Seatbelt Law Model 2

*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: Mon, 14 Dec 2009 12:23:19 -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/14/t1260818643jbap3dnp8kaey06.htm/, Retrieved Mon, 14 Dec 2009 20:24:16 +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/14/t1260818643jbap3dnp8kaey06.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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8.9 -3 8.8 -1 8.3 -3 7.5 -4 7.2 -6 7.4 0 8.8 -4 9.3 -2 9.3 -2 8.7 -6 8.2 -7 8.3 -6 8.5 -6 8.6 -3 8.5 -2 8.2 -5 8.1 -11 7.9 -11 8.6 -11 8.7 -10 8.7 -14 8.5 -8 8.4 -9 8.5 -5 8.7 -1 8.7 -2 8.6 -5 8.5 -4 8.3 -6 8 -2 8.2 -2 8.1 -2 8.1 -2 8 2 7.9 1 7.9 -8 8 -1 8 1 7.9 -1 8 2 7.7 2 7.2 1 7.5 -1 7.3 -2 7 -2 7 -1 7 -8 7.2 -4 7.3 -6 7.1 -3 6.8 -3 6.4 -7 6.1 -9 6.5 -11 7.7 -13 7.9 -11 7.5 -9 6.9 -17 6.6 -22 6.9 -25

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 7 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation TW[t] = + 8.0769536423841 + 0.0330160044150110CV[t] + 0.315300772626926M1[t] + 0.215871964679911M2[t] + 0.0354911699779244M3[t] -0.238096026490067M4[t] -0.398857615894041M5[t] -0.525080022075055M6[t] + 0.287745584988962M7[t] + 0.361332781456953M8[t] + 0.234539183222958M9[t] -0.0588576158940406M10[t] -0.159809602649007M11[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) 8.0769536423841 0.378928 21.3153 0 0 CV 0.0330160044150110 0.019386 1.7031 0.095151 0.047576 M1 0.315300772626926 0.482028 0.6541 0.516227 0.258114 M2 0.215871964679911 0.491891 0.4389 0.662773 0.331386 M3 0.0354911699779244 0.485059 0.0732 0.941982 0.470991 M4 -0.238096026490067 0.481076 -0.4949 0.62296 0.31148 M5 -0.398857615894041 0.471991 -0.8451 0.402362 0.201181 M6 -0.525080022075055 0.47676 -1.1014 0.276353 0.138176 M7 0.287745584988962 0.471433 0.6104 0.544561 0.272281 M8 0.361332781456953 0.47385 0.7625 0.449544 0.224772 M9 0.234539183222958 0.47258 0.4963 0.621999 0.311 M10 -0.0588576158940406 0.471991 -0.1247 0.901292 0.450646 M11 -0.159809602649007 0.466948 -0.3422 0.733694 0.366847 Multiple Linear Regression - Regression Statistics Multiple R 0.463186198470339 R-squared 0.214541454453404 Adjusted R-squared 0.0139988470798053 F-TEST (value) 1.06980485226128 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0.40595074698375 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.738080518422222 Sum Squared Residuals 25.6038540286975 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.9 8.29320640176602 0.606793598233977 2 8.8 8.259809602649 0.540190397350994 3 8.3 8.013396799117 0.286603200883003 4 7.5 7.706793598234 -0.206793598233995 5 7.2 7.48 -0.280000000000000 6 7.4 7.55187362030905 -0.151873620309050 7 8.8 8.23263520971302 0.567364790286977 8 9.3 8.37225441501104 0.927745584988963 9 9.3 8.24546081677704 1.05453918322296 10 8.7 7.82 0.88 11 8.2 7.68603200883002 0.513967991169977 12 8.3 7.87885761589404 0.42114238410596 13 8.5 8.19415838852097 0.305841611479034 14 8.6 8.19377759381898 0.406222406181015 15 8.5 8.046412803532 0.453587196467991 16 8.2 7.67377759381898 0.526222406181015 17 8.1 7.31491997792495 0.785080022075055 18 7.9 7.18869757174393 0.71130242825607 19 8.6 8.00152317880795 0.598476821192052 20 8.7 8.10812637969095 0.59187362030905 21 8.7 7.84926876379691 0.85073123620309 22 8.5 7.75396799116998 0.746032008830022 23 8.4 7.62 0.78 24 8.5 7.91187362030905 0.588126379690949 25 8.7 8.35923841059602 0.340761589403978 26 8.7 8.226793598234 