Paper. Multi Regression with Lineair Trend

*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, 12 Dec 2009 11:09:49 -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/12/t1260641460kyx427m82xalcj4.htm/, Retrieved Sat, 12 Dec 2009 19:11:12 +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/12/t1260641460kyx427m82xalcj4.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)

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Dataseries X:

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593530 3922 18004 707169 610763 3759 17537 703434 612613 4138 20366 701017 611324 4634 22782 696968 594167 3996 19169 688558 595454 4308 13807 679237 590865 4143 29743 677362 589379 4429 25591 676693 584428 5219 29096 670009 573100 4929 26482 667209 567456 5761 22405 662976 569028 5592 27044 660194 620735 4163 17970 652270 628884 4962 18730 648024 628232 5208 19684 629295 612117 4755 19785 624961 595404 4491 18479 617306 597141 5732 10698 607691 593408 5731 31956 596219 590072 5040 29506 591130 579799 6102 34506 584528 574205 4904 27165 576798 572775 5369 26736 575683 572942 5578 23691 574369 619567 4619 18157 566815 625809 4731 17328 573074 619916 5011 18205 567739 587625 5299 20995 571942 565742 4146 17382 570274 557274 4625 9367 568800 560576 4736 31124 558115 548854 4219 26551 550591 531673 5116 30651 548872 525919 4205 25859 547009 511038 4121 25100 545946 498662 5103 25778 539702 555362 4300 20418 542427 564591 4578 18688 542968 541657 3809 20424 536640 527070 5526 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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Werkzoekend[t] = + 434529.160361769 + 10.2471001027768Bouw[t] -1.70629265950963Auto[t] + 0.252198183143766Krediet[t] -1065.96796545348t + 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) 434529.160361769 70188.97008 6.1908 0 0 Bouw 10.2471001027768 6.152758 1.6654 0.101409 0.050704 Auto -1.70629265950963 0.590407 -2.89 0.005471 0.002736 Krediet 0.252198183143766 0.081582 3.0914 0.003103 0.001551 t -1065.96796545348 290.63326 -3.6677 0.000546 0.000273 Multiple Linear Regression - Regression Statistics Multiple R 0.82249180459475 R-squared 0.676492768625529 Adjusted R-squared 0.653385109241639 F-TEST (value) 29.2756941491503 F-TEST (DF numerator) 4 F-TEST (DF denominator) 56 p-value 3.77142761465166e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 25003.5073243768 Sum Squared Residuals 35009821197.1293 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 593530 621278.962933186 -27748.9629331861 2 610763 618397.596108931 -7634.5961089306 3 612613 615778.614140018 -3165.61414001825 4 611324 614651.654316618 -3327.65431661761 5 594167 611091.885144162 -16924.8851441618 6 595454 620021.414385982 -24567.4143859823 7 590865 589600.32348823 1264.67651176940 8 589379 598380.832689932 -9001.8326899321 9 584428 597743.825377958 -13315.8253779581 10 573100 597460.292481855 -24360.292481855 11 567456 610808.912065485 -43352.912065485 12 569028 599394.077189691 -30366.0771896911 13 620735 597169.484366529 23565.5156334712 14 628884 601923.333476338 26960.6665236618 15 628232 597026.929166896 31205.0708331039 16 612117 590053.662370529 22063.3376294709 17 595404 586580.301099297 8823.69890070334 18 597141 609082.762014106 -11941.7620141063 19 593408 568840.960035669 24567.039964331 20 590072 563591.226360977 26480.7736390233 21 579799 563211.203002009 16587.7969979911 22 574205 560445.611571188 13759.3884288123 23 572775 564595.34373025 8179.6562697502 24 572942 570535.292421833 2406.70757816740 25 619567 567179.873960075 52387.1260399255 26 625809 570254.606249162 55554.3937508377 27 619916 569215.930343024 50700.0696569756 28 587625 567400.559650892 20224.440349108 29 565742 560263.854076261 5478.14592373859 30 557274 577410.442604054 -20136.4426040538 31 560576 537663.355770166 22912.6442298337 32 548854 537204.974253541 11649.0257464589 33 531673 537901.326499465 -6228.32649946477 34 525919 535206.959549555 -9287.95954955498 35 511038 534307.224635354 -23269.2246353542 36 498662 540572.31709213 -41910.3170921304 37 