend = c("Death Sentence","Unemployment", "Life Sentence","Execution","Conviction"), pch = c(15,1,5,4,6), cex=0.72)
mtext("%",at=0.4)

2 Analysis of the Deterrent Effect on the Robbery-Homicide Rate (Section 3.4 of the Paper)

2.1 Descriptive Statistics of Variables (Section 3.4.1)

2.1.1 Preparing for variables

capital$K4_N <- capital$K4/capital$N*100
capital$A4_K4 <- capital$A4/capital$K4*100
capital$C4_A4 <- capital$C4/capital$A4*100
capital$I4_C4 <- capital$I4/capital$C4*100
capital$S4_C4 <- capital$S4/capital$C4*100
capital$X4_C4 <- capital$X4/capital$C4*100

capital$LK4_N <- log(capital$K4_N)
capital$LA4_K4 <- log(capital$A4_K4)
capital$LC4_A4 <- log(capital$C4_A4)
capital$LI4_C4 <- log(capital$I4_C4)

capital$S4_C4_1 <- (capital$S4+1)/capital$C4*100
capital$X4_C4_1 <- (capital$X4+1)/capital$C4*100
capital$LS4_C4 <- log(capital$S4_C4_1)
capital$LX4_C4 <- log(capital$X4_C4_1)

2.1.2 Descriptive Statistics of Variables (Table 6)

sum_K4 <- numSummary(capital[,c("K4_N","A4_K4","C4_A4","I4_C4","S4_C4","X4_C4"),drop=F],statistics=c("mean", "sd", "quantiles"), quantiles=c(0,.5,1))
sum_K4a <- cbind(sum_K4$table,sum_K4$n)
print(round(sum_K4a,3))
##         mean     sd     0%    50%   100%   
## K4_N   1.364  0.607  0.576  1.194  2.664 71
## A4_K4 78.502 10.011 51.811 82.839 90.755 71
## C4_A4 48.651 10.027 30.610 47.165 81.741 71
## I4_C4  6.748  1.797  2.643  6.677 13.710 71
## S4_C4  1.220  1.008  0.000  0.872  4.617 71
## X4_C4  1.074  1.211  0.000  0.559  4.075 71

2.1.3 Correlation Coefficients between Variables (Table 7)

cor_K4 <- cor(capital[,c("U_L","A4_K4","C4_A4","I4_C4","S4_C4","X4_C4")]) 
print(round(cor_K4,3))
##          U_L  A4_K4  C4_A4  I4_C4  S4_C4  X4_C4
## U_L    1.000 -0.839 -0.047  0.099 -0.390 -0.457
## A4_K4 -0.839  1.000 -0.114 -0.055  0.427  0.447
## C4_A4 -0.047 -0.114  1.000  0.119  0.489  0.243
## I4_C4  0.099 -0.055  0.119  1.000  0.119 -0.072
## S4_C4 -0.390  0.427  0.489  0.119  1.000  0.431
## X4_C4 -0.457  0.447  0.243 -0.072  0.431  1.000

2.1.4 Plot of time-series data of each variable (Figure 5)

par(mar = c(4, 4, 2, 1))
plot(capital$YEAR,capital$K4_N,type="l",xlab="",ylab="",main="Robbery-homicide rate")

plot(capital$YEAR,capital$A4_K4,type="l",xlab="",ylab="",main="Clearance rate")

plot(capital$YEAR,capital$C4_A4,type="l",xlab="",ylab="",main="Conviction rate")

plot(capital$YEAR,capital$I4_C4,type="l",xlab="",ylab="",main="Life Sentence rate")

plot(capital$YEAR,capital$S4_C4,type="l",xlab="",ylab="",main="Death Sentence rate")

plot(capital$YEAR,capital$X4_C4,type="l",xlab="",ylab="",main="Execution rate")

