*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, 19 Nov 2009 09:59:59 -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/19/t125865009015f62sevb7kejvp.htm/, Retrieved Thu, 19 Nov 2009 18:01:42 +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/19/t125865009015f62sevb7kejvp.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:

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No (this computation is public)

User-defined keywords:

Dataseries X:

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114 106.3 113.8 107.2 113.6 107.8 113.7 109.2 114.2 109.7 114.8 108.7 115.2 109.3 115.3 110.4 114.9 111.1 115.1 110.1 116 109.5 116 109 116 108.5 115.9 108.8 115.6 109.8 116.6 110.7 116.9 110.6 117.9 111.2 117.9 112 117.7 111.1 117.4 111.6 117.3 110.2 119 111.5 119.1 110.6 119 110.6 118.5 110.3 117 111.7 117.5 113.8 118.2 113.9 118.2 114.3 118.3 113.8 118.2 114.3 117.9 116.4 117.8 115.6 118.6 115.2 118.9 113.6 120.8 115.5 121.8 115.6 121.3 115.3 121.9 117.3 122 118.7 121.9 118.3 122 120.6 122.2 119.3 123 121.8 123.1 120.8 124.9 121.6 125.4 121.6 124.7 121.1 124.4 122.4 124 121.9 125 125.1 125.1 124.5 125.4 123.5 125.7 124.9 126.4 125.2 125.7 125.7 125.4 124.5 126.4 124.7 126.2 122.9

