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multi regression "t"

*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: Thu, 19 Nov 2009 13:09:01 -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/19/t1258661563dr5fzr6368uocdr.htm/, Retrieved Thu, 19 Nov 2009 21:12:55 +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/19/t1258661563dr5fzr6368uocdr.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 «
8,9 6,3 8,2 6,2 7,6 6,1 7,7 6,3 8,1 6,5 8,3 6,6 8,3 6,5 7,9 6,2 7,8 6,2 8 5,9 8,5 6,1 8,6 6,1 8,5 6,1 8 6,1 7,8 6,1 8 6,4 8,2 6,7 8,3 6,9 8,2 7 8,1 7 8 6,8 7,8 6,4 7,8 5,9 7,7 5,5 7,6 5,5 7,6 5,6 7,6 5,8 7,8 5,9 8 6,1 8 6,1 7,9 6 7,7 6 7,4 5,9 6,9 5,5 6,7 5,6 6,5 5,4 6,4 5,2 6,7 5,2 6,8 5,2 6,9 5,5 6,9 5,8 6,7 5,8 6,4 5,5 6,2 5,3 5,9 5,1 6,1 5,2 6,7 5,8 6,8 5,8 6,6 5,5 6,4 5 6,4 4,9 6,7 5,3 7,1 6,1 7,1 6,5 6,9 6,8 6,4 6,6 6 6,4 6 6,4
 
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
X[t] = + 5.84831039520222 + 0.448276619582171Y[t] + 0.0237866131382666M1[t] -0.117936154009505M2[t] -0.224486583115485M3[t] -0.12758893331283M4[t] -0.015518945468392M5[t] -0.0248281013158757M6[t] -0.122412998030213M7[t] -0.306204700394689M8[t] -0.449996402759166M9[t] -0.386891507948712M10[t] -0.0756910638313447M11[t] -0.0334495708940197t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)5.848310395202220.7742917.553100
Y0.4482766195821710.1220453.6730.0006460.000323
M10.02378661313826660.2360360.10080.9201870.460093
M2-0.1179361540095050.236405-0.49890.6203530.310177
M3-0.2244865831154850.236168-0.95050.347030.173515
M4-0.127588933312830.236315-0.53990.5919810.29599
M5-0.0155189454683920.244118-0.06360.9495990.4748
M6-0.02482810131587570.249726-0.09940.9212550.460628
M7-0.1224129980302130.249527-0.49060.6261620.313081
M8-0.3062047003946890.244954-1.250.217890.108945
M9-0.4499964027591660.241298-1.86490.0688730.034437
M10-0.3868915079487120.237591-1.62840.1105820.055291
M11-0.07569106383134470.248838-0.30420.7624260.381213
t-0.03344957089401970.003317-10.085400


Multiple Linear Regression - Regression Statistics
Multiple R0.920223356592374
R-squared0.846811026018135
Adjusted R-squared0.801550647341675
F-TEST (value)18.7097644955976
F-TEST (DF numerator)13
F-TEST (DF denominator)44
p-value8.81517081552374e-14
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.351111530753584
Sum Squared Residuals5.42428950923751


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
18.98.66279014081410.237209859185902
28.28.44279014081413-0.242790140814131
37.68.25796247885591-0.657962478855914
47.78.41106588168098-0.711065881680984
58.18.57934162254784-0.479341622547837
68.38.58141055776455-0.281410557764549
78.38.40554842819798-0.105548428197976
87.98.05382416906483-0.153824169064829
97.87.87658289580633-0.076582895806333
1087.771755233848120.228244766151884
118.58.13916143098790.360838569012104
128.68.181402923925220.418597076074778
138.58.171739966169470.328260033830531
1487.996567628127680.00343237187232173
157.87.85656762812768-0.056567628127678
1688.05449869291096-0.054498692910965
178.28.26760209573603-0.0676020957360353
188.38.31449869291097-0.0144986929109648
198.28.22829188726083-0.0282918872608263
208.18.011050614002330.0889493859976702
2187.74415401682740.255845983172601
227.87.594498692910960.205501307089035
237.87.648111256343230.151888743656773
247.77.511042101447680.188957898552316
257.67.501379143691930.0986208563080689
267.67.371034467608360.228965532391643
277.67.32068979152480.279310208475209
287.87.428965532391640.371034467608356
2987.59724127325850.402758726741504
3087.554482546516990.445517453483008
317.97.378620416950420.521379583049582
327.77.161379143691920.538620856308078
337.46.939310208475210.460689791524791
346.96.789654884558780.110345115441225
356.77.11223341974034-0.412233419740339
366.57.06481958876123-0.564819588761231
376.46.96550130708904-0.565501307089043
386.76.79032896904725-0.090328969047252
396.86.650328969047250.149671030952747
406.96.848260033830540.0517399661694614
416.97.06136343665561-0.161363436655608
426.77.0186047099141-0.318604709914105
436.46.7530872564311-0.353087256431097
446.26.44619065925617-0.246190659256167
455.96.17929406208124-0.279294062081236
466.16.25377704795589-0.153777047955888
476.76.80049389292854-0.100493892928537
486.86.84273538586586-0.042735385865863
496.66.69858944223546-0.098589442235459
506.46.299278794402580.100721205597419
516.46.114451132444360.285548867555635
526.76.357209859185870.342790140814131
537.16.794451571802020.305548428197976
547.16.931003492893390.168996507106611
556.96.93445201115968-0.034452011159683
566.46.62755541398475-0.227555413984753
5766.36065881680982-0.360658816809822
5866.39031414072626-0.390314140726257


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
170.1264311605134780.2528623210269570.873568839486522
180.1411744297399710.2823488594799430.858825570260029
190.1518074447687310.3036148895374630.848192555231269
200.08610391369916130.1722078273983230.913896086300839
210.04338148344968580.08676296689937150.956618516550314
220.04059239248159520.08118478496319030.959407607518405
230.08778236628416920.1755647325683380.912217633715831
240.08698802308150240.1739760461630050.913011976918498
250.06410467571078430.1282093514215690.935895324289216
260.05441858196179610.1088371639235920.945581418038204
270.07816874227224860.1563374845444970.921831257727751
280.1110097410862100.2220194821724210.88899025891379
290.09658433492732130.1931686698546430.903415665072679
300.07495003396994770.1499000679398950.925049966030052
310.07828509781895080.1565701956379020.921714902181049
320.1138235805323260.2276471610646510.886176419467674
330.3830072254103850.766014450820770.616992774589615
340.8786925207620690.2426149584758620.121307479237931
350.947667690295340.1046646194093190.0523323097046596
360.97980663565930.04038672868139830.0201933643406991
370.9842682001371320.03146359972573520.0157317998628676
380.9653046210746480.06939075785070360.0346953789253518
390.9485070277997150.1029859444005690.0514929722002845
400.9311037651230040.1377924697539920.0688962348769959
410.8515847589377220.2968304821245550.148415241062278


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level20.08NOK
10% type I error level50.2NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/10rnh21258661337.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/1ivby1258661337.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/1ivby1258661337.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/2kos71258661337.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/2kos71258661337.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/3h2e81258661337.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/5v4lp1258661337.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/6y2ue1258661337.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/7q0ji1258661337.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/7q0ji1258661337.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/8wtkm1258661337.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/8wtkm1258661337.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/9zw701258661337.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/19/t1258661563dr5fzr6368uocdr/9zw701258661337.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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