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: Fri, 18 Dec 2009 07:05:22 -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/18/t12611452363tmszsvh0fczb6u.htm/, Retrieved Fri, 18 Dec 2009 15:07:31 +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/18/t12611452363tmszsvh0fczb6u.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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100.0 114.1 141.7 100.0 93.5 110.3 153.4 117.8 88.2 103.9 145 95.7 89.2 101.6 137.7 100.5 91.4 94.6 148.3 105.1 92.5 95.9 152.2 116.2 91.4 104.7 169.4 125.3 88.2 102.8 168.6 130.2 87.1 98.1 161.1 137.1 84.9 113.9 174.1 136.3 92.5 80.9 179 107.8 93.5 95.7 190.6 118.1 93.5 113.2 190 119.5 91.4 105.9 181.6 124.1 90.3 108.8 174.8 114.0 91.4 102.3 180.5 132.2 93.5 99 196.8 160.0 93.5 100.7 193.8 124.6 92.5 115.5 197 138.7 91.4 100.7 216.3 105.1 89.2 109.9 221.4 132.3 86.0 114.6 217.9 118.4 88.2 85.4 229.7 114.2 87.1 100.5 227.4 106.7 87.1 114.8 204.2 110.7 86.0 116.5 196.6 115.3 84.9 112.9 198.8 95.7 84.9 102 207.5 106.0 86.0 106 190.7 109.3 86.0 105.3 201.6 105.9 84.9 118.8 210.5 118.5 86.0 106.1 223.5 107.5 82.8 109.3 223.8 102.4 77.4 117.2 231.2 126.7 80.6 92.5 244 112.0 78.5 104.2 234.7 99.5 75.3 112.5 250.2 88.3 75.3 122.4 265.7 118.0 75.3 113.3 287.6 96.2 77.4 100 283.3 96.0 78.5 110.7 295.4 117.5 76.3 112.8 312.3 113.5 73.1 109.8 333.8 101.9 68.8 117.3 347.7 130.1 etc...

