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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: Tue, 15 Dec 2009 08:08:14 -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/15/t1260889790c2e5s4uudl6gn6s.htm/, Retrieved Wed, 19 Jun 2013 07:14:13 +0000

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

System-generated keywords (parent):

t1258546792kjxr91v06m32f9k (pk = 57439)

Estimated Impact

Dataseries X:

» Textfile « » CSV « » Correlation Matrix « » Notched Boxplots «

8.5	0	8.5	8.3	8.2	8.7
8.6	0	8.6	8.5	8.3	8.2
8.5	0	8.5	8.6	8.5	8.3
8.2	0	8.2	8.5	8.6	8.5
8.1	0	8.1	8.2	8.5	8.6
7.9	0	7.9	8.1	8.2	8.5
8.6	0	8.6	7.9	8.1	8.2
8.7	0	8.7	8.6	7.9	8.1
8.7	0	8.7	8.7	8.6	7.9
8.5	0	8.5	8.7	8.7	8.6
8.4	0	8.4	8.5	8.7	8.7
8.5	0	8.5	8.4	8.5	8.7
8.7	0	8.7	8.5	8.4	8.5
8.7	0	8.7	8.7	8.5	8.4
8.6	0	8.6	8.7	8.7	8.5
8.5	0	8.5	8.6	8.7	8.7
8.3	0	8.3	8.5	8.6	8.7
8	0	8	8.3	8.5	8.6
8.2	0	8.2	8	8.3	8.5
8.1	0	8.1	8.2	8	8.3
8.1	0	8.1	8.1	8.2	8
8	0	8	8.1	8.1	8.2
7.9	0	7.9	8	8.1	8.1
7.9	0	7.9	7.9	8	8.1
8	0	8	7.9	7.9	8
8	0	8	8	7.9	7.9
7.9	0	7.9	8	8	7.9
8	0	8	7.9	8	8
7.7	0	7.7	8	7.9	8
7.2	0	7.2	7.7	8	7.9
7.5	0	7.5	7.2	7.7	8
7.3	0	7.3	7.5	7.2	7.7
7	0	7	7.3	7.5	7.2
7	0	7	7	7.3	7.5
7	0	7	7	7	7.3
7.2	0	7.2	7	7	7
7.3	0	7.3	7.2	7	7
7.1	0	7.1	7.3	7.2	7
6.8	0	6.8	7.1	7.3	7.2
6.4	0	6.4	6.8	7.1	7.3
6.1	0	6.1	6.4	6.8	7.1
6.5	0	6.5	6.1	6.4	6.8
7.7	0	7.7	6.5	6.1	6.4
7.9	0	7.9	7.7	6.5	6.1
7.5	0	7.5	7.9	7.7	6.5
6.9	1	6.9	7.5	7.9	7.7
6.6	1	6.6	6.9	7.5	7.9
6.9	1	6.9	6.6	6.9	7.5
7.7	1	7.7	6.9	6.6	6.9
8	1	8	7.7	6.9	6.6
8	1	8	8	7.7	6.9
7.7	1	7.7	8	8	7.7
7.3	1	7.3	7.7	8	8
7.4	1	7.4	7.3	7.7	8
8.1	1	8.1	7.4	7.3	7.7
8.3	1	8.3	8.1	7.4	7.3
8.2	1	8.2	8.3	8.1	7.4

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 Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	12 seconds
R Server	'Gwilym Jenkins' @ 72.249.127.135

Multiple Linear Regression - Estimated Regression Equation

Y[t] = -3.03750610467177e-17 + 1.44281539971909e-16X[t] + 1Y1[t] + 1.80407309583598e-16Y2[t] -1.34871633389605e-16Y3[t] + 1.25242085042886e-17Y4[t] -1.18735169270795e-17M1[t] -4.29542034326534e-17M2[t] + 1.71068599226658e-16M3[t] -2.53276152719436e-17M4[t] -3.45439284701359e-17M5[t] -1.26583126706111e-17M6[t] + 7.21379251138558e-17M7[t] -3.62069740559273e-17M8[t] -2.76317048100518e-17M9[t] -3.29487196717181e-17M10[t] -1.64934388509514e-17M11[t] -4.39099679542517e-18t + e[t]

Multiple Linear Regression - Ordinary Least Squares
Variable	Parameter	S.D.	T-STAT H0: parameter = 0	2-tail p-value	1-tail p-value
(Intercept)	-3.03750610467177e-17	0	-0.0513	0.95932	0.47966
X	1.44281539971909e-16	0	1.6918	0.098658	0.049329
Y1	1	0	8017416243731516	0	0
Y2	1.80407309583598e-16	0	0.7839	0.437815	0.218908
Y3	-1.34871633389605e-16	0	-0.5951	0.555196	0.277598
Y4	1.25242085042886e-17	0	0.1014	0.919765	0.459882
M1	-1.18735169270795e-17	0	-0.1279	0.898895	0.449447
M2	-4.29542034326534e-17	0	-0.4253	0.672957	0.336478
M3	1.71068599226658e-16	0	1.7027	0.096591	0.048296
M4	-2.53276152719436e-17	0	-0.2642	0.793024	0.396512
M5	-3.45439284701359e-17	0	-0.3574	0.722727	0.361363
M6	-1.26583126706111e-17	0	-0.1384	0.890627	0.445314
M7	7.21379251138558e-17	0	0.6876	0.495752	0.247876
M8	-3.62069740559273e-17	0	-0.2724	0.786738	0.393369
M9	-2.76317048100518e-17	0	-0.2365	0.814295	0.407148
M10	-3.29487196717181e-17	0	-0.3279	0.744737	0.372368
M11	-1.64934388509514e-17	0	-0.1703	0.865629	0.432815
t	-4.39099679542517e-18	0	-1.3506	0.184613	0.092306

