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: Mon, 23 Nov 2009 12:03:44 -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/Nov/23/t12590031179pteovlld0n477z.htm/, Retrieved Mon, 23 Nov 2009 20:05:29 +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/Nov/23/t12590031179pteovlld0n477z.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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93.3 121.8 97.3 127.6 127 129.9 111.7 128 96.4 123.5 133 124 72.2 127.4 95.8 127.6 124.1 128.4 127.6 131.4 110.7 135.1 104.6 134 112.7 144.5 115.3 147.3 139.4 150.9 119 148.7 97.4 141.4 154 138.9 81.5 139.8 88.8 145.6 127.7 147.9 105.1 148.5 114.9 151.1 106.4 157.5 104.5 167.5 121.6 172.3 141.4 173.5 99 187.5 126.7 205.5 134.1 195.1 81.3 204.5 88.6 204.5 132.7 201.7 132.9 207 134.4 206.6 103.7 210.6 119.7 211.1 115 215 132.9 223.9 108.5 238.2 113.9 238.9 142 229.6 97.7 232.2 92.2 222.1 128.8 221.6 134.9 227.3 128.2 221 114.8 213.6 117.9 243.4 119.1 253.8 120.7 265.3 129.1 268.2 117.6 268.5 129.2 266.9 100 268.4 87 250.8 128 231.2 127.7 192 93.4 171.4 84.1 160 71.7 148.1

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation IPtran[t] = + 97.3828106445095 + 0.0857405341783205IGpic[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) 97.3828106445095 9.733742 10.0047 0 0 IGpic 0.0857405341783205 0.050892 1.6847 0.097321 0.048661 Multiple Linear Regression - Regression Statistics Multiple R 0.214242028795646 R-squared 0.0458996469024745 Adjusted R-squared 0.0297284544770927 F-TEST (value) 2.83835883558172 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 0.0973214844633553 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 18.4121694602166 Sum Squared Residuals 20001.4710696723 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 93.3 107.826007707429 -14.5260077074290 2 97.3 108.323302805663 -11.0233028056633 3 127 108.520506034273 18.4794939657266 4 111.7 108.357599019335 3.34240098066542 5 96.4 107.971766615532 -11.5717666155321 6 133 108.014636882621 24.9853631173787 7 72.2 108.306154698828 -36.1061546988276 8 95.8 108.323302805663 -12.5233028056633 9 124.1 108.391895233006 15.7081047669941 10 127.6 108.649116835541 18.9508831644591 11 110.7 108.966356812001 1.73364318799934 12 104.6 108.872042224405 -4.27204222440451 13 112.7 109.772317833277 2.92768216672313 14 115.3 110.012391328976 5.28760867102383 15 139.4 110.321057252018 29.0789427479819 16 119 110.132428076826 8.86757192317418 17 97.4 109.506522177324 -12.1065221773241 18 154 109.292170841878 44.7078291581217 19 81.5 109.369337322639 -27.8693373226388 20 88.8 109.866632420873 -21.0666324208730 21 127.7 110.063835649483 17.6361643505168 22 105.1 110.115279969990 -5.01527996999016 23 114.9 110.338205358854 4.56179464114622 24 106.4 110.886944777595 -4.48694477759503 25 104.5 111.744350119378 -7.24435011937824 26 121.6 112.155904683434 9.4440953165658 27 141.4 112.258793324448 29.1412066755518 28 99 113.459160802945 -14.4591608029447 29 126.7 115.002490418154 11.6975095818456 30 134.1 114.1107888627 19.9892111373001 31 81.3 114.916749883976 -33.6167498839761 32 88.6 114.916749883976 -26.3167498839761 33 132.7 114.676676388277 18.0233236117232 34 132.9 115.131101219422 17.7688987805781 35 134.4 115.096805005751 19.3031949942494 36 103.7 115.439767142464 -11.7397671424639 37 119.7 115.482637409553 4.21736259044698 38 115 115.817025492848 -0.817025492848469 39 132.9 116.580116247036 16.3198837529645 40 108.5 117.806205885786 -9.3062058857855 41 113.9 117.866224259710 -3.96622425971032 42 142 117.068837291852 24.9311627081481 43 97.7 117.291762680716 -19.5917626807156 44 92.2 116.425783285515 -24.2257832855145 45 128.8 116.382913018425 12.4170869815746 46 134.9 116.871634063242 18.0283659367582 47 128.2 