WS7: Mini-tutorial 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: Fri, 19 Nov 2010 11:46:42 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Nov/19/t1290167609nb4jhrxp3f41unr.htm/, Retrieved Fri, 19 Nov 2010 12:53:39 +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/2010/Nov/19/t1290167609nb4jhrxp3f41unr.htm/}, year = {2010}, } @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 = {2010}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

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Dataseries X:

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73 2 71,91 5,11 50 3 28 6 6,06 3,53 48 5 40 5 8,1 4,52 63 11 79 3 79,38 3,72 113 13 75 3 65,34 5,99 128 11 21 3 34,62 3,15 52 7 16 2 26,26 3,17 104 1 81 2 60,92 3,5 40 1 90 2 39,56 3,39 89 11 87 5 65,61 4,15 97 3 99 3 56,49 4,5 29 9 54 3 56,19 3,31 36 5 53 5 80,3 3,09 114 11 6 4 61,2 5,31 49 9 71 5 58,2 4,24 57 7 93 6 75,91 5,06 82 4 82 3 73,66 4,72 34 10 32 4 73,87 4,58 36 13 93 4 87,21 5,3 89 9 24 4 64,29 5,11 69 5 96 5 71,82 4,05 35 8 88 4 89,31 4,62 65 12 83 2 1,41 4,66 70 8 23 6 35,17 4,66 60 5 23 5 34,68 2,76 57 9 20 5 41,08 5,1 127 11 33 3 30,57 4,97 96 8 88 2 68,84 2,87 61 9 42 6 7,17 5,14 127 10 98 2 71,05 4,98 36 1 34 4 23,32 4,55 55 9 59 3 61,39 5,45 75 2 26 6 8,41 4,36 42 3 64 4 65,88 4,78 64 4 13 1 64,06 4,74 83 3 6 2 26,8 5,44 56 1 49 4 12,78 5,78 114 5 3 5 23,84 2,92 33 4 87 6 42,69 4,22 91 2 77 2 54,94 3,93 127 2 70 4 89,99 3,01 45 10 76 4 5,68 3,22 80 6 82 4 72,64 5,12 40 9 12 2 45,92 3,04 115 7 44 3 24,96 5,82 33 1 63 5 18,17 3,11 127 13 35 1 29, etc...

