*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, 19 Nov 2010 14:23:14 +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/Nov/19/t12901765178kndu6v4qjzx0wg.htm/, Retrieved Fri, 19 Nov 2010 15:22:08 +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/Nov/19/t12901765178kndu6v4qjzx0wg.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:

IsPrivate?

No (this computation is public)

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Dataseries X:

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101,82 107,34 93,63 101,76 101,68 107,34 93,63 102,37 101,68 107,34 93,63 102,38 102,45 107,34 96,13 102,86 102,45 107,34 96,13 102,87 102,45 107,34 96,13 102,92 102,45 107,34 96,13 102,95 102,45 107,34 96,13 103,02 102,45 112,60 96,13 104,08 102,52 112,60 96,13 104,16 102,52 112,60 96,13 104,24 102,85 112,60 96,13 104,33 102,85 112,61 96,13 104,73 102,85 112,61 96,13 104,86 103,25 112,61 96,13 105,03 103,25 112,61 98,73 105,62 103,25 112,61 98,73 105,63 103,25 112,61 98,73 105,63 104,45 112,61 98,73 105,94 104,45 112,61 98,73 106,61 104,45 118,65 98,73 107,69 104,80 118,65 98,73 107,78 104,80 118,65 98,73 107,93 105,29 118,65 98,73 108,48 105,29 114,29 98,73 108,14 105,29 114,29 98,73 108,48 105,29 114,29 98,73 108,48 106,04 114,29 101,67 108,89 105,94 114,29 101,67 108,93 105,94 114,29 101,67 109,21 105,94 114,29 101,67 109,47 106,28 114,29 101,67 109,80 106,48 123,33 101,67 111,73 107,19 123,33 101,67 111,85 108,14 123,33 101,67 112,12 108,22 123,33 101,67 112,15 108,22 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 6 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation And.dienstenrecr.&cultuur[t] = + 61.3522977549418 + 0.104860262173398Bioscoop[t] + 0.167292816816152Schouwburgabonnement[t] + 0.121128872847212Eendagsattracties[t] + 0.178769949566188t + 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) 61.3522977549418 3.419552 17.9416 0 0 Bioscoop 0.104860262173398 0.017876 5.866 0 0 Schouwburgabonnement 0.167292816816152 0.018921 8.8416 0 0 Eendagsattracties 0.121128872847212 0.031641 3.8282 0.000343 0.000171 t 0.178769949566188 0.012691 14.0861 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.998649583453835 R-squared 0.997300990532518 Adjusted R-squared 0.997097291704784 F-TEST (value) 4895.95841873246 F-TEST (DF numerator) 4 F-TEST (DF denominator) 53 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.300357119404692 Sum Squared Residuals 4.78136315638548 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 101.76 101.506446920734 0.253553079266132 2 102.37 101.670536433596 0.699463566404492 3 102.38 101.849306383162 0.530693616838284 4 102.86 102.411640916719 0.448359083280561 5 102.87 102.590410866286 0.279589133714374 6 102.92 102.769180815852 0.150819184148186 7 102.95 102.947950765418 0.00204923458199948 8 103.02 103.126720714984 -0.106720714984195 9 104.08 104.185450881003 -0.105450881003340 10 104.16 104.371561048922 -0.211561048921668 11 104.24 104.550330998488 -0.310330998487857 12 104.33 104.763704834571 -0.433704834571263 13 104.73 104.944147712306 -0.214147712305608 14 104.86 105.122917661872 -0.2629176618718 15 105.03 105.343631716307 -0.313631716307346 16 105.62 105.837336735276 -0.217336735276282 17 105.63 106.016106684842 -0.386106684842479 18 105.63 106.194876634409 -0.564876634408667 19 105.94 106.499478898583 -0.559478898582932 20 106.61 106.678248848149 -0.068248848149118 21 107.69 107.867467411285 -0.177467411284868 22 107.78 108.082938452612 -0.302938452611742 23 107.93 108.261708402178 -0.331708402177924 24 108.48 108.491859880209 -0.0118598802090809 25 108.14 107.941233148457 0.198766851543152 26 108.48 108.120003098023 0.359996901976967 27 108.48 108.298773047589 0.181226952410779 28 108.89 108.912307079956 -0.0223070799562636 29 108.93 109.080591003305 -0.150591003305104 30 109.21 109.259360952871 -0.0493609528713051 31 109.47 109.438130902437 0.031869097562512 32 109.8 109.652553341143 0.147446658857367 33 111.73 111.364622407162 0.365377592838490 34 111.85 111.617843142871 0.23215685712918 35 112.12 111.896230341502 