*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 09:49:54 -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/t1260636695top0260fzi69mjq.htm/, Retrieved Sat, 12 Dec 2009 17:51:47 +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/t1260636695top0260fzi69mjq.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)

User-defined keywords:

Dataseries X:

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467 98,8 460 100,5 448 110,4 443 96,4 436 101,9 431 106,2 484 81 510 94,7 513 101 503 109,4 471 102,3 471 90,7 476 96,2 475 96,1 470 106 461 103,1 455 102 456 104,7 517 86 525 92,1 523 106,9 519 112,6 509 101,7 512 92 519 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

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 Y[t] = + 586.7804012508 -1.42669564850463X[t] + 12.0494979595140M1[t] + 8.44581355654682M2[t] + 14.6086479566102M3[t] -5.90728858834744M4[t] -14.4134526969954M5[t] -3.74449074871385M6[t] + 13.1997510290256M7[t] + 38.1263347080594M8[t] + 56.1630560508234M9[t] + 42.5115962852315M10[t] + 11.7074756774717M11[t] + 2.38749164561372t + 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) 586.7804012508 66.421625 8.8342 0 0 X -1.42669564850463 0.703358 -2.0284 0.047119 0.02356 M1 12.0494979595140 10.86877 1.1086 0.272164 0.136082 M2 8.44581355654682 10.890023 0.7756 0.441163 0.220581 M3 14.6086479566102 13.912429 1.05 0.298054 0.149027 M4 -5.90728858834744 11.311053 -0.5223 0.60348 0.30174 M5 -14.4134526969954 11.245576 -1.2817 0.205048 0.102524 M6 -3.74449074871385 13.812299 -0.2711 0.787278 0.393639 M7 13.1997510290256 13.904571 0.9493 0.346402 0.173201 M8 38.1263347080594 10.925984 3.4895 0.000931 0.000466 M9 56.1630560508234 14.006585 4.0098 0.000176 8.8e-05 M10 42.5115962852315 14.464508 2.939 0.00472 0.00236 M11 11.7074756774717 11.934092 0.981 0.330662 0.165331 t 2.38749164561372 0.126694 18.8445 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.952013796598217 R-squared 0.906330268913351 Adjusted R-squared 0.885335329187034 F-TEST (value) 43.1689864666408 F-TEST (DF numerator) 13 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 18.7094421164429 Sum Squared Residuals 20302.5070098946 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 467 460.259860783672 6.74013921632807 2 460 456.61828542386 3.38171457614026 3 448 451.044324549341 -3.04432454934088 4 443 452.889618729062 -9.88961872906187 5 436 438.924120199252 -2.92412019925211 6 431 445.845782504578 -14.8457825045775 7 484 501.130246270247 -17.1302462702472 8 510 508.898591210381 1.10140878961863 9 513 520.33462161318 -7.33462161318013 10 503 497.086410045763 5.91358995423699 11 471 478.799320188 -7.79932018799969 12 471 486.029005678795 -15.0290056787953 13 476 492.619169217148 -16.6191692171476 14 475 491.545646024645 -16.5456460246447 15 470 485.971685150126 -15.9716851501259 16 461 471.980657631445 -10.9806576314454 17 455 467.431350381766 -12.4313503817663 18 456 476.635725724699 -20.6357257246990 19 517 522.646667775089 -5.64666777508872 20 525 541.257899643858 -16.257899643858 21 523 540.567017034367 -17.5670170343673 22 519 521.170883717913 -2.17088371791268 23 509 508.305237324467 0.69476267553299 24 512 512.824201083104 -0.824201083103979 25 519 519.557034186307 -0.557034186306691 26 517 518.911519688355 -1.91151968835504 27 510 515.477602286593 -5.47760228659326 28 509 501.201235638212 7.79876436178816 29 501 501.645363158299 -0.64536315829888 30 507 505.570964601765 1.42903539823544 31 569 549.299193614547 19.7008063854532 32 580 573.046529817933 6.95347018206725 33 578 565.079499401068 12.9205005989315 34 565 551.104809548931 13.8951904510686 35 547 541.377893582196 5.62210641780406 36 555 534.483292152796 20.5167078472041 37 562 551.488333925232 10.5116660747681 38 561 547.56141943572 13.4385805642804 39 555 533.712623799874 21.2873762001260 40 544 529.565796255876 14.4342037441245 41 537 531.293949859617 