multiple regression analysis: lineaire 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: Sat, 14 Nov 2009 05:58:47 -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/14/t1258203655l4oc6wzpz7vxkin.htm/, Retrieved Sat, 14 Nov 2009 14:01:07 +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/14/t1258203655l4oc6wzpz7vxkin.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:

» Textbox « » Textfile « » CSV «

8,4 99 8,4 98.6 8,4 98.6 8,6 98.5 8,9 98.9 8,8 99.4 8,3 99.8 7,5 99.9 7,2 100 7,4 100.1 8,8 100.1 9,3 100.2 9,3 100.3 8,7 100 8,2 99.9 8,3 99.4 8,5 99.8 8,6 99.6 8,5 100 8,2 99.9 8,1 100.3 7,9 100.6 8,6 100.7 8,7 100.8 8,7 100.8 8,5 100.6 8,4 101.1 8,5 101.1 8,7 100.9 8,7 101.1 8,6 101.2 8,5 101.4 8,3 101.9 8 102.1 8,2 102.1 8,1 103 8,1 103.4 8 103.2 7,9 103.1 7,9 103 8 103.7 8 103.4 7,9 103.5 8 103.8 7,7 104 7,2 104.2 7,5 104.4 7,3 104.4 7 104.9 7 105.3 7 105.2 7,2 105.4 7,3 105.4 7,1 105.5 6,8 105.7 6,4 105.6 6,1 105.8 6,5 105.4 7,7 105.5 7,9 105.8 7,5 106.1 6,9 106 6,6 105.5 6,9 105.4 7,7 106 8 106.1 8 106.4 7,7 106 7,3 106 7,4 106 8,1 106 8,3 106.1 8,2 106.1

