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Multiple regression (MR)

*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: Tue, 14 Dec 2010 21:27:06 +0000
 
Cite this page as follows:
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y.htm/, Retrieved Tue, 14 Dec 2010 22:26:08 +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/2010/Dec/14/t1292361958306fov635e81a6y.htm/},
    year = {2010},
}
@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 = {2010},
    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 «
3.18 0.22 6.62 3.64 3.14 0.22 6.56 3.62 3.02 0.23 6.59 3.61 3.02 0.24 6.56 3.6 3.03 0.25 6.57 3.6 3.04 0.25 6.62 3.63 3.09 0.24 6.69 3.59 3.06 0.24 6.69 3.55 3.06 0.22 6.64 3.54 3.09 0.21 6.6 3.53 3.11 0.21 6.66 3.53 3.1 0.21 6.62 3.53 3.09 0.2 6.64 3.52 3.19 0.2 6.64 3.52 3.22 0.2 6.73 3.48 3.22 0.2 6.73 3.49 3.25 0.2 6.69 3.47 3.25 0.2 6.78 3.46 3.27 0.2 6.77 3.4 3.28 0.2 6.8 3.36 3.24 0.2 6.8 3.3 3.23 0.2 6.74 3.28 3.2 0.2 6.84 3.28 3.19 0.2 6.83 3.24 3.23 0.2 6.89 3.23 3.19 0.2 6.9 3.2 3.16 0.2 6.86 3.15 3.11 0.2 6.78 3.1 3.11 0.2 6.82 3.07 3.07 0.2 6.81 3.03 3.05 0.21 6.81 2.96 3 0.2 6.78 2.88 2.95 0.2 6.79 2.83 2.9 0.19 6.83 2.8 2.88 0.18 6.9 2.8 2.9 0.18 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 time10 seconds
R Server'George Udny Yule' @ 72.249.76.132


Multiple Linear Regression - Estimated Regression Equation
Mayonaise[t] = + 2.47865060894444 -0.465444129700099Eieren[t] -0.0988998564557219Olijfolie[t] + 0.413491668082886Mosterd[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)2.478650608944440.8180873.02980.0038060.001903
Eieren-0.4654441297000991.326341-0.35090.7270640.363532
Olijfolie-0.09889985645572190.098111-1.0080.3181040.159052
Mosterd0.4134916680828860.0911714.53543.4e-051.7e-05


Multiple Linear Regression - Regression Statistics
Multiple R0.831522318599948
R-squared0.691429366329834
Adjusted R-squared0.673627214387324
F-TEST (value)38.8396508783175
F-TEST (DF numerator)3
F-TEST (DF denominator)52
p-value2.58681964737661e-13
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.112734946676432
Sum Squared Residuals0.660876746511169


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
13.183.22664552249524-0.0466455224952405
23.143.22430968052093-0.08430968052093
33.023.21255332684943-0.192553326849429
43.023.20673096456527-0.186730964565271
53.033.20108752470371-0.171087524703713
63.043.20854728192341-0.168547281923413
73.093.1897390665452-0.099739066545198
83.063.17319939982188-0.113199399821882
93.063.18331835855784-0.123318358557842
103.093.18779387743224-0.0977938774322427
113.113.1818598860449-0.0718598860448994
123.13.18581588030313-0.085815880303128
133.093.18435740779019-0.094357407790186
143.193.184357407790190.00564259220981405
153.223.158916753985860.0610832460141448
163.223.163051670666680.0569483293333158
173.253.158737831563260.0912621684367445
183.253.145701927801410.104298072198588
193.273.121881426281000.148118573719004
203.283.102374763864010.177625236135991
213.243.077565263779040.162434736220965
223.233.075229421804720.154770578195279
233.23.065339436159150.134660563840851
243.193.049788768000390.140211231999609
253.233.039719859932220.190280140067781
263.193.026326111325170.163673888674825
273.163.009607522179260.150392477820741
283.112.996844927291570.113155072708427
293.112.980484182990860.129515817009143
303.072.96493351483210.105066485167901
313.052.931334656769300.118665343230703
3232.905876760313340.094123239686662
332.952.884213178344640.0657868216553636
342.92.872506875340920.0274931246590779
352.882.870238326686020.00976167331397747
362.92.876982394215320.0230176057846768
372.892.872735848431310.0172641515686910
382.892.867611933185920.0223880668140772
392.912.869768852737640.0402311472623627
402.92.863655938927690.0363440610723058
412.92.867611933185920.032388066814077
422.882.844847723549480.0351522764505207
432.832.83145397494244-0.00145397494243526
442.82.80933816167698-0.00933816167697668
452.772.79531325935058-0.0253132593505756
462.782.78740127083412-0.00740127083411803
472.752.78146727944677-0.0314672794467746
482.742.79124563598916-0.0512456359891613
492.732.78711071930833-0.0571107193083328
502.692.79421263168283-0.104212631682833
512.672.78711071930833-0.117110719308333
522.662.79106671356656-0.131066713566561
532.672.77463867594643-0.104638675946431
542.652.77642050800132-0.126420508001318
552.642.78523695911129-0.14523695911129
562.632.79378010134711-0.163780101347105


