workshop 7,1

*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, 20 Nov 2009 06:10:43 -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/20/t1258722746ceibaq879luhq2q.htm/, Retrieved Fri, 20 Nov 2009 14:12:38 +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/20/t1258722746ceibaq879luhq2q.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:

» Textbox « » Textfile « » CSV «

8,9 11,1 8,9 10,9 8,6 10 8,3 9,2 8,3 9,2 8,3 9,5 8,4 9,6 8,5 9,5 8,4 9,1 8,6 8,9 8,5 9 8,5 10,1 8,4 10,3 8,5 10,2 8,5 9,6 8,5 9,2 8,5 9,3 8,5 9,4 8,5 9,4 8,5 9,2 8,5 9 8,6 9 8,4 9 8,1 9,8 8,0 10 8,0 9,8 8,0 9,3 8,0 9 7,9 9 7,8 9,1 7,8 9,1 7,9 9,1 8,1 9,2 8,0 8,8 7,6 8,3 7,3 8,4 7,0 8,1 6,8 7,7 7,0 7,9 7,1 7,9 7,2 8 7,1 7,9 6,9 7,6 6,7 7,1 6,7 6,8 6,6 6,5 6,9 6,9 7,3 8,2 7,5 8,7 7,3 8,3 7,1 7,9 6,9 7,5 7,1 7,8 7,5 8,3 7,7 8,4 7,8 8,2 7,8 7,7 7,7 7,2 7,8 7,3 7,8 8,1 7,9 8,5

