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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: Wed, 18 Nov 2009 08:16:02 -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/18/t1258557400qphyxrw7uia0zx6.htm/, Retrieved Wed, 18 Nov 2009 16:16:52 +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/18/t1258557400qphyxrw7uia0zx6.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 «
7.2 97.78 7.5 8.3 8.9 7.4 97.69 7.2 7.5 8.8 8.8 96.67 7.4 7.2 8.3 9.3 98.29 8.8 7.4 7.5 9.3 98.2 9.3 8.8 7.2 8.7 98.71 9.3 9.3 7.4 8.2 98.54 8.7 9.3 8.8 8.3 98.2 8.2 8.7 9.3 8.5 96.92 8.3 8.2 9.3 8.6 99.06 8.5 8.3 8.7 8.5 99.65 8.6 8.5 8.2 8.2 99.82 8.5 8.6 8.3 8.1 99.99 8.2 8.5 8.5 7.9 100.33 8.1 8.2 8.6 8.6 99.31 7.9 8.1 8.5 8.7 101.1 8.6 7.9 8.2 8.7 101.1 8.7 8.6 8.1 8.5 100.93 8.7 8.7 7.9 8.4 100.85 8.5 8.7 8.6 8.5 100.93 8.4 8.5 8.7 8.7 99.6 8.5 8.4 8.7 8.7 101.88 8.7 8.5 8.5 8.6 101.81 8.7 8.7 8.4 8.5 102.38 8.6 8.7 8.5 8.3 102.74 8.5 8.6 8.7 8 102.82 8.3 8.5 8.7 8.2 101.72 8 8.3 8.6 8.1 103.47 8.2 8 8.5 8.1 102.98 8.1 8.2 8.3 8 102.68 8.1 8.1 8 7.9 102.9 8 8.1 8.2 7.9 103.03 7.9 8 8.1 8 101.29 7.9 7.9 8.1 8 103.69 8 7.9 8 7.9 103.68 8 8 7.9 8 104.2 7.9 8 7.9 7.7 104.08 8 7.9 8 7.2 104.16 7.7 8 8 7.5 103.05 7.2 7.7 7.9 7.3 104.66 7.5 7.2 8 7 104.46 7.3 7.5 7.7 7 104.95 7 7.3 7.2 7 105.85 7 7 7.5 7.2 106.23 7 7 7.3 7.3 104.86 7.2 7 7 7.1 107.44 7.3 7.2 7 6.8 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 time3 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
Y[t] = -3.92515432159056 + 0.0512518252555557X[t] + 1.51663485005415Y1[t] -0.91199180830936Y2[t] + 0.278978156355666Y4[t] -0.142293504458614M1[t] -0.115723656701091M2[t] + 0.67913254261951M3[t] -0.426949905291582M4[t] + 0.0675867935318781M5[t] + 0.106854305203872M6[t] + 0.0377162842122586M7[t] + 0.195531686682516M8[t] + 0.128227851596769M9[t] -0.0737310721580348M10[t] + 0.000156731586821268M11[t] -0.0170088270929127t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)-3.925154321590563.178199-1.2350.224210.112105
X0.05125182525555570.0294581.73990.0897720.044886
Y11.516634850054150.09913715.298400
Y2-0.911991808309360.110388-8.261700
Y40.2789781563556660.0681774.0920.0002080.000104
M1-0.1422935044586140.101081-1.40770.1671340.083567
M2-0.1157236567010910.103819-1.11470.2718140.135907
M30.679132542619510.1098946.179900
M4-0.4269499052915820.130658-3.26770.0022670.001133
M50.06758679353187810.1039880.64990.5195380.259769
M60.1068543052038720.1084520.98530.3305660.165283
M70.03771628421225860.0993790.37950.7063610.35318
M80.1955316866825160.1022141.9130.0631150.031558
M90.1282278515967690.1262951.01530.3162170.158109
M10-0.07373107215803480.108096-0.68210.4992130.249607
M110.0001567315868212680.1042280.00150.9988080.499404
t-0.01700882709291270.006222-2.73350.0093730.004686