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 6 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation slaagkans[t] = + 21.2435270406808 -0.189021294324605verzekeraar[t] + 0.546658570171708kost[t] + 0.78868266792152grootte[t] + 0.0270026552096039snelheid[t] + 0.345099845668746maand[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) 21.2435270406808 24.710473 0.8597 0.394616 0.197308 verzekeraar -0.189021294324605 2.766775 -0.0683 0.945842 0.472921 kost 0.546658570171708 0.158908 3.4401 0.001284 0.000642 grootte 0.78868266792152 4.447924 0.1773 0.860075 0.430037 snelheid 0.0270026552096039 0.134153 0.2013 0.841405 0.420703 maand 0.345099845668746 1.134591 0.3042 0.762438 0.381219 Multiple Linear Regression - Regression Statistics Multiple R 0.479791384134689 R-squared 0.23019977228988 Adjusted R-squared 0.142722473686458 F-TEST (value) 2.63153727841424 F-TEST (DF numerator) 5 F-TEST (DF denominator) 44 p-value 0.0363218943091438 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 27.7904116050644 Sum Squared Residuals 33981.5069958716 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 73 66.5913029636445 6.4086970363555 2 28 29.2278267061414 -1.22782670614142 3 40 33.7884662270152 6.21153377298483 4 79 74.5417180149841 4.45828198501586 5 75 68.3717814827618 6.62821851723822 6 21 45.9059702515849 -24.9059702515849 7 16 40.8742385495194 -24.8742385495194 8 81 58.3535199386703 22.6464800613297 9 90 51.3642663482893 38.6357336517107 10 87 63.0922795222356 23.9077204777644 11 99 58.9952534044508 40.0047465955492 12 54 56.701342662365 -2.70134266236495 13 53 73.5065341939744 -20.5065341939744 14 6 62.5598800408435 -56.5598800408435 15 71 59.4128141316671 11.5871858683329 16 93 69.1916027460129 23.8083972539871 17 82 69.0350043629585 12.9649956370415 18 32 69.9396706422864 -37.9396706422864 19 93 77.8506888327145 15.1493111672855 20 24 63.2509722106068 -39.2509722106068 21 96 66.4594955815581 29.5405044184419 22 88 78.8496034278642 9.15039657213577 23 83 29.9625188985102 53.0374811014898 24 23 46.3563009611064 -23.3563009611064 25 23 46.0783539040421 -23.0783539040421 26 20 54.0028717520872 -34.0028717520872 27 33 46.660622172898 -13.660622172898 28 88 65.5140402583913 22.4859597416087 29 42 34.9631058042881 7.0368941957119 30 98 64.9504109821951 33.0495890178049 31 34 41.4150705063765 -7.41507050637646 32 59 61.2495521527782 -2.2495521527782 33 26 30.3148653378247 -4.31486533782468 34 64 63.379780935049 0.620219064951014 35 13 63.0883295169072 -50.0883295169072 36 6 41.663616383533 -35.663616383533 37 49 36.8361261330018 12.1638738669982 38 3 37.905201276874 -34.905201276874 39 87 49.9219358094036 37.0780641905964 40 77 58.117966085154 18.882033914846 41 70 76.7212993646978 -6.72129936469776 42 76 30.3628322234458 45.6371677765542 43 82 68.4207804798163 13.5792195201837 44 12 53.8866455735835 -41.8866455735835 45 44 40.1473816640817 3.85261833591825 46 63 40.599645091627 22.400354908373 47 35 44.3464233300635 -9.3464233300635 48 69 51.7164360314109 17.2835639685892 49 10 30.9765395379208 -20.9765395379208 50 36 58.5771155907836 -22.5771155907836 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 9 0.537032818778377 0.925934362443245 0.462967181221623 10 0.487884027537456 0.975768055074911 0.512115972462544 11 0.411889610180891 0.823779220361782 0.588110389819109 12 0.32829621448542 0.65659242897084 0.67170378551458 13 0.29377220143728 0.587544402874559 0.70622779856272 14 0.73244412037935 0.5351117592413 0.26755587962065 15 0.653719444377638 0.692561111244723 0.346280555622362 16 0.614620997907952 0.770758004184097 0.385379002092049 17 0.521757142936794 0.956485714126413 0.478242857063206 18 0.583784142951404 0.832431714097191 0.416215857048596 19 0.50429737582904 0.991405248341921 0.495702624170961 20 0.586223215888362 0.827553568223277 0.413776784111638 21 0.580486125822995 0.83902774835401 0.419513874177005 22 0.497453739674429 0.994907479348857 0.502546260325571 23 0.669542517240056 0.660914965519889 0.330457482759944 24 0.638018530533255 0.72396293893349 0.361981469466745 25 0.601550256169436 0.796899487661129 0.398449743830564 26 0.643174623320532 0.713650753358935 0.356825376679468 27 0.57580084971642 0.84839830056716 0.42419915028358 28 0.54026254580832 0.91947490838336 0.45973745419168 29 0.470684012688482 0.941368025376965 0.529315987311518 30 0.548578505079109 0.902842989841783 0.451421494920891 31 0.459533155585378 0.919066311170757 0.540466844414621 32 0.368513731662366 0.737027463324733 0.631486268337634 33 0.285493211950997 0.570986423901994 0.714506788049003 34 0.204934796032072 0.409869592064144 0.795065203967928 35 0.317779778895449 0.635559557790898 0.682220221104551 36 0.340149363958446 0.680298727916891 0.659850636041554 37 0.258475040950477 0.516950081900954 0.741524959049523 38 0.330395065139085 0.66079013027817 0.669604934860915 39 0.27007958131117 0.54015916262234 0.72992041868883 40 0.367744887303539 0.735489774607078 0.632255112696461 41 0.26297622798848 0.52595245597696 0.73702377201152 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 0 0 OK 10% type I error level 0 0 OK

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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Software written by Ed van Stee & Patrick Wessa

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