2.2 Unit Root Test (Section 3.4.2)

2.2.1 ADF Test (Table 8)

2.2.1.1 Level variables

2.2.1.1.1 Trend and Constant
print(CADFtest(capital$LK4_N, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LK4_N
## ADF(1) = -1.411, p-value = 0.849
## alternative hypothesis: true delta is less than 0
## sample estimates:
##    delta 
## -0.04369
print(CADFtest(capital$LA4_K4, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LA4_K4
## ADF(0) = -1.396, p-value = 0.853
## alternative hypothesis: true delta is less than 0
## sample estimates:
##      delta 
## -0.0912374
print(CADFtest(capital$LC4_A4, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LC4_A4
## ADF(1) = -2.303, p-value = 0.426
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.211811
print(CADFtest(capital$LI4_C4, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LI4_C4
## ADF(5) = -1.679, p-value = 0.749
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.414913
print(CADFtest(capital$LS4_C4, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LS4_C4
## ADF(4) = -2.575, p-value = 0.293
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.577755
print(CADFtest(capital$LX4_C4, type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LX4_C4
## ADF(2) = -2.852, p-value = 0.185
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.479578
2.2.1.1.2 Constant
print(CADFtest(capital$LK4_N, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LK4_N
## ADF(1) = -1.189, p-value = 0.674
## alternative hypothesis: true delta is less than 0
## sample estimates:
##      delta 
## -0.0351153
print(CADFtest(capital$LA4_K4, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LA4_K4
## ADF(0) = -1.65, p-value = 0.452
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.078448
print(CADFtest(capital$LC4_A4, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LC4_A4
## ADF(1) = -2.156, p-value = 0.224
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.190156
print(CADFtest(capital$LI4_C4, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LI4_C4
## ADF(5) = -1.667, p-value = 0.443
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.408702
print(CADFtest(capital$LS4_C4, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LS4_C4
## ADF(4) = -2.492, p-value = 0.122
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.376591
print(CADFtest(capital$LX4_C4, type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  capital$LX4_C4
## ADF(2) = -2.741, p-value = 0.0727
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.362308

2.2.1.2 First-differenced variables

2.2.1.2.1 Trend and Constant
print(CADFtest(diff(capital$LK4_N), type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LK4_N)
## ADF(2) = -2.967, p-value = 0.15
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.432255
print(CADFtest(diff(capital$LA4_K4), type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LA4_K4)
## ADF(3) = -3.026, p-value = 0.133
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.716112
print(CADFtest(diff(capital$LC4_A4), type="trend", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LC4_A4)
## ADF(1) = -6.048, p-value = 1.95e-05
## alternative hypothesis: true delta is less than 0
## sample estimates:
##    delta 
## -1.23779
print(CADFtest(diff(capital$LI4_C4), type="trend", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LI4_C4)
## ADF(0) = -14.914, p-value = 1e-04
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.610214
print(CADFtest(diff(capital$LS4_C4), type="trend", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LS4_C4)
## ADF(0) = -13.304, p-value = 1e-04
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.503655
print(CADFtest(diff(capital$LX4_C4), type="trend", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LX4_C4)
## ADF(0) = -12.639, p-value = 3.416e-05
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.446717
2.2.1.2.2 Constant
print(CADFtest(diff(capital$LK4_N), type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LK4_N)
## ADF(2) = -2.998, p-value = 0.0404
## alternative hypothesis: true delta is less than 0
## sample estimates:
##    delta 
## -0.43247
print(CADFtest(diff(capital$LA4_K4), type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LA4_K4)
## ADF(3) = -2.983, p-value = 0.0418
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -0.700026
print(CADFtest(diff(capital$LC4_A4), type="drift", max.lag.y=5, criterion="MAIC"),digits=6)
## 
##  ADF test
## 
## data:  diff(capital$LC4_A4)
## ADF(1) = -6.081, p-value = 1.98e-06
## alternative hypothesis: true delta is less than 0
## sample estimates:
##    delta 
## -1.23534
print(CADFtest(diff(capital$LI4_C4), type="drift", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LI4_C4)
## ADF(0) = -15.019, p-value = 1.655e-12
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.611149
print(CADFtest(diff(capital$LS4_C4), type="drift", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LS4_C4)
## ADF(0) = -13.396, p-value = 3.141e-13
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.503467
print(CADFtest(diff(capital$LX4_C4), type="drift", max.lag.y=5, criterion="MAIC"),digits=7)
## 
##  ADF test
## 
## data:  diff(capital$LX4_C4)
## ADF(0) = -12.662, p-value = 2.953e-13
## alternative hypothesis: true delta is less than 0
## sample estimates:
##     delta 
## -1.441917

