WS 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, 27 Nov 2009 11:28:39 -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/27/t125934761645j6zli0ym7ctiu.htm/, Retrieved Fri, 27 Nov 2009 19:47:07 +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/27/t125934761645j6zli0ym7ctiu.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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100,00 100,00 94,97 106,73 107,50 104,81 124,27 96,15 107,06 88,46 79,71 88,46 163,41 91,35 144,83 92,31 166,82 91,35 154,26 87,50 132,60 85,58 157,51 86,54 104,02 97,12 106,03 99,04 113,23 98,08 117,64 92,31 113,34 88,46 66,62 89,42 185,99 90,38 174,57 90,38 208,19 88,46 163,81 86,54 162,46 86,54 148,16 86,54 113,41 94,23 105,63 96,15 111,79 94,23 132,36 89,42 110,75 86,54 67,37 86,54 178,29 87,50 156,38 87,50 189,71 87,50 152,80 88,46 150,80 84,62 160,40 79,81 127,25 80,77 108,47 77,88 117,09 74,04 147,25 75,96 116,19 75,96 75,83 76,92 181,94 75,96 179,12 73,08 183,15 68,27 197,90 65,38 155,42 62,50 162,54 66,35 125,90 78,85 105,50 83,65 121,11 79,81 137,51 75,96 97,20 72,12 69,74 75,00 152,58 79,81 146,59 80,77 161,16 78,85 152,84 74,04 121,95 69,23 140,12 70,19

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 214.097601517345 -0.93455974132132X[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) 214.097601517345 37.270637 5.7444 0 0 X -0.93455974132132 0.438498 -2.1313 0.037316 0.018658 Multiple Linear Regression - Regression Statistics Multiple R 0.269496193141837 R-squared 0.0726281981179421 Adjusted R-squared 0.0566390291199758 F-TEST (value) 4.54233726137856 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.0373157680746052 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 32.992520766721 Sum Squared Residuals 63133.3727394659 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 100 120.641627385213 -20.6416273852127 2 94.97 114.352040326120 -19.3820403261202 3 107.5 116.146395029457 -8.64639502945716 4 124.27 124.239682389300 0.0303176107002066 5 107.06 131.426446800061 -24.3664468000607 6 79.71 131.426446800061 -51.7164468000608 7 163.41 128.725569147642 34.6844308523579 8 144.83 127.828391795974 17.0016082040264 9 166.82 128.725569147642 38.0944308523579 10 154.26 132.323624151729 21.9363758482708 11 132.6 134.117978855066 -1.51797885506616 12 157.51 133.220801503398 24.2891984966023 13 104.02 123.333159440218 -19.3131594402181 14 106.03 121.538804736881 -15.5088047368812 15 113.23 122.435982088550 -9.20598208854964 16 117.64 127.828391795974 -10.1883917959737 17 113.34 131.426446800061 -18.0864468000607 18 66.62 130.529269448392 -63.9092694483923 19 185.99 129.632092096724 56.3579079032762 20 174.57 129.632092096724 44.9379079032762 21 208.19 131.426446800061 76.7635531999392 22 163.81 133.220801503398 30.5891984966023 23 162.46 133.220801503398 29.2391984966023 24 148.16 133.220801503398 14.9391984966023 25 113.41 126.034037092637 -12.6240370926367 26 105.63 124.239682389300 -18.6096823892998 27 111.79 126.034037092637 -14.2440370926367 28 132.36 130.529269448392 1.83073055160774 29 110.75 133.220801503398 -22.4708015033977 30 67.37 133.220801503398 -65.8508015033977 31 178.29 132.323624151729 45.9663758482708 32 156.38 132.323624151729 24.0563758482708 33 189.71 132.323624151729 57.3863758482708 34 152.8 131.426446800061 21.3735531999393 35 150.8 135.015156206735 15.7848437932654 36 160.4 139.51038856249 20.8896114375098 37 127.25 138.613211210822 -11.3632112108217 38 108.47 141.314088863240 -32.8440888632403 39 117.09 144.902798269914 -27.8127982699142 40 147.25 143.108443566577 4.14155643342275 41 116.19 143.108443566577 -26.9184435665773 42 75.83 142.211266214909 -66.3812662149088 43 181.94 143.108443566577 38.8315564334227 44 179.12 145.799975621583 33.3200243784174 45 183.15 150.295207977338 32.8547920226618 46 197.9 