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*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: Wed, 02 Dec 2009 11:40:51 -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/Dec/02/t1259779364ezcm4x77gcewj71.htm/, Retrieved Wed, 02 Dec 2009 19:42:56 +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/02/t1259779364ezcm4x77gcewj71.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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825 696 627 0 677 825 696 0 656 677 825 0 785 656 677 0 412 785 656 0 352 412 785 0 839 352 412 0 729 839 352 0 696 729 839 0 641 696 729 0 695 641 696 0 638 695 641 0 762 638 695 0 635 762 638 0 721 635 762 0 854 721 635 0 418 854 721 0 367 418 854 0 824 367 418 0 687 824 367 0 601 687 824 0 676 601 687 0 740 676 601 0 691 740 676 0 683 691 740 0 594 683 691 0 729 594 683 0 731 729 594 0 386 731 729 0 331 386 731 0 707 331 386 0 715 707 331 0 657 715 707 0 653 657 715 0 642 653 657 0 643 642 653 0 718 643 642 0 654 718 643 0 632 654 718 0 731 632 654 0 392 731 632 0 344 392 731 0 792 344 392 0 852 792 344 0 649 852 792 0 629 649 852 0 685 629 649 1 617 685 629 1 715 617 685 1 715 715 617 1 629 715 715 1 916 629 715 1 531 916 629 1 357 531 916 1 917 357 531 1 828 917 357 1 708 828 917 1 858 708 828 1

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y(t)[t] = + 545.186455397854 + 0.141348092963167`Y(t-1)`[t] + 0.00880922248867556`Y(t-2)`[t] + 71.4208577566432X[t] + 98.2230519212738M1[t] + 1.62679385532139M2[t] + 32.0269502909186M3[t] + 160.81742550343M4[t] -232.68270559757M5[t] -257.700350126129M6[t] + 222.814216532563M7[t] + 104.723350792037M8[t] + 8.83524526213836M9[t] + 53.2794395618514M10[t] + 48.363914887757M11[t] -0.637210678003292t + 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) 545.186455397854 149.255417 3.6527 0.000714 0.000357 `Y(t-1)` 0.141348092963167 0.16342 0.8649 0.391987 0.195993 `Y(t-2)` 0.00880922248867556 0.160074 0.055 0.956374 0.478187 X 71.4208577566432 28.221974 2.5307 0.015219 0.007609 M1 98.2230519212738 37.66051 2.6081 0.012554 0.006277 M2 1.62679385532139 37.747257 0.0431 0.965828 0.482914 M3 32.0269502909186 40.301539 0.7947 0.431266 0.215633 M4 160.81742550343 36.947052 4.3526 8.4e-05 4.2e-05 M5 -232.68270559757 41.349124 -5.6273 1e-06 1e-06 M6 -257.700350126129 63.495252 -4.0586 0.00021 0.000105 M7 222.814216532563 71.811288 3.1028 0.003423 0.001711 M8 104.723350792037 65.462672 1.5997 0.117152 0.058576 M9 8.83524526213836 46.796198 0.1888 0.851157 0.425578 M10 53.2794395618514 41.860811 1.2728 0.210098 0.105049 M11 48.363914887757 39.410049 1.2272 0.226583 0.113291 t -0.637210678003292 0.641103 -0.9939 0.32595 0.162975 Multiple Linear Regression - Regression Statistics Multiple R 0.947365382278292 R-squared 0.897501167539294 Adjusted R-squared 0.860894441660471 F-TEST (value) 24.5173843328744 F-TEST (DF numerator) 15 F-TEST (DF denominator) 42 p-value 4.44089209850063e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 54.8915474089605 Sum Squared Residuals 126549.443031907 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 825 746.673951843892 78.3260481561084 2 677 668.2822234439 8.71777655609997 3 656 678.262041143984 -22.2620411439842 4 785 802.143230797942 -17.1432307979421 5 412 426.054799338925 -14.0547993389251 6 352 348.813495158141 3.18650484185928 7 839 816.924125572763 22.0758744272371 8 729 766.504017077975 -37.5040170779754 9 696 658.72050199611 37.2794980038895 10 641 696.893984076281 -55.8939840762814 