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multiple regression with linear trend

*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: Mon, 14 Dec 2009 12:49:08 -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/Dec/14/t1260820174wcejlx4mr6b0f9z.htm/, Retrieved Mon, 14 Dec 2009 20:49:46 +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/Dec/14/t1260820174wcejlx4mr6b0f9z.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 «
441 1919 449 1911 452 1870 462 2263 455 1802 461 1863 461 1989 463 2197 462 2409 456 2502 455 2593 456 2598 472 2053 472 2213 471 2238 465 2359 459 2151 465 2474 468 3079 467 2312 463 2565 460 1972 462 2484 461 2202 476 2151 476 1976 471 2012 453 2114 443 1772 442 1957 444 2070 438 1990 427 2182 424 2008 416 1916 406 2397 431 2114 434 1778 418 1641 412 2186 404 1773 409 1785 412 2217 406 2153 398 1895 397 2475 385 1793 390 2308 413 2051 413 1898 401 2142 397 1874 397 1560 409 1808 419 1575 424 1525 428 1997 430 1753 424 1623 433 2251 456 1890
 
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 time3 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
wkl[t] = + 433.120040994186 + 0.0131096616397323bvg[t] + 18.3563768945723M1[t] + 15.1404600533549M2[t] + 9.57257124330357M3[t] + 3.39628227004529M4[t] + 2.71829725161415M5[t] + 7.10981194734444M6[t] + 8.9402331248942M7[t] + 10.6796447634358M8[t] + 5.36103830139233M9[t] + 5.01234802383615M10[t] + 1.76664625014594M11[t] -0.965096595597911t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)433.12004099418630.31546814.287100
bvg0.01310966163973230.0108691.20620.2337950.116897
M118.356376894572312.1897161.50590.1387870.069394
M215.140460053354913.1681951.14980.2560540.128027
M39.5725712433035713.0160140.73540.4657220.232861
M43.3962822700452912.3982740.27390.7853370.392669
M52.7182972516141513.6947990.19850.8435170.421758
M67.1098119473444412.9008650.55110.5841680.292084
M78.940233124894212.255590.72950.4693270.234663
M810.679644763435812.6226470.84610.4018010.2009
M95.3610383013923312.1720660.44040.6616390.33082
M105.0123480238361512.2805960.40820.6850160.342508
M111.7666462501459412.4038880.14240.8873520.443676
t-0.9650965955979110.164695-5.859900


Multiple Linear Regression - Regression Statistics
Multiple R0.765394087726793
R-squared0.58582810952713
Adjusted R-squared0.47126992705591
F-TEST (value)5.11380415514454
F-TEST (DF numerator)13
F-TEST (DF denominator)47
p-value1.53292904616631e-05
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation19.0230958277144
Sum Squared Residuals17008.2742189089


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
1441475.668761979807-34.6687619798071
2449471.382871249874-22.3828712498737
3452464.312389716995-12.3123897169955
4462462.323101172554-0.323101172554079
5455454.6364655426080.363534457391566
6461458.8625730027642.13742699723552
7461461.379714951323-0.379714951322605
8463464.880839615331-1.88083961533059
9462461.3763848253120.623615174687509
10456461.281796484654-5.28179648465351
11455458.263977324581-3.26397732458102
12456455.5977827870360.402217212964203
13472465.8442974923566.15570250764386
14472463.7608299178988.23917008210207
15471457.55558605324213.4444139467580
16465452.00046954279312.9995304572065
17459447.630578307711.3694216922999
18465455.2914171174669.708582882534
19468464.0880869914563.91191300854411
20467454.80729155672512.1927084432751
21463451.84033289393611.1596671060642
22460442.7525166684217.2474833315796
23462445.25386505867516.7461349413247
24461438.82519763052722.1748023694731
25476455.54788518587520.4521148141251
26476449.07268096210626.9273190378936
27471443.01164337548827.9883566245124
28453437.20744329388415.7925567061159
29443431.08085739906711.9191426009334
30442436.9325629025495.06743709745054
31444439.2792792497914.72072075020895
32438439.004821361556-1.00482136155615
33427435.238173338743-8.23817333874338
34424431.643305340276-7.64330534027587
35416426.226418100132-10.2264181001324
36406429.8004225031-23.8004225030997
37431443.48166855803-12.4816685580299
38434434.895808810264-0.8958088102645
39418426.566799759972-8.56679975997198
40412426.57017978477-14.5701797847699
41404419.512807913531-15.5128079135314
42409423.096541953341-14.0965419533406
43412429.625240363657-17.6252403636568
44406429.560537061658-23.5605370616576
45398419.894541300965-21.8945413009653
46397426.184358178856-29.1843581788559
47385413.032770571270-28.0327705712704
48390417.052503469989-27.0525034699886
49413431.074600727552-18.0746007275519
50413424.887809059857-11.8878090598574
51401421.553581094303-20.5535810943029
52397410.898806205998-13.8988062059985
53397405.139290837093-8.13929083709349
54409411.816905023879-2.81690502387947
55419409.6276784437749.37232155622631
56424409.74651040473114.2534895952692
57428409.65056764104318.3494323589570
58430405.13802332779424.8619766722057
59424399.22296894534124.7770310546590
60433404.72409360934928.275906390651
61456417.3827860563838.61721394362


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
170.1244905396962230.2489810793924460.875509460303777
180.05220701184719270.1044140236943850.947792988152807
190.02067205152887150.0413441030577430.979327948471128
200.01117375393940890.02234750787881770.98882624606059
210.006523140538140180.01304628107628040.99347685946186
220.002675259733876940.005350519467753890.997324740266123
230.001409290563859880.002818581127719760.99859070943614
240.0004996949121759370.0009993898243518730.999500305087824
250.0001991031823996720.0003982063647993440.9998008968176
269.78424699412255e-050.0001956849398824510.999902157530059
279.91733485521008e-050.0001983466971042020.999900826651448
280.0009538539505601920.001907707901120380.99904614604944
290.004211127058623340.008422254117246690.995788872941377
300.01608391340895180.03216782681790360.983916086591048
310.02575147344570330.05150294689140660.974248526554297
320.05430613282109260.1086122656421850.945693867178907
330.1085628104864790.2171256209729580.891437189513521
340.1315947130953770.2631894261907550.868405286904623
350.1654018949679960.3308037899359920.834598105032004
360.3020019655817590.6040039311635170.697998034418241
370.2798828921260420.5597657842520840.720117107873958
380.2985596473170720.5971192946341440.701440352682928
390.3683511866256150.736702373251230.631648813374385
400.5591686670775070.8816626658449850.440831332922493
410.7206222305610770.5587555388778460.279377769438923
420.9263331357503830.1473337284992330.0736668642496167
430.9526758198239720.09464836035205570.0473241801760279
440.9664082154751760.06718356904964730.0335917845248236


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level80.285714285714286NOK
5% type I error level120.428571428571429NOK
10% type I error level150.535714285714286NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/10wr6n1260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/10wr6n1260820143.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/1p52h1260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/1p52h1260820143.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/212m61260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/212m61260820143.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/38oln1260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/38oln1260820143.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/4qqab1260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/4qqab1260820143.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/55yda1260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/55yda1260820143.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/6sx6q1260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/6sx6q1260820143.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/7l9tt1260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/7l9tt1260820143.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/8mvzd1260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/8mvzd1260820143.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/9emhk1260820143.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/14/t1260820174wcejlx4mr6b0f9z/9emhk1260820143.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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