2.2.2 KPSS Test (Table 8)

2.2.2.1 Level variables

2.2.2.1.1 Trend and Constant
summary(ur.kpss(capital$LK4_N,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.2495 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LA4_K4,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.1062 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LC4_A4,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.1736 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LI4_C4,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.1623 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LS4_C4,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.1981 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(capital$LX4_C4,type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.1571 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
2.2.2.1.2 Constant
summary(ur.kpss(capital$LK4_N,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.5175 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LA4_K4,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.999 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LC4_A4,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.5412 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LI4_C4,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.2096 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LS4_C4,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 1.0753 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(capital$LX4_C4,type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.897 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739

2.2.2.2 First-differenced variables

2.2.2.2.1 Trend and Constant
summary(ur.kpss(diff(capital$LK4_N),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.1423 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LA4_K4),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.0917 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LC4_A4),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.0612 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LI4_C4),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.0681 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LS4_C4),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.0293 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
summary(ur.kpss(diff(capital$LX4_C4),type="tau",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: tau with 3 lags. 
## 
## Value of test-statistic is: 0.0326 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.119 0.146  0.176 0.216
2.2.2.2.2 Constant
summary(ur.kpss(diff(capital$LK4_N),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.138 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LA4_K4),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.1084 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LC4_A4),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.0816 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LI4_C4),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.1336 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LS4_C4),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.092 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739
summary(ur.kpss(diff(capital$LX4_C4),type="mu",lags="short"))
## 
## ####################### 
## # KPSS Unit Root Test # 
## ####################### 
## 
## Test is of type: mu with 3 lags. 
## 
## Value of test-statistic is: 0.0596 
## 
## Critical value for a significance level of: 
##                 10pct  5pct 2.5pct  1pct
## critical values 0.347 0.463  0.574 0.739

2.3 Cointegration Test (Section 3.4.3)

2.3.1 Preparing for variables

dat_K4_1 <- capital[,c("LU_L","LK4_N","LA4_K4","LC4_A4","LI4_C4","LS4_C4","LX4_C4")]
dum <- capital[,c("Y1955","Y1972","Y2005")]

2.3.2 Determining the lag length

VARselect(dat_K4_1, lag.max=5, type="const", exogen=dum)
## $selection
## AIC(n)  HQ(n)  SC(n) FPE(n) 
##      1      1      1      1 
## 
## $criteria
##                    1             2             3             4             5
## AIC(n) -2.539298e+01 -2.463197e+01 -2.404783e+01 -2.361504e+01 -2.430604e+01
## HQ(n)  -2.438353e+01 -2.298016e+01 -2.175365e+01 -2.067848e+01 -2.072710e+01
## SC(n)  -2.283838e+01 -2.045172e+01 -1.824193e+01 -1.618349e+01 -1.524883e+01
## FPE(n)  9.583241e-12  2.215751e-11  4.752089e-11  1.033733e-10  9.549952e-11

