WS7 Lineair trend

*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: Wed, 18 Nov 2009 08:53:19 -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/18/t1258559668lm5v4mhodbw1yw9.htm/, Retrieved Wed, 18 Nov 2009 16:54:40 +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/18/t1258559668lm5v4mhodbw1yw9.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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7,55 42,97 7,55 42,98 7,59 43,01 7,59 43,09 7,59 43,14 7,57 43,39 7,57 43,46 7,59 43,54 7,6 43,62 7,64 44,01 7,64 44,5 7,76 44,73 7,76 44,89 7,76 45,09 7,77 45,17 7,83 45,24 7,94 45,42 7,94 45,67 7,94 45,68 8,09 46,56 8,18 46,72 8,26 47,01 8,28 47,26 8,28 47,49 8,28 47,51 8,29 47,52 8,3 47,66 8,3 47,71 8,31 47,87 8,33 48 8,33 48 8,34 48,05 8,48 48,25 8,59 48,72 8,67 48,94 8,67 49,16 8,67 49,18 8,71 49,25 8,72 49,34 8,72 49,49 8,72 49,57 8,74 49,63 8,74 49,67 8,74 49,7 8,74 49,8 8,79 50,09 8,85 50,49 8,86 50,73 8,87 51,12 8,92 51,15 8,96 51,41 8,97 51,61 8,99 52,06 8,98 52,17 8,98 52,18 9,01 52,19 9,01 52,74 9,03 53,05 9,05 53,38 9,05 53,78