116.331468697918 11.8685313020816 48 114.8 115.696988744999 -0.896988744998822 49 117.9 118.252056663513 -0.352056663512765 50 119.1 119.143758218967 -0.0437582189673105 51 120.7 120.129774362018 0.570225637982011 52 129.1 120.378421911135 8.72157808886488 53 117.6 120.404144071389 -2.80414407138862 54 129.2 120.266959216703 8.93304078329669 55 100 120.395570017971 -20.3955700179708 56 87 118.886536616432 -31.8865366164323 57 128 117.206022146537 10.7939778534627 58 127.7 113.844993206747 13.8550067932529 59 93.4 112.078738202674 -18.6787382026737 60 84.1 111.101296113041 -27.0012961130408 61 71.7 110.080983756319 -38.3809837563188 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.131410588925575 0.262821177851151 0.868589411074425 6 0.509659679135319 0.980680641729362 0.490340320864681 7 0.81480031002353 0.370399379952939 0.185199689976470 8 0.744448263624338 0.511103472751324 0.255551736375662 9 0.714528083945006 0.570943832109988 0.285471916054994 10 0.658682452635752 0.682635094728495 0.341317547364248 11 0.585699800958922 0.828600398082156 0.414300199041078 12 0.511580015564176 0.976839968871648 0.488419984435824 13 0.425231391648809 0.850462783297619 0.57476860835119 14 0.335380318812535 0.67076063762507 0.664619681187465 15 0.344625284657924 0.689250569315848 0.655374715342076 16 0.275045220580559 0.550090441161117 0.724954779419442 17 0.274977356262681 0.549954712525361 0.725022643737319 18 0.614173616549744 0.771652766900512 0.385826383450256 19 0.747274166648743 0.505451666702515 0.252725833351257 20 0.79156199357425 0.416876012851500 0.208438006425750 21 0.769946140361413 0.460107719277175 0.230053859638587 22 0.722655344935323 0.554689310129355 0.277344655064677 23 0.660408619560799 0.679182760878403 0.339591380439202 24 0.606140722032167 0.787718555935666 0.393859277967833 25 0.556213368363208 0.887573263273584 0.443786631636792 26 0.496897995925313 0.993795991850626 0.503102004074687 27 0.588999788630009 0.822000422739983 0.411000211369991 28 0.595306468215345 0.80938706356931 0.404693531784655 29 0.54605861482833 0.90788277034334 0.45394138517167 30 0.557638794471835 0.884722411056331 0.442361205528165 31 0.737178101169349 0.525643797661302 0.262821898830651 32 0.778714322713027 0.442571354573945 0.221285677286973 33 0.794430240923467 0.411139518153066 0.205569759076533 34 0.806313340415143 0.387373319169715 0.193686659584857 35 0.832501116104103 0.334997767791794 0.167498883895897 36 0.797541188079425 0.40491762384115 0.202458811920575 37 0.75027363586992 0.499452728260161 0.249726364130081 38 0.687164122660346 0.625671754679307 0.312835877339654 39 0.690971503643914 0.618056992712172 0.309028496356086 40 0.637221765924141 0.725556468151718 0.362778234075859 41 0.56189610705135 0.8762077858973 0.43810389294865 42 0.66361799658984 0.672764006820321 0.336382003410160 43 0.661769690706391 0.676460618587218 0.338230309293609 44 0.690543841492885 0.61891231701423 0.309456158507115 45 0.67674430227635 0.6465113954473 0.32325569772365 46 0.72129133163708 0.557417336725839 0.278708668362920 47 0.729721689151849 0.540556621696302 0.270278310848151 48 0.664897678454344 0.670204643091312 0.335102321545656 49 0.574674889369099 0.850650221261802 0.425325110630901 50 0.474260446816282 0.948520893632564 0.525739553183718 51 0.370189615079285 0.74037923015857 0.629810384920715 52 0.300727137852107 0.601454275704215 0.699272862147893 53 0.207703323355713 0.415406646711425 0.792296676644287 54 0.171433461348764 0.342866922697528 0.828566538651236 55 0.135021686581601 0.270043373163201 0.864978313418399 56 0.685020850909161 0.629958298181678 0.314979149090839 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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