Multiple Linear Regression - Regression Statistics
Multiple R	1
R-squared	1
Adjusted R-squared	1
F-TEST (value)	8.20700486715373e+31
F-TEST (DF numerator)	17
F-TEST (DF denominator)	39
p-value	0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation	1.32513466822904e-16
Sum Squared Residuals	6.84832936687566e-31

Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or Index	Actuals	Interpolation Forecast	Residuals Prediction Error
1	8.5	8.5	-7.6991384674347e-17
2	8.6	8.6	-6.43709360977525e-17
3	8.5	8.5	6.5756488250235e-16
4	8.2	8.2	-5.67866034633681e-17
5	8.1	8.1	-6.34876583350881e-18
6	7.9	7.9	-6.39936729524118e-17
7	8.6	8.6	-3.07938464271981e-17
8	8.7	8.7	-1.50779618795439e-16
9	8.7	8.7	-7.60896371306682e-17
10	8.5	8.5	-1.12544189552567e-17
11	8.4	8.4	-1.87973185793847e-17
12	8.5	8.5	-9.52569968939496e-18
13	8.7	8.7	-7.26912276957142e-17
14	8.7	8.7	-5.85614221220454e-17
15	8.6	8.6	-1.61756676361002e-16
16	8.5	8.5	2.42587675254507e-17
17	8.3	8.3	3.73154830395712e-17
18	8	8	1.31687079577835e-18
19	8.2	8.2	-3.86450271428722e-17
20	8.1	8.1	-2.49932245985543e-18
21	8.1	8.1	4.20887252772615e-17
22	8	8	2.63137569411794e-17
23	7.9	7.9	2.40516497712711e-17
24	7.9	7.9	1.65027753351442e-17
25	8	8	3.00235215224727e-17
26	8	8	4.87068947155409e-17
27	7.9	7.9	-1.56928722762741e-16
28	8	8	7.01377735925727e-17
29	7.7	7.7	2.37442644288036e-17
30	7.2	7.2	-2.09157096049553e-17
31	7.5	7.5	3.80917633257233e-17
32	7.3	7.3	8.34339014046975e-17
33	7	7	-3.98905873177544e-17
34	7	7	-6.79197201479108e-18
35	7	7	-5.68129043561564e-17
36	7.2	7.2	-4.54277053024157e-18
37	7.3	7.3	1.16970240403136e-16
38	7.1	7.1	1.00760206093542e-16
39	6.8	6.8	-1.52730786210754e-16
40	6.4	6.4	-4.72787562303053e-17
41	6.1	6.1	-9.03881407145042e-17
42	6.5	6.5	1.72786690122292e-17
43	7.7	7.7	2.64171238617347e-17
44	7.9	7.9	-6.47885859535384e-19
45	7.5	7.5	9.18365444334975e-17
46	6.9	6.9	-8.2673659711316e-18
47	6.6	6.6	5.15585731642701e-17
48	6.9	6.9	-2.43430511550802e-18
49	7.7	7.7	2.68885044445246e-18
50	8	8	-2.65347425892852e-17
51	8	8	-1.86148697167854e-16
52	7.7	7.7	9.6688185756501e-18
53	7.3	7.3	3.56771590796384e-17
54	7.4	7.4	6.63138427493595e-17
55	8.1	8.1	4.92998638261233e-18
56	8.3	8.3	7.04929257101326e-17
57	8.2	8.2	-1.79450452623363e-17

Goldfeld-Quandt test for Heteroskedasticity
p-values	Alternative Hypothesis
breakpoint index	greater	2-sided	less
21	0.158294841147041	0.316589682294082	0.841705158852959
22	0.472545462453909	0.945090924907818	0.527454537546091
23	0.0113339049571471	0.0226678099142943	0.988666095042853
24	0.00327250994365368	0.00654501988730735	0.996727490056346
25	3.15455655968536e-05	6.30911311937072e-05	0.999968454434403
26	0.800661376100752	0.398677247798496	0.199338623899248
27	4.1744642165599e-13	8.3489284331198e-13	0.999999999999583
28	8.48769212319241e-06	1.69753842463848e-05	0.999991512307877
29	0.000103362618003300	0.000206725236006601	0.999896637381997
30	0	0	1
31	3.04215469987899e-20	6.08430939975798e-20	1
32	0.887069516552428	0.225860966895143	0.112930483447572
33	3.59512425445158e-05	7.19024850890317e-05	0.999964048757455
34	0.980176438260019	0.0396471234799623	0.0198235617399812
35	0.0192602136189009	0.0385204272378017	0.9807397863811
36	0.000106107607578102	0.000212215215156204	0.999893892392422

Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description	# significant tests	% significant tests	OK/NOK
1% type I error level	9	0.5625	NOK
5% type I error level	12	0.75	NOK
10% type I error level	12	0.75	NOK

Charts produced by software:

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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')
}