*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, 15 Dec 2009 08:12:35 -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/Dec/15/t1260890046nk7b1b9ixmswhab.htm/, Retrieved Tue, 15 Dec 2009 16:14:18 +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/Dec/15/t1260890046nk7b1b9ixmswhab.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:

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Dataseries X:

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8.3 0 8.2 8.7 8.5 0 8.3 8.2 8.6 0 8.5 8.3 8.5 0 8.6 8.5 8.2 0 8.5 8.6 8.1 0 8.2 8.5 7.9 0 8.1 8.2 8.6 0 7.9 8.1 8.7 0 8.6 7.9 8.7 0 8.7 8.6 8.5 0 8.7 8.7 8.4 0 8.5 8.7 8.5 0 8.4 8.5 8.7 0 8.5 8.4 8.7 0 8.7 8.5 8.6 0 8.7 8.7 8.5 0 8.6 8.7 8.3 0 8.5 8.6 8 0 8.3 8.5 8.2 0 8 8.3 8.1 0 8.2 8 8.1 0 8.1 8.2 8 0 8.1 8.1 7.9 0 8 8.1 7.9 0 7.9 8 8 0 7.9 7.9 8 0 8 7.9 7.9 0 8 8 8 0 7.9 8 7.7 0 8 7.9 7.2 0 7.7 8 7.5 0 7.2 7.7 7.3 0 7.5 7.2 7 0 7.3 7.5 7 0 7 7.3 7 0 7 7 7.2 0 7 7 7.3 0 7.2 7 7.1 0 7.3 7.2 6.8 0 7.1 7.3 6.4 0 6.8 7.1 6.1 0 6.4 6.8 6.5 0 6.1 6.4 7.7 0 6.5 6.1 7.9 0 7.7 6.5 7.5 0 7.9 7.7 6.9 1 7.5 7.9 6.6 1 6.9 7.5 6.9 1 6.6 6.9 7.7 1 6.9 6.6 8 1 7.7 6.9 8 1 8 7.7 7.7 1 8 8 7.3 1 7.7 8 7.4 1 7.3 7.7 8.1 1 7.4 7.3 8.3 1 8.1 7.4 8.2 1 8.3 8.1

