case: seatbelt law: Q3 multiple regression

*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: Sun, 23 Nov 2008 13:31:17 -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/23/t1227472331ek75gjoiljomsk7.htm/, Retrieved Sun, 23 Nov 2008 20:32:20 +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/23/t1227472331ek75gjoiljomsk7.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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Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

multiple regression

Dataseries X:

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25 0 23.6 0 22.3 0 21.8 0 20.8 0 19.7 0 18.3 0 17.4 0 17 0 18.1 0 23.9 0 25.6 0 25.3 0 23.6 0 21.9 0 21.4 0 20.6 0 20.5 0 20.2 0 20.6 0 19.7 0 19.3 0 22.8 0 23.5 0 23.8 0 22.6 0 22 0 21.7 0 20.7 0 20.2 0 19.1 0 19.5 0 18.7 0 18.6 0 22.2 0 23.2 0 23.5 0 21.3 0 20 0 18.7 0 18.9 0 18.3 0 18.4 0 19.9 0 19.2 0 18.5 0 20.9 1 20.5 1 19.4 1 18.1 1 17 1 17 1 17.3 1 16.7 1 15.5 1 15.3 1 13.7 1 14.1 1 17.3 1 18.1 1 18.1 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 werklozen[t] = + 20.9108695652174 -3.64420289855072jobtonic[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) 20.9108695652174 0.321358 65.0703 0 0 jobtonic -3.64420289855072 0.64805 -5.6233 1e-06 0 Multiple Linear Regression - Regression Statistics Multiple R 0.590713985271516 R-squared 0.348943012395357 Adjusted R-squared 0.337908148198669 F-TEST (value) 31.6218673991531 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 5.40300283180528e-07 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 2.17955794985863 Sum Squared Residuals 280.277898550725 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 25 20.9108695652174 4.08913043478261 2 23.6 20.9108695652174 2.68913043478261 3 22.3 20.9108695652174 1.38913043478261 4 21.8 20.9108695652174 0.88913043478261 5 20.8 20.9108695652174 -0.110869565217391 6 19.7 20.9108695652174 -1.21086956521739 7 18.3 20.9108695652174 -2.61086956521739 8 17.4 20.9108695652174 -3.51086956521739 9 17 20.9108695652174 -3.91086956521739 10 18.1 20.9108695652174 -2.81086956521739 11 23.9 20.9108695652174 2.98913043478261 12 25.6 20.9108695652174 4.68913043478261 13 25.3 20.9108695652174 4.38913043478261 14 23.6 20.9108695652174 2.68913043478261 15 21.9 20.9108695652174 0.989130434782607 16 21.4 20.9108695652174 0.489130434782607 17 20.6 20.9108695652174 -0.31086956521739 18 20.5 20.9108695652174 -0.410869565217391 19 20.2 20.9108695652174 -0.710869565217392 20 20.6 20.9108695652174 -0.31086956521739 21 19.7 20.9108695652174 -1.21086956521739 22 19.3 20.9108695652174 -1.61086956521739 23 22.8 20.9108695652174 1.88913043478261 24 23.5 20.9108695652174 2.58913043478261 25 23.8 20.9108695652174 2.88913043478261 26 22.6 20.9108695652174 1.68913043478261 27 22 20.9108695652174 1.08913043478261 28 21.7 20.9108695652174 0.789130434782608 29 20.7 20.9108695652174 -0.210869565217392 30 20.2 20.9108695652174 -0.710869565217392 31 19.1 20.9108695652174 -1.81086956521739 32 19.5 20.9108695652174 -1.41086956521739 33 18.7 20.9108695652174 -2.21086956521739 34 18.6 20.9108695652174 -2.31086956521739 35 22.2 20.9108695652174 1.28913043478261 36 23.2 20.9108695652174 2.28913043478261 37 23.5 20.9108695652174 2.58913043478261 38 21.3 20.9108695652174 0.389130434782609 39 20 20.9108695652174 -0.910869565217391 40 18.7 20.9108695652174 -2.21086956521739 41 18.9 20.9108695652174 -2.01086956521739 42 18.3 20.9108695652174 -2.61086956521739 43 18.4 20.9108695652174 -2.51086956521739 44 19.9 20.9108695652174 -1.01086956521739 45 19.2 20.9108695652174 -1.71086956521739 46 18.5 20.9108695652174 -2.41086956521739 47 20.9 17.2666666666667 3.63333333333333 48 20.5 17.2666666666667 3.23333333333333 49 19.4 17.2666666666667 2.13333333333333 50 18.1 17.2666666666667 0.833333333333334 