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*The author of this computation has been verified*
R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)
Title produced by software: Multiple Regression
Date of computation: Fri, 20 Nov 2009 06:18:27 -0700
 
Cite this page as follows:
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5.htm/, Retrieved Fri, 20 Nov 2009 14:22:25 +0100
 
BibTeX entries for LaTeX users:
@Manual{KEY,
    author = {{YOUR NAME}},
    publisher = {Office for Research Development and Education},
    title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5.htm/},
    year = {2009},
}
@Manual{R,
    title = {R: A Language and Environment for Statistical Computing},
    author = {{R Development Core Team}},
    organization = {R Foundation for Statistical Computing},
    address = {Vienna, Austria},
    year = {2009},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}
 
Original text written by user:
 
IsPrivate?
No (this computation is public)
 
User-defined keywords:
 
Dataseries X:
» Textbox « » Textfile « » CSV «
280 1258 557 1199 831 1158 1081 1427 1318 934 1578 709 1859 1186 2141 986 2428 1033 2715 1257 3004 1105 3309 1179 269 1092 537 1092 813 1087 1068 2028 1411 2039 1675 2010 1958 754 2242 760 2524 715 2836 855 3143 971 3522 815 285 915 574 843 865 761 1147 1858 1516 2968 1789 4061 2087 3661 2372 3269 2669 2857 2966 2568 3270 2274 3652 1987 329 683 658 381 988 71 1303 1772 1603 3485 1929 5181 2235 4479 2544 3782 2872 3067 3198 2489 3544 1903 3903 1330 332 736 665 483 1001 242 1329 1334 1639 2423 1975 3523 2304 2986 2640 2462 2992 1908 3330 1575 3690 1237 4063 904
 
Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits!


Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time6 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
Y[t] = + 3354.30967526023 + 0.0263231337674931X[t] -3290.22659632555M1[t] -2995.82535874672M2[t] -2699.26097355517M3[t] -2448.52086637209M4[t] -2163.18883251067M5[t] -1898.93604713372M6[t] -1595.21647601783M7[t] -1294.91359184834M8[t] -985.284579903292M9[t] -677.293648311444M10[t] -357.902102736634M11[t] + 8.41029637407708t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)3354.3096752602347.11012771.201500
X0.02632313376749310.0142951.84140.0720210.03601
M1-3290.2265963255556.484817-58.249800
M2-2995.8253587467256.500306-53.023200
M3-2699.2609735551756.609733-47.681900
M4-2448.5208663720956.870559-43.054300
M5-2163.1888325106758.989503-36.670700
M6-1898.9360471337262.774943-30.249900
M7-1595.2164760178359.861038-26.648700
M8-1294.9135918483458.148033-22.269300
M9-985.28457990329256.978523-17.292200
M10-677.29364831144456.532069-11.980700
M11-357.90210273663456.120128-6.377400
t8.410296374077080.72533911.59500


Multiple Linear Regression - Regression Statistics
Multiple R0.997355409116792
R-squared0.994717812094524
Adjusted R-squared0.993225019860367
F-TEST (value)666.347124090398
F-TEST (DF numerator)13
F-TEST (DF denominator)46
p-value0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation88.5111261077739
Sum Squared Residuals360374.094463848


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
1280105.607877588275174.392122411725
2557406.866346648887150.133653351113
3831710.761779730045120.238220269955
41081976.993106270657104.006893729343
513181257.7581315587860.2418684412153
615781524.4985082121353.5014917878694
718591849.184510509189.81548949081726
821412152.63306429926-11.6330642992569
924282471.90955990545-43.9095599054542
1027152794.20716983530-79.2071698352957
1130043118.00789545153-114.007895451526
1233093486.26820646103-177.268206461031
13269202.16179387178866.8382061282116
14537504.9733278246932.0266721753097
15813809.8163937214783.18360627852161
1610681093.73686615385-25.7368661538465
1714111387.7687508607923.2312491392118
1816751659.6684617325615.3315382674365
1919581938.7364732105519.2635267894478
2022422247.60759255673-5.60759255672876
2125242564.46235985632-40.4623598563159
2228362884.54882654969-48.54882654969
2331433215.40415201561-72.4041520156063
2435223577.61014225859-55.6101422585885
25285298.426155683867-13.4261556838671
26574599.34242400551-25.3424240055096
27865902.1586086022-37.1586086022
2811471190.18548990230-43.1854899022977
2915161513.146498619712.85350138028586
3017891814.58076557862-25.5807655786167
3120872116.18137956158-29.1813795615795
3223722414.57589166829-42.575891668294
3326692721.77006887521-52.7700688752107
3429663030.56391118233-64.5639111823307
3532703350.62675180357-80.6267518035745
3636523709.38441152301-57.3844115230150
37329393.242745138733-64.2427451387334
38658688.104692693852-30.1046926938525
39988984.9192027915553.08079720844485
4013031288.8452568872214.1547431127822
4116031627.67911526643-24.679115266433
4219291944.98623188713-15.9862318871341
4322352238.63725947231-3.63725947231396
4425442529.0032157799414.9967842200572
4528722828.2214834553143.7785165446904
4631983129.4079401036268.5920598963763
4735443441.78442566476102.215574335241
4839033793.0136691267109.986330873303
49332495.561427717336-163.561427717336
50665791.713208827061-126.713208827061
5110011090.34401515472-89.3440151547216
5213291378.23928078598-49.2392807859809
5316391700.64750369428-61.6475036942798
5419752002.26603258956-27.2660325895552
5523042300.260377246373.73962275362838
5626402595.1802356957844.8197643042226
5729922898.6365279077193.3634720922902
5833303206.27215232906123.727847670940
5936903525.17677506453164.823224935466
6040633882.72357063067180.276429369330


