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Multiple Regression

*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, 18 Dec 2009 08:13:02 -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/Dec/18/t126114926451welxecwf90ubp.htm/, Retrieved Fri, 18 Dec 2009 16:14:36 +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/18/t126114926451welxecwf90ubp.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 «
12.1 0 0 12 0 0 11.8 0 0 12.7 0 0 12.3 0 0 11.9 0 0 12 0 0 12.3 0 0 12.8 0 0 12.4 0 0 12.3 0 0 12.7 0 0 12.7 0 0 12.9 0 0 13 0 0 12.2 0 0 12.3 0 0 12.8 0 0 12.8 0 0 12.8 0 0 12.2 0 0 12.6 0 0 12.8 0 0 12.5 0 0 12.4 0 0 12.3 1 0 11.9 1 0 11.7 1 0 12 1 0 12.1 1 0 11.7 1 0 11.8 1 0 11.8 1 0 11.8 1 0 11.3 1 0 11.3 1 0 11.3 1 0 11.2 0 1 11.4 0 1 12.2 0 1 12.9 0 1 13.1 0 1 13.5 0 1 13.6 0 1 14.4 0 1 14.1 0 1 15.1 0 1 15.8 0 1 15.9 0 1 15.4 0 1 15.5 0 1 14.8 0 1 13.2 0 1 12.7 0 1 12.1 0 1 11.9 0 1 10.6 0 1 10.7 0 1 9.8 0 1 9 0 1 8.3 0 1 9.3 0 1 9 0 1 9.1 0 1 10 0 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 time4 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
Gzhdsidx[t] = + 14.1235749185668 + 1.14362276058632Vr_crisis[t] + 3.74368892508143NA_crisis[t] -0.354213179741224M1[t] -0.811778693901559M2[t] -0.795396053881653M3[t] -0.679013413861747M4[t] -0.579297440508507M5[t] -0.338295840119437M6[t] -0.338579866766198M7[t] -0.178863893412958M8[t] -0.199147920059719M9[t] -0.13943194670648M10[t] -0.09971597335324M11[t] -0.0997159733532392t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)14.12357491856680.87812116.083900
Vr_crisis1.143622760586320.7627341.49940.1400650.070032
NA_crisis3.743688925081431.191913.14090.0028280.001414
M1-0.3542131797412240.944799-0.37490.7093140.354657
M2-0.8117786939015590.965996-0.84040.4047110.202355
M3-0.7953960538816530.960251-0.82830.4114250.205713
M4-0.6790134138617470.955328-0.71080.4805330.240266
M5-0.5792974405085070.95124-0.6090.5452870.272643
M6-0.3382958401194370.999318-0.33850.7363840.368192
M7-0.3385798667661980.994807-0.34030.7350210.36751
M8-0.1788638934129580.991101-0.18050.8575130.428757
M9-0.1991479200597190.988209-0.20150.8411060.420553
M10-0.139431946706480.986138-0.14140.8881290.444064
M11-0.099715973353240.984894-0.10120.9197610.45988
t-0.09971597335323920.028596-3.48710.0010270.000514


Multiple Linear Regression - Regression Statistics
Multiple R0.473575913072725
R-squared0.224274145442665
Adjusted R-squared0.00707090616661099
F-TEST (value)1.03255433109644
F-TEST (DF numerator)14
F-TEST (DF denominator)50
p-value0.438381636751226
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation1.55659732147622
Sum Squared Residuals121.149761061346


