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Multuple 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: Tue, 17 Nov 2009 12:25:44 -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/17/t1258486210w5nwy33rgcglrld.htm/, Retrieved Tue, 17 Nov 2009 20:30:22 +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/17/t1258486210w5nwy33rgcglrld.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:
JSSHWWS7P1
 
Dataseries X:
» Textbox « » Textfile « » CSV «
8 11,1 8,1 10,9 7,7 10 7,5 9,2 7,6 9,2 7,8 9,5 7,8 9,6 7,8 9,5 7,5 9,1 7,5 8,9 7,1 9 7,5 10,1 7,5 10,3 7,6 10,2 7,7 9,6 7,7 9,2 7,9 9,3 8,1 9,4 8,2 9,4 8,2 9,2 8,2 9 7,9 9 7,3 9 6,9 9,8 6,6 10 6,7 9,8 6,9 9,3 7 9 7,1 9 7,2 9,1 7,1 9,1 6,9 9,1 7 9,2 6,8 8,8 6,4 8,3 6,7 8,4 6,6 8,1 6,4 7,7 6,3 7,9 6,2 7,9 6,5 8 6,8 7,9 6,8 7,6 6,4 7,1 6,1 6,8 5,8 6,5 6,1 6,9 7,2 8,2 7,3 8,7 6,9 8,3 6,1 7,9 5,8 7,5 6,2 7,8 7,1 8,3 7,7 8,4 7,9 8,2 7,7 7,7 7,4 7,2 7,5 7,3 8 8,1
 
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 time6 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
Y[t] = + 2.10224383044583 + 0.926017592780038X[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)2.102243830445831.1400871.84390.0703030.035151
X0.9260175927800380.1583185.849100


Multiple Linear Regression - Regression Statistics
Multiple R0.609110084951344
R-squared0.371015095589434
Adjusted R-squared0.36017052827201
F-TEST (value)34.2120699452285
F-TEST (DF numerator)1
F-TEST (DF denominator)58
p-value2.41376559961815e-07
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.799807543855793
Sum Squared Residuals37.1021422181009


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
111.19.510384572686121.58961542731388
210.99.602986331964141.29701366803586
3109.232579294852120.76742070514788
49.29.047375776296110.152624223703887
59.29.139977535574120.0600224644258833
69.59.325181054130120.174818945869876
79.69.325181054130120.274818945869876
89.59.325181054130120.174818945869876
99.19.047375776296110.052624223703887
108.99.04737577629611-0.147375776296112
1198.67696873918410.323031260815903
1210.19.047375776296111.05262422370389
1310.39.047375776296111.25262422370389
1410.29.139977535574121.06002246442588
159.69.232579294852120.367420705147879
169.29.23257929485212-0.032579294852121
179.39.41778281340813-0.117782813408127
189.49.60298633196414-0.202986331964135
199.49.69558809124214-0.295588091242138
209.29.69558809124214-0.495588091242139
2199.69558809124214-0.695588091242138
2299.41778281340813-0.417782813408128
2398.86217225774010.137827742259895
249.88.491765220628091.30823477937191
25108.213959942794081.78604005720592
269.88.306561702072081.49343829792792
279.38.491765220628090.80823477937191
2898.58436697990610.415633020093906
2998.67696873918410.323031260815903
309.18.76957049846210.330429501537898
319.18.67696873918410.423031260815903
329.18.491765220628090.608234779371909
339.28.58436697990610.615633020093906
348.88.399163461350090.400836538649915
358.38.028756424238070.271243575761929
368.48.306561702072080.0934382979279179
378.18.21395994279408-0.113959942794079
387.78.02875642423807-0.328756424238071
397.97.93615466496007-0.0361546649600666
407.97.843552905682060.0564470943179368
4188.12135818351607-0.121358183516075
427.98.39916346135009-0.499163461350086
437.68.39916346135009-0.799163461350086
447.18.02875642423807-0.928756424238071
456.87.75095114640406-0.95095114640406
466.57.47314586857005-0.973145868570048
476.97.75095114640406-0.850951146404059
488.28.7695704984621-0.569570498462102
498.78.8621722577401-0.162172257740106
508.38.49176522062809-0.191765220628090
517.97.750951146404060.149048853595941
527.57.473145868570050.0268541314299519
537.87.84355290568206-0.0435529056820637
548.38.6769687391841-0.376968739184096
558.49.23257929485212-0.83257929485212
568.29.41778281340813-1.21778281340813
577.79.23257929485212-1.53257929485212
587.28.95477401701811-1.75477401701811
597.39.04737577629611-1.74737577629611
608.19.51038457268613-1.41038457268613