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 Y[t] = + 2.4321893216906 + 0.305335987791798X[t] + 1.39865251096371Y1[t] -0.676438938461147Y2[t] + 0.207220269727292M1[t] + 0.168459387450287M2[t] -0.0761236072848482M3[t] -0.050328548004132M4[t] -0.0430944321998975M5[t] -0.0921983456723488M6[t] + 0.0485017764361306M7[t] + 0.644831160482746M8[t] -0.237639033210727M9[t] -0.0218547348532218M10[t] -0.0836582576862774M11[t] -0.0123382569501433t + 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) 2.4321893216906 0.622279 3.9085 0.000332 0.000166 X 0.305335987791798 0.100615 3.0347 0.004121 0.00206 Y1 1.39865251096371 0.113901 12.2795 0 0 Y2 -0.676438938461147 0.121961 -5.5464 2e-06 1e-06 M1 0.207220269727292 0.119015 1.7411 0.088984 0.044492 M2 0.168459387450287 0.126392 1.3328 0.189773 0.094887 M3 -0.0761236072848482 0.131337 -0.5796 0.565277 0.282639 M4 -0.050328548004132 0.123219 -0.4084 0.68502 0.34251 M5 -0.0430944321998975 0.120444 -0.3578 0.722287 0.361144 M6 -0.0921983456723488 0.118936 -0.7752 0.44257 0.221285 M7 0.0485017764361306 0.118346 0.4098 0.684013 0.342007 M8 0.644831160482746 0.119648 5.3894 3e-06 1e-06 M9 -0.237639033210727 0.151134 -1.5724 0.123368 0.061684 M10 -0.0218547348532218 0.126701 -0.1725 0.86388 0.43194 M11 -0.0836582576862774 0.1251 -0.6687 0.507328 0.253664 t -0.0123382569501433 0.003569 -3.4575 0.001262 0.000631 Multiple Linear Regression - Regression Statistics Multiple R 0.973328878979688 R-squared 0.947369106655857 Adjusted R-squared 0.928572359032948 F-TEST (value) 50.4006930167676 F-TEST (DF numerator) 15 F-TEST (DF denominator) 42 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.176104199514564 Sum Squared Residuals 1.30253294163994 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.3 8.21100315975823 0.0889968402417674 2 8.5 8.63798874085802 -0.137988740858020 3 8.6 8.59315409751937 0.00684590248063174 4 8.5 8.61118836325408 -0.111188363254083 5 8.2 8.39857507716569 -0.198575077165688 6 8.1 7.9851810473001 0.114818952699907 7 7.9 8.1766093429004 -0.276609342900402 8 8.6 8.54851386165025 0.0514861383497519 9 8.7 8.76804995637346 -0.0680499563734595 10 8.7 8.63785399195439 0.0621460080456102 11 8.5 8.49606831832508 0.00393168167492438 12 8.4 8.28765781686847 0.112342183131532 13 8.5 8.47796236624147 0.0220376337585254 14 8.7 8.63437237195681 0.0656276280431873 15 8.7 8.58953772861816 0.110462271381838 16 8.6 8.4677067432565 0.132293256743495 17 8.5 8.32273735101423 0.177262648985775 18 8.3 8.18907382334137 0.110926176658627 19 8 8.10534908015308 -0.105349080153083 20 8.2 8.40503224165267 -0.20503224165267 21 8.1 7.99288597474014 0.107114025259860 22 8.1 7.9211789773589 0.178821022641098 23 8 7.91468109142182 0.0853189085781831 24 7.9 7.84613584106158 0.0538641589384202 25 7.9 7.96879649658847 -0.0687964965884718 26 8 7.98534125120744 0.0146587487925612 27 8 7.86828525061853 0.131714749381469 28 7.9 7.81409815910299 0.085901840897011 29 8 7.66912876686071 0.33087123313929 30 7.7 7.8151957413806 -0.115195741380600 31 7.2 7.4563179594037 -0.256317959403708 32 7.5 7.54391451255667 -0.0439145125566678 33 7.3 7.40692128443274 -0.106921284432739 34 7 7.12770514210901 -0.127705142109014 35 7 6.76925539672893 0.230744603271069 36 7 7.0435070790034 -0.0435070790034087 37 7.2 7.23838909178056 -0.0383890917805569 38 7.3 7.46702045474615 -0.167020454746152 39 7.1 7.21467666646501 -0.114676666465015 40 6.8 6.88075907275673 -0.0807590727567306 41 6.4 6.59134696601394 -0.191346966013937 42 6.1 6.1733754727442 -0.073375472744202 43 6.5 6.15271715999788 0.347282840002119 44 7.7 7.49910097301818 0.200899026981816 45 7.9 8.01209996014657 -0.112099960146566 46 7.5 7.6835497775933 -0.183549777593294 47 6.9 7.21999519352418 -0.319995193524177 48 6.6 6.72269926306654 -0.122699263066543 49 6.9 6.90384888563126 -0.00384888563126394 50 7.7 7.47527718123158 0.224722818768424 51 8 8.13434625677892 -0.134346256778924 52 8 8.0262476616297 -0.0262476616296928 53 7.7 7.81821183894544 -0.11821183894544 54 7.3 7.33717391523373 -0.0371739152337316 55 7.4 7.10900645754493 0.290993542455075 56 8.1 8.10343841112223 -0.00343841112222986 57 8.3 8.1200428243071 0.179957175692904 58 8.2 8.1297121109844 0.070287889015599 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.0506658865541022 0.101331773108204 0.949334113445898 20 0.511275355080137 0.977449289839726 0.488724644919863 21 0.357742386036734 0.715484772073467 0.642257613963267 22 0.235943415047963 0.471886830095925 0.764056584952037 23 0.143141297306238 0.286282594612476 0.856858702693762 24 0.0891895340992182 0.178379068198436 0.910810465900782 25 0.0574049335490349 0.114809867098070 0.942595066450965 26 0.0295005032240758 0.0590010064481517 0.970499496775924 27 0.020927990603178 0.041855981206356 0.979072009396822 28 0.0118158295951030 0.0236316591902061 0.988184170404897 29 0.120106659640725 0.240213319281449 0.879893340359275 30 0.136131483967661 0.272262967935323 0.863868516032339 31 0.170349875977737 0.340699751955473 0.829650124022263 32 0.117370190740401 0.234740381480802 0.8826298092596 33 0.0863492926995539 0.172698585399108 0.913650707300446 34 0.0727974508927255 0.145594901785451 0.927202549107274 35 0.435927022512143 0.871854045024287 0.564072977487857 36 0.427193742332341 0.854387484664682 0.572806257667659 37 0.527823976181017 0.944352047637966 0.472176023818983 38 0.398377939369357 0.796755878738713 0.601622060630643 39 0.449857167267898 0.899714334535797 0.550142832732102 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 2 0.0952380952380952 NOK 10% type I error level 3 0.142857142857143 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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