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation x[t] = + 88.8882218596472 + 0.227427671589674y[t] + 0.148419980617825t + 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) 88.8882218596472 7.612749 11.6762 0 0 y 0.227427671589674 0.072136 3.1528 0.002579 0.00129 t 0.148419980617825 0.023891 6.2125 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.975357223639291 R-squared 0.951321713705346 Adjusted R-squared 0.94961370365992 F-TEST (value) 556.976650256076 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.891308699612329 Sum Squared Residuals 45.2825782862634 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 114 113.212203330247 0.787796669752694 2 113.8 113.565308215296 0.234691784704195 3 113.6 113.850184798867 -0.250184798867435 4 113.7 114.317003519711 -0.617003519710796 5 114.2 114.579137336123 -0.379137336123458 6 114.8 114.500129645152 0.299870354848385 7 115.2 114.785006228723 0.414993771276764 8 115.3 115.183596648090 0.116403351910290 9 114.9 115.491215998820 -0.591215998820295 10 115.1 115.412208307848 -0.312208307848457 11 116 115.424171685512 0.575828314487526 12 116 115.458877830335 0.541122169664538 13 116 115.493583975158 0.50641602484155 14 115.9 115.710232257253 0.189767742746829 15 115.6 116.086079909461 -0.48607990946068 16 116.6 116.439184794509 0.160815205490788 17 116.9 116.564862007968 0.335137992031943 18 117.9 116.849738591540 1.05026140846031 19 117.9 117.180100709429 0.71989929057075 20 117.7 117.123835785616 0.57616421438363 21 117.4 117.385969602029 0.0140303979709707 22 117.3 117.215990842421 0.084009157578678 23 119 117.660066796106 1.33993320389428 24 119.1 117.603801872293 1.49619812770716 25 119 117.752221852911 1.24777814708934 26 118.5 117.832413532052 0.667586467948416 27 117 118.299232252895 -1.29923225289495 28 117.5 118.925250343851 -1.42525034385109 29 118.2 119.096413091628 -0.896413091627882 30 118.2 119.335804140882 -1.13580414088157 31 118.3 119.370510285705 -1.07051028570457 32 118.2 119.632644102117 -1.43264410211722 33 117.9 120.258662193073 -2.35866219307336 34 117.8 120.225140036419 -2.42514003641945 35 118.6 120.282588948401 -1.68258894840141 36 118.9 120.067124654476 -1.16712465447575 37 120.8 120.647657211114 0.152342788886038 38 121.8 120.818819958891 0.981180041109247 39 121.3 120.899011638032 0.400988361968324 40 121.9 121.502286961829 0.39771303817116 41 122 121.969105682672 0.0308943173277856 42 121.9 122.026554594654 -0.126554594654163 43 122 122.698058219928 -0.698058219928242 44 122.2 122.550822227479 -0.350822227479489 45 123 123.267811387072 -0.2678113870715 46 123.1 123.188803696100 -0.0888036960996574 47 124.9 123.519165813989 1.38083418601079 48 125.4 123.667585794607 1.73241420539297 49 124.7 123.70229193943 0.997708060569976 50 124.4 124.146367893114 0.253632106885575 51 124 124.181074037937 -0.181074037937418 52 125 125.057262567642 -0.0572625676421956 53 125.1 125.069225945306 0.0307740546937768 54 125.4 124.990218254334 0.409781745665636 55 125.7 125.457036975178 0.242963024822265 56 126.4 125.673685257272 0.726314742727542 57 125.7 125.935819073685 -0.235819073685123 58 125.4 125.811325848395 -0.411325848395336 59 126.4 126.005231363331 0.394768636668904 60 126.2 125.744281535088 0.455718464912489 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.0354089622041121 0.0708179244082242 0.964591037795888 7 0.0109046933700085 0.0218093867400170 0.989095306629991 8 0.00360181131151107 0.00720362262302214 0.996398188688489 9 0.00176550418656520 0.00353100837313041 0.998234495813435 10 0.00422130689434465 0.0084426137886893 0.995778693105655 11 0.00136487538194401 0.00272975076388801 0.998635124618056 12 0.000611381225310139 0.00122276245062028 0.99938861877469 13 0.000384997523926913 0.000769995047853826 0.999615002476073 14 0.000301083126807713 0.000602166253615426 0.999698916873192 15 0.000550141760063188 0.00110028352012638 0.999449858239937 16 0.000225267153680344 0.000450534307360688 0.99977473284632 17 9.78705086959258e-05 0.000195741017391852 0.999902129491304 18 0.000445884269612252 0.000891768539224503 0.999554115730388 19 0.000433761498973297 0.000867522997946594 0.999566238501027 20 0.000225804504298313 0.000451609008596626 0.999774195495702 21 0.000147539054593082 0.000295078109186165 0.999852460945407 22 0.000133414984829712 0.000266829969659424 0.99986658501517 23 0.000561787925515613 0.00112357585103123 0.999438212074484 24 0.00171558822989977 0.00343117645979953 0.9982844117701 25 0.00349652925956023 0.00699305851912046 0.99650347074044 26 0.00638794436416176 0.0127758887283235 0.993612055635838 27 0.102036424197103 0.204072848394205 0.897963575802897 28 0.164002697008041 0.328005394016082 0.835997302991959 29 0.14137946925893 0.28275893851786 0.85862053074107 30 0.118641173754538 0.237282347509076 0.881358826245462 31 0.101293723993982 0.202587447987965 0.898706276006017 32 0.0961715051165223 0.192343010233045 0.903828494883478 33 0.144300059616738 0.288600119233477 0.855699940383262 34 0.373516556824766 0.747033113649533 0.626483443175234 35 0.546507462997124 0.906985074005753 0.453492537002876 36 0.73508833102953 0.529823337940941 0.264911668970471 37 0.775352497321146 0.449295005357708 0.224647502678854 38 0.882032895070942 0.235934209858117 0.117967104929058 39 0.864654472323194 0.270691055353611 0.135345527676805 40 0.881640321327847 0.236719357344306 0.118359678672153 41 0.877114641966813 0.245770716066375 0.122885358033187 42 0.853408434878448 0.293183130243103 0.146591565121552 43 0.878288023928744 0.243423952142512 0.121711976071256 44 0.908352361077403 0.183295277845193 0.0916476389225966 45 0.927091408932829 0.145817182134343 0.0729085910671714 46 0.963275139966122 0.0734497200677562 0.0367248600338781 47 0.968105351175462 0.0637892976490766 0.0318946488245383 48 0.99427194774829 0.0114561045034211 0.00572805225171056 49 0.995843732827945 0.00831253434411072 0.00415626717205536 50 0.990080876938869 0.0198382461222627 0.00991912306113133 51 0.98232944733957 0.0353411053208621 0.0176705526604311 52 0.957769006583817 0.0844619868323663 0.0422309934161832 53 0.911962243378874 0.176075513242252 0.088037756621126 54 0.81188939864559 0.376221202708820 0.188110601354410 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 19 0.387755102040816 NOK 5% type I error level 24 0.489795918367347 NOK 10% type I error level 28 0.571428571428571 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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