2.3.3 Johansen Cointegration Test (Table 9)

2.3.3.1 Constant

rank_K4_const_1 <- ca.jo(dat_K4_1, ecdet=c("const"),type="trace",K=2,spec="longrun",dumvar=dum)
summary(rank_K4_const_1)
## 
## ###################### 
## # Johansen-Procedure # 
## ###################### 
## 
## Test type: trace statistic , without linear trend and constant in cointegration 
## 
## Eigenvalues (lambda):
## [1]  6.559731e-01  4.176390e-01  3.732293e-01  3.099152e-01  2.360455e-01
## [6]  8.803051e-02  2.501138e-02 -5.933362e-16
## 
## Values of teststatistic and critical values of test:
## 
##            test  10pct   5pct   1pct
## r <= 6 |   1.75   7.52   9.24  12.97
## r <= 5 |   8.11  17.85  19.96  24.60
## r <= 4 |  26.68  32.00  34.91  41.07
## r <= 3 |  52.28  49.65  53.12  60.16
## r <= 2 |  84.51  71.86  76.07  84.45
## r <= 1 | 121.82  97.18 102.14 111.01
## r = 0  | 195.45 126.58 131.70 143.09
## 
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
## 
##              LU_L.l2   LK4_N.l2    LA4_K4.l2  LC4_A4.l2  LI4_C4.l2    LS4_C4.l2
## LU_L.l2    1.0000000   1.000000  1.000000000  1.0000000  1.0000000   1.00000000
## LK4_N.l2  -0.7633787  -6.391299  1.442850194 -1.6588591 -2.2668217   3.88587942
## LA4_K4.l2  0.6214023 -25.059805  1.777648446 -6.3620929 -1.1919151  17.63483288
## LC4_A4.l2  0.8658186   6.596472 -2.891706118  1.1811123  1.6214239   2.39613615
## LI4_C4.l2 -0.7558959 -22.114828 -0.340844706 -1.3672581  2.9390301  -0.02665562
## LS4_C4.l2 -0.4930297  11.456098  0.108523151  0.3947668  0.4111308   0.66443577
## LX4_C4.l2 -0.1345719  -5.077050  0.006060826  0.8510883 -0.1210375  -0.45566816
## constant  -4.1075872 123.713870  1.974550574 25.8878111 -7.0581885 -90.31638326
##             LX4_C4.l2   constant
## LU_L.l2     1.0000000   1.000000
## LK4_N.l2   -6.2356697  -1.739161
## LA4_K4.l2  -6.3737246 -35.125059
## LC4_A4.l2 -23.1768615  -6.297913
## LI4_C4.l2   0.1377409   2.742130
## LS4_C4.l2  -0.2060730  -1.029825
## LX4_C4.l2   1.0626976   1.321556
## constant  138.3460615 184.825540
## 
## Weights W:
## (This is the loading matrix)
## 
##              LU_L.l2      LK4_N.l2   LA4_K4.l2    LC4_A4.l2    LI4_C4.l2
## LU_L.d   -0.07654587 -0.0004480034 -0.02839327 -0.025986112 -0.032558038
## LK4_N.d   0.11241595  0.0002503249  0.04540905  0.014670703  0.005796459
## LA4_K4.d -0.06447764  0.0018165251 -0.03336651  0.007170122  0.001691396
## LC4_A4.d -0.12416726 -0.0041263579  0.08827012 -0.025227909 -0.025503238
## LI4_C4.d  0.18972108  0.0163018319 -0.02606956  0.104017751 -0.115811969
## LS4_C4.d  1.64785466 -0.0295001823 -0.13504648 -0.101205888 -0.089764443
## LX4_C4.d  0.15177255  0.0651001317 -0.21138314 -0.591048630  0.105474467
##              LS4_C4.l2     LX4_C4.l2      constant
## LU_L.d    0.0007498904  0.0020510118  2.889056e-16
## LK4_N.d   0.0020707994  0.0022278772  2.198385e-15
## LA4_K4.d -0.0090838241  0.0001877385  4.048139e-15
## LC4_A4.d -0.0042259154 -0.0011838570  1.117836e-15
## LI4_C4.d  0.0033947640 -0.0013179631  2.604560e-14
## LS4_C4.d -0.0470183018 -0.0003867474 -3.239142e-14
## LX4_C4.d  0.0016535376  0.0028003634 -2.947397e-14