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
70.1460026699274340.2920053398548680.853997330072566
80.0971446991413510.1942893982827020.90285530085865
90.07413050999853050.1482610199970610.92586949000147
100.05102929586987480.1020585917397500.948970704130125
110.0380824014954650.076164802990930.961917598504535
120.04688028075337370.09376056150674750.953119719246626
130.1102685387917760.2205370775835530.889731461208224
140.3145060755581270.6290121511162530.685493924441873
150.473290715883290.946581431766580.52670928411671
160.5649942849141150.870011430171770.435005715085885
170.8607260746820.2785478506360010.139273925318000
180.8946280354402560.2107439291194870.105371964559743
190.9267340018247480.1465319963505030.0732659981752517
200.9119287080756150.176142583848770.088071291924385
210.8745231314087350.2509537371825290.125476868591265
220.8445503019167430.3108993961665130.155449698083257
230.895897889537750.2082042209245000.104102110462250
240.9068782351036240.1862435297927530.0931217648963763
250.8792108539635840.2415782920728320.120789146036416
260.8798424595403260.2403150809193470.120157540459674
270.8578034928745580.2843930142508840.142196507125442
280.861295237061170.2774095258776610.138704762938831
290.8505657946630010.2988684106739980.149434205336999
300.9045745235111610.1908509529776770.0954254764888386
310.8885593553507570.2228812892984860.111440644649243
320.8479361425287350.3041277149425300.152063857471265
330.8181027367824690.3637945264350620.181897263217531
340.9220767596938220.1558464806123550.0779232403061775
350.9941732423528020.01165351529439630.00582675764719816
360.9960274737431520.00794505251369650.00397252625684825
370.9981711390018430.003657721996313540.00182886099815677
380.998538011256710.00292397748658180.0014619887432909
390.9982758928403080.003448214319384710.00172410715969235
400.9981587481452030.00368250370959440.0018412518547972
410.9992341279829650.001531744034069910.000765872017034953
420.9989546424820630.002090715035874270.00104535751793714
430.9985227769231680.002954446153664550.00147722307683228
440.9981480072056530.003703985588694420.00185199279434721
450.996598989089740.006802021820520020.00340101091026001
460.994858431895060.01028313620988160.00514156810494078
470.9920970261087050.01580594778259020.00790297389129508
480.9906113653108280.01877726937834480.00938863468917238
490.997919082094030.004161835811939430.00208091790596972


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level110.255813953488372NOK
5% type I error level150.348837209302326NOK
10% type I error level170.395348837209302NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/10j1ll1292362015.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/10j1ll1292362015.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/1v0o91292362015.png (open in new window)
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http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/2nrou1292362015.png (open in new window)
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http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/3nrou1292362015.png (open in new window)
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http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/6g15f1292362015.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/6g15f1292362015.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/79s401292362015.png (open in new window)
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http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/89s401292362015.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/89s401292362015.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/9j1ll1292362015.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/14/t1292361958306fov635e81a6y/9j1ll1292362015.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;
 
Parameters (R input):
par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No 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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Software written by Ed van Stee & Patrick Wessa


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