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
112.113.6696457654723-1.56964576547231
21213.1123642779587-1.11236427795874
311.813.0290309446254-1.22903094462541
412.713.0456976112921-0.345697611292075
512.313.0456976112921-0.745697611292073
611.913.1869832383279-1.28698323832790
71213.0869832383279-1.08698323832790
812.313.1469832383279-0.846983238327904
912.813.0269832383279-0.226983238327904
1012.412.9869832383279-0.586983238327904
1112.312.9269832383279-0.626983238327904
1212.712.9269832383279-0.226983238327906
1312.712.47305408523340.226945914766557
1412.911.91577259771990.984227402280131
151311.83243926438651.16756073561346
1612.211.84910593105320.350894068946797
1712.311.84910593105320.450894068946797
1812.811.99039155808900.809608441910967
1912.811.89039155808900.909608441910967
2012.811.95039155808900.849608441910966
2112.211.83039155808900.369608441910965
2212.611.79039155808900.809608441910965
2312.811.73039155808901.06960844191097
2412.511.73039155808900.769608441910964
2512.411.27646240499461.12353759500543
2612.311.86280367806730.437196321932684
2711.911.77947034473400.120529655266017
2811.711.7961370114007-0.096137011400651
291211.79613701140070.203862988599348
3012.111.93742263843650.162577361563518
3111.711.8374226384365-0.137422638436483
3211.811.8974226384365-0.0974226384364817
3311.811.77742263843650.0225773615635184
3411.811.73742263843650.0625773615635187
3511.311.6774226384365-0.377422638436482
3611.311.6774226384365-0.377422638436483
3711.311.22349348534200.0765065146579809
3811.213.2662781623236-2.06627816232356
3911.413.1829448289902-1.78294482899023
4012.213.1996114956569-0.999611495656894
4112.913.1996114956569-0.299611495656894
4213.113.3408971226927-0.240897122692725
4313.513.24089712269270.259102877307275
4413.613.30089712269270.299102877307274
4514.413.18089712269271.21910287730728
4614.113.14089712269270.959102877307275
4715.113.08089712269272.01910287730727
4815.813.08089712269272.71910287730727
4915.912.62696796959833.27303203040174
5015.412.06968648208473.33031351791531
5115.511.98635314875143.51364685124864
5214.812.00301981541802.79698018458198
5313.212.00301981541801.19698018458198
5412.712.14430544245390.555694557546145
5512.112.04430544245390.0556945575461452
5611.912.1043054424539-0.204305442453855
5710.611.9843054424539-1.38430544245385
5810.711.9443054424539-1.24430544245385
599.811.8843054424539-2.08430544245385
60911.8843054424539-2.88430544245386
618.311.4303762893594-3.13037628935939
629.310.8730948018458-1.57309480184582
63910.7897614685125-1.78976146851249
649.110.8064281351792-1.70642813517915
651010.8064281351792-0.806428135179155


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
180.05251141108380330.1050228221676070.947488588916197
190.01518617553463120.03037235106926240.984813824465369
200.003764222022489590.007528444044979180.99623577797751
210.002911138039248410.005822276078496830.997088861960752
220.0007764766880112990.001552953376022600.999223523311989
230.0001863908067187660.0003727816134375320.999813609193281
246.18628731656522e-050.0001237257463313040.999938137126834
251.94693720460980e-053.89387440921959e-050.999980530627954
264.11468038336778e-068.22936076673556e-060.999995885319617
279.14167094156606e-071.82833418831321e-060.999999085832906
282.20229721397927e-074.40459442795853e-070.999999779770279
294.17067864395790e-088.34135728791579e-080.999999958293214
307.55283373741456e-091.51056674748291e-080.999999992447166
311.48242957741428e-092.96485915482856e-090.99999999851757
322.80690555070802e-105.61381110141605e-100.99999999971931
334.62643356937396e-119.25286713874791e-110.999999999953736
347.15441217555591e-121.43088243511118e-110.999999999992846
353.17719617874822e-126.35439235749644e-120.999999999996823
361.23674037443182e-122.47348074886365e-120.999999999998763
373.53246745482788e-137.06493490965576e-130.999999999999647
382.97632862910995e-135.95265725821989e-130.999999999999702
397.76944712071632e-131.55388942414326e-120.999999999999223
401.70443753469765e-113.40887506939529e-110.999999999982956
413.87611781565107e-097.75223563130213e-090.999999996123882
421.43117096916203e-072.86234193832406e-070.999999856882903
436.05289451548266e-061.21057890309653e-050.999993947105485
440.0002686345249889810.0005372690499779610.99973136547501
450.001500380373766360.003000760747532720.998499619626234
460.008505902140257060.01701180428051410.991494097859743
470.01347298163922950.02694596327845890.98652701836077


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level260.866666666666667NOK
5% type I error level290.966666666666667NOK
10% type I error level290.966666666666667NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Dec/18/t126114926451welxecwf90ubp/10n4aw1261149177.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Dec/18/t126114926451welxecwf90ubp/98jhs1261149177.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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Software written by Ed van Stee & Patrick Wessa


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