2.3.3.2 Trend

rank_K4_trend_1 <- ca.jo(dat_K4_1, ecdet=c("trend"),type="trace",K=2,spec="longrun",dumvar=dum)
summary(rank_K4_trend_1)
## 
## ###################### 
## # Johansen-Procedure # 
## ###################### 
## 
## Test type: trace statistic , with linear trend in cointegration 
## 
## Eigenvalues (lambda):
## [1] 6.560762e-01 4.182975e-01 3.479068e-01 3.208137e-01 2.725254e-01
## [6] 1.813054e-01 3.193872e-02 1.693301e-17
## 
## Values of teststatistic and critical values of test:
## 
##            test  10pct   5pct   1pct
## r <= 6 |   2.24  10.49  12.25  16.26
## r <= 5 |  16.04  22.76  25.32  30.45
## r <= 4 |  38.00  39.06  42.44  48.45
## r <= 3 |  64.69  59.14  62.99  70.05
## r <= 2 |  94.19  83.20  87.31  96.58
## r <= 1 | 131.58 110.42 114.90 124.75
## r = 0  | 205.22 141.01 146.76 158.49
## 
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
## 
##                LU_L.l2     LK4_N.l2   LA4_K4.l2   LC4_A4.l2   LI4_C4.l2
## LU_L.l2    1.000000000    1.0000000  1.00000000  1.00000000  1.00000000
## LK4_N.l2  -0.760778238  -27.7592491  1.56373119 -0.52850803 -1.19867272
## LA4_K4.l2  0.752788153 -108.3912983  1.63991656 -1.93818834 -9.33664569
## LC4_A4.l2  0.965065430   61.0449086 -3.82543909  3.14474277 -3.05832493
## LI4_C4.l2 -0.768644470 -132.1021293 -0.28524537 -0.89627782 -2.74098819
## LS4_C4.l2 -0.498657753   74.2214071 -0.08951528  0.88488533 -0.32153806
## LX4_C4.l2 -0.135800391  -29.1759793 -0.11252956  1.30592806  0.40950809
## trend.l2   0.002375558    0.8662285 -0.02290114  0.08725219 -0.09055825
##             LS4_C4.l2   LX4_C4.l2     trend.l2
## LU_L.l2    1.00000000  1.00000000  1.000000000
## LK4_N.l2  -1.54307933  0.88019421  0.006631826
## LA4_K4.l2 -5.10769637  2.33288525  4.885518860
## LC4_A4.l2 -1.52420171  1.30885953 -1.149810061
## LI4_C4.l2  0.43449231 -0.05908823 -0.439725012
## LS4_C4.l2 -0.05192552  0.10852771  0.163724198
## LX4_C4.l2  0.05381763 -0.23072374 -0.248383595
## trend.l2  -0.04875911 -0.04930501 -0.054704007
## 
## Weights W:
## (This is the loading matrix)
## 
##              LU_L.l2      LK4_N.l2    LA4_K4.l2    LC4_A4.l2   LI4_C4.l2
## LU_L.d   -0.07689673 -0.0001340966 -0.007709578 -0.018466026  0.01476587
## LK4_N.d   0.11240175  0.0001299907  0.035744015  0.020276004 -0.01530963
## LA4_K4.d -0.06458751  0.0002406082 -0.034253366 -0.006442816  0.01610107
## LC4_A4.d -0.12043580 -0.0006574796  0.106522939 -0.012050745  0.02280892
## LI4_C4.d  0.17890187  0.0025367830 -0.021279749  0.024227817  0.15707483
## LS4_C4.d  1.58586253 -0.0052891014 -0.043503378 -0.115916577  0.13198500
## LX4_C4.d  0.13443813  0.0101237493 -0.088226087 -0.341461636 -0.28080786
##              LS4_C4.l2    LX4_C4.l2      trend.l2
## LU_L.d   -0.0768062024 -0.019651269  2.334001e-14
## LK4_N.d  -0.0133852528 -0.020034433 -2.821232e-14
## LA4_K4.d  0.0383584400 -0.004699274  9.548578e-15
## LC4_A4.d -0.0077228267  0.009296940  9.868233e-14
## LI4_C4.d -0.1510505707  0.012139385 -4.226426e-13
## LS4_C4.d  0.0414716074 -0.013604254 -5.326142e-13
## LX4_C4.d  0.0007542011 -0.026089988 -2.244985e-13

