Case: the Seatbelt Law & Tutorial - Q3

*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, 26 Nov 2008 09:42:24 -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/26/t12277178123xlryn4ty5br4d4.htm/, Retrieved Wed, 26 Nov 2008 16:43:32 +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/26/t12277178123xlryn4ty5br4d4.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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4.8 19.2 5.5 26.6 5.4 26.6 5.9 31.4 5.8 31.2 5.1 26.4 4.1 20.7 4.4 20.7 3.6 15 3.5 13.3 3.1 8.7 2.9 10.2 2.2 4.3 1.4 -0.1 1.2 -4.6 1.3 -3.9 1.3 -3.5 1.3 -3.4 1.8 -2.5 1.8 -1.1 1.8 0.3 1.7 -0.9 2.1 3.6 2 2.7 1.7 -0.2 1.9 -1 2.3 5.8 2.4 6.4 2.5 9.6 2.8 13.2 2.6 10.6 2.2 10.9 2.8 12.9 2.8 15.9 2.8 12.2 2.3 9.1 2.2 9 3 17.4 2.9 14.7 2.7 17 2.7 13.7 2.3 9.5 2.4 14.8 2.8 13.6 2.3 12.6 2 8.9 1.9 10.2 2.3 12.7 2.7 16 1.8 10.4 2 9.9 2.1 9.5 2 8.6 2.4 10 1.7 3.5 1 -4.2 1.2 -4.4 1.4 -1.5 1.7 -0.1 1.8 0.8

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 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Inflatie_België[t] = + 1.47130483639317 + 0.118306870310270Inflatie_energiedragers[t] + 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) 1.47130483639317 0.089396 16.4582 0 0 Inflatie_energiedragers 0.118306870310270 0.006903 17.1385 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.913837761374175 R-squared 0.835099454113363 Adjusted R-squared 0.832256341253249 F-TEST (value) 293.727155832904 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.481063202894402 Sum Squared Residuals 13.4224647003832 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 4.8 3.74279674635036 1.05720325364964 2 5.5 4.61826758664636 0.881732413353644 3 5.4 4.61826758664635 0.781732413353646 4 5.9 5.18614056413565 0.713859435864349 5 5.8 5.1624791900736 0.637520809926402 6 5.1 4.5946062125843 0.5053937874157 7 4.1 3.92025705181576 0.17974294818424 8 4.4 3.92025705181576 0.479742948184241 9 3.6 3.24590789104722 0.354092108952781 10 3.5 3.04478621151976 0.45521378848024 11 3.1 2.50057460809252 0.599425391907483 12 2.9 2.67803491355792 0.221965086442078 13 2.2 1.98002437872733 0.219975621272672 14 1.4 1.45947414936214 -0.0594741493621391 15 1.2 0.927093232965923 0.272906767034077 16 1.3 1.00990804218311 0.290091957816888 17 1.3 1.05723079030722 0.24276920969278 18 1.3 1.06906147733825 0.230938522661753 19 1.8 1.17553766061749 0.62446233938251 20 1.8 1.34116727905187 0.458832720948131 21 1.8 1.50679689748625 0.293203102513753 22 1.7 1.36482865311392 0.335171346886077 23 2.1 1.89720956951014 0.202790430489861 24 2 1.79073338623090 0.209266613769104 25 1.7 1.44764346233111 0.252356537668888 26 1.9 1.35299796608290 0.547002033917104 27 2.3 2.15748468419273 0.142515315807267 28 2.4 2.22846880637890 0.171531193621104 29 2.5 2.60705079137176 -0.107050791371760 30 2.8 3.03295552448873 -0.232955524488733 31 2.6 2.72535766168203 -0.125357661682030 32 2.2 2.76084972277511 -0.560849722775111 33 2.8 2.99746346339565 -0.197463463395652 34 2.8 3.35238407432646 -0.552384074326463 35 2.8 2.91464865417846 -0.114648654178463 36 2.3 2.54789735621662 -0.247897356216625 37 2.2 2.5360666691856 -0.336066669185598 38 3 3.52984437979187 -0.529844379791868 39 2.9 3.21041582995414 -0.310415829954138 40 2.7 3.48252163166776 -0.78252163166776 41 2.7 3.09210895964387 -0.392108959643868 42 2.3 2.59522010434073 -0.295220104340733 43 2.4 3.22224651698517 -0.822246516985166 44 2.8 3.08027827261284 -0.280278272612841 45 2.3 2.96197140230257 -0.661971402302571 46 2 2.52423598215457 -0.524235982154571 47 1.9 2.67803491355792 -0.778034913557922 48 2.3 2.9738020893336 -0.673802089333598 49 2.7 3.36421476135749 -0.66421476135749 50 1.8 2.70169628761998 -0.901696287619976 51 2 2.64254285246484 -0.642542852464841 