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Model4

*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 16:04:12 -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/21/t1258758384vlb06yj9dniay4u.htm/, Retrieved Sat, 21 Nov 2009 00:06:35 +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/21/t1258758384vlb06yj9dniay4u.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 «
537 20,1 544 555 561 562 543 19,9 537 544 555 561 594 20 543 537 544 555 611 22,6 594 543 537 544 613 20,6 611 594 543 537 611 20,1 613 611 594 543 594 20,2 611 613 611 594 595 21,8 594 611 613 611 591 22 595 594 611 613 589 19,5 591 595 594 611 584 17,5 589 591 595 594 573 18,2 584 589 591 595 567 18,8 573 584 589 591 569 19,7 567 573 584 589 621 18,8 569 567 573 584 629 18,5 621 569 567 573 628 18,7 629 621 569 567 612 18,5 628 629 621 569 595 19,3 612 628 629 621 597 18,9 595 612 628 629 593 21,4 597 595 612 628 590 22,5 593 597 595 612 580 25 590 593 597 595 574 22,9 580 590 593 597 573 22,9 574 580 590 593 573 21,3 573 574 580 590 620 22,3 573 573 574 580 626 20,9 620 573 573 574 620 19,9 626 620 573 573 588 20,2 620 626 620 573 566 19,8 588 620 626 620 557 17,7 566 588 620 626 561 18,1 557 566 588 620 549 17,6 561 557 566 588 532 18,2 549 561 557 566 526 16 532 549 561 557 511 16,3 526 532 549 561 499 17,3 511 526 532 549 555 19 499 511 526 532 565 18,6 555 499 511 etc...
 
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 time4 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
Y[t] = + 58.2998057129362 -0.668637279101902X[t] + 0.961166527491067Y1[t] + 0.0391683255838521Y2[t] + 0.0478769811627857Y3[t] -0.127097092762909Y4[t] -4.4972676862563M1[t] + 6.69352067183267M2[t] + 56.6788489417115M3[t] + 16.6872113068170M4[t] -4.55219702469572M5[t] -14.5874408744148M6[t] -9.10435377427196M7[t] + 9.94245587206924M8[t] + 9.88391350018948M9[t] + 2.43364712654867M10[t] -5.18399963045349M11[t] -0.254141845139373t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)58.299805712936227.4585052.12320.0401370.020069
X-0.6686372791019020.344845-1.9390.0597670.029884
Y10.9611665274910670.1613955.95541e-060
Y20.03916832558385210.2244540.17450.8623710.431186
Y30.04787698116278570.2229870.21470.8311140.415557
Y4-0.1270970927629090.165009-0.77020.4457970.222898
M1-4.49726768625634.845255-0.92820.3590240.179512
M26.693520671832675.0633931.32190.1938890.096944
M356.67884894171155.33530210.623400
M416.687211306817011.2126491.48820.1447270.072364
M5-4.5521970246957210.654938-0.42720.6715550.335778
M6-14.587440874414810.20374-1.42960.1607890.080395
M7-9.104353774271965.147178-1.76880.0847470.042374
M89.942455872069245.1801071.91940.0622780.031139
M99.883913500189485.871441.68340.100290.050145
M102.433647126548676.0639020.40130.6903650.345183
M11-5.183999630453494.995633-1.03770.3057980.152899
t-0.2541418451393730.110433-2.30130.0268030.013401


Multiple Linear Regression - Regression Statistics
Multiple R0.991290024248878
R-squared0.98265591217534
Adjusted R-squared0.975095668764592
F-TEST (value)129.976755877757
F-TEST (DF numerator)17
F-TEST (DF denominator)39
p-value0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation6.82215927009146
Sum Squared Residuals1815.1324271533


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
1537540.152218825339-3.15221882533875
2543543.903410726035-0.903410726035087
3594599.296490072511-5.29649007251066
4611607.6296856746123.37031432538824
5613606.9877671646056.01223283539486
6611601.300038181939.69996181807006
7594598.950080254068-4.95008025406777
8595598.189864175547-3.18986417554682
9591597.688809347421-6.68880934742142
10589587.2907820477741.70921795222589
11584580.8857892046513.11421079534892
12573580.144826588557-7.14482658855679
13567564.6361956681052.36380433189497
14569568.7880271632060.211972836793970
15621620.917148911640.0828510883598304
16629632.041762829455-3.04176282945526
17628620.9989068661757.00109313382533
18612612.430837539255-0.430837539255436
19595595.481007231167-0.481007231167442
20597596.5099520440540.490047955945787
21593595.143211543495-2.14321154349531
22590584.1566176633465.84338233665436
23580573.8294675179366.17053248206424
24574569.9885912275254.0114087724753
25573559.44325670290713.5567432970926
26573570.1560678480872.84393215191329
27620620.163157708793-0.163157708792835
28626626.743002786996-0.743002786996366
29620613.6530974495976.34690255040347
30588599.881349474215-11.8813494742152
31566568.699109334762-2.69910933476215
32557565.447020955037-8.44702095503706
33561554.5851990754816.41480092451887
34549553.721074058793-4.72107405879328
35532536.436021271952-4.43602127195213
36526527.362411956453-1.36241195645339
37511514.89463839645-3.8946383964502
38499511.221396197816-12.2213961978163
39555549.5677647244255.43223527557525
40565562.9891736276612.01082637233926
41542555.033829943566-13.0338299435664
42527527.302437158433-0.302437158433059
43510510.641209357457-0.641209357457052
44514509.6663337178084.33366628219154
45517515.4730334989451.52696650105468
46508510.831526230087-2.83152623008697
47493497.848722005461-4.84872200546102
48490485.5041702274654.49582977253489
49469477.873690407199-8.87369040719865
50478467.93109806485610.0689019351442
51528528.055438582632-0.0554385826315939
52534535.596375081276-1.59637508127588
53518524.326398576057-6.32639857605727
54506503.0853376461662.91466235383363
55502493.2285938225468.77140617745442
56516509.1868291075536.81317089244655
57528527.1097465346570.890253465343169


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
210.2109263101645940.4218526203291880.789073689835406
220.1203440480479370.2406880960958740.879655951952063
230.06040298678582780.1208059735716560.939597013214172
240.03903244282114120.07806488564228230.96096755717886
250.08629661000432070.1725932200086410.91370338999568
260.08099943786147310.1619988757229460.919000562138527
270.04458464102007450.08916928204014890.955415358979925
280.02450776630271790.04901553260543590.975492233697282
290.1822780514348180.3645561028696360.817721948565182
300.6265563965402060.7468872069195880.373443603459794
310.7138290454645490.5723419090709010.286170954535451
320.6202231755566690.7595536488866610.379776824443331
330.5430800487580.9138399024840010.456919951242000
340.4184510844133370.8369021688266730.581548915586664
350.4793007107124570.9586014214249130.520699289287543
360.4351617077967650.870323415593530.564838292203235


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level10.0625NOK
10% type I error level30.1875NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/109k831258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/109k831258758247.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/16h3u1258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/16h3u1258758247.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/21zqm1258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/21zqm1258758247.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/3di8m1258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/3di8m1258758247.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/4x10f1258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/4x10f1258758247.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/5zddx1258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/5zddx1258758247.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/6ue7k1258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/6ue7k1258758247.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/7202a1258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/7202a1258758247.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/8l3wg1258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/8l3wg1258758247.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/992oc1258758247.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/21/t1258758384vlb06yj9dniay4u/992oc1258758247.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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