Workshop 10; PLC: 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: Tue, 14 Dec 2010 10:19:58 +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/Dec/14/t129232203604hdfp36e6mz1w4.htm/, Retrieved Tue, 14 Dec 2010 11:20:47 +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/Dec/14/t129232203604hdfp36e6mz1w4.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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107.11 236.67 8.92 1 122.23 258.1 9.32 2 134.69 241.52 8.9 3 128.79 190.71 8.53 4 126.16 200.32 8.51 5 119.98 223.41 9.03 6 108.45 201.38 9.6 7 108.43 211.83 9.88 8 98.17 224.41 10.81 9 106.09 211.57 11.61 10 108.81 194.77 11.81 11 103.03 201.86 13.93 12 124.36 225 16.19 1 118.52 278.9 18.05 2 112.2 259.74 17.08 3 114.71 230.45 17.46 4 107.96 238.26 16.9 5 101.21 250.14 15.69 6 102.77 263.81 15.86 7 112.13 247.22 12.98 8 109.36 229.81 12.31 9 110.91 224.27 11.51 10 123.57 213.23 11.73 11 129.95 239.57 11.7 12 124.46 249.7 10.9 1 122.34 212.5 10.57 2 116.61 203.27 10.37 3 114.59 192.05 9.59 4 112.52 190.04 9.09 5 118.67 202.05 9.26 6 116.8 211.91 9.9 7 123.63 210.39 9.61 8 128.04 231.25 9.85 9 134.57 224.3 9.99 10 130.33 209.64 9.9 11 136.47 206.05 10.45 12 139.05 229.7 11.66 1 158.21 264.67 13.61 2 148.07 246.29 12.88 3 137.74 260.91 12.52 4 139.74 265.14 10.93 5 144.08 284.52 12.07 6 145.35 287.48 13.21 7 145.77 321.9 13.68 8 140.56 321.59 14.02 9 121.41 282.39 11 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 7 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Coffee[t] = + 68.0723686397847 + 0.304500792349745Tea[t] -1.25713504067567Sugar[t] -0.262795671803581Month[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) 68.0723686397847 8.810102 7.7266 0 0 Tea 0.304500792349745 0.053903 5.6491 1e-06 0 Sugar -1.25713504067567 0.637807 -1.971 0.053671 0.026835 Month -0.262795671803581 0.432824 -0.6072 0.546196 0.273098 Multiple Linear Regression - Regression Statistics Multiple R 0.690489041401806 R-squared 0.476775116295986 Adjusted R-squared 0.448745211811842 F-TEST (value) 17.0095162673742 F-TEST (DF numerator) 3 F-TEST (DF denominator) 56 p-value 5.65215638737016e-08 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 11.3971446966654 Sum Squared Residuals 7274.11480525678 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 107.11 128.662130930568 -21.5521309305683 2 122.23 134.421933222549 -12.1919332225493 3 134.69 129.638511130671 5.05148886932928 4 128.79 114.369170164627 14.4208298353734 5 126.16 117.057769808118 9.10223019188241 6 119.98 123.172187210518 -3.19218721051825 7 108.45 115.484672110065 -7.03467211006466 8 108.43 118.051911906927 -9.62191190692672 9 98.17 120.450600615055 -22.2806006150546 10 106.09 115.272306736940 -9.18230673693971 11 108.81 109.642470745525 -0.832470745525286 12 103.03 108.873459405249 -5.84345940524896 13 124.36 115.969234938134 8.39076506186558 14 118.52 129.780760798325 -11.2607607983253 15 112.2 124.903150934556 -12.7031509345560 16 114.71 115.243815739372 -0.533815739371678 17 107.96 118.063166878598 -10.1031668785980 18 101.21 122.938974019127 -21.7289740191269 19 102.77 126.624991221830 -23.8549912218295 20 112.13 124.931076322090 -12.8010763220896 21 109.36 120.209202332730 -10.8492023327297 22 110.91 119.265180303849 -8.35518030384904 23 123.57 115.364126175556 8.20587382444437 24 129.95 123.159595425465 6.7904045745354 25 124.46 130.140648874347 -5.68064887434744 26 122.34 118.965278290556 3.37472170944368 27 116.61 116.143367313500 0.466632686500259 28 114.59 113.444638083259 1.14536191674095 29 112.52 113.198363339170 -0.67836333917032 30 118.67 116.378909226572 2.29109077342769 31 116.8 118.313924941305 -1.51392494130478 32 123.63 117.952857226926 5.67714277307447 33 128.04 123.740235673775 4.29976432622453 34 134.57 121.185160589447 13.3848394105534 35 130.33 116.571525455457 13.7584745445435 36 136.47 114.524147666746 21.9458523332542 37 139.05 123.095210396439 15.954789603561 38 158.21 131.029394103788 27.1806058962115 39 148.07 