Multiple Linear 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: Thu, 27 Nov 2008 03:30:18 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Nov/27/t1227782057n4p8u2asoypl3cw.htm/, Retrieved Thu, 27 Nov 2008 10:34:18 +0000

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/2008/Nov/27/t1227782057n4p8u2asoypl3cw.htm/}, year = {2008}, } @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 = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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Dataseries X:

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4,25 101,8 0 4,5 108,3 0 4,7 106,7 0 4,75 108,2 0 4,75 94,2 0 4,75 95,1 0 4,75 98,1 0 4,75 93,2 0 4,75 94 0 4,58 97,2 0 4,5 95 0 4,5 90,5 0 4,49 91,6 0 4,03 90,5 0 3,75 79,9 0 3,39 74,9 0 3,25 74,3 0 3,25 75,9 1 3,25 77,7 1 3,25 86,9 1 3,25 90,7 1 3,25 91 1 3,25 89,5 1 3,25 92,5 1 3,25 94,1 1 3,25 98,5 1 3,25 96,8 1 3,25 91,2 1 2,85 97,1 1 2,75 104,9 1 2,75 110,9 1 2,55 104,8 1 2,5 94,1 1 2,5 95,8 1 2,1 99,3 1 2 101,1 1 2 104 1 2 99 1 2 105,4 1 2 107,1 1 2 110,7 1 2 117,1 1 2 118,7 1 2 126,5 1 2 127,5 1 2 134,6 1 2 131,8 1 2 135,9 1 2 142,7 1 2 141,7 1 2 153,4 1 2 145 1 2 137,7 1 2 148,3 1 2 152,2 1 2 169,4 1 2 168,6 1 2 161,1 1 2 174,1 1 2 179 1 2 190,6 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 Input view raw input (R code) Raw Output view raw output of R engine Computing time 7 seconds R Server 'Herman Ole Andreas Wold' @ 193.190.124.10:1001 Multiple Linear Regression - Estimated Regression Equation rentetarief[t] = + 3.69690495663196 + 0.0144572867995880grondstofprijs[t] -0.327115197475534dummy[t] -0.145460946079961M1[t] -0.154487406440737M2[t] -0.115487200762331M3[t] -0.0646578478855724M4[t] -0.0696594500325334M5[t] -0.0360288698731171M6[t] -0.0160152982497135M7[t] -0.0559527824097417M8[t] + 0.0182511426038327M9[t] + 0.0375164738662887M10[t] -0.0202537731428269M11[t] -0.0671443265900604t + 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) 3.69690495663196 0.250192 14.7763 0 0 grondstofprijs 0.0144572867995880 0.002741 5.2745 3e-06 2e-06 dummy -0.327115197475534 0.156742 -2.087 0.042461 0.021231 M1 -0.145460946079961 0.16072 -0.9051 0.370152 0.185076 M2 -0.154487406440737 0.167684 -0.9213 0.3617 0.18085 M3 -0.115487200762331 0.167488 -0.6895 0.493957 0.246978 M4 -0.0646578478855724 0.168063 -0.3847 0.702215 0.351108 M5 -0.0696594500325334 0.169563 -0.4108 0.683114 0.341557 M6 -0.0360288698731171 0.16757 -0.215 0.830711 0.415356 M7 -0.0160152982497135 0.167322 -0.0957 0.924163 0.462081 M8 -0.0559527824097417 0.167431 -0.3342 0.73976 0.36988 M9 0.0182511426038327 0.166831 0.1094 0.913362 0.456681 M10 0.0375164738662887 0.166657 0.2251 0.822889 0.411445 M11 -0.0202537731428269 0.166571 -0.1216 0.903752 0.451876 t -0.0671443265900604 0.006309 -10.6422 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.97490116193405 R-squared 0.95043227554036 Adjusted R-squared 0.935346446356991 F-TEST (value) 63.001659636194 F-TEST (DF numerator) 14 F-TEST (DF denominator) 46 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.263326786223516 Sum Squared Residuals 3.18968583176904 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 4.25 4.95605148015999 -0.706051480159995 2 4.5 4.97385305740648 -0.473853057406483 3 4.7 4.92257727761549 -0.222577277615487 4 4.75 4.92794823410157 -0.177948234101568 5 4.75 4.65340029017031 0.0965997098296858 6 4.75 4.6328981018593 0.117101898140701 7 4.75 4.62913920729141 0.120860792708594 8 4.75 4.45121669122334 0.298783308776663 9 4.75 4.46984211908652 0.280157880913479 10 4.58 4.4682264415176 0.111773558482402 11 4.5 4.31150583695933 0.188494163040672 12 4.5 4.19955749291395 0.300442507086052 13 4.49 4.00285523572347 0.487144764276526 14 4.03 3.91078143329309 0.11921856670691 