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
170.04324201559157320.08648403118314650.956757984408427
180.01242042026212250.02484084052424510.987579579737877
190.1410496681555280.2820993363110560.858950331844472
200.1447219860305960.2894439720611910.855278013969404
210.1124583042990740.2249166085981490.887541695700926
220.1002765186635540.2005530373271090.899723481336446
230.10533071214270.21066142428540.8946692878573
240.2285569820152820.4571139640305640.771443017984718
250.307524874224070.615049748448140.69247512577593
260.2991935247696130.5983870495392260.700806475230387
270.2424448033862220.4848896067724450.757555196613778
280.1699310595010390.3398621190020770.830068940498961
290.1623990002433280.3247980004866560.837600999756672
300.1156200306911760.2312400613823530.884379969308824
310.07808728285590320.1561745657118060.921912717144097
320.04879898652663990.09759797305327970.95120101347336
330.03528019394264940.07056038788529880.96471980605735
340.03719752841428340.07439505682856680.962802471585717
350.128272969520740.256545939041480.87172703047926
360.7812569946013290.4374860107973420.218743005398671
370.7504154816640470.4991690366719070.249584518335953
380.7165155368292690.5669689263414630.283484463170731
390.8130870433433340.3738259133133330.186912956656666
400.976010911621520.04797817675696090.0239890883784804
410.999632306120480.0007353877590420330.000367693879521017
420.9992103911422560.001579217715487360.00078960885774368
430.99913098053630.001738038927398760.000869019463699378


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level30.111111111111111NOK
5% type I error level50.185185185185185NOK
10% type I error level90.333333333333333NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/100nwy1258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/100nwy1258723100.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/17m0l1258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/17m0l1258723100.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/2sbpd1258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/2sbpd1258723100.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/3lot41258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/3lot41258723100.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/4ce4n1258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/4ce4n1258723100.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/59f0g1258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/59f0g1258723100.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/6pfuf1258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/6pfuf1258723100.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/7366w1258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/7366w1258723100.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/8wzka1258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/8wzka1258723100.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/9drhs1258723100.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258723327zr3jwpfzwgyxxm5/9drhs1258723100.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;
 
Parameters (R input):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;
 
