Lineaire trend - paper

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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Tue, 21 Dec 2010 18:49:04 +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/Dec/21/t1292957231uii56xz4wvw011u.htm/, Retrieved Tue, 21 Dec 2010 19:47:14 +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/Dec/21/t1292957231uii56xz4wvw011u.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}, }

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Dataseries X:

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8.1 9.9 11.5 23.4 25.4 27.9 26.1 18.8 14.1 11.5 15.8 12.4 4.5 -2.2 -4.2 -9.4 -14.5 -17.9 -15.1 -15.2 -15.7 -18 -18.1 -13.5 -9.9 -4.8 -1.7 -0.1 2.2 10.2 7.6 10.8 3.8 11 10.8 20.1 14.9 13 10.9 9.6 4 -1.1 -7.7 -8.9 -8 -7.1 -5.3 -2.5 -2.4 -2.9 -4.8 -7.2 1.7 2.2 13.4 12.3 13.7 4.4 -2.5

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 5 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation registraties_personenwagens[t] = + 8.35652173913044 -1.79025362318841M1[t] -2.08920289855073M2[t] -2.20815217391305M3[t] -1.14710144927537M4[t] -0.506050724637686M5[t] + 0.134999999999994M6[t] + 0.876050724637677M7[t] -0.282898550724641M8[t] -2.12184782608696M9[t] -3.20079710144928M10[t] -3.2797463768116M11[t] -0.141050724637681t + 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) 8.35652173913044 7.24513 1.1534 0.254705 0.127352 M1 -1.79025362318841 8.824591 -0.2029 0.840131 0.420065 M2 -2.08920289855073 8.819279 -0.2369 0.813794 0.406897 M3 -2.20815217391305 8.815145 -0.2505 0.80332 0.40166 M4 -1.14710144927537 8.81219 -0.1302 0.896998 0.448499 M5 -0.506050724637686 8.810417 -0.0574 0.954445 0.477223 M6 0.134999999999994 8.809826 0.0153 0.98784 0.49392 M7 0.876050724637677 8.810417 0.0994 0.921226 0.460613 M8 -0.282898550724641 8.81219 -0.0321 0.974529 0.487264 M9 -2.12184782608696 8.815145 -0.2407 0.810854 0.405427 M10 -3.20079710144928 8.819279 -0.3629 0.718318 0.359159 M11 -3.2797463768116 8.824591 -0.3717 0.711853 0.355926 t -0.141050724637681 0.102054 -1.3821 0.173611 0.086806 Multiple Linear Regression - Regression Statistics Multiple R 0.231935098563939 R-squared 0.053793889945864 Adjusted R-squared -0.193042486589998 F-TEST (value) 0.217933396612021 F-TEST (DF numerator) 12 F-TEST (DF denominator) 46 p-value 0.996729660179765 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 13.1329135293497 Sum Squared Residuals 7933.7772173913 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.1 6.42521739130433 1.67478260869567 2 9.9 5.98521739130435 3.91478260869565 3 11.5 5.72521739130435 5.77478260869565 4 23.4 6.64521739130435 16.7547826086957 5 25.4 7.14521739130435 18.2547826086957 6 27.9 7.64521739130435 20.2547826086956 7 26.1 8.24521739130435 17.8547826086957 8 18.8 6.94521739130435 11.8547826086957 9 14.1 4.96521739130435 9.13478260869565 10 11.5 3.74521739130435 7.75478260869565 11 15.8 3.52521739130435 12.2747826086957 12 12.4 6.66391304347826 5.73608695652174 13 4.5 4.73260869565218 -0.232608695652179 14 -2.2 4.29260869565217 -6.49260869565217 15 -4.2 4.03260869565217 -8.23260869565217 16 -9.4 4.95260869565218 -14.3526086956522 17 -14.5 5.45260869565217 -19.9526086956522 18 -17.9 5.95260869565217 -23.8526086956522 19 -15.1 6.55260869565218 -21.6526086956522 20 -15.2 