Multiple Linear Regression - Regression Statistics
Multiple R0.986192861958267
R-squared0.972576360977437
Adjusted R-squared0.961325637275873
F-TEST (value)86.4456711208912
F-TEST (DF numerator)16
F-TEST (DF denominator)39
p-value0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.146278312041805
Sum Squared Residuals0.83449643837819


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
17.27.21508177835-0.0150817783500001
27.47.4667353107373-0.0667353107373064
38.88.629741255530130.170258744469869
49.39.30738584076954-0.0073858407695366
59.39.178136494714360.121863505285645
68.78.82633333729022-0.126333337290224
78.28.21206218777769-0.0120621877776920
88.38.263809880704520.0361901192954789
98.58.72155427135885-0.221554271358849
108.68.65700632192451-0.0570063219245135
118.58.57389992064295-0.073899920642945
128.28.35048232205587-0.150482322055872
138.17.891897157883610.208102842116388
147.98.06871567225807-0.168715672258072
158.68.554260577909630.0457394220903664
168.78.683259379906150.0167406200938494
178.78.646158655190.0538413448100059
188.58.51270971737356-0.0127097173735626
198.48.314420462706730.0855795372932725
208.58.51795987639654-0.0179598763965408
218.78.608344952464340.0916550475356572
228.78.662563521108050.0374364788919462
238.68.505558692694670.0944413073053296
248.58.393841005040750.106158994959246
258.38.248320657677880.0516793423221192
2688.04985403518304-0.0498540351830433
278.28.47083449063968-0.270834490639679
288.17.986460606700970.113539393299032
298.18.049017606117870.0509823938821255
3088.06340647704453-0.0634064770445249
317.97.892667176781940.00733282321806117
327.97.95177436963246-0.0517743696324604
3387.869482712340070.130517287659930
3487.897285011475540.102714988524465
357.97.834554473408420.0654455265915794
3687.692376378856160.307623621143839
377.77.79768430974588-0.0976843097458844
387.27.26515584058376-0.0651558405837586
397.57.473495988607950.0265040113920546
407.37.37180432707188-0.0718043270718777
4177.17846387434098-0.178463874340975
4276.813754781763070.186245218236926
4377.13102556580805-0.131025565808055
447.27.23551220351138-0.0355122035113778
457.37.30061806383674-0.000618063836738301
467.17.1831451454919-0.0831451454918977
476.86.88598691325396-0.085986913253964
486.46.66330029404721-0.263300294047212
496.16.24701609634262-0.147016096342623
506.56.149539141237820.35046085876218
517.77.671667687312610.0283323126873895
527.97.95108984555147-0.0510898455514668
537.57.5482233696368-0.0482233696368014
546.96.883795686528620.016204313471385
556.66.549824606925590.050175393074413
566.96.83094366975510.0690563302449001


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
200.1185970394808650.2371940789617290.881402960519135
210.1586419433682550.317283886736510.841358056631745
220.08519168584335060.1703833716867010.91480831415665
230.03755343201305330.07510686402610660.962446567986947
240.0613723483513530.1227446967027060.938627651648647
250.03188573211196790.06377146422393590.968114267888032
260.01689459587690790.03378919175381580.983105404123092
270.2829560974383290.5659121948766590.71704390256167
280.194847895356710.389695790713420.80515210464329
290.1564320841275820.3128641682551630.843567915872418
300.1809036132766430.3618072265532860.819096386723357
310.1401968538540470.2803937077080940.859803146145953
320.1234838676335370.2469677352670750.876516132366463
330.095532463632690.191064927265380.90446753636731
340.06625228657139560.1325045731427910.933747713428604
350.03357353921491520.06714707842983030.966426460785085
360.1788377486765530.3576754973531060.821162251323447


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level10.0588235294117647NOK
10% type I error level40.235294117647059NOK
 
Charts produced by software:
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557400qphyxrw7uia0zx6/8p92j1258557357.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557400qphyxrw7uia0zx6/8p92j1258557357.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557400qphyxrw7uia0zx6/9e7fm1258557357.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557400qphyxrw7uia0zx6/9e7fm1258557357.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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