R code (references can be found in the software module):
library(lattice)
library(lmtest)
n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test
par1 <- as.numeric(par1)
x <- t(y)
k <- length(x[1,])
n <- length(x[,1])
x1 <- cbind(x[,par1], x[,1:k!=par1])
mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1])
colnames(x1) <- mycolnames #colnames(x)[par1]
x <- x1
if (par3 == 'First Differences'){
x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep='')))
for (i in 1:n-1) {
for (j in 1:k) {
x2[i,j] <- x[i+1,j] - x[i,j]
}
}
x <- x2
}
if (par2 == 'Include Monthly Dummies'){
x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep ='')))
for (i in 1:11){
x2[seq(i,n,12),i] <- 1
}
x <- cbind(x, x2)
}
if (par2 == 'Include Quarterly Dummies'){
x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep ='')))
for (i in 1:3){
x2[seq(i,n,4),i] <- 1
}
x <- cbind(x, x2)
}
k <- length(x[1,])
if (par3 == 'Linear Trend'){
x <- cbind(x, c(1:n))
colnames(x)[k+1] <- 't'
}
x
k <- length(x[1,])
df <- as.data.frame(x)
(mylm <- lm(df))
(mysum <- summary(mylm))
if (n > n25) {
kp3 <- k + 3
nmkm3 <- n - k - 3
gqarr <- array(NA, dim=c(nmkm3-kp3+1,3))
numgqtests <- 0
numsignificant1 <- 0
numsignificant5 <- 0
numsignificant10 <- 0
for (mypoint in kp3:nmkm3) {
j <- 0
numgqtests <- numgqtests + 1
for (myalt in c('greater', 'two.sided', 'less')) {
j <- j + 1
gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value
}
if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1
if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1
if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1
}
gqarr
}
bitmap(file='test0.png')
plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index')
points(x[,1]-mysum$resid)
grid()
dev.off()
bitmap(file='test1.png')
plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index')
grid()
dev.off()
bitmap(file='test2.png')
hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals')
grid()
dev.off()
bitmap(file='test3.png')
densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals')
dev.off()
bitmap(file='test4.png')
qqnorm(mysum$resid, main='Residual Normal Q-Q Plot')
qqline(mysum$resid)
grid()
dev.off()
(myerror <- as.ts(mysum$resid))
bitmap(file='test5.png')
dum <- cbind(lag(myerror,k=1),myerror)
dum
dum1 <- dum[2:length(myerror),]
dum1
z <- as.data.frame(dum1)
z
plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals')
lines(lowess(z))
abline(lm(z))
grid()
dev.off()
bitmap(file='test6.png')
acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function')
grid()
dev.off()
bitmap(file='test7.png')
pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function')
grid()
dev.off()
bitmap(file='test8.png')
opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0))
plot(mylm, las = 1, sub='Residual Diagnostics')
par(opar)
dev.off()
if (n > n25) {
bitmap(file='test9.png')
plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint')
grid()
dev.off()
}
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE)
a<-table.row.end(a)
myeq <- colnames(x)[1]
myeq <- paste(myeq, '[t] = ', sep='')
for (i in 1:k){
if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '')
myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ')
if (rownames(mysum$coefficients)[i] != '(Intercept)') {
myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='')
if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='')
}
}
myeq <- paste(myeq, ' + e[t]')
a<-table.row.start(a)
a<-table.element(a, myeq)
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable1.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Variable',header=TRUE)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'S.D.',header=TRUE)
a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE)
a<-table.element(a,'2-tail p-value',header=TRUE)
a<-table.element(a,'1-tail p-value',header=TRUE)
a<-table.row.end(a)
for (i in 1:k){
a<-table.row.start(a)
a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE)
a<-table.element(a,mysum$coefficients[i,1])
a<-table.element(a, round(mysum$coefficients[i,2],6))
a<-table.element(a, round(mysum$coefficients[i,3],4))
a<-table.element(a, round(mysum$coefficients[i,4],6))
a<-table.element(a, round(mysum$coefficients[i,4]/2,6))
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable2.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Multiple R',1,TRUE)
a<-table.element(a, sqrt(mysum$r.squared))
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'R-squared',1,TRUE)
a<-table.element(a, mysum$r.squared)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Adjusted R-squared',1,TRUE)
a<-table.element(a, mysum$adj.r.squared)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'F-TEST (value)',1,TRUE)
a<-table.element(a, mysum$fstatistic[1])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE)
a<-table.element(a, mysum$fstatistic[2])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE)
a<-table.element(a, mysum$fstatistic[3])
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'p-value',1,TRUE)
a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3]))
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Residual Standard Deviation',1,TRUE)
a<-table.element(a, mysum$sigma)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Sum Squared Residuals',1,TRUE)
a<-table.element(a, sum(myerror*myerror))
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable3.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a, 'Time or Index', 1, TRUE)
a<-table.element(a, 'Actuals', 1, TRUE)
a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE)
a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE)
a<-table.row.end(a)
for (i in 1:n) {
a<-table.row.start(a)
a<-table.element(a,i, 1, TRUE)
a<-table.element(a,x[i])
a<-table.element(a,x[i]-mysum$resid[i])
a<-table.element(a,mysum$resid[i])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable4.tab')
if (n > n25) {
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'p-values',header=TRUE)
a<-table.element(a,'Alternative Hypothesis',3,header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'breakpoint index',header=TRUE)
a<-table.element(a,'greater',header=TRUE)
a<-table.element(a,'2-sided',header=TRUE)
a<-table.element(a,'less',header=TRUE)
a<-table.row.end(a)
for (mypoint in kp3:nmkm3) {
a<-table.row.start(a)
a<-table.element(a,mypoint,header=TRUE)
a<-table.element(a,gqarr[mypoint-kp3+1,1])
a<-table.element(a,gqarr[mypoint-kp3+1,2])
a<-table.element(a,gqarr[mypoint-kp3+1,3])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file='mytable5.tab')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Description',header=TRUE)
a<-table.element(a,'# significant tests',header=TRUE)
a<-table.element(a,'% significant tests',header=TRUE)
a<-table.element(a,'OK/NOK',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'1% type I error level',header=TRUE)
a<-table.element(a,numsignificant1)
a<-table.element(a,numsignificant1/numgqtests)
if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK'
a<-table.element(a,dum)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'5% type I error level',header=TRUE)
a<-table.element(a,numsignificant5)
a<-table.element(a,numsignificant5/numgqtests)
if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK'
a<-table.element(a,dum)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'10% type I error level',header=TRUE)
a<-table.element(a,numsignificant10)
a<-table.element(a,numsignificant10/numgqtests)
if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK'
a<-table.element(a,dum)
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file='mytable6.tab')
}
 





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Software written by Ed van Stee & Patrick Wessa


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