5.25260869565217 -20.4526086956522 21 -15.7 3.27260869565217 -18.9726086956522 22 -18 2.05260869565218 -20.0526086956522 23 -18.1 1.83260869565218 -19.9326086956522 24 -13.5 4.97130434782609 -18.4713043478261 25 -9.9 3.04 -12.94 26 -4.8 2.6 -7.4 27 -1.7 2.34 -4.04 28 -0.1 3.26 -3.36 29 2.2 3.76 -1.56 30 10.2 4.26 5.94 31 7.6 4.86 2.74 32 10.8 3.56 7.24 33 3.8 1.58 2.22 34 11 0.359999999999999 10.64 35 10.8 0.140000000000001 10.66 36 20.1 3.27869565217392 16.8213043478261 37 14.9 1.34739130434783 13.5526086956522 38 13 0.907391304347822 12.0926086956522 39 10.9 0.647391304347822 10.2526086956522 40 9.6 1.56739130434783 8.03260869565217 41 4 2.06739130434782 1.93260869565218 42 -1.1 2.56739130434782 -3.66739130434783 43 -7.7 3.16739130434783 -10.8673913043478 44 -8.9 1.86739130434782 -10.7673913043478 45 -8 -0.112608695652177 -7.88739130434782 46 -7.1 -1.33260869565218 -5.76739130434782 47 -5.3 -1.55260869565217 -3.74739130434783 48 -2.5 1.58608695652174 -4.08608695652174 49 -2.4 -0.345217391304347 -2.05478260869565 50 -2.9 -0.785217391304349 -2.11478260869565 51 -4.8 -1.04521739130435 -3.75478260869565 52 -7.2 -0.12521739130435 -7.07478260869565 53 1.7 0.374782608695649 1.32521739130435 54 2.2 0.87478260869565 1.32521739130435 55 13.4 1.47478260869565 11.9252173913044 56 12.3 0.174782608695651 12.1252173913043 57 13.7 -1.80521739130435 15.5052173913043 58 4.4 -3.02521739130435 7.42521739130435 59 -2.5 -3.24521739130435 0.745217391304348 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.395272114058481 0.790544228116961 0.60472788594152 17 0.535750678531894 0.928498642936211 0.464249321468106 18 0.650256648186114 0.699486703627772 0.349743351813886 19 0.625529246429432 0.748941507141135 0.374470753570568 20 0.543293504627206 0.913412990745588 0.456706495372794 21 0.454273861304534 0.908547722609069 0.545726138695466 22 0.393476947967838 0.786953895935675 0.606523052032162 23 0.357236048116345 0.71447209623269 0.642763951883655 24 0.359655886353192 0.719311772706384 0.640344113646808 25 0.608354078511745 0.783291842976511 0.391645921488255 26 0.777302913771828 0.445394172456345 0.222697086228172 27 0.844713468183765 0.310573063632469 0.155286531816235 28 0.848643137774833 0.302713724450334 0.151356862225167 29 0.851844663470458 0.296310673059085 0.148155336529542 30 0.871375703897164 0.257248592205671 0.128624296102836 31 0.854917161564009 0.290165676871982 0.145082838435991 32 0.8526743247754 0.294651350449201 0.147325675224601 33 0.821017547206566 0.357964905586868 0.178982452793434 34 0.817685511393437 0.364628977213126 0.182314488606563 35 0.80977982986559 0.380440340268821 0.19022017013441 36 0.864601640052461 0.270796719895078 0.135398359947539 37 0.880592357587191 0.238815284825618 0.119407642412809 38 0.89198861585421 0.21602276829158 0.10801138414579 39 0.915940086735909 0.168119826528183 0.0840599132640913 40 0.97205657023252 0.0558868595349604 0.0279434297674802 41 0.972802153645305 0.0543956927093909 0.0271978463546955 42 0.962428846261902 0.0751423074761963 0.0375711537380981 43 0.912128827012848 0.175742345974303 0.0878711729871517 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 3 0.107142857142857 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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