555362 541110.896448186 14251.1035518145 38 564591 545981.947829336 18609.0521706636 39 541657 532477.925725005 9179.07427499488 40 527070 540827.127008783 -13757.1270087830 41 509846 536983.880547292 -27137.8805472923 42 514258 543032.460466743 -28774.460466743 43 516922 514538.216471078 2383.78352892249 44 507561 520302.862415197 -12741.8624151969 45 492622 513829.088024233 -21207.0880242327 46 490243 511616.392791107 -21373.392791107 47 469357 515718.931057956 -46361.9310579564 48 477580 517657.763052394 -40077.7630523942 49 528379 518959.379964803 9419.6200351969 50 533590 520640.262303227 12949.737696773 51 517945 520869.950465061 -2924.95046506124 52 506174 514023.272985714 -7849.27298571422 53 501866 528439.970399457 -26573.9703994567 54 516141 544951.379178959 -28810.3791789592 55 528222 506235.23069208 21986.7693079198 56 532638 516183.039345462 16454.9606545381 57 536322 515484.337340401 20837.6626595988 58 536535 521328.453261719 15206.5467382811 59 523597 535113.501282521 -11516.5012825209 60 536214 539482.800828889 -3268.80082888884 61 586570 543703.005493048 42866.9945069522 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.073436405855734 0.146872811711468 0.926563594144266 9 0.0284711344887235 0.056942268977447 0.971528865511276 10 0.0278738622093116 0.0557477244186232 0.972126137790688 11 0.0206373756351975 0.041274751270395 0.979362624364803 12 0.0132841451571472 0.0265682903142945 0.986715854842853 13 0.040493346550204 0.080986693100408 0.959506653449796 14 0.109199108475342 0.218398216950683 0.890800891524658 15 0.186569701016654 0.373139402033308 0.813430298983346 16 0.140983648457482 0.281967296914964 0.859016351542518 17 0.203180427237826 0.406360854475651 0.796819572762174 18 0.181040216316976 0.362080432633952 0.818959783683024 19 0.144514194545983 0.289028389091965 0.855485805454017 20 0.105337218279214 0.210674436558429 0.894662781720786 21 0.0701655183609203 0.140331036721841 0.92983448163908 22 0.0872059570089358 0.174411914017872 0.912794042991064 23 0.0782480275283909 0.156496055056782 0.921751972471609 24 0.0701573465763624 0.140314693152725 0.929842653423638 25 0.0689438080933335 0.137887616186667 0.931056191906667 26 0.0831196025866649 0.166239205173330 0.916880397413335 27 0.134519148971220 0.269038297942441 0.86548085102878 28 0.178632958404967 0.357265916809934 0.821367041595033 29 0.354718296125004 0.709436592250007 0.645281703874996 30 0.484239741142803 0.968479482285605 0.515760258857197 31 0.472861994000333 0.945723988000665 0.527138005999667 32 0.457569131927338 0.915138263854676 0.542430868072662 33 0.437935693445563 0.875871386891125 0.562064306554437 34 0.42181893472493 0.84363786944986 0.57818106527507 35 0.468783545510404 0.937567091020807 0.531216454489596 36 0.558588518880511 0.882822962238978 0.441411481119489 37 0.547500262695162 0.904999474609676 0.452499737304838 38 0.675457741927061 0.649084516145877 0.324542258072939 39 0.661482364460475 0.67703527107905 0.338517635539525 40 0.708036857870193 0.583926284259615 0.291963142129807 41 0.631761188477681 0.736477623044637 0.368238811522319 42 0.566617524304563 0.866764951390873 0.433382475695437 43 0.588772660829171 0.822454678341659 0.411227339170829 44 0.613270680324571 0.773458639350858 0.386729319675429 45 0.537247403549767 0.925505192900466 0.462752596450233 46 0.493873072010079 0.987746144020157 0.506126927989921 47 0.543430706442489 0.913138587115021 0.456569293557511 48 0.584853191930117 0.830293616139765 0.415146808069883 49 0.518478472431952 0.963043055136097 0.481521527568049 50 0.594250150830763 0.811499698338474 0.405749849169237 51 0.716507242382752 0.566985515234496 0.283492757617248 52 0.758308639920363 0.483382720159274 0.241691360079637 53 0.610155892520195 0.77968821495961 0.389844107479805 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 2 0.0434782608695652 OK 10% type I error level 5 0.108695652173913 NOK

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 = 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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