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 3.099991115504 + 0.544364175491007X[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) 3.099991115504 0.380236 8.1528 0 0 X 0.544364175491007 0.043235 12.5907 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.853678892468338 R-squared 0.728767651445969 Adjusted R-squared 0.7241704929959 F-TEST (value) 158.525676102188 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.332225836176954 Sum Squared Residuals 6.51206636718511 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.9 9.14243346345418 -0.242433463454176 2 8.9 9.03356062835598 -0.133560628355982 3 8.6 8.54363287041407 0.0563671295859263 4 8.3 8.10814153002127 0.191858469978734 5 8.3 8.10814153002127 0.191858469978734 6 8.3 8.27145078266857 0.0285492173314311 7 8.4 8.32588720021767 0.0741127997823302 8 8.5 8.27145078266857 0.228549217331430 9 8.4 8.05370511247217 0.346294887527834 10 8.6 7.94483227737397 0.655167722626034 11 8.5 7.99926869492307 0.500731305076934 12 8.5 8.59806928796317 -0.0980692879631738 13 8.4 8.70694212306138 -0.306942123061375 14 8.5 8.65250570551227 -0.152505705512274 15 8.5 8.32588720021767 0.17411279978233 16 8.5 8.10814153002127 0.391858469978733 17 8.5 8.16257794757037 0.337422052429631 18 8.5 8.21701436511947 0.282985634880531 19 8.5 8.21701436511947 0.282985634880531 20 8.5 8.10814153002127 0.391858469978733 21 8.5 7.99926869492307 0.500731305076934 22 8.6 7.99926869492307 0.600731305076934 23 8.4 7.99926869492307 0.400731305076934 24 8.1 8.43476003531587 -0.334760035315873 25 8 8.54363287041407 -0.543632870414073 26 8 8.43476003531587 -0.434760035315872 27 8 8.16257794757037 -0.162577947570369 28 8 7.99926869492307 0.000731305076934089 29 7.9 7.99926869492307 -0.0992686949230656 30 7.8 8.05370511247217 -0.253705112472167 31 7.8 8.05370511247217 -0.253705112472167 32 7.9 8.05370511247217 -0.153705112472166 33 8.1 8.10814153002127 -0.00814153002126736 34 8 7.89039585982486 0.109604140175135 35 7.6 7.61821377207936 -0.0182137720793615 36 7.3 7.67265018962846 -0.372650189628462 37 7 7.50934093698116 -0.509340936981159 38 6.8 7.29159526678476 -0.491595266784757 39 7 7.40046810188296 -0.400468101882958 40 7.1 7.40046810188296 -0.300468101882958 41 7.2 7.45490451943206 -0.254904519432058 42 7.1 7.40046810188296 -0.300468101882958 43 6.9 7.23715884923566 -0.337158849235655 44 6.7 6.96497676149015 -0.264976761490151 45 6.7 6.80166750884285 -0.101667508842850 46 6.6 6.63835825619555 -0.0383582561955479 47 6.9 6.85610392639195 0.0438960736080499 48 7.3 7.56377735453026 -0.26377735453026 49 7.5 7.83595944227576 -0.335959442275763 50 7.3 7.61821377207936 -0.318213772079361 51 7.1 7.40046810188296 -0.300468101882958 52 6.9 7.18272243168655 -0.282722431686554 53 7.1 7.34603168433386 -0.246031684333857 54 7.5 7.61821377207936 -0.118213772079361 55 7.7 7.67265018962846 0.0273498103715385 56 7.8 7.56377735453026 0.23622264546974 57 7.8 7.29159526678476 0.508404733215243 58 7.7 7.01941317903925 0.680586820960748 59 7.8 7.07384959658835 0.726150403411646 60 7.8 7.50934093698116 0.290659063018841 61 7.9 7.72708660717756 0.172913392822438 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.00113581722388712 0.00227163444777423 0.998864182776113 6 0.00127398965471256 0.00254797930942513 0.998726010345288 7 0.000178645537032669 0.000357291074065338 0.999821354462967 8 0.000166713909722544 0.000333427819445089 0.999833286090277 9 0.000133993230828706 0.000267986461657412 0.999866006769171 10 0.00429735544283069 0.00859471088566137 0.99570264455717 11 0.00313102414938009 0.00626204829876019 0.99686897585062 12 0.00162996269393847 0.00325992538787693 0.998370037306062 13 0.00228198335150269 0.00456396670300539 0.997718016648497 14 0.00111209263787796 0.00222418527575592 0.998887907362122 15 0.000445674153359218 0.000891348306718436 0.99955432584664 16 0.000247553578502042 0.000495107157004083 0.999752446421498 17 0.00012201913636438 0.00024403827272876 0.999877980863636 18 5.48303340620725e-05 0.000109660668124145 0.999945169665938 19 2.50119703961384e-05 5.00239407922768e-05 0.999974988029604 20 1.53430066422522e-05 3.06860132845045e-05 0.999984656993358 21 1.48981532684439e-05 2.97963065368878e-05 0.999985101846732 22 4.80286206037526e-05 9.60572412075051e-05 0.999951971379396 23 4.57025665403578e-05 9.14051330807156e-05 0.99995429743346 24 0.000740630070869017 0.00148126014173803 0.999259369929131 25 0.00969420723793727 0.0193884144758745 0.990305792762063 26 0.0294328575722987 0.0588657151445975 0.97056714242770 27 0.0438173536574026 0.0876347073148053 0.956182646342597 28 0.0518569384045606 0.103713876809121 0.94814306159544 29 0.0688195552953434 0.137639110590687 0.931180444704657 30 0.105324830318129 0.210649660636258 0.894675169681871 31 0.134321800583289 0.268643601166578 0.86567819941671 32 0.130285405094817 0.260570810189633 0.869714594905183 33 0.107233603513326 0.214467207026653 0.892766396486674 34 0.0967008675520962 0.193401735104192 0.903299132447904 35 0.101848810821931 0.203697621643862 0.898151189178069 36 0.167595805773268 0.335191611546535 0.832404194226732 37 0.307516403810907 0.615032807621814 0.692483596189093 38 0.432187864168415 0.86437572833683 0.567812135831585 39 0.461780405195224 0.923560810390448 0.538219594804776 40 0.436979164200397 0.873958328400793 0.563020835799603 41 0.392122423942691 0.784244847885382 0.607877576057309 42 0.364668749381315 0.72933749876263 0.635331250618685 43 0.36152644301761 0.72305288603522 0.63847355698239 44 0.354054743231733 0.708109486463467 0.645945256768267 45 0.323167423887878 0.646334847775757 0.676832576112122 46 0.343409139504988 0.686818279009977 0.656590860495012 47 0.366313134005784 0.732626268011567 0.633686865994217 48 0.323058624681501 0.646117249363002 0.676941375318499 49 0.262594281005242 0.525188562010485 0.737405718994758 50 0.243775071719186 0.487550143438373 0.756224928280814 51 0.304712675021597 0.609425350043193 0.695287324978403 52 0.662295422004347 0.675409155991305 0.337704577995653 53 0.98150343331585 0.0369931333683009 0.0184965666841504 54 0.998091869504788 0.00381626099042467 0.00190813049521233 55 0.999122191116719 0.00175561776656174 0.000877808883280871 56 0.995666696224416 0.00866660755116881 0.00433330377558441 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 23 0.442307692307692 NOK 5% type I error level 25 0.480769230769231 NOK 10% type I error level 27 0.519230769230769 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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