workshop 7,2

*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 06:14:25 -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/t1258722989467ginllt5eam9n.htm/, Retrieved Fri, 20 Nov 2009 14:16:41 +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/t1258722989467ginllt5eam9n.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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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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 1.93696698747801 + 0.657290696471075X[t] -0.198364069129669M1[t] -0.202353720376696M2[t] + 0.026854186070578M3[t] + 0.196624650729587M4[t] + 0.170895581082480M5[t] + 0.0925832557176857M6[t] + 0.125729069647108M7[t] + 0.277187208941323M8[t] + 0.468082790023802M9[t] + 0.652124185035703M10[t] + 0.578978371106282M11[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) 1.93696698747801 0.338009 5.7305 1e-06 0 X 0.657290696471075 0.035783 18.3689 0 0 M1 -0.198364069129669 0.151798 -1.3068 0.19752 0.09876 M2 -0.202353720376696 0.158165 -1.2794 0.206911 0.103456 M3 0.026854186070578 0.157308 0.1707 0.865168 0.432584 M4 0.196624650729587 0.157833 1.2458 0.218892 0.109446 M5 0.170895581082480 0.157581 1.0845 0.283563 0.141781 M6 0.0925832557176857 0.157332 0.5885 0.558984 0.279492 M7 0.125729069647108 0.157347 0.7991 0.428193 0.214097 M8 0.277187208941323 0.157672 1.758 0.085123 0.042561 M9 0.468082790023802 0.158577 2.9518 0.004877 0.002438 M10 0.652124185035703 0.160152 4.0719 0.000174 8.7e-05 M11 0.578978371106282 0.160019 3.6182 0.000712 0.000356 Multiple Linear Regression - Regression Statistics Multiple R 0.93612045291971 R-squared 0.876321502374602 Adjusted R-squared 0.845401877968253 F-TEST (value) 28.3419193861440 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.248722542545239 Sum Squared Residuals 2.96941935216807 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.9 9.03452964917727 -0.134529649177271 2 8.9 8.89908185863604 0.000918141363964084 3 8.6 8.53672813825934 0.0632718617406598 4 8.3 8.18066604574149 0.119333954258513 5 8.3 8.15493697609438 0.145063023905620 6 8.3 8.27381185967091 0.0261881403290909 7 8.4 8.37268674324744 0.0273132567525615 8 8.5 8.45841581289455 0.0415841871054534 9 8.4 8.3863951153886 0.0136048846114047 10 8.6 8.43897837110628 0.161021628893719 11 8.5 8.43156162682397 0.0684383731760322 12 8.5 8.57560302183587 -0.0756030218358685 13 8.4 8.50869709200042 -0.108697092000415 14 8.5 8.43897837110628 0.0610216288937193 15 8.5 8.27381185967091 0.226188140329091 16 8.5 8.18066604574149 0.319333954258512 17 8.5 8.22066604574149 0.279333954258511 18 8.5 8.2080827900238 0.291917209976198 19 8.5 8.24122860395322 0.258771396046776 20 8.5 8.26122860395322 0.238771396046776 21 8.5 8.32066604574149 0.179333954258512 22 8.6 8.50470744075339 0.0952925592466107 23 8.4 8.43156162682397 -0.0315616268239674 24 8.1 8.37841581289455 -0.278415812894547 25 8 8.3115098830591 -0.311509883059092 26 8 8.17606209251785 -0.176062092517852 27 8 8.07662465072959 -0.0766246507295872 28 8 8.04920790644727 -0.0492079064472736 29 7.9 8.02347883680017 -0.123478836800165 30 7.8 8.01089558108248 -0.210895581082479 31 7.8 8.0440413950119 -0.244041395011901 32 7.9 8.19549953430612 -0.295499534306116 33 8.1 8.4521241850357 -0.352124185035703 34 8 8.37324930145918 -0.373249301459175 35 7.6 7.97145813929422 -0.371458139294216 36 7.3 7.45820883783504 -0.158208837835041 37 7 7.06265755976405 -0.062657559764049 38 6.8 6.79575162992859 0.0042483700714069 39 7 7.15641767567008 -0.156417675670082 40 7.1 7.32618814032909 -0.226188140329091 41 7.2 7.36618814032909 -0.166188140329090 42 7.1 7.22214674531719 -0.122146745317190 43 6.9 7.05810535030529 -0.158105350305288 44 6.7 6.88091814136397 -0.180918141363965 45 6.7 6.87462651350512 -0.174626513505122 46 6.6 6.8614806995757 -0.261480699575701 47 6.9 7.05125116423471 -0.151251164234709 48 7.3 7.32675069854082 -0.0267506985408253 49 7.5 7.4570319776467 0.0429680223533061 50 7.3 7.19012604781124 0.109873952188761 51 7.1 7.15641767567008 -0.056417675670082 52 6.9 7.06327186174066 -0.16327186174066 53 7.1 7.23473000103488 -0.134730001034876 54 7.5 7.48506302390562 0.0149369760943803 55 7.7 7.58393790748215 0.116062092517851 56 7.8 7.60393790748215 0.196062092517852 57 7.8 7.46618814032909 0.333811859670909 58 7.7 7.32158418710545 0.378415812894547 59 7.8 7.31416744282314 0.48583255717686 60 7.8 7.26102162889372 0.538978371106282 61 7.9 7.32557383835248 0.574426161647521 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0531170011327804 0.106234002265561 0.94688299886722 17 0.0285600670852211 0.0571201341704421 0.971439932914779 18 0.0368668186732730 0.0737336373465459 0.963133181326727 19 0.0303728049845187 0.0607456099690373 0.969627195015481 20 0.0198468891846181 0.0396937783692362 0.980153110815382 21 0.0125646747047455 0.0251293494094911 0.987435325295254 22 0.00605042536159747 0.0121008507231949 0.993949574638402 23 0.00283386047369268 0.00566772094738537 0.997166139526307 24 0.00316367125579150 0.00632734251158299 0.996836328744209 25 0.00443100701124474 0.00886201402248947 0.995568992988755 26 0.00332869236471632 0.00665738472943263 0.996671307635284 27 0.00240081182182157 0.00480162364364315 0.997599188178178 28 0.00275037054404603 0.00550074108809206 0.997249629455954 29 0.00372723571865105 0.00745447143730209 0.99627276428135 30 0.00448213928387817 0.00896427856775634 0.995517860716122 31 0.00477081299065301 0.00954162598130602 0.995229187009347 32 0.00685295175248257 0.0137059035049651 0.993147048247517 33 0.0170353204696957 0.0340706409393913 0.982964679530304 34 0.0437523981382714 0.0875047962765429 0.956247601861729 35 0.178794881020581 0.357589762041161 0.82120511897942 36 0.326236447420482 0.652472894840963 0.673763552579518 37 0.323405886229708 0.646811772459416 0.676594113770292 38 0.256542822012115 0.513085644024231 0.743457177987885 39 0.191154783942183 0.382309567884367 0.808845216057817 40 0.171412731783793 0.342825463567586 0.828587268216207 41 0.121168400409862 0.242336800819724 0.878831599590138 42 0.0720232551954663 0.144046510390933 0.927976744804534 43 0.040294524072307 0.080589048144614 0.959705475927693 44 0.0244396651579274 0.0488793303158547 0.975560334842073 45 0.0106259472711783 0.0212518945423566 0.989374052728822 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 9 0.3 NOK 5% type I error level 16 0.533333333333333 NOK 10% type I error level 21 0.7 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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