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 20 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 R Framework error message The field 'Names of X columns' contains a hard return which cannot be interpreted. Please, resubmit your request without hard returns in the 'Names of X columns'. Multiple Linear Regression - Estimated Regression Equation wrkl[t] = + 96.0103100392773 -0.162234536415654ind[t] -0.0647807862517575gron[t] + 0.177593971047212`bouw `[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) 96.0103100392773 7.729542 12.4212 0 0 ind -0.162234536415654 0.073695 -2.2014 0.031843 0.015922 gron -0.0647807862517575 0.010285 -6.2986 0 0 `bouw ` 0.177593971047212 0.044526 3.9885 0.000195 9.7e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.745775531648637 R-squared 0.556181143605807 Adjusted R-squared 0.532405133441833 F-TEST (value) 23.3925347343825 F-TEST (DF numerator) 3 F-TEST (DF denominator) 56 p-value 6.053550993812e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.03255307031101 Sum Squared Residuals 1418.28906270782 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 100 86.0793091270984 13.9206908729016 2 93.5 89.0990378509726 4.40096214902736 3 88.2 86.7566707284042 1.44332927159580 4 89.2 88.4551609628246 0.74483903717537 5 91.4 89.7210586502828 1.67894134971723 6 92.5 91.2288017651846 1.27119823481537 7 91.4 90.3030134577263 1.09698654227373 8 88.2 91.5332941640488 -3.33329416404876 9 87.1 94.0070507823163 -6.9070507823163 10 84.9 90.4595197088383 -5.55951970883833 11 92.5 90.4344053830758 2.06559461692425 12 93.5 89.11109502539 4.38890497461003 13 93.5 86.5594906693332 6.94050933066683 14 91.4 89.1048936564994 2.29510634350061 15 90.3 87.2812237398291 3.0187762601709 16 91.4 91.1987080179551 0.201291982044904 17 93.5 95.6152675673356 -2.11526756733562 18 93.5 89.246984639113 4.25301536088704 19 92.5 89.1426899759213 3.35731002407865 20 91.4 84.3263345130278 7.07366548697224 21 89.2 87.333950780604 1.86604921939604 22 86 84.3296250137753 1.67037498622471 23 88.2 87.5565655209434 0.643434479056639 24 87.1 83.923865046592 3.17613495340806 25 87.1 83.8172013010777 3.28279869892229 26 86 84.8506688315016 1.14933116849838 27 84.9 81.8113536003187 3.08864639968126 28 84.9 84.8453351086454 0.0546648913546286 29 86 85.870774276468 0.129225723531912 30 86 84.6744083802544 1.32559161974563 31 84.9 84.1453771761973 0.754622823802738 32 86 83.4100718858839 2.58992811411611 33 82.8 81.9657578811375 0.83424211886251 34 77.4 84.520260721638 -7.12026072163807 35 80.6 85.0876283326882 -4.48762833268822 36 78.5 81.5720209306762 -3.07202093067625 37 75.3 77.2323196157953 -1.9323196157953 38 75.3 79.8966364584803 -4.59663645848029 39 75.3 76.08272295212 -0.782722952120023 40 77.4 78.4834808731213 -1.08348087312133 41 78.5 79.7819941973426 -1.28199419734264 42 76.3 77.6361304990262 -1.33613049902621 43 73.1 74.6699571397127 -1.56995713971273 44 68.8 77.5608951712273 -8.76089517122728 45 65.6 70.6243430307119 -5.02434303071192 46 69.9 72.5370524475818 -2.63705244758181 47 82.8 72.538459869071 10.2615401309290 48 84.9 74.0471904419943 10.8528095580057 49 80.6 76.2764998198231 4.32350018017688 50 74.2 79.4196134636177 -5.21961346361768 51 71 82.6065544844886 -11.6065544844886 52 74.2 87.5571980882456 -13.3571980882456 53 82.8 85.4376051430436 -2.63760514304364 54 86 85.0834703839902 0.916529616009756 55 86 87.0581732371102 -1.05817323711021 56 82.8 85.5116047884892 -2.71160478848916 57 78.5 83.3666372216097 -4.86663722160976 58 79.6 85.5627008972895 -5.9627008972895 59 87.1 86.0616186976552 1.03838130234475 60 89.2 86.2882659048737 2.91173409512627 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.449112540219449 0.898225080438897 0.550887459780551 8 0.357411145909158 0.714822291818317 0.642588854090842 9 0.283862505123179 0.567725010246359 0.71613749487682 10 0.37594878638314 0.75189757276628 0.62405121361686 11 0.326009001342933 0.652018002685865 0.673990998657067 12 0.24420540764916 0.48841081529832 0.75579459235084 13 0.188107270302170 0.376214540604341 0.81189272969783 14 0.124882793771879 0.249765587543758 0.875117206228121 15 0.0991373149707507 0.198274629941501 0.90086268502925 16 0.0677769021788928 0.135553804357786 0.932223097821107 17 0.101251353821326 0.202502707642651 0.898748646178674 18 0.0706191437274763 0.141238287454953 0.929380856272524 19 0.0475263148134027 0.0950526296268055 0.952473685186597 20 0.0523601402659306 0.104720280531861 0.94763985973407 21 0.0420322153038795 0.084064430607759 0.95796778469612 22 0.0570575529762149 0.114115105952430 0.942942447023785 23 0.0405615916916785 0.081123183383357 0.959438408308321 24 0.0341012702964229 0.0682025405928458 0.965898729703577 25 0.0336330205378608 0.0672660410757215 0.96636697946214 26 0.0344841394569458 0.0689682789138916 0.965515860543054 27 0.0439298458546819 0.0878596917093638 0.956070154145318 28 0.0400963669283592 0.0801927338567185 0.95990363307164 29 0.0362423480817673 0.0724846961635346 0.963757651918233 30 0.0326166889399510 0.0652333778799021 0.967383311060049 31 0.039022847022414 0.078045694044828 0.960977152977586 32 0.0409112774804884 0.0818225549609767 0.959088722519512 33 0.0462929800524028 0.0925859601048056 0.953707019947597 34 0.0813915679874818 0.162783135974964 0.918608432012518 35 0.0837768983237418 0.167553796647484 0.916223101676258 36 0.0779641627413424 0.155928325482685 0.922035837258658 37 0.0713371834632371 0.142674366926474 0.928662816536763 38 0.0596587795088786 0.119317559017757 0.940341220491121 39 0.0442727123844478 0.0885454247688956 0.955727287615552 40 0.0282920593483832 0.0565841186967664 0.971707940651617 41 0.0189828284923038 0.0379656569846076 0.981017171507696 42 0.0121929548872422 0.0243859097744845 0.987807045112758 43 0.00699206356347909 0.0139841271269582 0.993007936436521 44 0.00932106270464812 0.0186421254092962 0.990678937295352 45 0.0187793954320762 0.0375587908641524 0.981220604567924 46 0.0559863752555673 0.111972750511135 0.944013624744433 47 0.16075596191727 0.32151192383454 0.83924403808273 48 0.176617806360351 0.353235612720702 0.823382193639649 49 0.168629844090672 0.337259688181344 0.831370155909328 50 0.155037427518307 0.310074855036615 0.844962572481693 51 0.277625994141713 0.555251988283426 0.722374005858287 52 0.937898604457236 0.124202791085529 0.0621013955427643 53 0.89816618739287 0.203667625214262 0.101833812607131 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 5 0.106382978723404 NOK 10% type I error level 20 0.425531914893617 NOK

Charts produced by software:

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Parameters (Session):

par1 = 3 ; par2 = TRUE ; par3 = TRUE ;

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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