*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, 07 Dec 2010 13:11:12 +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/07/t1291727396jhx1fx7iy8yusy3.htm/, Retrieved Tue, 07 Dec 2010 14:10:10 +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/07/t1291727396jhx1fx7iy8yusy3.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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No (this computation is public)

User-defined keywords:

Dataseries X:

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112.3 0 117.2 96.8 117.3 0 112.3 117.2 111.1 1 117.3 112.3 102.2 1 111.1 117.3 104.3 1 102.2 111.1 122.9 1 104.3 102.2 107.6 1 122.9 104.3 121.3 1 107.6 122.9 131.5 1 121.3 107.6 89 1 131.5 121.3 104.4 1 89 131.5 128.9 1 104.4 89 135.9 1 128.9 104.4 133.3 1 135.9 128.9 121.3 1 133.3 135.9 120.5 0 121.3 133.3 120.4 0 120.5 121.3 137.9 0 120.4 120.5 126.1 0 137.9 120.4 133.2 0 126.1 137.9 151.1 0 133.2 126.1 105 0 151.1 133.2 119 0 105 151.1 140.4 0 119 105 156.6 0 140.4 119 137.1 0 156.6 140.4 122.7 0 137.1 156.6 125.8 0 122.7 137.1 139.3 0 125.8 122.7 134.9 0 139.3 125.8 149.2 0 134.9 139.3 132.3 0 149.2 134.9 149 0 132.3 149.2 117.2 0 149 132.3 119.6 0 117.2 149 152 0 119.6 117.2 149.4 0 152 119.6 127.3 0 149.4 152 114.1 0 127.3 149.4 102.1 0 114.1 127.3 107.7 0 102.1 114.1 104.4 0 107.7 102.1 102.1 0 104.4 107.7 96 1 102.1 104.4 109.3 0 96 102.1 90 1 109.3 96 83.9 1 90 109.3 112 1 83.9 90 114.3 1 112 83.9 103.6 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 X[t] = + 39.5718545572897 + 1.44282250927461Y[t] + 0.462651817415592`y(t)`[t] + 0.456836619187372`y(t-1)`[t] -12.0357634082612M1[t] -35.1685550498614M2[t] -43.9263346792956M3[t] -39.5414071812871M4[t] -25.9417353094251M5[t] -15.6128526883177M6[t] -27.0108528497098M7[t] -31.7675430580503M8[t] -12.4005152354837M9[t] -53.5642426604347M10[t] -38.3567051102191M11[t] -0.0981315636876867t + 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) 39.5718545572897 14.916987 2.6528 0.011062 0.005531 Y 1.44282250927461 3.352062 0.4304 0.668985 0.334493 `y(t)` 0.462651817415592 0.134518 3.4393 0.001287 0.000644 `y(t-1)` 0.456836619187372 0.155403 2.9397 0.005218 0.002609 M1 -12.0357634082612 5.673619 -2.1214 0.039562 0.019781 M2 -35.1685550498614 5.876229 -5.9849 0 0 M3 -43.9263346792956 6.513124 -6.7443 0 0 M4 -39.5414071812871 5.828352 -6.7843 0 0 M5 -25.9417353094251 5.317747 -4.8783 1.4e-05 7e-06 M6 -15.6128526883177 5.046647 -3.0937 0.003429 0.001715 M7 -27.0108528497098 5.191112 -5.2033 5e-06 2e-06 M8 -31.7675430580503 5.75193 -5.5229 2e-06 1e-06 M9 -12.4005152354837 5.350115 -2.3178 0.025172 0.012586 M10 -53.5642426604347 5.698954 -9.399 0 0 M11 -38.3567051102191 7.794011 -4.9213 1.2e-05 6e-06 t -0.0981315636876867 0.067336 -1.4573 0.152124 0.076062 Multiple Linear Regression - Regression Statistics Multiple R 0.932787352234213 R-squared 0.870092244488114 Adjusted R-squared 0.825805509654516 F-TEST (value) 19.6467914773439 F-TEST (DF numerator) 15 F-TEST (DF denominator) 44 p-value 1.04360964314765e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 7.91699716280576 Sum Squared Residuals 2757.86913933847 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 112.3 125.882537323786 -13.5825373237857 2 117.3 109.704087244584 7.59591275541616 3 111.1 102.365758213796 8.73424178620355 4 102.2 106.068295976078 -3.86829597607751 5 104.3 112.619848070291 -8.31984807029128 6 122.9 119.756322033516 3.14367796648385 7 107.6 117.82487101266 -10.2248710126599 8 121.3 114.388637551058 6.9113624489418 9 131.5 133.006263434964 -1.50626343496396 10 89 102.722114666831 -13.7221146668313 11 104.4 102.828551928908 1.57144807109218 12 128.9 128.796407148176 0.103592851823989 13 135.9 135.032765638395 