JJ Workshop 7, optimaliseren model (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 13:10:08 -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/t12587480390r6ewzr9sabtnt0.htm/, Retrieved Fri, 20 Nov 2009 21:14:12 +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/t12587480390r6ewzr9sabtnt0.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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95,1 93,8 111,7 97 93,8 98,6 112,7 107,6 96,9 102,9 101 95,1 97,4 95,4 97 111,4 96,5 112,7 87,4 89,2 102,9 96,8 87,1 97,4 114,1 110,5 111,4 110,3 110,8 87,4 103,9 104,2 96,8 101,6 88,9 114,1 94,6 89,8 110,3 95,9 90 103,9 104,7 93,9 101,6 102,8 91,3 94,6 98,1 87,8 95,9 113,9 99,7 104,7 80,9 73,5 102,8 95,7 79,2 98,1 113,2 96,9 113,9 105,9 95,2 80,9 108,8 95,6 95,7 102,3 89,7 113,2 99 92,8 105,9 100,7 88 108,8 115,5 101,1 102,3 100,7 92,7 99 109,9 95,8 100,7 114,6 103,8 115,5 85,4 81,8 100,7 100,5 87,1 109,9 114,8 105,9 114,6 116,5 108,1 85,4 112,9 102,6 100,5 102 93,7 114,8 106 103,5 116,5 105,3 100,6 112,9 118,8 113,3 102 106,1 102,4 106 109,3 102,1 105,3 117,2 106,9 118,8 92,5 87,3 106,1 104,2 93,1 109,3 112,5 109,1 117,2 122,4 120,3 92,5 113,3 104,9 104,2 100 92,6 112,5 110,7 109,8 122,4 112,8 111,4 113,3 109,8 117,9 100 117,3 121,6 110,7 109,1 117,8 112,8 115,9 124,2 109,8 96 106,8 117,3 99,8 102,7 109,1 116,8 116,8 115,9 115,7 113,6 96 99,4 96,1 99 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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation TIA[t] = + 32.5049364211236 + 0.303638779125189IAidM[t] + 0.354779288294477`TIA(t-3)`[t] -1.15156326894194M1[t] + 2.55491853186197M2[t] + 11.9498804134043M3[t] + 6.94062493998272M4[t] + 5.91613478259514M5[t] + 10.2762807456465M6[t] -8.00791974071992M7[t] + 2.74328260551863M8[t] + 8.67593779647815M9[t] + 17.3117411390319M10[t] + 9.64099946115172M11[t] -0.0091814120305812t + 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) 32.5049364211236 14.565839 2.2316 0.030668 0.015334 IAidM 0.303638779125189 0.075723 4.0099 0.000226 0.000113 `TIA(t-3)` 0.354779288294477 0.144699 2.4518 0.018158 0.009079 M1 -1.15156326894194 2.402733 -0.4793 0.634066 0.317033 M2 2.55491853186197 2.549927 1.002 0.321722 0.160861 M3 11.9498804134043 3.475651 3.4382 0.001272 0.000636 M4 6.94062493998272 3.218519 2.1565 0.036428 0.018214 M5 5.91613478259514 3.032412 1.951 0.057304 0.028652 M6 10.2762807456465 2.655537 3.8698 0.000349 0.000174 M7 -8.00791974071992 2.475079 -3.2354 0.00228 0.00114 M8 2.74328260551863 2.590436 1.059 0.295249 0.147625 M9 8.67593779647815 2.634067 3.2937 0.001931 0.000966 M10 17.3117411390319 4.998995 3.463 0.001183 0.000592 M11 9.64099946115172 3.39272 2.8417 0.006721 0.00336 t -0.0091814120305812 0.040456 -0.2269 0.821492 0.410746 Multiple Linear Regression - Regression Statistics Multiple R 0.93894829808783 R-squared 0.881623906482032 Adjusted R-squared 0.844795788498664 F-TEST (value) 23.9388802566611 F-TEST (DF numerator) 14 F-TEST (DF denominator) 45 p-value 2.22044604925031e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.54551443401741 Sum Squared Residuals 565.68026708216 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 95.1 99.4543557245867 -4.35435572458669 2 97 98.5040474367025 -1.50404743670252 3 112.7 111.476918268041 1.22308173195868 4 102.9 103.815862721433 -0.915862721432816 5 97.4 101.755894636673 -4.35589463667311 6 111.4 112.010896670955 -0.610896670954898 7 87.4 88.0241146596581 -0.624114659658122 8 96.8 96.1772080720836 0.622791927916421 9 114.1 114.172739318665 -0.0727393186646354 10 110.3 114.375749963858 -4.07574996385792 11 103.9 108.026736241689 -4.12673624168897 12 101.6 99.8685637353857 1.73143626461426 13 94.6 97.6329326601069 -3.03293266010687 