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
50.06756043959725890.1351208791945180.932439560402741
60.09233880458839860.1846776091767970.907661195411601
70.06321583089633980.1264316617926800.93678416910366
80.04719524841154620.09439049682309240.952804751588454
90.02188244778434380.04376489556868760.978117552215656
100.009203016269270510.01840603253854100.99079698373073
110.03129435183339180.06258870366678360.968705648166608
120.0528159997560530.1056319995121060.947184000243947
130.1012586373331520.2025172746663050.898741362666848
140.1069524433008150.2139048866016300.893047556699185
150.0784469646668720.1568939293337440.921553035333128
160.0738529372178550.147705874435710.926147062782145
170.08580662865817030.1716132573163410.91419337134183
180.1012840564451670.2025681128903340.898715943554833
190.1042314252730940.2084628505461880.895768574726906
200.1055581541647680.2111163083295360.894441845835232
210.1087215283249880.2174430566499760.891278471675012
220.09473498013584720.1894699602716940.905265019864153
230.07439591052738140.1487918210547630.925604089472619
240.09353626762606840.1870725352521370.906463732373932
250.1708108049051660.3416216098103320.829189195094834
260.2556084357978390.5112168715956770.744391564202161
270.2836048176366680.5672096352733350.716395182363333
280.2973712591500130.5947425183000260.702628740849987
290.3079705158317220.6159410316634430.692029484168278
300.3250492871262670.6500985742525340.674950712873733
310.3628032543390270.7256065086780530.637196745660973
320.4413146899968470.8826293799936940.558685310003153
330.5770737952574780.8458524094850430.422926204742522
340.6715238061892830.6569523876214340.328476193810717
350.7267838600421010.5464322799157970.273216139957899
360.76907552137190.4618489572562020.230924478628101
370.7892306489430670.4215387021138660.210769351056933
380.7999282707524870.4001434584950270.200071729247513
390.7886315147793210.4227369704413570.211368485220679
400.7757117391252870.4485765217494260.224288260874713
410.7649098537659430.4701802924681130.235090146234057
420.7477978755537830.5044042488924340.252202124446217
430.7402675410302020.5194649179395960.259732458969798
440.7548646318848650.490270736230270.245135368115135
450.7857897300740870.4284205398518250.214210269925913
460.8749092437691250.250181512461750.125090756230875
470.9230649974602270.1538700050795470.0769350025397734
480.8932986139887920.2134027720224160.106701386011208
490.9267941247375060.1464117505249870.0732058752624936
500.9139773791757110.1720452416485780.0860226208242888
510.8650693662812450.2698612674375110.134930633718756
520.7816001782716890.4367996434566220.218399821728311
530.6750954403386360.6498091193227290.324904559661364
540.8998084745925720.2003830508148560.100191525407428
550.992517013466220.01496597306756030.00748298653378016


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level30.0588235294117647NOK
10% type I error level50.0980392156862745OK
 
Charts produced by software:
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http://www.freestatistics.org/blog/date/2009/Nov/17/t1258486210w5nwy33rgcglrld/9u6mn1258485936.png (open in new window)
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Parameters (Session):
par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;
 
Parameters (R input):
par1 = 2 ; 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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