0.867234361605393 14 133.3 126.232902325106 7.06709767489354 15 121.3 119.371952741016 1.92804725898431 16 120.5 115.476329147188 5.02367085281237 17 120.4 123.125708571181 -2.72570857118095 18 137.9 132.944725151509 4.95527484849077 19 126.1 129.499316569284 -3.39931656928364 20 133.2 127.179844187530 6.0201558124696 21 151.1 144.342896243649 6.75710375635096 22 105 114.606044782980 -9.60604478297976 23 119 116.564577470103 2.43542252989709 24 140.4 140.240108315915 0.159891684085290 25 156.6 144.402674905283 12.1973250947173 26 137.1 138.443014792737 -1.34301479273716 27 122.7 127.966146390847 -5.26614639084666 28 125.8 116.682442080229 9.11755791977079 29 139.3 125.039755706094 14.2602442939064 30 134.9 142.932499818105 -8.03249981810475 31 149.2 135.567994455426 13.6320055445740 32 132.3 135.319012548016 -3.0190125480162 33 149 153.301856746951 -4.30185674695106 34 117.2 112.045744244886 5.15425575511383 35 119.6 120.071993978027 -0.471993978027433 36 152 144.913527396198 7.08647260380223 37 149.4 148.865959194564 0.534040805436267 38 127.3 139.233647725666 -11.9336477256662 39 114.1 118.965356157773 -4.86535615777256 40 102.1 107.049058818167 -4.94905881816663 41 107.7 108.968533944080 -1.26853394408049 42 104.4 116.308095748779 -11.9080957487791 43 102.1 105.843498093677 -3.74349809367719 44 96 99.8598388075494 -3.85983880754935 45 109.3 113.813012246788 -4.51301224678758 46 90 77.3605415620079 12.6394584379921 47 83.9 89.616694507607 -5.71669450760701 48 112 116.236145217587 -4.23614521758699 49 114.3 114.316062937973 -0.0160629379732478 50 103.6 104.986347911906 -1.38634791190638 51 91.7 92.2307864965686 -0.530786496568626 52 80.8 86.123873978339 -5.32387397833903 53 87.2 89.1461537083536 -1.94615370835363 54 109.2 97.3583572480908 11.8416427519092 55 102.7 98.9643198689533 3.73568013104673 56 95.1 101.152666905846 -6.05266690584586 57 117.5 113.935971327648 3.56402867235164 58 85.1 79.5655547432949 5.53444525670511 59 92.1 89.9181821153548 2.18181788464517 60 113.5 116.613811922124 -3.11381192212448 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.0103166245804496 0.0206332491608991 0.98968337541955 20 0.00435753517348285 0.0087150703469657 0.995642464826517 21 0.00202856761597115 0.0040571352319423 0.997971432384029 22 0.000944964467037748 0.00188992893407550 0.999055035532962 23 0.000209186042294786 0.000418372084589572 0.999790813957705 24 0.000368108455547483 0.000736216911094966 0.999631891544453 25 0.000134538808541603 0.000269077617083207 0.999865461191458 26 0.00111688794899438 0.00223377589798875 0.998883112051006 27 0.0781291569964196 0.156258313992839 0.92187084300358 28 0.221697272512878 0.443394545025757 0.778302727487122 29 0.237274728586044 0.474549457172089 0.762725271413956 30 0.514763555358583 0.970472889282834 0.485236444641417 31 0.601748122144277 0.796503755711446 0.398251877855723 32 0.574134690284328 0.851730619431344 0.425865309715672 33 0.668014633116487 0.663970733767027 0.331985366883513 34 0.576248945884112 0.847502108231775 0.423751054115888 35 0.51333913961099 0.97332172077802 0.48666086038901 36 0.655682687559171 0.688634624881658 0.344317312440829 37 0.634153740103837 0.731692519792326 0.365846259896163 38 0.713955851244151 0.572088297511698 0.286044148755849 39 0.66564656535192 0.668706869296161 0.334353434648080 40 0.656596651836924 0.686806696326152 0.343403348163076 41 0.948782465762291 0.102435068475418 0.0512175342377091 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 7 0.304347826086957 NOK 5% type I error level 8 0.347826086956522 NOK 10% type I error level 8 0.347826086956522 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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