152.996085629757 44.9039143702432 47 155.42 155.687617684762 -0.267617684762230 48 162.54 152.089562680675 10.4504373193249 49 125.9 140.407565914159 -14.5075659141586 50 105.5 135.921679155816 -30.4216791558163 51 121.11 139.51038856249 -18.4003885624902 52 137.51 143.108443566577 -5.59844356657726 53 97.2 146.697152973251 -49.4971529732511 54 69.74 144.005620918246 -74.2656209182457 55 152.58 139.51038856249 13.0696114375099 56 146.59 138.613211210822 7.9767887891783 57 161.16 140.407565914159 20.7524340858414 58 152.84 144.902798269914 7.93720173008583 59 121.95 149.398030625670 -27.4480306256697 60 140.12 148.500853274001 -8.38085327400126 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0477520540492521 0.0955041080985041 0.952247945950748 6 0.0851848682978938 0.170369736595788 0.914815131702106 7 0.403438196816777 0.806876393633554 0.596561803183223 8 0.368203324201228 0.736406648402456 0.631796675798772 9 0.452302765535845 0.90460553107169 0.547697234464155 10 0.375649281310077 0.751298562620154 0.624350718689923 11 0.278950090826277 0.557900181652554 0.721049909173723 12 0.223602418513333 0.447204837026666 0.776397581486667 13 0.170981395253633 0.341962790507267 0.829018604746367 14 0.120003735466361 0.240007470932723 0.879996264533639 15 0.0787799319409556 0.157559863881911 0.921220068059044 16 0.0522526748755487 0.104505349751097 0.94774732512445 17 0.041187199351846 0.082374398703692 0.958812800648154 18 0.167481031229127 0.334962062458254 0.832518968770873 19 0.310190031085964 0.620380062171928 0.689809968914036 20 0.365043129971266 0.730086259942532 0.634956870028734 21 0.649669276300311 0.700661447399378 0.350330723699689 22 0.611544284107687 0.776911431784625 0.388455715892313 23 0.569998603591855 0.860002792816289 0.430001396408145 24 0.501266511733506 0.997466976532988 0.498733488266494 25 0.433943749081998 0.867887498163996 0.566056250918002 26 0.376361304184344 0.752722608368689 0.623638695815655 27 0.31857245765379 0.63714491530758 0.68142754234621 28 0.254666987320817 0.509333974641634 0.745333012679183 29 0.247601489725774 0.495202979451548 0.752398510274226 30 0.522607807223761 0.954784385552478 0.477392192776239 31 0.551646616469542 0.896706767060916 0.448353383530458 32 0.499516731571934 0.999033463143868 0.500483268428066 33 0.635730249535567 0.728539500928865 0.364269750464432 34 0.61312341423793 0.773753171524141 0.386876585762071 35 0.581220843359083 0.837558313281834 0.418779156640917 36 0.560245747634492 0.879508504731016 0.439754252365508 37 0.511711929245317 0.976576141509366 0.488288070754683 38 0.524324160236344 0.951351679527311 0.475675839763656 39 0.51271420479959 0.97457159040082 0.48728579520041 40 0.439208234272266 0.878416468544532 0.560791765727734 41 0.402545863267512 0.805091726535024 0.597454136732488 42 0.60597348209512 0.788053035809761 0.394026517904880 43 0.659483147426795 0.681033705146409 0.340516852573205 44 0.673936132344232 0.652127735311536 0.326063867655768 45 0.671678700964059 0.656642598071882 0.328321299035941 46 0.773543569162332 0.452912861675336 0.226456430837668 47 0.71220769441226 0.57558461117548 0.28779230558774 48 0.71668499734322 0.56663000531356 0.28331500265678 49 0.627366364588773 0.745267270822454 0.372633635411227 50 0.651138829283553 0.697722341432894 0.348861170716447 51 0.589292485140439 0.821415029719122 0.410707514859561 52 0.468395223732779 0.936790447465558 0.531604776267221 53 0.449871223872662 0.899742447745324 0.550128776127338 54 0.985283129054709 0.0294337418905818 0.0147168709452909 55 0.947937899823475 0.104124200353051 0.0520621001765254 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 1 0.0196078431372549 OK 10% type I error level 3 0.0588235294117647 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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