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W7: Linear Trend

*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: Sat, 21 Nov 2009 07:22:35 -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/21/t12588134670coiglzelq0oaut.htm/, Retrieved Sat, 21 Nov 2009 15:24:39 +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/21/t12588134670coiglzelq0oaut.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:
cvm
 
Dataseries X:
» Textbox « » Textfile « » CSV «
6,3 2 6,2 1,8 6,1 2,7 6,3 2,3 6,5 1,9 6,6 2 6,5 2,3 6,2 2,8 6,2 2,4 5,9 2,3 6,1 2,7 6,1 2,7 6,1 2,9 6,1 3 6,1 2,2 6,4 2,3 6,7 2,8 6,9 2,8 7 2,8 7 2,2 6,8 2,6 6,4 2,8 5,9 2,5 5,5 2,4 5,5 2,3 5,6 1,9 5,8 1,7 5,9 2 6,1 2,1 6,1 1,7 6 1,8 6 1,8 5,9 1,8 5,5 1,3 5,6 1,3 5,4 1,3 5,2 1,2 5,2 1,4 5,2 2,2 5,5 2,9 5,8 3,1 5,8 3,5 5,5 3,6 5,3 4,4 5,1 4,1 5,2 5,1 5,8 5,8 5,8 5,9 5,5 5,4 5 5,5 4,9 4,8 5,3 3,2 6,1 2,7 6,5 2,1 6,8 1,9 6,6 0,6 6,4 0,7 6,4 -0,2 6,6 -1 6,7 -1,7
 
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
WMan>25[t] = + 6.59619621729017 -0.1480526479753Infl[t] -0.202067711946028M1[t] -0.297369689987745M2[t] -0.286749562110452M3[t] -0.042778910868714M4[t] + 0.324880164049072M5[t] + 0.460695027128835M6[t] + 0.460198313884645M7[t] + 0.313052124004902M8[t] + 0.177750145963182M9[t] -0.0205128850380429M10[t] + 0.110107242839249M11[t] -0.0106201278772925t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)6.596196217290170.23613627.933800
Infl-0.14805264797530.039238-3.77320.000460.00023
M1-0.2020677119460280.271928-0.74310.4612040.230602
M2-0.2973696899877450.271368-1.09580.2788620.139431
M3-0.2867495621104520.270993-1.05810.2955150.147757
M4-0.0427789108687140.27011-0.15840.8748540.437427
M50.3248801640490720.2697641.20430.2346290.117315
M60.4606950271288350.2692961.71070.0938670.046933
M70.4601983138846450.2691971.70950.0940930.047046
M80.3130521240049020.2688021.16460.2501780.125089
M90.1777501459631820.268610.66170.5114390.25572
M10-0.02051288503804290.268449-0.07640.9394220.469711
M110.1101072428392490.2683890.41030.6835250.341763
t-0.01062012787729250.003229-3.28880.0019340.000967


Multiple Linear Regression - Regression Statistics
Multiple R0.714346509058098
R-squared0.510290935003491
Adjusted R-squared0.371894894895783
F-TEST (value)3.68717872712506
F-TEST (DF numerator)13
F-TEST (DF denominator)46
p-value0.000506791225271641
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.424236906606732
Sum Squared Residuals8.27893983465347


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
16.36.087403081516260.212596918483744
26.26.01109150519230.188908494807704
36.15.877844122014530.222155877985473
46.36.170415704569090.129584295430907
56.56.58667571079971-0.086675710799707
66.66.69706518120465-0.0970651812046475
76.56.64153254569058-0.141532545690575
86.26.40973990394589-0.209739903945889
96.26.323038857217-0.123038857216996
105.96.12896096313601-0.228960963136009
116.16.18973990394589-0.0897399039458891
126.16.069012533229350.0309874667706529
136.15.826714163810970.273285836189033
146.15.705986793094430.394013206905573
156.15.824428911474670.275571088525333
166.46.042974170041580.357025829958418
176.76.325986793094430.374013206905574
186.96.45118152829690.448818471703104
1976.440064687175410.559935312824586
2076.371129958203560.628870041796442
216.86.165986793094430.634013206905573
226.45.927493104620850.472506895379152
235.96.09190889901344-0.191908899013438
245.55.98598679309443-0.485986793094426
255.55.78810421806864-0.288104218068636
265.65.74140317133975-0.141403171339746
275.85.77101370093480.0289862990651938
285.95.95994842990666-0.0599484299066617
296.16.30218211214963-0.202182112149626
306.16.48659790654222-0.386597906542216
3166.4606758006232-0.460675800623204
3266.30290948286617-0.302909482866168
335.96.15698737694716-0.256987376947156
345.56.02213054205629-0.522130542056288
355.66.14213054205629-0.542130542056288
365.46.02140317133975-0.621403171339746
375.25.82352059631396-0.623520596313956
385.25.68798796079989-0.487987960799885
395.25.56954584241965-0.369545842419645
405.55.69925951220138-0.199259512201382
415.86.02668792964682-0.226687929646815
425.86.09266160565917-0.292661605659166
435.56.06673949974015-0.566739499740154
445.35.79053106360288-0.490531063602878
455.15.68902475207646-0.589024752076456
465.25.33208894522264-0.132088945222638
475.85.348452091639930.451547908360072
485.85.212919456125860.587080543874144
495.55.074257940290190.425742059709814
5054.953530569573640.0464694304263549
514.95.05716742315636-0.157167423156355
525.35.52740218328128-0.227402183281281
536.15.958467454309420.141532545690575
546.56.172493778297080.327506221702925
556.86.190987466770650.609012533229347
566.66.22568959138150.374310408618493
576.46.064962220664960.335037779335035
586.45.989326444964220.410673555035783
596.66.227768563344460.372231436655543
606.76.210678046210630.489321953789375


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
170.02282373858911840.04564747717823680.977176261410882
180.01708702242546270.03417404485092540.982912977574537
190.02637431782676070.05274863565352130.97362568217324
200.0509101102645360.1018202205290720.949089889735464
210.06809211845205170.1361842369041030.931907881547948
220.08809842621431420.1761968524286280.911901573785686
230.1284962830921880.2569925661843750.871503716907812
240.2745346803927680.5490693607855350.725465319607232
250.4453232167243250.890646433448650.554676783275675
260.456598198565320.913196397130640.54340180143468
270.493879952947830.987759905895660.50612004705217
280.5584831913780760.8830336172438490.441516808621924
290.5543322919225160.8913354161549670.445667708077484
300.5035710433742520.9928579132514960.496428956625748
310.4499633120954540.8999266241909080.550036687904546
320.4437584645269930.8875169290539860.556241535473007
330.551453535211460.897092929577080.44854646478854
340.4810078510651490.9620157021302970.518992148934851
350.3949981336226090.7899962672452180.605001866377391
360.3336887483819360.6673774967638710.666311251618064
370.2960740914950320.5921481829900630.703925908504968
380.2146232834525520.4292465669051050.785376716547448
390.228983597637090.457967195274180.77101640236291
400.5084497669397610.9831004661204780.491550233060239
410.7872277275421070.4255445449157870.212772272457893
420.986091182244820.02781763551035860.0139088177551793
430.9807423588543640.03851528229127120.0192576411456356


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level40.148148148148148NOK
10% type I error level50.185185185185185NOK
 
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')
}
 





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