Q3 - Bob Leysen - Seatbelt Law

*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, 22 Nov 2008 08:57:05 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Nov/22/t1227369454r0348709xqvi48q.htm/, Retrieved Sat, 22 Nov 2008 15:57:43 +0000

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/2008/Nov/22/t1227369454r0348709xqvi48q.htm/}, year = {2008}, } @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 = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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User-defined keywords:

Dataseries X:

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15107 0 15024 0 12083 0 15761 0 16943 0 15070 0 13660 0 14769 0 14725 0 15998 0 15371 0 14957 0 15470 0 15102 0 11704 0 16284 0 16727 0 14969 0 14861 0 14583 0 15306 0 17904 0 16379 0 15420 0 17871 0 15913 0 13867 0 17823 0 17872 0 17422 0 16705 0 15991 0 16584 0 19124 0 17839 0 17209 0 18587 0 16258 0 15142 1 19202 1 17747 1 19090 1 18040 1 17516 1 17752 1 21073 1 17170 1 19440 1 19795 1 17575 1 16165 1 19465 1 19932 1 19961 1 17343 1 18924 1 18574 1 21351 1 18595 1 19823 1 20844 1 19640 1 17735 1 19814 1 22239 1 20682 1 17819 1 21872 1 22117 1 21866 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 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation x[t] = + 14076.3826086956 + 229.952898550719y[t] + 1035.84078099838M1[t] -413.421336553946M2[t] -2676.67560386473M3[t] + 843.228945249598M4[t] + 1272.80016103060M5[t] + 472.871376811594M6[t] -1077.05740740741M7[t] -294.819524959742M8[t] -149.914975845411M9[t] + 1804.15623993559M10[t] -210.071215780999M11[t] + 88.9287842190017t + 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) 14076.3826086956 404.507584 34.7988 0 0 y 229.952898550719 372.136901 0.6179 0.539129 0.269565 M1 1035.84078099838 465.143914 2.2269 0.029991 0.014995 M2 -413.421336553946 464.753566 -0.8895 0.377513 0.188757 M3 -2676.67560386473 468.539967 -5.7128 0 0 M4 843.228945249598 467.458052 1.8039 0.076633 0.038316 M5 1272.80016103060 466.55407 2.7281 0.008494 0.004247 M6 472.871376811594 465.829057 1.0151 0.314416 0.157208 M7 -1077.05740740741 465.28385 -2.3148 0.024312 0.012156 M8 -294.819524959742 464.91908 -0.6341 0.528577 0.264289 M9 -149.914975845411 464.735172 -0.3226 0.748214 0.374107 M10 1804.15623993559 464.732342 3.8821 0.000275 0.000138 M11 -210.071215780999 485.074324 -0.4331 0.666627 0.333314 t 88.9287842190017 9.174418 9.6931 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.956290780826021 R-squared 0.914492057492842 Adjusted R-squared 0.894641999410823 F-TEST (value) 46.0699940380139 F-TEST (DF numerator) 13 F-TEST (DF denominator) 56 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 766.832656869349 Sum Squared Residuals 32929810.1239131 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 15107 15201.1521739131 -94.1521739130608 2 15024 13840.8188405797 1183.18115942029 3 12083 11666.4933574879 416.506642512088 4 15761 15275.3266908213 485.673309178745 5 16943 15793.8266908213 1149.17330917875 6 15070 15082.8266908213 -12.8266908212572 7 13660 13621.8266908213 38.1733091787446 8 14769 14492.9933574879 276.006642512078 9 14725 14726.8266908213 -1.82669082125556 10 15998 16769.8266908213 -771.826690821257 11 15371 14844.5280193237 526.471980676328 12 14957 15143.5280193237 -186.528019323673 13 15470 16268.2975845411 -798.297584541058 14 15102 14907.9642512077 194.035748792267 15 11704 12733.6387681159 -1029.63876811594 16 16284 16342.4721014493 -58.4721014492754 17 16727 16860.9721014493 -133.972101449276 18 14969 16149.9721014493 -1180.97210144927 19 14861 14688.9721014493 172.027898550724 20 14583 15560.1387681159 -977.138768115942 21 15306 15793.9721014493 -487.972101449275 22 17904 17836.9721014493 67.027898550725 23 16379 15911.6734299517 467.326570048309 24 15420 16210.6734299517 -790.673429951692 25 17871 17335.4429951691 535.557004830921 26 15913 15975.1096618357 -62.1096618357488 27 13867 13800.7841787440 66.2158212560356 28 17823 17409.6175120773 413.382487922705 29 17872 17928.1175120773 -56.1175120772951 30 17422 17217.1175120773 204.882487922705 31 16705 15756.1175120773 