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: Sat, 21 Nov 2009 07:56:47 -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/21/t1258815523x9wa8iwzb9oyef2.htm/, Retrieved Sat, 21 Nov 2009 15:58:55 +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/21/t1258815523x9wa8iwzb9oyef2.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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562000 4814 561000 3908 555000 5250 544000 3937 537000 4004 543000 5560 594000 3922 611000 3759 613000 4138 611000 4634 594000 3996 595000 4308 591000 4143 589000 4429 584000 5219 573000 4929 567000 5755 569000 5592 621000 4163 629000 4962 628000 5208 612000 4755 595000 4491 597000 5732 593000 5731 590000 5040 580000 6102 574000 4904 573000 5369 573000 5578 620000 4619 626000 4731 620000 5011 588000 5299 566000 4146 557000 4625 561000 4736 549000 4219 532000 5116 526000 4205 511000 4121 499000 5103 555000 4300 565000 4578 542000 3809 527000 5526 510000 4247 514000 3830 517000 4394 508000 4826 493000 4409 490000 4569 469000 4106 478000 4794 528000 3914 534000 3793 518000 4405 506000 4022 502000 4100 516000 4788

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 werkloos[t] = + 475949.678886236 + 29.5723106155533bouw[t] -11842.4901659986M1[t] -7378.78713939663M2[t] -38101.4070769657M3[t] -22886.1237129371M4[t] -36075.6385920404M5[t] -52820.644756119M6[t] + 33752.1334074593M7[t] + 39406.6590887836M8[t] + 27789.7553234363M9[t] + 4149.28979119656M10[t] + 9613.89236678436M11[t] -1607.11390273949t + 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) 475949.678886236 27487.144689 17.3154 0 0 bouw 29.5723106155533 5.300663 5.579 1e-06 1e-06 M1 -11842.4901659986 12629.736113 -0.9377 0.353313 0.176656 M2 -7378.78713939663 12656.745614 -0.583 0.562746 0.281373 M3 -38101.4070769657 12894.807876 -2.9548 0.004919 0.00246 M4 -22886.1237129371 12610.279469 -1.8149 0.076066 0.038033 M5 -36075.6385920404 12560.793694 -2.8721 0.006148 0.003074 M6 -52820.644756119 13005.694619 -4.0613 0.000188 9.4e-05 M7 33752.1334074593 12808.504325 2.6351 0.011424 0.005712 M8 39406.6590887836 12636.769149 3.1184 0.003134 0.001567 M9 27789.7553234363 12551.185649 2.2141 0.031817 0.015908 M10 4149.28979119656 12557.245067 0.3304 0.742575 0.371288 M11 9613.89236678436 12757.192604 0.7536 0.454926 0.227463 t -1607.11390273949 152.058078 -10.5691 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.909203821455076 R-squared 0.826651588948513 Adjusted R-squared 0.777661820607875 F-TEST (value) 16.8739640326650 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 2.53130849614536e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 19789.9972825814 Sum Squared Residuals 18015623652.4507 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 562000 604861.178120772 -42861.1781207716 2 561000 580925.253826943 -19925.2538269429 3 555000 588281.560832707 -33281.5608327068 4 544000 563061.286455774 -19061.2864557744 5 537000 550246.002485174 -13246.0024851737 6 543000 577908.397736157 -34908.3977361566 7 594000 614434.617208719 -20434.617208719 8 611000 613661.742356969 -2661.74235696869 9 613000 611645.630412177 1354.36958782336 10 611000 601065.917042512 9934.0829574882 11 594000 586056.271542637 7943.7284573629 12 595000 584061.826185166 10938.1738148341 13 591000 565732.790864862 25267.2091351385 14 589000 577047.060824772 11952.9391752277 15 584000 568079.452370751 15920.5476292492 16 573000 573111.651753529 -111.651753529381 17 567000 582741.751540134 -15741.7515401337 18 569000 559569.34484298 9430.6551570196 19 621000 602276.177234193 18723.8227658065 20 629000 629951.865194605 -951.865194605427 21 628000 624002.635937945 