2.4 Constructing VECM (Section 3.4.4)

2.4.1 Estimating the VECM equation (Equation (3))

2.4.1.1 Coefficients of the equation

vecm_K4 <- cajorls(rank_K4_const_1, r=3)
print(vecm_K4)
## $rlm
## 
## Call:
## lm(formula = substitute(form1), data = data.mat)
## 
## Coefficients:
##             LU_L.d     LK4_N.d    LA4_K4.d   LC4_A4.d   LI4_C4.d   LS4_C4.d 
## ect1        -0.105387   0.158075  -0.096028  -0.040024   0.179953   1.483308
## ect2         0.020330  -0.021897  -0.010532   0.248520  -0.286633  -1.264244
## ect3        -0.086812   0.144304  -0.144902   0.183161  -0.336970   1.523184
## Y1955       -0.025989  -0.043104  -0.008581   0.104687  -0.071707  -0.650867
## Y1972        0.137153  -0.056878   0.062868   0.166677  -0.235830  -1.930133
## Y2005        0.004488  -0.193506   0.078006   0.039598  -0.230276  -0.583895
## LU_L.dl1     0.165268   0.125937  -0.227323   0.184317   0.624526   0.321886
## LK4_N.dl1    0.248287  -0.089652   0.116766   0.272322  -0.209376   0.205218
## LA4_K4.dl1  -0.158433  -0.367154  -0.091505   0.440665  -0.487213   4.240481
## LC4_A4.dl1  -0.149772  -0.235426   0.132033  -0.462518   0.246989   1.910868
## LI4_C4.dl1  -0.011055  -0.140683   0.021083   0.125021  -0.887378  -0.336371
## LS4_C4.dl1   0.050404  -0.016255   0.016751   0.019205   0.130987  -1.129958
## LX4_C4.dl1   0.008233  -0.012282   0.001646   0.040373  -0.038316  -0.032347
##             LX4_C4.d 
## ect1         0.005490
## ect2        -0.836929
## ect3        -1.912850
## Y1955       -0.097909
## Y1972       -0.398633
## Y2005       -0.104716
## LU_L.dl1     0.244727
## LK4_N.dl1   -0.285533
## LA4_K4.dl1   3.386827
## LC4_A4.dl1   2.480289
## LI4_C4.dl1  -0.536461
## LS4_C4.dl1   0.518489
## LX4_C4.dl1  -0.598659
## 
## 
## $beta
##                    ect1       ect2          ect3
## LU_L.l2    1.000000e+00  0.0000000  0.000000e+00
## LK4_N.l2   0.000000e+00  1.0000000 -5.551115e-17
## LA4_K4.l2 -1.110223e-16  0.0000000  1.000000e+00
## LC4_A4.l2 -6.075316e-01 -1.7920097  1.695650e-01
## LI4_C4.l2 -1.524495e+00 -0.2798960  8.930330e-01
## LS4_C4.l2  3.210171e-01  0.5835283 -5.931644e-01
## LX4_C4.l2 -2.918869e-01 -0.0419351  2.016449e-01
## constant   4.437890e+00  6.0614448 -6.305575e+00