14 95.9 99.1203733596206 -3.22037335962058 15 104.7 108.874352704643 -4.1743527046433 16 102.8 100.582999975404 2.21700002459573 17 98.1 98.9478057538308 -0.84780575383078 18 113.9 110.034129513433 3.86587048656729 19 80.9 83.1113309541962 -2.21133095419624 20 95.7 93.9166302744337 1.78336972556626 21 113.2 110.820023198931 2.37997680106873 22 105.9 107.222742691224 -1.32274269122390 23 108.8 104.915008579721 3.88499142027855 24 102.3 99.6819964548539 2.61800354514611 25 99 96.8726431846198 2.12735681538024 26 100.7 100.141337369646 0.558662630353834 27 115.5 111.198720471784 4.30127952821619 28 100.7 102.458946190308 -1.75894619030826 29 109.9 102.969679626279 6.9303203737212 30 114.6 115.000487877059 -0.400487877059376 31 85.4 84.77631937115 0.623680628850072 32 100.5 100.391595287031 0.108404712969405 33 114.8 113.690940768497 1.10905923150287 34 116.5 112.626012794897 3.87398720510299 35 112.9 108.633243673044 4.26675632695571 36 102 101.354021488259 0.645978511741171 37 106 103.772061632814 2.22793836718623 38 105.3 105.311604124264 -0.0116041242639373 39 118.8 114.686502846256 4.11349715374419 40 106.1 107.777520421517 -1.67752042151698 41 109.3 106.404411716555 2.89558828344488 42 117.2 117.002362799352 0.197637200647743 43 92.5 88.2519638687617 4.24803613123832 44 104.2 101.890383444438 2.30961655556194 45 112.5 115.474834066896 -2.9748340668964 46 122.4 118.739161902748 3.66083809725188 47 113.3 110.534119287355 2.76588071264518 48 100 100.093849523777 -0.0938495237768465 49 110.7 107.668006797873 3.03199320212709 50 112.8 108.622637709767 4.1773622902332 51 109.8 115.263505709276 -5.46350570927576 52 117.3 115.164670691338 2.13532930866233 53 109.1 113.722208266662 -4.62220826666220 54 115.9 118.952123139201 -3.05212313920076 55 96 98.036271146234 -2.03627114623404 56 99.8 104.624182922014 -4.82418292201402 57 116.8 117.241462647011 -0.441462647010568 58 115.7 117.836332647273 -2.13633264727305 59 99.4 106.190892218190 -6.79089221819047 60 94.3 99.2015687977247 -4.90156879772469 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.240402767718595 0.48080553543719 0.759597232281405 19 0.156069323308035 0.312138646616071 0.843930676691965 20 0.0803744096858877 0.160748819371775 0.919625590314112 21 0.0892172002147046 0.178434400429409 0.910782799785295 22 0.0582838367120907 0.116567673424181 0.94171616328791 23 0.184366069286090 0.368732138572179 0.81563393071391 24 0.127193752895896 0.254387505791793 0.872806247104104 25 0.0799763713517787 0.159952742703557 0.920023628648221 26 0.0780215597470313 0.156043119494063 0.921978440252969 27 0.0603299849990349 0.12065996999807 0.939670015000965 28 0.104140720031313 0.208281440062626 0.895859279968687 29 0.204775686933931 0.409551373867863 0.795224313066069 30 0.246961059870721 0.493922119741443 0.753038940129279 31 0.224026936389358 0.448053872778717 0.775973063610642 32 0.168503905357824 0.337007810715648 0.831496094642176 33 0.139221676847252 0.278443353694504 0.860778323152748 34 0.106327115971249 0.212654231942497 0.893672884028751 35 0.0688646729432135 0.137729345886427 0.931135327056786 36 0.0889201504270339 0.177840300854068 0.911079849572966 37 0.0753102481224137 0.150620496244827 0.924689751877586 38 0.125754359132600 0.251508718265199 0.8742456408674 39 0.122766871073735 0.245533742147470 0.877233128926265 40 0.270492702129114 0.540985404258228 0.729507297870886 41 0.194740275555743 0.389480551111487 0.805259724444257 42 0.113701858018165 0.227403716036331 0.886298141981835 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 0 0 OK 10% type I error level 0 0 OK

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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