3997.36406205522 22 612000 585358.79979412 26641.2002058801 23 595000 581409.198464462 13590.8015355379 24 597000 606887.42966884 -9887.4296688399 25 593000 593408.253289486 -408.2532894863 26 590000 575830.375778001 14169.6242219986 27 580000 574906.43581141 5093.56418858954 28 574000 553086.977155267 20913.0228447333 29 573000 552041.472809656 20958.5271903438 30 573000 539869.965661489 33130.0343385113 31 620000 596475.784042012 23524.2159579881 32 626000 603835.294609539 22164.7053904613 33 620000 598891.523913807 21108.4760861931 34 588000 582160.769936107 5839.23006389302 35 566000 551921.384469222 14078.6155307777 36 557000 554865.514984549 2134.4850154515 37 561000 544698.437394137 16301.5626058631 38 549000 532266.141929758 16733.8580702417 39 532000 526462.770711601 5537.22928839899 40 526000 513130.565202121 12869.4347978790 41 511000 495849.862328572 15150.1376714282 42 499000 506537.751286227 -7537.75128622706 43 555000 567756.850122777 -12756.8501227765 44 565000 580025.364252485 -15025.3642524852 45 542000 544060.239721038 -2060.23972103791 46 527000 569588.317612964 -42588.3176129637 47 510000 535622.821008519 -25622.8210085193 48 514000 512070.16121231 1929.83878769025 49 517000 515299.340330744 1700.65966925625 50 508000 530931.167640525 -22931.1676405252 51 493000 486269.780273531 6730.21972646904 52 490000 504609.519433309 -14609.5194333085 53 469000 476120.910836465 -7120.91083646461 54 478000 478114.540473147 -114.540473147198 55 528000 537056.571392299 -9056.57139229907 56 534000 537525.733586402 -3525.73358640195 57 518000 542399.970015034 -24399.9700150338 58 506000 505826.195614298 173.804385702344 59 502000 511990.324515159 -9990.32451515911 60 516000 521115.067949136 -5115.06794913591 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 8.7270039363367e-05 0.000174540078726734 0.999912729960637 18 7.82089142366058e-05 0.000156417828473212 0.999921791085763 19 8.6912668201574e-06 1.73825336403148e-05 0.99999130873318 20 0.000746519037084482 0.00149303807416896 0.999253480962915 21 0.00183549307575444 0.00367098615150888 0.998164506924246 22 0.0414453633418403 0.0828907266836806 0.95855463665816 23 0.0819177037895992 0.163835407579198 0.9180822962104 24 0.109009352211357 0.218018704422714 0.890990647788643 25 0.0997598553223255 0.199519710644651 0.900240144677675 26 0.0803889157244211 0.160777831448842 0.919611084275579 27 0.0751012965347477 0.150202593069495 0.924898703465252 28 0.0513044630523868 0.102608926104774 0.948695536947613 29 0.0316585829898334 0.0633171659796668 0.968341417010167 30 0.0293179242006478 0.0586358484012957 0.970682075799352 31 0.0286132364195516 0.0572264728391032 0.971386763580448 32 0.0387193197295114 0.0774386394590228 0.961280680270489 33 0.175003658671792 0.350007317343583 0.824996341328208 34 0.621416910959826 0.757166178080347 0.378583089040174 35 0.82823755714298 0.343524885714041 0.171762442857020 36 0.863702664214664 0.272594671570671 0.136297335785336 37 0.864665374458713 0.270669251082575 0.135334625541287 38 0.89108045311356 0.217839093772881 0.108919546886441 39 0.877755728986324 0.244488542027352 0.122244271013676 40 0.877135907660646 0.245728184678708 0.122864092339354 41 0.941883963555556 0.116232072888888 0.058116036444444 42 0.896737812394296 0.206524375211408 0.103262187605704 43 0.842178852925989 0.315642294148022 0.157821147074011 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 5 0.185185185185185 NOK 5% type I error level 5 0.185185185185185 NOK 10% type I error level 10 0.370370370370370 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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