0.473206401766004 27 8.6 7.94736479028698 0.652635209713024 28 8.5 7.706793598234 0.793206401766004 29 8.3 7.48 0.82 30 8 7.48584161147903 0.514158388520971 31 8.2 8.29866721854305 -0.0986672185430473 32 8.1 8.37225441501104 -0.272254415011038 33 8.1 8.24546081677704 -0.145460816777042 34 8 8.08412803532009 -0.0841280353200881 35 7.9 7.9501600441501 -0.0501600441501097 36 7.9 7.81282560706402 0.0871743929359821 37 8 8.35923841059602 -0.359238410596021 38 8 8.32584161147903 -0.325841611479029 39 7.9 8.07942880794702 -0.179428807947020 40 8 7.90488962472406 0.095110375275938 41 7.7 7.74412803532009 -0.0441280353200881 42 7.2 7.58488962472406 -0.384889624724062 43 7.5 8.33168322295806 -0.831683222958057 44 7.3 8.37225441501104 -1.07225441501104 45 7 8.24546081677704 -1.24546081677704 46 7 7.98508002207506 -0.985080022075055 47 7 7.65301600441501 -0.653016004415011 48 7.2 7.94488962472406 -0.744889624724062 49 7.3 8.19415838852097 -0.894158388520966 50 7.1 8.19377759381898 -1.09377759381898 51 6.8 8.013396799117 -1.21339679911700 52 6.4 7.60774558498896 -1.20774558498896 53 6.1 7.38095198675497 -1.28095198675497 54 6.5 7.18869757174393 -0.68869757174393 55 7.7 7.93549116997792 -0.235491169977925 56 7.9 8.07511037527594 -0.175110375275937 57 7.5 8.01434878587196 -0.514348785871965 58 6.9 7.45682395143488 -0.556823951434878 59 6.6 7.19079194260486 -0.590791942604857 60 6.9 7.25155353200883 -0.35155353200883 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0905437923260304 0.181087584652061 0.909456207673970 17 0.0986213871374158 0.197242774274832 0.901378612862584 18 0.0468337972049642 0.0936675944099285 0.953166202795036 19 0.0320624519831754 0.0641249039663507 0.967937548016825 20 0.0373109884565413 0.0746219769130827 0.962689011543459 21 0.0376312771609689 0.0752625543219378 0.962368722839031 22 0.0255907849250844 0.0511815698501689 0.974409215074916 23 0.0176152707528054 0.0352305415056108 0.982384729247195 24 0.0101509107662319 0.0203018215324638 0.989849089233768 25 0.00572643228655548 0.0114528645731110 0.994273567713445 26 0.00387995288194981 0.00775990576389961 0.99612004711805 27 0.00391928271385136 0.00783856542770272 0.996080717286149 28 0.0103044969518619 0.0206089939037239 0.989695503048138 29 0.0330626023724212 0.0661252047448424 0.966937397627579 30 0.0374904298154384 0.0749808596308769 0.962509570184562 31 0.0368368043822846 0.0736736087645692 0.963163195617715 32 0.0599149149256733 0.119829829851347 0.940085085074327 33 0.0925118156129695 0.185023631225939 0.90748818438703 34 0.0820576578312827 0.164115315662565 0.917942342168717 35 0.0599412159881346 0.119882431976269 0.940058784011865 36 0.0602184422758688 0.120436884551738 0.939781557724131 37 0.0588477024837732 0.117695404967546 0.941152297516227 38 0.0653928170508678 0.130785634101736 0.934607182949132 39 0.0895646398462824 0.179129279692565 0.910435360153718 40 0.208816020828253 0.417632041656506 0.791183979171747 41 0.651771100391394 0.696457799217212 0.348228899608606 42 0.712912817785643 0.574174364428714 0.287087182214357 43 0.647547221554577 0.704905556890847 0.352452778445423 44 0.804694649859458 0.390610700281085 0.195305350140542 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 2 0.0689655172413793 NOK 5% type I error level 6 0.206896551724138 NOK 10% type I error level 14 0.482758620689655 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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