CVM Paper: Multiple Linear Regression (gewoon)

*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, 17 Dec 2009 04:25:28 -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/Dec/17/t1261049189g6m3zi8l04aejh6.htm/, Retrieved Thu, 17 Dec 2009 12:26:42 +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/Dec/17/t1261049189g6m3zi8l04aejh6.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:

Dataseries X:

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25,6 7,4 1,8 23,7 7,1 2,7 22 6,8 2,3 21,3 6,9 1,9 20,7 7,2 2 20,4 7,4 2,3 20,3 7,3 2,8 20,4 6,9 2,4 19,8 6,9 2,3 19,5 6,8 2,7 23,1 7,1 2,7 23,5 7,2 2,9 23,5 7,1 3 22,9 7 2,2 21,9 6,9 2,3 21,5 7,1 2,8 20,5 7,3 2,8 20,2 7,5 2,8 19,4 7,5 2,2 19,2 7,5 2,6 18,8 7,3 2,8 18,8 7 2,5 22,6 6,7 2,4 23,3 6,5 2,3 23 6,5 1,9 21,4 6,5 1,7 19,9 6,6 2 18,8 6,8 2,1 18,6 6,9 1,7 18,4 6,9 1,8 18,6 6,8 1,8 19,9 6,8 1,8 19,2 6,5 1,3 18,4 6,1 1,3 21,1 6,1 1,3 20,5 5,9 1,2 19,1 5,7 1,4 18,1 5,9 2,2 17 5,9 2,9 17,1 6,1 3,1 17,4 6,3 3,5 16,8 6,2 3,6 15,3 5,9 4,4 14,3 5,7 4,1 13,4 5,4 5,1 15,3 5,6 5,8 22,1 6,2 5,9 23,7 6,3 5,4 22,2 6 5,5 19,5 5,6 4,8 16,6 5,5 3,2 17,3 5,9 2,7 19,8 6,5 2,1 21,2 6,8 1,9 21,5 6,8 0,6 20,6 6,5 0,7 19,1 6,2 -0,2 19,6 6,2 -1 23,5 6,5 -1,7 24 6,7 -0,7