0.223769658498273 36 112.15 112.083389112042 0.0666108879582143 37 112.17 112.262159061608 -0.0921590616079776 38 112.67 112.481824513422 0.188175486578210 39 112.8 112.660594462988 0.139405537012017 40 113.44 113.598842445306 -0.158842445306188 41 113.53 113.777612394872 -0.247612394872372 42 114.53 114.003569462417 0.526430537583410 43 114.51 114.182339411983 0.327660588017226 44 115.05 114.76691857616 0.283081423839978 45 116.67 116.204400944503 0.465599055497137 46 117.07 116.624270349262 0.445729650737927 47 116.92 116.803040298828 0.116959701171748 48 117 117.012219724425 -0.0122197244247277 49 117.02 117.203572905452 -0.183572905451728 50 117.35 117.412752331048 -0.0627523310482026 51 117.36 117.604105512075 -0.244105512075195 52 117.82 118.079225627778 -0.259225627778146 53 117.88 118.268481603562 -0.388481603561673 54 118.24 118.502827492080 -0.262827492079763 55 118.5 118.683694646889 -0.183694646889414 56 118.8 118.862464596456 -0.0624645964556048 57 119.76 119.688657747100 0.0713422528997051 58 120.09 119.867427696666 0.222572303333515 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.00106771540000924 0.00213543080001848 0.99893228459999 9 8.29172946740711e-05 0.000165834589348142 0.999917082705326 10 0.0202576939307905 0.040515387861581 0.97974230606921 11 0.0126024341060529 0.0252048682121058 0.987397565893947 12 0.109040383060299 0.218080766120599 0.8909596169397 13 0.172617050094518 0.345234100189036 0.827382949905482 14 0.143086676535707 0.286173353071413 0.856913323464293 15 0.0979231660618269 0.195846332123654 0.902076833938173 16 0.138726408742863 0.277452817485726 0.861273591257137 17 0.0936401032911739 0.187280206582348 0.906359896708826 18 0.0775673263447698 0.155134652689540 0.92243267365523 19 0.0676213465226017 0.135242693045203 0.932378653477398 20 0.173807302023062 0.347614604046124 0.826192697976938 21 0.234267235231381 0.468534470462762 0.765732764768619 22 0.22173611529168 0.44347223058336 0.77826388470832 23 0.284629378857554 0.569258757715108 0.715370621142446 24 0.331591495396557 0.663182990793113 0.668408504603443 25 0.414759987357777 0.829519974715554 0.585240012642223 26 0.54527944896475 0.9094411020705 0.45472055103525 27 0.504071627341166 0.991856745317667 0.495928372658834 28 0.424631387716153 0.849262775432305 0.575368612283848 29 0.365112399557182 0.730224799114364 0.634887600442818 30 0.305597259502631 0.611194519005262 0.694402740497369 31 0.265512919455488 0.531025838910976 0.734487080544512 32 0.280294367856035 0.56058873571207 0.719705632143965 33 0.530698358582418 0.938603282835164 0.469301641417582 34 0.452862877824313 0.905725755648625 0.547137122175687 35 0.421833990343651 0.843667980687303 0.578166009656349 36 0.401252951931182 0.802505903862363 0.598747048068818 37 0.449286159598301 0.898572319196603 0.550713840401699 38 0.372923504996769 0.745847009993538 0.627076495003231 39 0.332927668002649 0.665855336005298 0.667072331997351 40 0.403951625790861 0.807903251581722 0.596048374209139 41 0.869209695252928 0.261580609494143 0.130790304747072 42 0.870847204475137 0.258305591049725 0.129152795524863 43 0.88811077111343 0.223778457773140 0.111889228886570 44 0.89853710617474 0.202925787650521 0.101462893825260 45 0.858229888192529 0.283540223614943 0.141770111807471 46 0.968357375520594 0.0632852489588116 0.0316426244794058 47 0.97991817473241 0.0401636505351799 0.0200818252675900 48 0.985794984605446 0.0284100307891076 0.0142050153945538 49 0.962950697620354 0.0740986047592917 0.0370493023796458 50 0.966943886863082 0.0661122262738351 0.0330561131369175 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 2 0.0465116279069767 NOK 5% type I error level 6 0.139534883720930 NOK 10% type I error level 9 0.209302325581395 NOK

Charts produced by software:

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Parameters (Session):

par1 = 4 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 4 ; par2 = Do not include Seasonal 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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