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WS7

*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 10:00:23 -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/t1258736644r6tv9k8webn8cy1.htm/, Retrieved Fri, 20 Nov 2009 18:04:15 +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/t1258736644r6tv9k8webn8cy1.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 «
363 14.3 364 14.2 363 15.9 358 15.3 357 15.5 357 15.1 380 15 378 12.1 376 15.8 380 16.9 379 15.1 384 13.7 392 14.8 394 14.7 392 16 396 15.4 392 15 396 15.5 419 15.1 421 11.7 420 16.3 418 16.7 410 15 418 14.9 426 14.6 428 15.3 430 17.9 424 16.4 423 15.4 427 17.9 441 15.9 449 13.9 452 17.8 462 17.9 455 17.4 461 16.7 461 16 463 16.6 462 19.1 456 17.8 455 17.2 456 18.6 472 16.3 472 15.1 471 19.2 465 17.7 459 19.1 465 18 468 17.5 467 17.8 463 21.1 460 17.2 462 19.4 461 19.8 476 17.6 476 16.2 471 19.5 453 19.9 443 20 442 17.3
 
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
WK>25j[t] = + 390.495158429888 -2.19378103570580ExpBE[t] + 10.6069652991252M1[t] + 10.2304297706313M2[t] + 11.8414563135492M3[t] + 2.98448805864265M4[t] -0.0308036769922362M5[t] + 1.30892941593749M6[t] + 14.246841747458M7[t] + 8.873604871128M8[t] + 14.0824323126034M9[t] + 9.7110161976826M10[t] + 0.0233314613383325M11[t] + 2.19079421849136t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)390.49515842988841.9935219.298900
ExpBE-2.193781035705803.135346-0.69970.4876420.243821
M110.60696529912529.5255581.11350.2712670.135634
M210.23042977063139.5772691.06820.2910050.145502
M311.841456313549212.5382440.94440.3498860.174943
M42.984488058642659.9261010.30070.765020.38251
M5-0.03080367699223629.914293-0.00310.9975340.498767
M61.3089294159374910.9256940.11980.9051610.452581
M714.2468417474589.4726881.5040.1394190.06971
M88.87360487112811.3078120.78470.4366330.218317
M914.082432312603411.060171.27330.2093240.104662
M109.711016197682611.0865580.87590.3856210.19281
M110.023331461338332510.2402840.00230.9981920.499096
t2.190794218491360.2815847.780300


Multiple Linear Regression - Regression Statistics
Multiple R0.937327976220717
R-squared0.878583735006024
Adjusted R-squared0.844270442725118
F-TEST (value)25.604763536343
F-TEST (DF numerator)13
F-TEST (DF denominator)46
p-value1.11022302462516e-16
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation14.8903066765169
Sum Squared Residuals10199.1767143533


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
1363371.921849136911-8.92184913691128
2364373.95548593048-9.95548593048008
3363374.027878931189-11.0278789311894
4358368.677973516198-10.6779735161977
5357367.414719791913-10.4147197919130
6357371.822759517616-14.8227595176165
7380387.170844171199-7.17084417119895
8378390.350366516907-12.3503665169071
9376389.632998344762-13.6329983447624
10380385.039217309057-5.03921730905656
11379381.491132655474-2.49113265547411
12384386.729888862615-2.72988886261529
13392397.114489240955-5.11448924095547
14394399.148126034524-5.14812603452353
15392400.098031449515-8.09803144951522
16396394.7481260345241.25187396547647
17392394.801140931662-2.80114093166232
18396397.234777725231-1.23477772523054
19419413.2409966895255.75900331047528
20421417.5174095530863.4825904469142
21420414.8256384488065.17436155119415
22418411.7675041380946.23249586190588
23410408.0000413809411.99995861905893
24418410.3868822416657.6131177583353
25426423.8427760699932.15722393000698
26428424.1213880349963.87861196500358
27430422.2193781035717.7806218964294
28424418.8438756207145.15612437928589
29423420.2131591392762.78684086072363
30427418.2592338614338.74076613856701
31441437.7755024828563.22449751714354
32449438.98062189642910.0193781035706
33452437.82449751714414.1755024828565
34462435.42449751714426.5755024828565
35455429.02449751714425.9755024828565
36461432.72760699929128.2723930007094
37461447.06101324190113.9389867580987
38463447.55900331047515.4409966895247
39462445.8763714826216.12362851738
40456442.06211279262213.9378872073776
41455442.55388389690212.4461161030977
42456443.01311775833512.9868822416647
43472463.1875206904718.8124793095295
44472462.6376152754799.36238472452115
45471461.0427346890529.9572653109482
46465462.1527843461812.84721565381893
47459451.584600378347.41539962165995
48465456.1652222747698.83477772523053
49468470.059872310239-2.05987231023896
50467471.215996689525-4.21599668952469
51463467.778340033105-4.77834003310478
52460469.667912035942-9.66791203594225
53462464.017096240246-2.01709624024594
54461466.670111137385-5.67011113738472
55476486.625135965949-10.6251359659494
56476486.513986758099-10.5139867580988
57471486.674131000236-15.6741310002364
58453483.615996689525-30.6159966895247
59443475.899728068101-32.8997280681012
60442483.99039962166-41.9903996216599


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
170.01318510262980010.02637020525960020.9868148973702
180.01056257556569150.02112515113138310.989437424434308
190.004722780051688320.009445560103376640.995277219948312
200.002048354812701200.004096709625402410.997951645187299
210.003042320771984550.00608464154396910.996957679228015
220.001155143681993350.002310287363986710.998844856318007
230.0005495414425328970.001099082885065790.999450458557467
240.0001821229773598630.0003642459547197250.99981787702264
250.0001175800081121250.000235160016224250.999882419991888
265.74731260398207e-050.0001149462520796410.99994252687396
272.64589758728815e-055.2917951745763e-050.999973541024127
282.75281750802481e-055.50563501604961e-050.99997247182492
292.26462357928356e-054.52924715856713e-050.999977353764207
302.85361502470566e-055.70723004941132e-050.999971463849753
310.0008084164456358740.001616832891271750.999191583554364
320.003008594170200580.006017188340401160.9969914058298
330.0189600424686340.0379200849372680.981039957531366
340.04591131284600170.09182262569200340.954088687153998
350.04777238255631650.0955447651126330.952227617443683
360.03765732620294240.07531465240588480.962342673797058
370.02689852810948060.05379705621896120.97310147189052
380.01711555984160640.03423111968321270.982884440158394
390.009099063069474080.01819812613894820.990900936930526
400.02094169556851610.04188339113703230.979058304431484
410.01290610392447130.02581220784894260.987093896075529
420.01416865494653060.02833730989306110.98583134505347
430.02020786470785710.04041572941571420.979792135292143


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level140.518518518518518NOK
5% type I error level230.851851851851852NOK
10% type I error level271NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/104mok1258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/104mok1258736416.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/1zl301258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/1zl301258736416.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/2wqmg1258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/2wqmg1258736416.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/3lrsi1258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/3lrsi1258736416.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/422l31258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/422l31258736416.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/5yoq11258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/5yoq11258736416.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/6vkug1258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/6vkug1258736416.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/7733z1258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/7733z1258736416.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/8xzw81258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/8xzw81258736416.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/9kzgs1258736416.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258736644r6tv9k8webn8cy1/9kzgs1258736416.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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