2.4.1.2 The t values of coefficients

print(coef(summary(vecm_K4$rlm)),digits=3)
## Response LU_L.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1       -0.10539     0.0577 -1.8258   0.0732
## ect2        0.02033     0.0561  0.3621   0.7186
## ect3       -0.08681     0.0788 -1.1019   0.2752
## Y1955      -0.02599     0.0330 -0.7875   0.4343
## Y1972       0.13715     0.0692  1.9806   0.0526
## Y2005       0.00449     0.0497  0.0902   0.9284
## LU_L.dl1    0.16527     0.1527  1.0824   0.2837
## LK4_N.dl1   0.24829     0.2048  1.2124   0.2304
## LA4_K4.dl1 -0.15843     0.2774 -0.5712   0.5701
## LC4_A4.dl1 -0.14977     0.1623 -0.9229   0.3600
## LI4_C4.dl1 -0.01105     0.0484 -0.2284   0.8202
## LS4_C4.dl1  0.05040     0.0252  2.0011   0.0502
## LX4_C4.dl1  0.00823     0.0163  0.5035   0.6166
## 
## Response LK4_N.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1         0.1581     0.0515   3.071 3.29e-03
## ect2        -0.0219     0.0501  -0.437 6.64e-01
## ect3         0.1443     0.0703   2.054 4.47e-02
## Y1955       -0.0431     0.0294  -1.465 1.49e-01
## Y1972       -0.0569     0.0618  -0.921 3.61e-01
## Y2005       -0.1935     0.0444  -4.363 5.57e-05
## LU_L.dl1     0.1259     0.1362   0.925 3.59e-01
## LK4_N.dl1   -0.0897     0.1826  -0.491 6.25e-01
## LA4_K4.dl1  -0.3672     0.2473  -1.484 1.43e-01
## LC4_A4.dl1  -0.2354     0.1447  -1.627 1.09e-01
## LI4_C4.dl1  -0.1407     0.0432  -3.259 1.90e-03
## LS4_C4.dl1  -0.0163     0.0225  -0.724 4.72e-01
## LX4_C4.dl1  -0.0123     0.0146  -0.842 4.03e-01
## 
## Response LA4_K4.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1       -0.09603    0.02828  -3.396 0.001265
## ect2       -0.01053    0.02751  -0.383 0.703242
## ect3       -0.14490    0.03860  -3.754 0.000415
## Y1955      -0.00858    0.01617  -0.531 0.597713
## Y1972       0.06287    0.03393   1.853 0.069147
## Y2005       0.07801    0.02437   3.201 0.002255
## LU_L.dl1   -0.22732    0.07481  -3.039 0.003605
## LK4_N.dl1   0.11677    0.10033   1.164 0.249427
## LA4_K4.dl1 -0.09151    0.13588  -0.673 0.503449
## LC4_A4.dl1  0.13203    0.07951   1.661 0.102377
## LI4_C4.dl1  0.02108    0.02371   0.889 0.377725
## LS4_C4.dl1  0.01675    0.01234   1.357 0.180093
## LX4_C4.dl1  0.00165    0.00801   0.205 0.837936
## 
## Response LC4_A4.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1        -0.0400     0.0571  -0.701 4.86e-01
## ect2         0.2485     0.0556   4.474 3.81e-05
## ect3         0.1832     0.0780   2.350 2.23e-02
## Y1955        0.1047     0.0327   3.206 2.23e-03
## Y1972        0.1667     0.0685   2.433 1.82e-02
## Y2005        0.0396     0.0492   0.805 4.24e-01
## LU_L.dl1     0.1843     0.1511   1.220 2.28e-01
## LK4_N.dl1    0.2723     0.2026   1.344 1.84e-01
## LA4_K4.dl1   0.4407     0.2744   1.606 1.14e-01
## LC4_A4.dl1  -0.4625     0.1606  -2.880 5.62e-03
## LI4_C4.dl1   0.1250     0.0479   2.611 1.16e-02
## LS4_C4.dl1   0.0192     0.0249   0.771 4.44e-01
## LX4_C4.dl1   0.0404     0.0162   2.496 1.55e-02
## 
## Response LI4_C4.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1         0.1800     0.1568   1.147 2.56e-01
## ect2        -0.2866     0.1525  -1.879 6.55e-02
## ect3        -0.3370     0.2141  -1.574 1.21e-01
## Y1955       -0.0717     0.0897  -0.800 4.27e-01
## Y1972       -0.2358     0.1882  -1.253 2.15e-01
## Y2005       -0.2303     0.1351  -1.704 9.39e-02
## LU_L.dl1     0.6245     0.4149   1.505 1.38e-01
## LK4_N.dl1   -0.2094     0.5564  -0.376 7.08e-01
## LA4_K4.dl1  -0.4872     0.7536  -0.647 5.21e-01
## LC4_A4.dl1   0.2470     0.4410   0.560 5.78e-01
## LI4_C4.dl1  -0.8874     0.1315  -6.748 9.03e-09
## LS4_C4.dl1   0.1310     0.0684   1.914 6.07e-02
## LX4_C4.dl1  -0.0383     0.0444  -0.863 3.92e-01
## 
## Response LS4_C4.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1         1.4833     0.2750   5.394 1.44e-06
## ect2        -1.2642     0.2675  -4.727 1.58e-05
## ect3         1.5232     0.3753   4.058 1.55e-04
## Y1955       -0.6509     0.1572  -4.140 1.18e-04
## Y1972       -1.9301     0.3299  -5.851 2.66e-07
## Y2005       -0.5839     0.2369  -2.464 1.68e-02
## LU_L.dl1     0.3219     0.7274   0.443 6.60e-01
## LK4_N.dl1    0.2052     0.9756   0.210 8.34e-01
## LA4_K4.dl1   4.2405     1.3213   3.209 2.20e-03
## LC4_A4.dl1   1.9109     0.7731   2.472 1.65e-02
## LI4_C4.dl1  -0.3364     0.2306  -1.459 1.50e-01
## LS4_C4.dl1  -1.1300     0.1200  -9.417 3.85e-13
## LX4_C4.dl1  -0.0323     0.0779  -0.415 6.80e-01
## 
## Response LX4_C4.d :
##            Estimate Std. Error t value Pr(>|t|)
## ect1        0.00549      0.492  0.0112 9.91e-01
## ect2       -0.83693      0.479 -1.7484 8.59e-02
## ect3       -1.91285      0.672 -2.8477 6.15e-03
## Y1955      -0.09791      0.281 -0.3480 7.29e-01
## Y1972      -0.39863      0.590 -0.6751 5.02e-01
## Y2005      -0.10472      0.424 -0.2469 8.06e-01
## LU_L.dl1    0.24473      1.302  0.1880 8.52e-01
## LK4_N.dl1  -0.28553      1.746 -0.1635 8.71e-01
## LA4_K4.dl1  3.38683      2.365  1.4322 1.58e-01
## LC4_A4.dl1  2.48029      1.384  1.7925 7.85e-02
## LI4_C4.dl1 -0.53646      0.413 -1.3000 1.99e-01
## LS4_C4.dl1  0.51849      0.215  2.4143 1.91e-02
## LX4_C4.dl1 -0.59866      0.139 -4.2945 7.03e-05