5.70605014038332 42 543 524.376664374447 18.6233356255528 43 594 577.949093361911 16.0509066380886 44 611 591.852229590615 19.1477704093846 45 613 587.594607859863 25.4053921401368 46 611 581.752083204203 29.2479167957975 47 594 562.466306392486 31.533693607514 48 595 556.427722352189 38.5722776478113 49 591 580.851581496849 10.1484185031512 50 589 577.780684396439 11.2193156035607 51 584 573.776088735276 10.2239112647243 52 573 558.35836556809 14.6416344319094 53 567 558.945162653028 8.05483734697193 54 569 549.45982500055 19.5401749994498 55 621 615.872514824556 5.12748517544391 56 629 622.071494551335 6.92850544866493 57 628 617.528533690882 10.4714663091181 58 612 616.679443804987 -4.67944380498741 59 595 584.125397462178 10.8746025378221 60 597 584.0789351456 12.9210648543999 61 593 603.224020390793 -10.2240203907930 62 590 599.582445030982 -9.58244503098164 63 580 587.01767547879 -7.01767547879024 64 574 590.004326177315 -16.0043261773148 65 573 570.760053748038 2.23994625196203 66 573 577.111037793962 -4.11103779396155 67 620 638.10228415365 -18.1022841536498 68 626 643.873255185877 -17.8732551858774 69 620 643.895720400639 -23.8957204006391 70 588 630.206369678203 -42.206369678203 71 566 606.925845050674 -40.9258450506735 72 557 613.156843587516 -56.156843587516 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0109086222103468 0.0218172444206935 0.989091377789653 18 0.00524911246947433 0.0104982249389487 0.994750887530526 19 0.004631789273461 0.009263578546922 0.99536821072654 20 0.00145754088455363 0.00291508176910726 0.998542459115446 21 0.00169651537569875 0.00339303075139749 0.998303484624301 22 0.000557471703950607 0.00111494340790121 0.99944252829605 23 0.00180092392328063 0.00360184784656126 0.99819907607672 24 0.00392344787453324 0.00784689574906649 0.996076552125467 25 0.00410392640060009 0.00820785280120019 0.9958960735994 26 0.00428611375719737 0.00857222751439473 0.995713886242803 27 0.00482659848239559 0.00965319696479118 0.995173401517604 28 0.00670809883423725 0.0134161976684745 0.993291901165763 29 0.00796815794345321 0.0159363158869064 0.992031842056547 30 0.0183020345643516 0.0366040691287032 0.981697965435648 31 0.0304236075193163 0.0608472150386325 0.969576392480684 32 0.0352910817662505 0.0705821635325011 0.96470891823375 33 0.0349272256229785 0.069854451245957 0.965072774377022 34 0.0257304227461135 0.051460845492227 0.974269577253887 35 0.0313800723737206 0.0627601447474413 0.96861992762628 36 0.0296818421174464 0.0593636842348929 0.970318157882554 37 0.0265457225440852 0.0530914450881704 0.973454277455915 38 0.0244250438933834 0.0488500877867669 0.975574956106616 39 0.0194568362678570 0.0389136725357141 0.980543163732143 40 0.0181115949653338 0.0362231899306676 0.981888405034666 41 0.0413472168140153 0.0826944336280306 0.958652783185985 42 0.0761551610237206 0.152310322047441 0.92384483897628 43 0.117762110999070 0.235524221998139 0.88223788900093 44 0.181738249330519 0.363476498661039 0.81826175066948 45 0.284273798492761 0.568547596985522 0.715726201507239 46 0.240415761783845 0.480831523567691 0.759584238216155 47 0.196203890952877 0.392407781905754 0.803796109047123 48 0.142183645180205 0.28436729036041 0.857816354819795 49 0.112067738177282 0.224135476354563 0.887932261822718 50 0.084541915615606 0.169083831231212 0.915458084384394 51 0.0510994929703597 0.102198985940719 0.94890050702964 52 0.0501014662125156 0.100202932425031 0.949898533787484 53 0.046280126185852 0.092560252371704 0.953719873814148 54 0.0728243324450573 0.145648664890115 0.927175667554943 55 0.0896965841079253 0.179393168215851 0.910303415892075 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 9 0.230769230769231 NOK 5% type I error level 17 0.435897435897436 NOK 10% type I error level 26 0.666666666666667 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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