15 3.75 3.7293900723058 0.0206099276941964 16 3.39 3.64078866459456 -0.250788664594561 17 3.25 3.55996836377779 -0.309968363777787 18 3.25 3.22247107875095 0.0275289212490499 19 3.25 3.20136344002355 0.0486365599764484 20 3.25 3.22728866782967 0.0227113321703271 21 3.25 3.28928595609162 -0.0392859560916213 22 3.25 3.24574414680389 0.00425585319610673 23 3.25 3.09914364300533 0.150856356994665 24 3.25 3.09562494995687 0.154375050043134 25 3.25 2.90615133616618 0.343848663833815 26 3.25 2.89359261113354 0.356407388866465 27 3.25 2.84087110266258 0.409128897337418 28 3.25 2.74359532287159 0.506404677128413 29 2.85 2.75674738625213 0.0932526137478654 30 2.75 2.83600047685828 -0.0860004768582776 31 2.75 2.87561344268915 -0.125613442689149 32 2.55 2.68034218246157 -0.130342182461573 33 2.5 2.53270881212950 -0.0327088121294952 34 2.5 2.50940720436119 -0.00940720436119051 35 2.1 2.43509313456057 -0.335093134560572 36 2 2.41422569735260 -0.414225697352597 37 2 2.24354655640138 -0.243546556401381 38 2 2.09508933545260 -0.0950893354526038 39 2 2.15947185005831 -0.159471850058313 40 2 2.16773426390431 -0.167734263904311 41 2 2.14763456764581 -0.147634567645806 42 2 2.20664745673253 -0.206647456732526 43 2 2.18264836064521 -0.18264836064521 44 2 2.18833338693191 -0.188333386931908 45 2 2.20985027215501 -0.20985027215501 46 2 2.26461801310448 -0.264618013104481 47 2 2.09922303646646 -0.0992230364664581 48 2 2.11160735889754 -0.111607358897535 49 2 1.99731163646471 0.00268836353528802 50 2 1.90668356271429 0.0933164372857127 51 2 2.04768969735781 -0.0476896973578137 52 2 1.90993351452797 0.0900664854720276 53 2 1.73224939215396 0.267750607846042 54 2 1.85198288579895 0.148017114201053 55 2 1.86123554935068 0.138764450649317 56 2 2.00281907155351 -0.00281907155350926 57 2 1.99831284053735 0.00168715946264738 58 2 1.84200419421284 0.157995805787162 59 2 1.90503434900831 0.0949656509916937 60 2 1.92898450087905 0.0710154991209456 61 2 1.88408375508425 0.115916244915746 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.98432743611728 0.0313451277654382 0.0156725638827191 19 0.962984823535815 0.0740303529283695 0.0370151764641848 20 0.978399040878588 0.0432019182428246 0.0216009591214123 21 0.976892632688376 0.0462147346232489 0.0231073673116245 22 0.960115895260275 0.07976820947945 0.039884104739725 23 0.930443810066572 0.139112379866857 0.0695561899334283 24 0.904064564783522 0.191870870432955 0.0959354352164775 25 0.89959121848289 0.200817563034222 0.100408781517111 26 0.8614135134697 0.277172973060601 0.138586486530301 27 0.905836037743154 0.188327924513691 0.0941639622568456 28 0.99041083704018 0.0191783259196388 0.00958916295981941 29 0.994330103700191 0.0113397925996173 0.00566989629980865 30 0.998992265672483 0.00201546865503333 0.00100773432751666 31 0.999025513184109 0.00194897363178286 0.000974486815891428 32 0.999465696172985 0.00106860765403078 0.000534303827015388 33 0.999937968604724 0.000124062790552424 6.20313952762122e-05 34 0.999999999997803 4.3938113323499e-12 2.19690566617495e-12 35 1 1.76776044251982e-183 8.83880221259911e-184 36 1 6.2491000182163e-153 3.12455000910815e-153 37 1 2.58579907074711e-132 1.29289953537356e-132 38 1 9.72098692463402e-120 4.86049346231701e-120 39 1 1.25376488372334e-102 6.2688244186167e-103 40 1 3.57791955418495e-102 1.78895977709247e-102 41 1 1.21122666211537e-78 6.05613331057687e-79 42 1 4.07200404225642e-59 2.03600202112821e-59 43 1 3.94586064374107e-44 1.97293032187053e-44 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 14 0.538461538461538 NOK 5% type I error level 19 0.73076923076923 NOK 10% type I error level 21 0.807692307692308 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') }

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