Case: the Seatbelt Law & Tutorial - Q3(b)

*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 09:44:09 -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/t1227717881lh90l2vgxheeovk.htm/, Retrieved Wed, 26 Nov 2008 16:44:54 +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/t1227717881lh90l2vgxheeovk.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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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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Inflatie_België[t] = + 1.42011577718145 + 0.118293552509654Inflatie_energiedragers[t] + 0.157168505575281M1[t] + 0.0388749530656324M2[t] + 0.100167792517370M3[t] + 0.030898108501924M4[t] + 0.0298250769034686M5[t] + 0.0420940478609984M6[t] -0.0144410418223971M7[t] + 0.0759016737915046M8[t] + 0.0587071605482621M9[t] + 0.0152682578996135M10[t] + 0.0812928394517373M11[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) 1.42011577718145 0.244267 5.8138 1e-06 0 Inflatie_energiedragers 0.118293552509654 0.007805 15.1555 0 0 M1 0.157168505575281 0.33703 0.4663 0.643131 0.321566 M2 0.0388749530656324 0.337583 0.1152 0.908811 0.454406 M3 0.100167792517370 0.33747 0.2968 0.767912 0.383956 M4 0.030898108501924 0.338675 0.0912 0.927696 0.463848 M5 0.0298250769034686 0.338534 0.0881 0.930171 0.465086 M6 0.0420940478609984 0.337912 0.1246 0.901394 0.450697 M7 -0.0144410418223971 0.336924 -0.0429 0.965994 0.482997 M8 0.0759016737915046 0.336508 0.2256 0.822524 0.411262 M9 0.0587071605482621 0.33644 0.1745 0.862226 0.431113 M10 0.0152682578996135 0.336438 0.0454 0.963995 0.481997 M11 0.0812928394517373 0.33644 0.2416 0.810121 0.40506 Multiple Linear Regression - Regression Statistics Multiple R 0.914661630495795 R-squared 0.836605898301226 Adjusted R-squared 0.794888255314305 F-TEST (value) 20.0540068518136 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 1.49880108324396e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.531954328560216 Sum Squared Residuals 13.2998441606757 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 4.8 3.84852049094212 0.951479509057882 2 5.5 4.60559922700388 0.894400772996119 3 5.4 4.66689206645562 0.733107933544381 4 5.9 5.16543143448651 0.734568565513489 5 5.8 5.14069969238613 0.659300307613875 6 5.1 4.58515961129732 0.514840388702683 7 4.1 3.85435127230889 0.245648727691105 8 4.4 3.9446939879228 0.455306012077204 9 3.6 3.25322622537453 0.346773774625473 10 3.5 3.00868828345947 0.491311716540533 11 3.1 2.53056252346718 0.569437476532816 12 2.9 2.62671001277993 0.273289987220073 13 2.2 2.08594655854825 0.11405344145175 14 1.4 1.44716137499613 -0.0471613749961258 15 1.2 0.97613322815442 0.223866771845579 16 1.3 0.989669030895733 0.310330969104267 17 1.3 1.03591342030114 0.264086579698862 18 1.3 1.06001174650963 0.239988253490367 19 1.8 1.10994085408493 0.690059145915074 20 1.8 1.36589454321234 0.434105456787656 21 1.8 1.51431100348262 0.285688996517383 22 1.7 1.32891983782238 0.371080162177617 23 2.1 1.92726540566795 0.172734594332051 24 2 1.73950836895752 0.260491631042477 25 1.7 1.55362557225481 0.146374427745192 26 1.9 1.34069717773744 0.559302822262563 27 2.3 2.20638617425482 0.09361382574518 28 2.4 2.20809262174517 0.191907378254833 29 2.5 2.58555895817760 -0.0855589581776032 30 2.8 3.02368471816989 -0.223684718169887 31 2.6 2.65958639196139 -0.0595863919613912 32 2.2 2.78541717332819 -0.585417173328189 33 2.8 3.00480976510425 -0.204809765104254 34 2.8 3.31625151998457 -0.516251519984567 35 2.8 2.94458995725097 -0.144589957250972 36 2.3 2.49658710501931 -0.196587105019308 37 2.2 2.64192625534362 -0.441926255343623 38 3 3.51729854391507 -0.517298543915066 39 2.9 3.25919879159074 -0.359198791590739 40 2.7 3.4620042783475 -0.762004278347497 41 2.7 3.07056252346718 -0.370562523467184 42 2.3 2.58599857388417 -0.285998573884168 43 2.4 3.15641931250194 -0.756419312501937 44 2.8 3.10480976510425 -0.304809765104255 45 2.3 2.96932169935136 -0.669321699351358 46 2 2.48819665241699 -0.48819665241699 47 1.9 2.70800285223166 -0.808002852231664 48 2.3 2.92244389405406 -0.622443894054062 49 2.7 3.4699811229112 -0.7699811229112 50 1.8 2.68924367634749 -0.88924367634749 51 2 2.6913897395444 -0.691389739544401 52 2.1 2.57480263452509 -0.474802634525093 53 2 2.46726540566795 -0.46726540566795 54 2.4 2.64514535013899 -0.245145350138995 55 1.7 1.81970216914285 -0.119702169142850 56 1 0.999184530432417 0.00081546956758323 57 1.2 0.958331306687243 0.241668693312756 58 1.4 1.25794370631659 0.142056293683409 59 1.7 1.48957926138223 0.210420738617770 60 1.8 1.51475061918918 0.285249380810819 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.698616695912173 0.602766608175654 0.301383304087827 17 0.621391021423344 0.757217957153313 0.378608978576656 18 0.542020808882708 0.915958382234584 0.457979191117292 19 0.802034472966898 0.395931054066204 0.197965527033102 20 0.75703058303965 0.485938833920699 0.242969416960350 21 0.668473493351429 0.663053013297142 0.331526506648571 22 0.574927453632275 0.850145092735449 0.425072546367725 23 0.534014737117672 0.931970525764657 0.465985262882328 24 0.441972772041828 0.883945544083655 0.558027227958172 25 0.377962333252612 0.755924666505223 0.622037666747388 26 0.44683243365532 0.89366486731064 0.55316756634468 27 0.448680964665624 0.897361929331249 0.551319035334376 28 0.498028609246841 0.996057218493681 0.501971390753159 29 0.553757948374588 0.892484103250825 0.446242051625412 30 0.629641320359888 0.740717359280224 0.370358679640112 31 0.715823807659271 0.568352384681457 0.284176192340729 32 0.866365369105111 0.267269261789778 0.133634630894889 33 0.879787244505023 0.240425510989954 0.120212755494977 34 0.919993893182165 0.160012213635669 0.0800061068178347 35 0.935400856821948 0.129198286356104 0.0645991431780521 36 0.906924262662709 0.186151474674583 0.0930757373372913 37 0.89755434103453 0.204891317930941 0.102445658965471 38 0.961204548472654 0.077590903054692 0.038795451527346 39 0.976207943614421 0.0475841127711575 0.0237920563855787 40 0.969180355927694 0.0616392881446128 0.0308196440723064 41 0.958814906409289 0.0823701871814223 0.0411850935907111 42 0.913630529037728 0.172738941924545 0.0863694709622724 43 0.852545681662423 0.294908636675154 0.147454318337577 44 0.951901863002157 0.0961962739956852 0.0480981369978426 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 1 0.0344827586206897 OK 10% type I error level 5 0.172413793103448 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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