11 695 683.276399269083 11.7236007309167 12 638 641.423563486457 -3.42356348645686 13 762 731.428261445215 30.5717385547847 14 635 651.219830546838 -16.2198305468378 15 721 664.123912086705 56.8760879132948 16 854 803.314341359984 50.6856586400158 17 418 428.733889079108 -10.7338890791081 18 367 342.622891931599 24.3771080684011 19 824 811.450674166103 12.5493258338969 20 687 756.869405884819 -69.8694058848186 21 601 645.005215618288 -44.0052156182879 22 676 675.449399764217 0.550600235783341 23 740 679.74017825033 60.2598217496695 24 691 640.446022320863 50.5539776791365 25 683 731.669597248214 -48.6695972482141 26 594 632.873691858608 -38.8736918586079 27 729 649.986183562570 79.0138164374295 28 731 796.437419845614 -65.4374198456143 29 386 403.772019288508 -17.7720192885085 30 331 329.369690454631 1.63030954536914 31 707 798.433719563752 -91.433719563752 32 715 732.368018862496 -17.3680188624962 33 657 640.285755054042 16.7142449459580 34 653 675.965023063797 -22.9650230637974 35 642 669.335960435504 -27.3359604355040 36 643 618.744768957194 24.2552310428059 37 718 716.375056846052 1.62494315394763 38 654 629.751504296823 24.2484957031771 39 632 651.128863791425 -19.1288637914246 40 731 775.608680041468 -44.6086800414681 41 392 395.270996571067 -3.27099657106742 42 344 322.571250876370 21.4287491236296 43 792 792.677571971166 -0.677571971165651 44 852 736.850598520679 115.149401479321 45 649 652.752699565494 -3.75269956549375 46 629 668.394573665001 -39.3945736650011 47 685 729.647462045082 -44.6474620450822 48 617 688.385645235486 -71.3856452354857 49 715 776.853132616627 -61.8531326166267 50 715 692.872749853831 22.1272501461686 51 629 723.498999415316 -94.4989994153155 52 916 839.496327954991 76.5036720450085 53 531 485.168295722391 45.8317042776090 54 357 407.622671579259 -50.6226715792592 55 917 859.513908726216 57.4860912737836 56 828 818.407959654031 9.5920403459689 57 708 714.235827766066 -6.2358277660659 58 858 740.297019430703 117.702980569297 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.160472088312954 0.320944176625908 0.839527911687046 20 0.0890915812943671 0.178183162588734 0.910908418705633 21 0.104043000923623 0.208086001847245 0.895956999076377 22 0.101778455909412 0.203556911818825 0.898221544090588 23 0.0684107915563523 0.136821583112705 0.931589208443648 24 0.0621117408361102 0.124223481672220 0.93788825916389 25 0.086790776011397 0.173581552022794 0.913209223988603 26 0.0482692594987604 0.0965385189975209 0.95173074050124 27 0.0728619904109107 0.145723980821821 0.92713800958909 28 0.163902627187174 0.327805254374349 0.836097372812826 29 0.112319101252584 0.224638202505168 0.887680898747416 30 0.0906570419370884 0.181314083874177 0.909342958062912 31 0.126593682121304 0.253187364242607 0.873406317878696 32 0.100344935163260 0.200689870326521 0.89965506483674 33 0.065065180386309 0.130130360772618 0.934934819613691 34 0.0377912551585088 0.0755825103170175 0.962208744841491 35 0.022281210201172 0.044562420402344 0.977718789798828 36 0.0211728285160158 0.0423456570320317 0.978827171483984 37 0.0194530017455784 0.0389060034911567 0.980546998254422 38 0.0166909044764902 0.0333818089529804 0.98330909552351 39 0.0443883366874526 0.0887766733749053 0.955611663312547 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 4 0.190476190476190 NOK 10% type I error level 7 0.333333333333333 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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