*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 10:59:51 -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/t1258653649x1vfg0xj5bluev0.htm/, Retrieved Thu, 19 Nov 2009 19:01:00 +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/t1258653649x1vfg0xj5bluev0.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:

» Textbox « » Textfile « » CSV «

519 97.4 517 97 510 105.4 509 102.7 501 98.1 507 104.5 569 87.4 580 89.9 578 109.8 565 111.7 547 98.6 555 96.9 562 95.1 561 97 555 112.7 544 102.9 537 97.4 543 111.4 594 87.4 611 96.8 613 114.1 611 110.3 594 103.9 595 101.6 591 94.6 589 95.9 584 104.7 573 102.8 567 98.1 569 113.9 621 80.9 629 95.7 628 113.2 612 105.9 595 108.8 597 102.3 593 99 590 100.7 580 115.5 574 100.7 573 109.9 573 114.6 620 85.4 626 100.5 620 114.8 588 116.5 566 112.9 557 102 561 106 549 105.3 532 118.8 526 106.1 511 109.3 499 117.2 555 92.5 565 104.2 542 112.5 527 122.4 510 113.3 514 100 517 110.7 508 112.8 493 109.8 490 117.3 469 109.1 478 115.9 528 96 534 99.8 518 116.8 506 115.7 502 99.4 516 94.3