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 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 4.22116168867431 + 0.0758549542204203X[t] + 0.0319607791880008M1[t] + 0.0312048573213947M2[t] + 0.0282010580184485M3[t] + 0.0179558082577059M4[t] + 0.0160972918846523M5[t] -0.0099407055871113M6[t] -0.0278141391933384M7[t] -0.0176448843761232M8[t] -0.00208246919267101M9[t] + 0.0154670920336852M10[t] + 0.00992691271068713M11[t] + 0.0159012047964963t + 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) 4.22116168867431 1.809593 2.3327 0.024096 0.012048 X 0.0758549542204203 0.042226 1.7964 0.078997 0.039499 M1 0.0319607791880008 0.059665 0.5357 0.59477 0.297385 M2 0.0312048573213947 0.060122 0.519 0.606231 0.303115 M3 0.0282010580184485 0.060453 0.4665 0.643061 0.321531 M4 0.0179558082577059 0.060981 0.2945 0.769738 0.384869 M5 0.0160972918846523 0.06084 0.2646 0.792513 0.396257 M6 -0.0099407055871113 0.060947 -0.1631 0.871152 0.435576 M7 -0.0278141391933384 0.06274 -0.4433 0.659609 0.329804 M8 -0.0176448843761232 0.062236 -0.2835 0.778054 0.389027 M9 -0.00208246919267101 0.061671 -0.0338 0.973209 0.486604 M10 0.0154670920336852 0.059932 0.2581 0.7975 0.39875 M11 0.00992691271068713 0.0591 0.168 0.867346 0.433673 t 0.0159012047964963 0.007468 2.1291 0.038629 0.019315 Multiple Linear Regression - Regression Statistics Multiple R 0.987232335545595 R-squared 0.97462768434681 Adjusted R-squared 0.967457247314387 F-TEST (value) 135.923051822328 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0932545584418886 Sum Squared Residuals 0.400034982828815 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.55 7.52851105551025 0.0214889444897528 2 7.55 7.54441488798236 0.00558511201764301 3 7.59 7.55958794210252 0.0304120578974805 4 7.59 7.57131229347591 0.0186877065240928 5 7.59 7.58914772961037 0.00085227038962934 6 7.57 7.5979746754902 -0.0279746754902079 7 7.57 7.60131229347591 -0.0313122934759064 8 7.59 7.63345114942725 -0.0434511494272518 9 7.6 7.67098316574483 -0.070983165744834 10 7.64 7.73401736391365 -0.0940173639136505 11 7.64 7.78154731695515 -0.141547316955155 12 7.76 7.80496824851166 -0.0449682485116601 13 7.76 7.86496702517142 -0.104967025171425 14 7.76 7.8952832989454 -0.135283298945399 15 7.77 7.91424910077658 -0.144249100776583 16 7.83 7.92521490260777 -0.0952149026077656 17 7.94 7.95291148279088 -0.0129114827908836 18 7.94 7.96173842867072 -0.0217384286707213 19 7.94 7.9605247494032 -0.0205247494031945 20 8.09 8.05334756873088 0.0366524312691235 21 8.18 8.0969479813861 0.0830520186139079 22 8.26 8.15239668413287 0.107603315867134 23 8.28 8.18172144816147 0.09827855183853 24 8.28 8.20514237971797 0.0748576202820239 25 8.28 8.25452146278688 0.0254785372131187 26 8.29 8.27042529525898 0.0195747047410238 27 8.3 8.29394239434338 0.00605760565661685 28 8.3 8.30339109709016 -0.00339109709015810 29 8.31 8.32957057818887 -0.019570578188868 30 8.33 8.32929492956226 0.00070507043774407 31 8.33 8.32732270075253 0.00267729924747491 32 8.34 8.35718590807726 -0.0171859080772575 33 8.48 8.40382051890129 0.0761794810987103 34 8.59 8.47292311340774 0.117076886592260 35 8.67 8.49997222880973 0.170027771190269 36 8.67 8.52263461082403 0.147365389175968 37 8.67 8.57201369389294 0.0979863061070621 38 8.71 8.59246882361826 0.117531176381743 39 8.72 8.61219317499165 0.107806825008355 40 8.72 8.62922737316046 0.0907726268395384 41 8.72 8.64933845792154 0.0706615420784622 42 8.74 8.6437529624995 0.0962470375005038 43 8.74 8.64481493185858 0.0951850681414178 44 8.74 8.6731610400989 0.0668389599010938 45 8.74 8.7122101555009 0.0277898444991037 46 8.79 8.76765885824767 0.0223411417523278 47 8.85 8.80836186540934 0.0416381345906621 48 8.86 8.83254134650805 0.0274586534919522 49 8.87 8.90998676263851 -0.0399867626385089 50 8.92 8.92740769419501 -0.00740769419501105 51 8.96 8.96002738778587 -2.73877858694475e-05 52 8.97 8.9808543336657 -0.0108543336657076 53 8.99 9.02903175148834 -0.03903175148834 54 8.98 9.02723900377732 -0.0472390037773186 55 8.98 9.0260253245098 -0.0460253245097919 56 9.01 9.0528543336657 -0.0428543336657079 57 9.01 9.12603817846689 -0.116038178466888 58 9.03 9.18300398029807 -0.153003980298071 59 9.05 9.21839714066431 -0.168397140664307 60 9.05 9.25471341443829 -0.204713414438284 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.217980429335175 0.43596085867035 0.782019570664825 18 0.215132823734107 0.430265647468213 0.784867176265893 19 0.272813795181605 0.54562759036321 0.727186204818395 20 0.232116806929576 0.464233613859151 0.767883193070424 21 0.152380629600453 0.304761259200906 0.847619370399547 22 0.154238475068633 0.308476950137267 0.845761524931367 23 0.33855804401265 0.6771160880253 0.66144195598735 24 0.247814116715856 0.495628233431712 0.752185883284144 25 0.222025466207575 0.444050932415151 0.777974533792425 26 0.254135332760566 0.508270665521132 0.745864667239434 27 0.219607069876699 0.439214139753398 0.780392930123301 28 0.203704976023105 0.40740995204621 0.796295023976895 29 0.246689077559443 0.493378155118886 0.753310922440557 30 0.305037050882074 0.610074101764148 0.694962949117926 31 0.509324331274381 0.981351337451238 0.490675668725619 32 0.985456945426939 0.0290861091461225 0.0145430545730612 33 0.998185707798422 0.00362858440315553 0.00181429220157777 34 0.998107816888668 0.00378436622266435 0.00189218311133217 35 0.999513325728787 0.00097334854242666 0.00048667427121333 36 0.999798139011306 0.000403721977387598 0.000201860988693799 37 0.999731028965552 0.000537942068895309 0.000268971034447655 38 0.999738251372968 0.000523497254063019 0.000261748627031510 39 0.999330747483951 0.00133850503209743 0.000669252516048715 40 0.99812980718608 0.00374038562783807 0.00187019281391903 41 0.993465068012914 0.013069863974173 0.0065349319870865 42 0.97819125602755 0.0436174879448986 0.0218087439724493 43 0.939616422263194 0.120767155473613 0.0603835777368064 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 8 0.296296296296296 NOK 5% type I error level 11 0.407407407407407 NOK 10% type I error level 11 0.407407407407407 NOK

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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