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WS 8 - 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: Mon, 29 Nov 2010 20:22:31 +0000
 
Cite this page as follows:
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9.htm/, Retrieved Mon, 29 Nov 2010 21:20:46 +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/2010/Nov/29/t1291062036anfpwg4qknq5hj9.htm/},
    year = {2010},
}
@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 = {2010},
    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 «
1576.23 1546.37 1545.05 1552.34 1594.3 1605.78 1673.21 1612.94 1566.34 1530.17 1582.54 1702.16 1701.93 1811.15 1924.2 2034.25 2011.13 2013.04 2151.67 1902.09 1944.01 1916.67 1967.31 2119.88 2216.38 2522.83 2647.64 2631.23 2693.41 3021.76 2953.67 2796.8 2672.05 2251.23 2046.08 2420.04 2608.89 2660.47 2493.98 2541.7 2554.6 2699.61 2805.48 2956.66 3149.51 3372.5 3379.33 3517.54 3527.34 3281.06 3089.65 3222.76 3165.76 3232.43 3229.54 3071.74 2850.17
 
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 time7 seconds
R Server'George Udny Yule' @ 72.249.76.132


Multiple Linear Regression - Estimated Regression Equation
PrijsCacao[t] = + 1384.86440476191 + 62.0890992063496M1[t] + 65.1430793650795M2[t] + 5.70305952380944M3[t] + 26.8870396825396M4[t] -0.89698015873049M5[t] + 74.6190000000001M6[t] + 87.6409801587303M7[t] -42.1950396825399M8[t] -108.993059523809M9[t] -101.926460317460M10[t] -160.92198015873M11[t] + 35.1680198412698t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)1384.86440476191141.2562269.803900
M162.0890992063496170.6330790.36390.7176940.358847
M265.1430793650795170.5206030.3820.7042820.352141
M35.70305952380944170.433070.03350.9734570.486729
M426.8870396825396170.3705190.15780.8753250.437662
M5-0.89698015873049170.332977-0.00530.9958220.497911
M674.6190000000001170.3204610.43810.663450.331725
M787.6409801587303170.3329770.51450.6094580.304729
M8-42.1950396825399170.370519-0.24770.8055450.402772
M9-108.993059523809170.43307-0.63950.5258090.262905
M10-101.926460317460179.581019-0.56760.5732060.286603
M11-160.92198015873179.545404-0.89630.3749840.187492
t35.16801984126982.06483217.031900


Multiple Linear Regression - Regression Statistics
Multiple R0.933280547277717
R-squared0.871012579926995
Adjusted R-squared0.835834192634357
F-TEST (value)24.7598780660784
F-TEST (DF numerator)12
F-TEST (DF denominator)44
p-value1.11022302462516e-15
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation253.898753064943
Sum Squared Residuals2836441.37954905


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
11576.231482.1215238095294.1084761904785
21546.371520.3435238095226.0264761904762
31545.051496.0715238095248.9784761904758
41552.341552.42352380952-0.0835238095243298
51594.31559.8075238095234.4924761904758
61605.781670.49152380952-64.7115238095235
71673.211718.68152380952-45.471523809524
81612.941624.01352380952-11.0735238095240
91566.341592.38352380952-26.0435238095238
101530.171634.61814285714-104.448142857143
111582.541610.79064285714-28.2506428571428
121702.161806.88064285714-104.720642857143
131701.931904.13776190476-202.207761904763
141811.151942.35976190476-131.209761904762
151924.21918.087761904766.11223809523815
162034.251974.4397619047659.8102380952381
172011.131981.8237619047629.3062380952381
182013.042092.50776190476-79.4677619047621
192151.672140.6977619047610.9722380952381
201902.092046.02976190476-143.939761904762
211944.012014.39976190476-70.3897619047622
221916.672056.63438095238-139.964380952381
231967.312032.80688095238-65.496880952381
242119.882228.89688095238-109.016880952381
252216.382326.154-109.774000000000
262522.832364.376158.454
272647.642340.104307.536
282631.232396.456234.774
292693.412403.84289.57
303021.762514.524507.236
312953.672562.714390.956
322796.82468.046328.754000000000
332672.052436.416235.634
342251.232478.65061904762-227.420619047619
352046.082454.82311904762-408.743119047619
362420.042650.91311904762-230.873119047619
372608.892748.17023809524-139.280238095239
382660.472786.39223809524-125.922238095238
392493.982762.12023809524-268.140238095238
402541.72818.47223809524-276.772238095238
412554.62825.85623809524-271.256238095238
422699.612936.54023809524-236.930238095238
432805.482984.73023809524-179.250238095238
442956.662890.0622380952466.597761904762
453149.512858.43223809524291.077761904762
463372.52900.66685714286471.833142857143
473379.332876.83935714286502.490642857143
483517.543072.92935714286444.610642857143
493527.343170.18647619048357.153523809523
503281.063208.4084761904872.6515238095239
513089.653184.13647619048-94.486476190476
523222.763240.48847619048-17.7284761904757
533165.763247.87247619048-82.1124761904755
543232.433358.55647619048-126.126476190476
553229.543406.74647619048-177.206476190476
563071.743312.07847619048-240.338476190477
572850.173280.44847619048-430.278476190476


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
160.05344086284638130.1068817256927630.946559137153619
170.01807305561491910.03614611122983830.98192694438508
180.005398979118823250.01079795823764650.994601020881177
190.002070348746216680.004140697492433360.997929651253783
200.0006208274845179410.001241654969035880.999379172515482
210.0001497683771258420.0002995367542516830.999850231622874
223.77219869932395e-057.5443973986479e-050.999962278013007
238.29500279427866e-061.65900055885573e-050.999991704997206
242.14612819336265e-064.29225638672529e-060.999997853871807
255.16074589248048e-071.03214917849610e-060.99999948392541
263.73500236851257e-067.47000473702514e-060.999996264997632
271.54093603165815e-053.0818720633163e-050.999984590639683
281.04637407524590e-052.09274815049180e-050.999989536259247
291.02893830125257e-052.05787660250513e-050.999989710616987
300.0003802959716277250.000760591943255450.999619704028372
310.0008737362432837370.001747472486567470.999126263756716
320.00161613931656660.00323227863313320.998383860683433
330.002265659004345710.004531318008691410.997734340995654
340.003490416946754530.006980833893509050.996509583053246
350.03907299833088260.07814599666176520.960927001669117
360.083767447418040.1675348948360800.91623255258196
370.1261502562450020.2523005124900050.873849743754998
380.108377791455570.216755582911140.89162220854443
390.1279562329549750.255912465909950.872043767045025
400.1661099920620360.3322199841240710.833890007937964
410.1999796768214160.3999593536428320.800020323178584


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level160.615384615384615NOK
5% type I error level180.692307692307692NOK
10% type I error level190.730769230769231NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/102tna1291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/102tna1291062143.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/1waqg1291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/1waqg1291062143.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/2waqg1291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/2waqg1291062143.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/3okpj1291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/3okpj1291062143.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/4okpj1291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/4okpj1291062143.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/5okpj1291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/5okpj1291062143.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/6hb641291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/6hb641291062143.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/7sk5p1291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/7sk5p1291062143.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/8sk5p1291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/8sk5p1291062143.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/9sk5p1291062143.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/29/t1291062036anfpwg4qknq5hj9/9sk5p1291062143.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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