51 17 17.2666666666667 -0.266666666666667 52 17 17.2666666666667 -0.266666666666667 53 17.3 17.2666666666667 0.0333333333333337 54 16.7 17.2666666666667 -0.566666666666668 55 15.5 17.2666666666667 -1.76666666666667 56 15.3 17.2666666666667 -1.96666666666667 57 13.7 17.2666666666667 -3.56666666666667 58 14.1 17.2666666666667 -3.16666666666667 59 17.3 17.2666666666667 0.0333333333333337 60 18.1 17.2666666666667 0.833333333333334 61 18.1 17.2666666666667 0.833333333333334 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.480757592332392 0.961515184664785 0.519242407667608 6 0.563457944664737 0.873084110670525 0.436542055335263 7 0.732655120457563 0.534689759084874 0.267344879542437 8 0.863319327666919 0.273361344666161 0.136680672333081 9 0.931301674798139 0.137396650403722 0.0686983252018611 10 0.932328003415436 0.135343993169128 0.0676719965845641 11 0.951931509456499 0.0961369810870024 0.0480684905435012 12 0.988189951176306 0.0236200976473888 0.0118100488236944 13 0.996515285240394 0.006969429519211 0.0034847147596055 14 0.996665969078962 0.00666806184207668 0.00333403092103834 15 0.994316220734841 0.0113675585303171 0.00568377926515853 16 0.990229108711803 0.0195417825763944 0.00977089128819722 17 0.984164873811934 0.0316702523761311 0.0158351261880655 18 0.975366082863716 0.0492678342725679 0.0246339171362840 19 0.964023806396733 0.0719523872065349 0.0359761936032675 20 0.9466338270643 0.106732345871400 0.0533661729356999 21 0.931048739609968 0.137902520780063 0.0689512603900315 22 0.918492094304942 0.163015811390117 0.0815079056950584 23 0.909575824051058 0.180848351897884 0.0904241759489418 24 0.921345205289833 0.157309589420335 0.0786547947101674 25 0.943300437308397 0.113399125383206 0.0566995626916031 26 0.939073944947206 0.121852110105588 0.0609260550527938 27 0.92620762213209 0.147584755735821 0.0737923778679107 28 0.90827507612444 0.183449847751119 0.0917249238755596 29 0.87843554271437 0.243128914571261 0.121564457285630 30 0.843423433162879 0.313153133674242 0.156576566837121 31 0.823220957438645 0.353558085122711 0.176779042561355 32 0.788003112980399 0.423993774039203 0.211996887019601 33 0.7761497519965 0.447700496006999 0.223850248003499 34 0.767433369671229 0.465133260657542 0.232566630328771 35 0.748442298127258 0.503115403745483 0.251557701872742 36 0.802614989936836 0.394770020126328 0.197385010063164 37 0.894864159857096 0.210271680285807 0.105135840142904 38 0.889996436595486 0.220007126809029 0.110003563404514 39 0.862070256775365 0.275859486449269 0.137929743224635 40 0.833357652237744 0.333284695524513 0.166642347762256 41 0.794750218721188 0.410499562557625 0.205249781278812 42 0.764037441167214 0.471925117665573 0.235962558832786 43 0.725923149823625 0.54815370035275 0.274076850176375 44 0.663522870319198 0.672954259361603 0.336477129680802 45 0.595489016916326 0.809021966167347 0.404510983083674 46 0.529764081366665 0.94047183726667 0.470235918633335 47 0.657123281006657 0.685753437986686 0.342876718993343 48 0.792316213749123 0.415367572501754 0.207683786250877 49 0.850439054918946 0.299121890162108 0.149560945081054 50 0.836675193518504 0.326649612962992 0.163324806481496 51 0.777612328108048 0.444775343783905 0.222387671891952 52 0.69993050844185 0.600138983116299 0.300069491558150 53 0.620440584935626 0.759118830128749 0.379559415064374 54 0.504513040737213 0.990973918525574 0.495486959262787 55 0.383009287899178 0.766018575798356 0.616990712100822 56 0.266772123801819 0.533544247603638 0.733227876198181 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 2 0.0384615384615385 NOK 5% type I error level 7 0.134615384615385 NOK 10% type I error level 9 0.173076923076923 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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