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 7 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 663.747235364029 -0.849955888088473X[t] -0.514576813656158t + 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) 663.747235364029 55.060133 12.055 0 0 X -0.849955888088473 0.549176 -1.5477 0.126271 0.063135 t -0.514576813656158 0.234269 -2.1965 0.031417 0.015709 Multiple Linear Regression - Regression Statistics Multiple R 0.373029972736068 R-squared 0.139151360559472 Adjusted R-squared 0.114199226082935 F-TEST (value) 5.57673174975546 F-TEST (DF numerator) 2 F-TEST (DF denominator) 69 p-value 0.00568819544894872 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 38.499212515376 Sum Squared Residuals 102271.066136982 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 519 580.446955050553 -61.4469550505531 2 517 580.272360592135 -63.272360592135 3 510 572.618154318536 -62.6181543185356 4 509 574.398458402718 -65.3984584027184 5 501 577.793678674269 -76.7936786742692 6 507 571.839384176847 -64.8393841768468 7 569 585.859053049503 -16.8590530495035 8 580 583.219586515626 -3.21958651562619 9 578 565.790887529009 12.2091124709906 10 565 563.661394527985 1.33860547201484 11 547 574.281239848288 -27.281239848288 12 555 575.211588044382 -20.2115880443822 13 562 576.226931829285 -14.2269318292854 14 561 574.097438828261 -13.0974388282611 15 555 560.238554571616 -5.2385545716159 16 544 568.053545461227 -24.0535454612268 17 537 572.213726032057 -35.2137260320572 18 543 559.799766785162 -16.7997667851624 19 594 579.684131285630 14.3158687143704 20 611 571.179969123942 39.8200308760582 21 613 555.961155446355 57.0388445536449 22 611 558.676411007435 52.3235889925649 23 594 563.601551877545 30.3984481224548 24 595 565.041873606493 29.9581263935075 25 591 570.476988009456 20.5230119905443 26 589 568.857468541285 20.1425314587155 27 584 560.86327991245 23.1367200875502 28 573 561.963619286162 11.0363807138383 29 567 565.443835146521 1.55616485347860 30 569 551.499955301067 17.5000446989326 31 621 579.033922794331 41.9660772056692 32 629 565.939998836965 63.0600011630347 33 628 550.551193981761 77.4488060182392 34 612 556.241295151151 55.7587048488495 35 595 553.261846262038 41.7381537379622 36 597 558.271982720957 38.7280172790433 37 593 560.562260337992 32.4377396620075 38 590 558.602758514586 31.3972414854141 39 580 545.50883455722 34.4911654427796 40 574 557.573604887274 16.4263951127264 41 573 549.239433903203 23.7605660967965 42 573 544.730064415532 28.2699355844685 43 620 569.034199534059 50.9658004659412 44 626 555.685288810267 70.3147111897333 45 620 543.016342796945 76.9836572030546 46 588 541.056840973539 46.9431590264612 47 566 543.602105357001 22.3978946429988 48 557 552.352047723509 4.64795227649065 49 561 548.437647357499 12.5623526425007 50 549 548.518039665505 0.481960334494919 51 532 536.529058362654 -4.52905836265454 52 526 546.808921327722 -20.808921327722 53 511 543.574485672183 -32.5744856721827 54 499 536.345257342628 -37.3452573426276 55 555 556.824590964757 -1.82459096475674 56 565 546.365530260465 18.6344697395346 57 542 538.796319575675 3.20368042432503 58 527 529.867179469943 -2.86717946994292 59 510 537.087201237892 -27.0872012378919 60 514 547.877037735812 -33.8770377358124 61 517 538.26793291961 -21.2679329196096 62 508 535.968448740968 -27.9684487409676 63 493 538.003739591577 -45.0037395915769 64 490 531.114493617257 -41.1144936172572 65 469 537.569555085926 -68.5695550859265 66 478 531.275278233269 -53.2752782332687 67 528 547.674823592573 -19.6748235925732 68 534 543.930414404181 -9.93041440418084 69 518 528.966587493021 -10.9665874930206 70 506 529.386962156262 -23.3869621562618 71 502 542.726666318448 -40.7266663184478 72 516 546.546864534043 -30.5468645340428 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.00262702682734828 0.00525405365469655 0.997372973172652 7 0.0556412182581413 0.111282436516283 0.944358781741859 8 0.0482245034275335 0.096449006855067 0.951775496572467 9 0.186898151165504 0.373796302331009 0.813101848834496 10 0.114909873128370 0.229819746256741 0.88509012687163 11 0.166414815532933 0.332829631065866 0.833585184467067 12 0.164213041907403 0.328426083814806 0.835786958092597 13 0.141735639979412 0.283471279958824 0.858264360020588 14 0.125948428698682 0.251896857397364 0.874051571301318 15 0.101705974930458 0.203411949860917 0.898294025069542 16 0.152173881966619 0.304347763933238 0.847826118033381 17 0.318258061807952 0.636516123615905 0.681741938192048 18 0.392569509524155 0.78513901904831 0.607430490475845 19 0.375310675524049 0.750621351048099 0.624689324475951 20 0.399919282311147 0.799838564622295 0.600080717688853 21 0.465162482671965 0.93032496534393 0.534837517328036 22 0.427774085105939 0.855548170211878 0.572225914894061 23 0.371867061446387 0.743734122892774 0.628132938553613 24 0.324599160868691 0.649198321737382 0.67540083913131 25 0.32083814848309 0.64167629696618 0.67916185151691 26 0.327675545441481 0.655351090882962 0.672324454558519 27 0.335168412813119 0.670336825626239 0.664831587186881 28 0.44019179724696 0.88038359449392 0.55980820275304 29 0.654257448148238 0.691485103703524 0.345742551851762 30 0.731182964033614 0.537634071932771 0.268817035966386 31 0.700819946842713 0.598360106314573 0.299180053157287 32 0.657241435136975 0.68551712972605 0.342758564863025 33 0.662764512229439 0.674470975541122 0.337235487770561 34 0.599520294155657 0.800959411688687 0.400479705844343 35 0.560133731960291 0.879732536079418 0.439866268039709 36 0.527593617813417 0.944812764373167 0.472406382186583 37 0.521645935122791 0.956708129754419 0.478354064877209 38 0.518299888776417 0.963400222447166 0.481700111223583 39 0.504525581722579 0.990948836554842 0.495474418277421 40 0.586990175295421 0.826019649409158 0.413009824704579 41 0.600289286239479 0.799421427521042 0.399710713760521 42 0.580658593819696 0.838682812360608 0.419341406180304 43 0.514404093522155 0.97119181295569 0.485595906477845 44 0.570050442995261 0.859899114009477 0.429949557004739 45 0.80586467945089 0.388270641098219 0.194135320549109 46 0.890353386961857 0.219293226076286 0.109646613038143 47 0.919763374863576 0.160473250272847 0.0802366251364237 48 0.936746816367436 0.126506367265128 0.0632531836325638 49 0.946309704734166 0.107380590531669 0.0536902952658345 50 0.952363542326623 0.095272915346753 0.0476364576733765 51 0.95794798808482 0.0841040238303585 0.0420520119151792 52 0.965600985591455 0.0687980288170908 0.0343990144085454 53 0.977913326752033 0.0441733464959343 0.0220866732479671 54 0.986351082810334 0.0272978343793315 0.0136489171896657 55 0.979225193774985 0.0415496124500301 0.0207748062250150 56 0.984153108364905 0.03169378327019 0.015846891635095 57 0.986392342924885 0.0272153141502297 0.0136076570751149 58 0.990617017640363 0.0187659647192742 0.00938298235963712 59 0.986116847684536 0.0277663046309275 0.0138831523154637 60 0.977233154927614 0.0455336901447723 0.0227668450723862 61 0.972010254818514 0.0559794903629727 0.0279897451814864 62 0.963768339762057 0.0724633204758863 0.0362316602379432 63 0.934812500529094 0.130374998941812 0.0651874994709058 64 0.883580578786232 0.232838842427535 0.116419421213768 65 0.909417448063266 0.181165103873468 0.0905825519367341 66 0.982739152389484 0.0345216952210322 0.0172608476105161 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.0163934426229508 NOK 5% type I error level 10 0.163934426229508 NOK 10% type I error level 16 0.262295081967213 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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