52 2.1 2.59522010434073 -0.495220104340733 53 2 2.48874392106149 -0.48874392106149 54 2.4 2.65437353949587 -0.254373539495868 55 1.7 1.88537888247911 -0.185378882479112 56 1 0.974415981090031 0.0255840189099690 57 1.2 0.950754607027977 0.249245392972023 58 1.4 1.29384453092776 0.106155469072239 59 1.7 1.45947414936214 0.240525850637861 60 1.8 1.56595033264138 0.234049667358618 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.00364598967906203 0.00729197935812406 0.996354010320938 6 0.0591244541194345 0.118248908238869 0.940875545880566 7 0.447050907210715 0.89410181442143 0.552949092789285 8 0.476723725614676 0.953447451229352 0.523276274385324 9 0.483664092480835 0.96732818496167 0.516335907519165 10 0.510519092365958 0.978961815268083 0.489480907634042 11 0.60981286774617 0.78037426450766 0.39018713225383 12 0.66873644094753 0.662527118104941 0.331263559052470 13 0.610628748375837 0.778742503248326 0.389371251624163 14 0.608565027181117 0.782869945637766 0.391434972818883 15 0.571645628382968 0.856708743234064 0.428354371617032 16 0.510924295308654 0.978151409382692 0.489075704691346 17 0.435203250077554 0.870406500155108 0.564796749922446 18 0.362635040858273 0.725270081716547 0.637364959141727 19 0.488047436427296 0.976094872854593 0.511952563572703 20 0.45869495800156 0.91738991600312 0.54130504199844 21 0.38766090864115 0.7753218172823 0.61233909135885 22 0.321212358950354 0.642424717900709 0.678787641049646 23 0.29364964992522 0.58729929985044 0.70635035007478 24 0.257816095321682 0.515632190643365 0.742183904678318 25 0.204776785924228 0.409553571848456 0.795223214075772 26 0.269345005993233 0.538690011986467 0.730654994006767 27 0.301723638267300 0.603447276534601 0.6982763617327 28 0.358118206352348 0.716236412704696 0.641881793647652 29 0.539403362984551 0.921193274030897 0.460596637015449 30 0.777924868297418 0.444150263405164 0.222075131702582 31 0.858411477591078 0.283177044817844 0.141588522408922 32 0.962331587242774 0.0753368255144525 0.0376684127572263 33 0.979140037702006 0.0417199245959871 0.0208599622979936 34 0.991996132651402 0.0160077346971963 0.00800386734859815 35 0.995910100207948 0.00817979958410391 0.00408989979205195 36 0.995560338530266 0.00887932293946704 0.00443966146973352 37 0.994953398778569 0.0100932024428621 0.00504660122143104 38 0.997254021453342 0.00549195709331574 0.00274597854665787 39 0.998677587087065 0.00264482582587032 0.00132241291293516 40 0.999048256777856 0.00190348644428706 0.000951743222143531 41 0.999203397721791 0.00159320455641791 0.000796602278208953 42 0.99886359161349 0.00227281677302189 0.00113640838651095 43 0.99881849911458 0.00236300177083837 0.00118150088541919 44 0.999599562093657 0.000800875812685835 0.000400437906342917 45 0.999299076667787 0.00140184666442695 0.000700923332213475 46 0.998595284237279 0.00280943152544272 0.00140471576272136 47 0.998683890096433 0.00263221980713320 0.00131610990356660 48 0.997292255603644 0.00541548879271259 0.00270774439635630 49 0.995996053719323 0.00800789256135325 0.00400394628067663 50 0.998374936127663 0.00325012774467375 0.00162506387233688 51 0.997649932626434 0.00470013474713214 0.00235006737356607 52 0.994483538520485 0.0110329229590310 0.00551646147951549 53 0.993243224387688 0.0135135512246250 0.00675677561231248 54 0.97699366143397 0.0460126771320594 0.0230063385660297 55 0.98764143004463 0.0247171399107387 0.0123585699553694 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 17 0.333333333333333 NOK 5% type I error level 24 0.470588235294118 NOK 10% type I error level 25 0.490196078431373 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 = No 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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