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SHW WS7

*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 03:54:07 -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/20/t1258714510oqkn6sqjvkro6eq.htm/, Retrieved Fri, 20 Nov 2009 11:55:22 +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/20/t1258714510oqkn6sqjvkro6eq.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 «
8.6 0 8.5 0 8.3 0 7.8 0 7.8 0 8 0 8.6 0 8.9 0 8.9 0 8.6 0 8.3 0 8.3 0 8.3 0 8.4 0 8.5 0 8.4 0 8.6 0 8.5 0 8.5 0 8.5 0 8.5 0 8.5 0 8.5 0 8.5 0 8.5 0 8.5 0 8.5 0 8.5 0 8.6 0 8.4 0 8.1 0 8 0 8 0 8 0 8 0 7.9 0 7.8 0 7.8 0 7.9 0 8.1 0 8 0 7.6 0 7.3 0 7 0 6.8 0 7 0 7.1 0 7.2 0 7.1 1 6.9 1 6.7 1 6.7 1 6.6 1 6.9 1 7.3 1 7.5 1 7.3 1 7.1 1 6.9 1 7.1 1
 
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] = + 8.14375 -1.13541666666667X[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)8.143750.068563118.777500
X-1.135416666666670.153312-7.405900


Multiple Linear Regression - Regression Statistics
Multiple R0.697161272237485
R-squared0.486033839507789
Adjusted R-squared0.477172353982061
F-TEST (value)54.8478963370955
F-TEST (DF numerator)1
F-TEST (DF denominator)58
p-value6.0921212519105e-10
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.475018904644978
Sum Squared Residuals13.0872916666667


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
18.68.143750000000020.456249999999984
28.58.143750.356250000000003
38.38.143750.156250000000001
47.88.14375-0.34375
57.88.14375-0.34375
688.14375-0.143750000000000
78.68.143750.45625
88.98.143750.75625
98.98.143750.75625
108.68.143750.45625
118.38.143750.156250000000001
128.38.143750.156250000000001
138.38.143750.156250000000001
148.48.143750.256250000000001
158.58.143750.35625
168.48.143750.256250000000001
178.68.143750.45625
188.58.143750.35625
198.58.143750.35625
208.58.143750.35625
218.58.143750.35625
228.58.143750.35625
238.58.143750.35625
248.58.143750.35625
258.58.143750.35625
268.58.143750.35625
278.58.143750.35625
288.58.143750.35625
298.68.143750.45625
308.48.143750.256250000000001
318.18.14375-0.0437500000000001
3288.14375-0.143750000000000
3388.14375-0.143750000000000
3488.14375-0.143750000000000
3588.14375-0.143750000000000
367.98.14375-0.243749999999999
377.88.14375-0.34375
387.88.14375-0.34375
397.98.14375-0.243749999999999
408.18.14375-0.0437500000000001
4188.14375-0.143750000000000
427.68.14375-0.54375
437.38.14375-0.84375
4478.14375-1.14375
456.88.14375-1.34375
4678.14375-1.14375
477.18.14375-1.04375
487.28.14375-0.94375
497.17.008333333333330.0916666666666664
506.97.00833333333333-0.108333333333333
516.77.00833333333333-0.308333333333333
526.77.00833333333333-0.308333333333333
536.67.00833333333333-0.408333333333334
546.97.00833333333333-0.108333333333333
557.37.008333333333330.291666666666667
567.57.008333333333330.491666666666667
577.37.008333333333330.291666666666667
587.17.008333333333330.0916666666666664
596.97.00833333333333-0.108333333333333
607.17.008333333333330.0916666666666664


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
50.511600001785270.976799996429460.48839999821473
60.3632242699992520.7264485399985050.636775730000748
70.3331517105573480.6663034211146960.666848289442652
80.4522477131569410.9044954263138830.547752286843059
90.514153078727140.971693842545720.48584692127286
100.4351934159592570.8703868319185140.564806584040743
110.3358061222384970.6716122444769930.664193877761504
120.2495912682212660.4991825364425320.750408731778734
130.1788588837456720.3577177674913450.821141116254328
140.124698360673830.249396721347660.87530163932617
150.08956427602260750.1791285520452150.910435723977392
160.05955703660106370.1191140732021270.940442963398936
170.04722577542463190.09445155084926380.952774224575368
180.03304158091800440.06608316183600870.966958419081996
190.02308632985921110.04617265971842220.976913670140789
200.01620148278607690.03240296557215380.983798517213923
210.01149709950691900.02299419901383790.988502900493081
220.008316546527516080.01663309305503220.991683453472484
230.006192075670120260.01238415134024050.99380792432988
240.004802295049668940.009604590099337880.995197704950331
250.003937779379977680.007875558759955350.996062220620022
260.003479098255664180.006958196511328360.996520901744336
270.003394183223737520.006788366447475050.996605816776262
280.003776435313366360.007552870626732730.996223564686634
290.007004529043337590.01400905808667520.992995470956662
300.009366649529613590.01873329905922720.990633350470386
310.01233290384413410.02466580768826810.987667096155866
320.01805261267254430.03610522534508860.981947387327456
330.02522105803641630.05044211607283260.974778941963584
340.03465953767459590.06931907534919180.965340462325404
350.04811976546466860.09623953092933730.951880234535331
360.06906205833925730.1381241166785150.930937941660743
370.09980313178277690.1996062635655540.900196868217223
380.1354385172925460.2708770345850930.864561482707454
390.1874418367802360.3748836735604730.812558163219764
400.3786283534832530.7572567069665060.621371646516747
410.7132475909187210.5735048181625580.286752409081279
420.8711223672055160.2577552655889690.128877632794485
430.932387303149950.1352253937001000.0676126968500502
440.9634388132484280.07312237350314310.0365611867515716
450.9850241265333750.02995174693324980.0149758734666249
460.9861103270464950.02777934590700890.0138896729535044
470.9827069479251110.03458610414977850.0172930520748892
480.9744543197889820.05109136042203680.0255456802110184
490.9528727037495080.09425459250098340.0471272962504917
500.917693034371520.1646139312569590.0823069656284795
510.8986911620198630.2026176759602730.101308837980137
520.8863105090191470.2273789819617060.113689490980853
530.9408869755979410.1182260488041180.0591130244020589
540.9202296296237330.1595407407525340.079770370376267
550.8316846074950080.3366307850099830.168315392504992


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level50.0980392156862745NOK
5% type I error level170.333333333333333NOK
10% type I error level250.490196078431373NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/10qela1258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/10qela1258714442.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/1h49w1258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/1h49w1258714442.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/2hjue1258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/2hjue1258714442.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/381981258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/381981258714442.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/4kp6b1258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/4kp6b1258714442.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/53rgw1258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/53rgw1258714442.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/65y621258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/65y621258714442.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/73n4u1258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/73n4u1258714442.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/8hd1u1258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/8hd1u1258714442.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/99n9x1258714442.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258714510oqkn6sqjvkro6eq/99n9x1258714442.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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