948.882487922705 32 15991 16627.2841787440 -636.284178743961 33 16584 16861.1175120773 -277.117512077295 34 19124 18904.1175120773 219.882487922706 35 17839 16978.8188405797 860.181159420291 36 17209 17277.8188405797 -68.8188405797107 37 18587 18402.5884057971 184.411594202902 38 16258 17042.2550724638 -784.255072463767 39 15142 15097.8824879227 44.1175120772932 40 19202 18706.7158212560 495.284178743962 41 17747 19225.2158212560 -1478.21582125604 42 19090 18514.2158212560 575.784178743962 43 18040 17053.2158212560 986.784178743962 44 17516 17924.3824879227 -408.382487922705 45 17752 18158.2158212560 -406.215821256038 46 21073 20201.2158212560 871.784178743962 47 17170 18275.9171497585 -1105.91714975845 48 19440 18574.9171497585 865.082850241545 49 19795 19699.6867149758 95.3132850241586 50 17575 18339.3533816425 -764.353381642512 51 16165 16165.0278985507 -0.0278985507267091 52 19465 19773.8612318841 -308.861231884058 53 19932 20292.3612318841 -360.361231884058 54 19961 19581.3612318841 379.638768115942 55 17343 18120.3612318841 -777.361231884058 56 18924 18991.5278985507 -67.5278985507247 57 18574 19225.3612318841 -651.361231884058 58 21351 21268.3612318841 82.6387681159416 59 18595 19343.0625603865 -748.062560386473 60 19823 19642.0625603865 180.937439613526 61 20844 20766.8321256039 77.1678743961387 62 19640 19406.4987922705 233.501207729469 63 17735 17232.1733091787 502.826690821254 64 19814 20841.0066425121 -1027.00664251208 65 22239 21359.5066425121 879.493357487922 66 20682 20648.5066425121 33.4933574879227 67 17819 19187.5066425121 -1368.50664251208 68 21872 20058.6733091787 1813.32669082126 69 22117 20292.5066425121 1824.49335748792 70 21866 22335.5066425121 -469.506642512077 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0706270911543349 0.141254182308670 0.929372908845665 18 0.025837840128451 0.051675680256902 0.974162159871549 19 0.0762672223522043 0.152534444704409 0.923732777647796 20 0.0439659980411562 0.0879319960823124 0.956034001958844 21 0.0243259111007940 0.0486518222015879 0.975674088899206 22 0.112792317262746 0.225584634525492 0.887207682737254 23 0.0850954854917458 0.170190970983492 0.914904514508254 24 0.0551516987575359 0.110303397515072 0.944848301242464 25 0.192596266306565 0.38519253261313 0.807403733693435 26 0.132704248865979 0.265408497731958 0.867295751134021 27 0.123587064155132 0.247174128310265 0.876412935844868 28 0.09905137852555 0.1981027570511 0.90094862147445 29 0.0642045196440858 0.128409039288172 0.935795480355914 30 0.072114685014847 0.144229370029694 0.927885314985153 31 0.0964011241262844 0.192802248252569 0.903598875873716 32 0.0739999182373505 0.147999836474701 0.92600008176265 33 0.0504110395722535 0.100822079144507 0.949588960427747 34 0.0387430005697441 0.0774860011394883 0.961256999430256 35 0.0490136705372855 0.0980273410745711 0.950986329462714 36 0.0341248646717049 0.0682497293434098 0.965875135328295 37 0.0226269519603833 0.0452539039207666 0.977373048039617 38 0.0247365592726754 0.0494731185453509 0.975263440727325 39 0.0143288082198439 0.0286576164396879 0.985671191780156 40 0.0131761107896089 0.0263522215792179 0.98682388921039 41 0.0442503327952324 0.0885006655904648 0.955749667204768 42 0.0454064620934052 0.0908129241868105 0.954593537906595 43 0.148126677880081 0.296253355760162 0.851873322119919 44 0.123772263907210 0.247544527814419 0.87622773609279 45 0.095292728441172 0.190585456882344 0.904707271558828 46 0.15599266411493 0.31198532822986 0.84400733588507 47 0.175565285474029 0.351130570948057 0.824434714525971 48 0.19525633352163 0.39051266704326 0.80474366647837 49 0.137499331753965 0.274998663507931 0.862500668246035 50 0.100795358244210 0.201590716488420 0.89920464175579 51 0.0566051054938698 0.113210210987740 0.94339489450613 52 0.0546331543541324 0.109266308708265 0.945366845645868 53 0.0286062902746122 0.0572125805492244 0.971393709725388 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 5 0.135135135135135 NOK 10% type I error level 13 0.351351351351351 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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