126.087582448290 21.9824175517102 40 137.74 130.729156975283 7.01084302471727 41 139.74 133.753244369793 5.98675563020711 42 144.08 137.958540107357 6.12145989264292 43 145.35 137.163932834538 8.1860671654615 44 145.77 146.791200966296 -1.02120096629554 45 140.56 146.006584135034 -5.44658413503381 46 121.41 136.723910697488 -15.3139106974878 47 120.44 123.694399675040 -3.25439967504047 48 116.97 120.260392953763 -3.29039295376268 49 128.03 132.076602340880 -4.04660234088049 50 128.51 133.029523839215 -4.5195238392153 51 127.76 131.070303825491 -3.31030382549129 52 134.58 134.559141760412 0.0208582395883736 53 147.64 135.006739398240 12.6332606017595 54 144.46 133.086653149866 11.3733468501339 55 137.6 146.670487750024 -9.07048775002383 56 146.87 138.657000373689 8.21299962631053 57 145.67 150.235532553376 -4.56553255337574 58 151.95 141.311902558122 10.6380974418783 59 150.23 146.496662451635 3.73333754836456 60 155.86 147.566444578439 8.29355542156121 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.492819412119448 0.985638824238895 0.507180587880552 8 0.327401748037203 0.654803496074405 0.672598251962798 9 0.246338347730052 0.492676695460103 0.753661652269949 10 0.436106638655858 0.872213277311717 0.563893361344142 11 0.413357717133063 0.826715434266126 0.586642282866937 12 0.360547402766308 0.721094805532617 0.639452597233692 13 0.28505544278446 0.57011088556892 0.71494455721554 14 0.22284707198364 0.44569414396728 0.77715292801636 15 0.173999659923807 0.347999319847614 0.826000340076193 16 0.11701686345221 0.23403372690442 0.88298313654779 17 0.0916475481218257 0.183295096243651 0.908352451878174 18 0.123472237774764 0.246944475549528 0.876527762225236 19 0.184167390228665 0.36833478045733 0.815832609771335 20 0.212327151541828 0.424654303083655 0.787672848458172 21 0.221618736361898 0.443237472723795 0.778381263638102 22 0.23545520162277 0.47091040324554 0.76454479837723 23 0.393336704148834 0.786673408297669 0.606663295851166 24 0.661559690637616 0.676880618724767 0.338440309362384 25 0.618190533336343 0.763618933327315 0.381809466663657 26 0.552072595477339 0.895854809045321 0.447927404522661 27 0.52180053276918 0.95639893446164 0.47819946723082 28 0.506975900811411 0.986048198377178 0.493024099188589 29 0.530502234861866 0.938995530276268 0.469497765138134 30 0.495439298414284 0.990878596828568 0.504560701585716 31 0.50506067626588 0.98987864746824 0.49493932373412 32 0.478869731981294 0.957739463962588 0.521130268018706 33 0.472527116871636 0.94505423374327 0.527472883128364 34 0.560913192974876 0.878173614050248 0.439086807025124 35 0.567923329743687 0.864153340512626 0.432076670256313 36 0.73753420845092 0.524931583098160 0.262465791549080 37 0.766294744188872 0.467410511622256 0.233705255811128 38 0.968394732007656 0.0632105359846883 0.0316052679923442 39 0.99165774360572 0.0166845127885581 0.00834225639427904 40 0.988360923718264 0.0232781525634723 0.0116390762817362 41 0.986958905737095 0.0260821885258094 0.0130410942629047 42 0.988969905020134 0.0220601899597327 0.0110300949798664 43 0.99408001244173 0.0118399751165406 0.00591998755827029 44 0.99569715097159 0.00860569805682074 0.00430284902841037 45 0.997524003708532 0.00495199258293566 0.00247599629146783 46 0.994886608407866 0.0102267831842679 0.00511339159213394 47 0.988241252890633 0.0235174942187335 0.0117587471093668 48 0.998124094407419 0.00375181118516282 0.00187590559258141 49 0.994524481637325 0.0109510367253506 0.00547551836267531 50 0.985074422405388 0.0298511551892250 0.0149255775946125 51 0.990725625532677 0.0185487489346465 0.00927437446732326 52 0.982491824492996 0.0350163510140072 0.0175081755070036 53 0.996330687444683 0.00733862511063342 0.00366931255531671 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.0851063829787234 NOK 5% type I error level 15 0.319148936170213 NOK 10% type I error level 16 0.340425531914894 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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