2.4.2 Diagnostic tests

var_K4_1 <- vec2var(rank_K4_const_1,r=3)
normality.test(var_K4_1)
## $JB
## 
##  JB-Test (multivariate)
## 
## data:  Residuals of VAR object var_K4_1
## Chi-squared = 14.812, df = 14, p-value = 0.3912
## 
## 
## $Skewness
## 
##  Skewness only (multivariate)
## 
## data:  Residuals of VAR object var_K4_1
## Chi-squared = 7.0819, df = 7, p-value = 0.4204
## 
## 
## $Kurtosis
## 
##  Kurtosis only (multivariate)
## 
## data:  Residuals of VAR object var_K4_1
## Chi-squared = 7.7296, df = 7, p-value = 0.357
serial.test(var_K4_1)
## 
##  Portmanteau Test (asymptotic)
## 
## data:  Residuals of VAR object var_K4_1
## Chi-squared = 675.76, df = 693, p-value = 0.6733
arch.test(var_K4_1)
## 
##  ARCH (multivariate)
## 
## data:  Residuals of VAR object var_K4_1
## Chi-squared = 1792, df = 3920, p-value = 1

2.5 Innovation Accounting (Section 3.4.5)

2.5.1 Cumulative impulse responses of the homicide rate (Figure 6)

cirf_LU_L_1 <- irf(var_K4_1, impulse=c("LU_L"), response=c("LK4_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LK4_N <- irf(var_K4_1, impulse=c("LK4_N"), response=c("LK4_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LA4_K4 <- irf(var_K4_1, impulse=c("LA4_K4"), response=c("LK4_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LC4_A4 <- irf(var_K4_1, impulse=c("LC4_A4"), response=c("LK4_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LI4_C4 <- irf(var_K4_1, impulse=c("LI4_C4"), response=c("LK4_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LS4_C4 <- irf(var_K4_1, impulse=c("LS4_C4"), response=c("LK4_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
cirf_LX4_C4 <- irf(var_K4_1, impulse=c("LX4_C4"), response=c("LK4_N"), cumulative=T, n.ahead=10, ci=0.95, runs=10000, seed=1)
plot(cirf_LU_L_1)

plot(cirf_LK4_N)

plot(cirf_LA4_K4)

plot(cirf_LC4_A4)

plot(cirf_LI4_C4)

plot(cirf_LS4_C4)

plot(cirf_LX4_C4)

2.5.2 Forecast error variance decomposition of the homicide rate (Figure 7)

fevd_K4 <- fevd(var_K4_1,n.ahead=10)
par(mar = c(4, 4, 1, 1))
plot(fevd_K4$LK4_N[,1]*100,type="o",pch=1,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
axis(1)
axis(2)
par(new=T)
plot(fevd_K4$LK4_N[,3]*100,type="o",pch=3,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
par(new=T)
plot(fevd_K4$LK4_N[,4]*100,type="o",pch=6,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
par(new=T)
plot(fevd_K4$LK4_N[,5]*100,type="o",pch=5,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
par(new=T)
plot(fevd_K4$LK4_N[,6]*100,type="o",pch=15,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
par(new=T)
plot(fevd_K4$LK4_N[,7]*100,type="o",pch=4,ylim=c(0,22),xlab="",ylab="",axes = FALSE)
box()
legend("topright", legend = c("Unemployment", "Life Sentence", "Conviction", "Death Sentence", "Execution", "Clearance"), pch = c(1,5,6,15,4,3), cex=0.72)
mtext("%",at=0.4)