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 Input view raw input (R code) Raw Output view raw output of R engine Computing time 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation W<25j[t] = + 6.48209224673715 + 2.17628821832016`W>25j`[t] -0.302658180265085Inflatie[t] + 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) 6.48209224673715 3.536478 1.8329 0.072038 0.036019 `W>25j` 2.17628821832016 0.512366 4.2475 8.1e-05 4e-05 Inflatie -0.302658180265085 0.195452 -1.5485 0.127036 0.063518 Multiple Linear Regression - Regression Statistics Multiple R 0.556061678884002 R-squared 0.309204590723295 Adjusted R-squared 0.284966155310077 F-TEST (value) 12.7567883591479 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 2.63911988551691e-05 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 2.14500650880961 Sum Squared Residuals 262.260016601628 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 25.6 22.0418403378292 3.55815966217077 2 23.7 21.1165615100946 2.58343848990541 3 22 20.5847383167046 1.41526168329543 4 21.3 20.9234304106426 0.37656958935738 5 20.7 21.5460510581122 -0.846051058112162 6 20.4 21.8905112476967 -1.49051124769667 7 20.3 21.5215533357321 -1.22155333573211 8 20.4 20.7721013205101 -0.372101320510080 9 19.8 20.8023671385366 -1.00236713853659 10 19.5 20.4636750445985 -0.963675044598535 11 23.1 21.1165615100946 1.98343848990542 12 23.5 21.2736586958736 2.22634130412641 13 23.5 21.0257640560151 2.47423594398494 14 22.9 21.0502617783951 1.84973822160489 15 21.9 20.8023671385366 1.09763286146341 16 21.5 21.0862956920681 0.413704307931924 17 20.5 21.5215533357321 -1.02155333573211 18 20.2 21.9568109793961 -1.75681097939614 19 19.4 22.1384058875552 -2.73840588755519 20 19.2 22.0173426154492 -2.81734261544916 21 18.8 21.5215533357321 -2.72155333573211 22 18.8 20.9594643243156 -2.15946432431558 23 22.6 20.3368436768460 2.26315632315396 24 23.3 19.9318518512085 3.36814814879148 25 23 20.0529151233146 2.94708487668545 26 21.4 20.1134467593676 1.28655324063243 27 19.9 20.2402781271201 -0.340278127120063 28 18.8 20.6452699527576 -1.84526995275758 29 18.6 20.9839620466956 -2.38396204669564 30 18.4 20.9536962286691 -2.55369622866913 31 18.6 20.7360674068371 -2.13606740683711 32 19.9 20.7360674068371 -0.836067406837113 33 19.2 20.2345100314736 -1.03451003147361 34 18.4 19.3639947441455 -0.96399474414554 35 21.1 19.3639947441455 1.73600525585446 36 20.5 18.9590029185080 1.54099708149198 37 19.1 18.4632136387910 0.636786361209036 38 18.1 18.6563447382429 -0.556344738242929 39 17 18.4444840120574 -1.44448401205737 40 17.1 18.8192100196684 -1.71921001966838 41 17.4 19.1334043912264 -1.73340439122639 42 16.8 18.8855097513679 -2.08550975136786 43 15.3 17.9904967416597 -2.69049674165974 44 14.3 17.6460365520752 -3.34603655207523 45 13.4 16.6904919063141 -3.2904919063141 46 15.3 16.9138888237926 -1.61388882379257 47 22.1 18.1893959367582 3.91060406324184 48 23.7 18.5583538487227 5.14164615127728 49 22.2 17.8752015652002 4.32479843479983 50 19.5 17.2165470040577 2.28345299594234 51 16.6 17.4831712706498 -0.883171270649777 52 17.3 18.5050156481104 -1.20501564811039 53 19.8 19.9923834872615 -0.192383487261536 54 21.2 20.7058015888106 0.494198411189396 55 21.5 21.0992572231552 0.400742776844786 56 20.6 20.4161049396327 0.183895060367345 57 19.1 20.0356108363752 -0.935610836375183 58 19.6 20.2777373805873 -0.677737380587251 59 23.5 21.1424845722689 2.35751542773114 60 24 21.2750840356678 2.72491596433219 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.738568476498445 0.52286304700311 0.261431523501555 7 0.647808478916879 0.704383042166243 0.352191521083121 8 0.529580995584778 0.940838008830444 0.470419004415222 9 0.448638093097259 0.897276186194517 0.551361906902741 10 0.336309249517422 0.672618499034844 0.663690750482578 11 0.336964248948503 0.673928497897006 0.663035751051497 12 0.32439780028426 0.64879560056852 0.67560219971574 13 0.309355263478211 0.618710526956422 0.690644736521789 14 0.269824620215346 0.539649240430692 0.730175379784654 15 0.205130204235269 0.410260408470537 0.794869795764731 16 0.148595450814443 0.297190901628886 0.851404549185557 17 0.135492083433873 0.270984166867747 0.864507916566126 18 0.132930457489568 0.265860914979136 0.867069542510432 19 0.157412876840685 0.314825753681369 0.842587123159315 20 0.165848252685423 0.331696505370846 0.834151747314577 21 0.189715808425585 0.37943161685117 0.810284191574415 22 0.223513227750634 0.447026455501267 0.776486772249366 23 0.186769871170587 0.373539742341174 0.813230128829413 24 0.183934788959314 0.367869577918629 0.816065211040686 25 0.168569153799962 0.337138307599924 0.831430846200038 26 0.141679621013023 0.283359242026047 0.858320378986977 27 0.131890284359623 0.263780568719245 0.868109715640377 28 0.15428241286324 0.30856482572648 0.84571758713676 29 0.186328042346873 0.372656084693746 0.813671957653127 30 0.237280231186236 0.474560462372472 0.762719768813764 31 0.280332901736235 0.560665803472471 0.719667098263765 32 0.258533166495206 0.517066332990413 0.741466833504794 33 0.230683486318700 0.461366972637399 0.7693165136813 34 0.201535777718629 0.403071555437257 0.798464222281371 35 0.177780823155805 0.35556164631161 0.822219176844195 36 0.166819365634578 0.333638731269157 0.833180634365422 37 0.169513209452244 0.339026418904489 0.830486790547756 38 0.162141114091693 0.324282228183386 0.837858885908307 39 0.167700652689719 0.335401305379437 0.832299347310281 40 0.165179535685007 0.330359071370014 0.834820464314993 41 0.185667876654985 0.371335753309971 0.814332123345014 42 0.224001996708077 0.448003993416154 0.775998003291923 43 0.270700891595440 0.541401783190881 0.729299108404559 44 0.337929037262894 0.675858074525788 0.662070962737106 45 0.407668522517907 0.815337045035815 0.592331477482093 46 0.540520122923509 0.918959754152982 0.459479877076491 47 0.612935370657437 0.774129258685127 0.387064629342563 48 0.738283606563806 0.523432786872387 0.261716393436194 49 0.88061133665047 0.238777326699061 0.119388663349530 50 0.978474312471665 0.0430513750566706 0.0215256875283353 51 0.979068880368768 0.0418622392624647 0.0209311196312323 52 0.986159244064315 0.0276815118713699 0.0138407559356850 53 0.981314030049233 0.0373719399015332 0.0186859699507666 54 0.947128399109084 0.105743201781831 0.0528716008909157 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 4 0.0816326530612245 NOK 10% type I error level 4 0.0816326530612245 OK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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