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation werkl[t] = + 68.14802305098 -0.600110625503214afzetp[t] -0.102420103023981M1[t] -0.495816819253388M2[t] -0.74391950407523M3[t] -0.705361053505565M4[t] -0.283423175642895M5[t] -0.278179620822697M6[t] -0.362915784660241M7[t] -0.714346271540255M8[t] -0.892417612469521M9[t] -0.95384072431599M10[t] -0.215263836162456M11[t] + 0.0514304868800148t + 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) 68.14802305098 8.045696 8.4701 0 0 afzetp -0.600110625503214 0.081786 -7.3376 0 0 M1 -0.102420103023981 0.20811 -0.4921 0.624445 0.312222 M2 -0.495816819253388 0.216436 -2.2908 0.025563 0.012782 M3 -0.74391950407523 0.216175 -3.4413 0.00107 0.000535 M4 -0.705361053505565 0.217236 -3.247 0.001925 0.000963 M5 -0.283423175642895 0.21591 -1.3127 0.19437 0.097185 M6 -0.278179620822697 0.215978 -1.288 0.202774 0.101387 M7 -0.362915784660241 0.215532 -1.6838 0.097502 0.048751 M8 -0.714346271540255 0.215729 -3.3113 0.001588 0.000794 M9 -0.892417612469521 0.215374 -4.1436 0.000111 5.5e-05 M10 -0.95384072431599 0.215406 -4.4281 4.2e-05 2.1e-05 M11 -0.215263836162456 0.215547 -0.9987 0.322024 0.161012 t 0.0514304868800148 0.010019 5.1332 3e-06 2e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.874136061372917 R-squared 0.764113853792555 Adjusted R-squared 0.712138940221424 F-TEST (value) 14.7015896957059 F-TEST (DF numerator) 13 F-TEST (DF denominator) 59 p-value 5.59552404411079e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.372864172098382 Sum Squared Residuals 8.20263375924211 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.4 8.68608151001797 -0.286081510017968 2 8.4 8.58415953086973 -0.184159530869726 3 8.4 8.3874873329279 0.0125126670720988 4 8.6 8.5374873329279 0.0625126670720989 5 8.9 8.7708114474693 0.129188552530704 6 8.8 8.5274301764179 0.2725698235821 7 8.3 8.2540802492591 0.0459197507409085 8 7.5 7.89406918670877 -0.394069186708767 9 7.2 7.7074172701092 -0.507417270109196 10 7.4 7.63741358259243 -0.237413582592426 11 8.8 8.42742095762597 0.372579042374027 12 9.3 8.63410421811812 0.665895781881884 13 9.3 8.52310353942383 0.776896460576166 14 8.7 8.3611704977254 0.338829502274594 15 8.2 8.2245093623339 -0.0245093623338956 16 8.3 8.61455361253518 -0.314553612535182 17 8.5 8.84787772707659 -0.347877727076587 18 8.6 9.02457389387744 -0.424573893877444 19 8.5 8.75122396671863 -0.251223966718626 20 8.2 8.51123502926894 -0.311235029268944 21 8.1 8.14454992501841 -0.0445499250184129 22 7.9 7.954524112401 -0.0545241124009965 23 8.6 8.68452042488422 -0.0845204248842189 24 8.7 8.89120368537637 -0.191203685376372 25 8.7 8.8402140692324 -0.140214069232406 26 8.5 8.61826996498366 -0.118269964983657 27 8.4 8.12154245429022 0.278457545709778 28 8.5 8.2115313917399 0.288468608260097 29 8.7 8.80492188158322 -0.104921881583225 30 8.7 8.7415737981828 -0.0415737981828007 31 8.6 8.64825705867494 -0.048257058674946 32 8.5 8.2282349335743 0.271765066425699 33 8.3 7.80153876677344 0.498461233226558 34 8 7.67152401670635 0.328475983293646 35 8.2 8.4615313917399 -0.261531391739903 36 8.1 8.18812615182948 -0.0881261518294768 37 8.1 7.89709228548422 0.202907714515778 38 8 7.67514818123547 0.324851818764526 39 7.9 7.53848704584397 0.361512954156027 40 7.9 7.68848704584397 0.211512954156029 41 8 7.7417779727344 0.258222027265595 42 8 7.97848520208558 0.0215147979144207 43 7.9 7.88516846257773 0.0148315374222672 44 8 7.40513527492677 0.594864725073230 45 7.7 7.15847229577688 0.541527704223125 46 7.2 7.02845754570978 0.171542454290223 47 7.5 7.69844279564268 -0.198442795642681 48 7.3 7.96513711868515 -0.665137118685151 49 7 7.61409218978958 -0.614092189789578 50 7 7.0320817102389 -0.0320817102389058 51 7 6.8954205748474 0.104579425152604 52 7.2 6.86538738719643 0.334612612803567 53 7.3 7.33875575193912 -0.0387557519391175 54 7.1 7.33541873108901 -0.235418731089011 55 6.8 7.18209092903084 -0.382090929030839 56 6.4 6.94210199158116 -0.542101991581165 57 6.1 6.69543901243127 -0.59543901243127 58 6.5 6.9254906376661 -0.425490637666097 59 7.7 7.65548695014933 0.0445130498506734 60 7.9 7.74214808554084 0.157851914459165 61 7.5 7.51112528174591 -0.0111252817459069 62 6.9 7.22917011494683 -0.329170114946832 63 6.6 7.33255322975661 -0.732553229756612 64 6.9 7.48255322975661 -0.58255322975661 65 7.7 7.59585521919737 0.104144780802630 66 8 7.59251819834726 0.407481801652736 67 8 7.37917933373877 0.620820666261235 68 7.7 7.31922358394005 0.380776416059947 69 7.3 7.1925827298908 0.107417270109197 70 7.4 7.18259010492435 0.217409895075650 71 8.1 7.9725974799579 0.127402520042102 72 8.3 8.17928074045005 0.120719259549950 73 8.2 8.12829112430608 0.0717088756939148 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.492815955231204 0.985631910462408 0.507184044768796 18 0.500742890009326 0.998514219981348 0.499257109990674 19 0.471163742357359 0.942327484714719 0.528836257642641 20 0.578021005609291 0.843957988781418 0.421978994390709 21 0.627281354694524 0.745437290610951 0.372718645305476 22 0.535504598931279 0.928990802137442 0.464495401068721 23 0.485794602486224 0.971589204972448 0.514205397513776 24 0.556982900532821 0.886034198934358 0.443017099467179 25 0.490068459886142 0.980136919772284 0.509931540113858 26 0.408533931085838 0.817067862171677 0.591466068914162 27 0.326056207643251 0.652112415286503 0.673943792356749 28 0.248758904005940 0.497517808011881 0.75124109599406 29 0.197331283140275 0.394662566280550 0.802668716859725 30 0.149851504244940 0.299703008489881 0.85014849575506 31 0.121006938139716 0.242013876279433 0.878993061860283 32 0.134739573866185 0.26947914773237 0.865260426133815 33 0.135723785181963 0.271447570363927 0.864276214818037 34 0.0984972304911074 0.196994460982215 0.901502769508893 35 0.138267971968037 0.276535943936074 0.861732028031963 36 0.187571297611305 0.37514259522261 0.812428702388695 37 0.159832285691049 0.319664571382098 0.840167714308951 38 0.118404573039519 0.236809146079038 0.881595426960481 39 0.0994942742144045 0.198988548428809 0.900505725785596 40 0.07054896697649 0.14109793395298 0.92945103302351 41 0.0477543371940507 0.0955086743881014 0.95224566280595 42 0.0320962470614855 0.064192494122971 0.967903752938514 43 0.0214978516597954 0.0429957033195908 0.978502148340205 44 0.0268897448290485 0.053779489658097 0.973110255170952 45 0.0769853825914163 0.153970765182833 0.923014617408584 46 0.135875712446772 0.271751424893543 0.864124287553228 47 0.149791375395332 0.299582750790663 0.850208624604668 48 0.211121638794446 0.422243277588893 0.788878361205554 49 0.248650396447646 0.497300792895291 0.751349603552354 50 0.344681934410925 0.689363868821849 0.655318065589075 51 0.544415151536721 0.911169696926557 0.455584848463279 52 0.750363900545709 0.499272198908583 0.249636099454291 53 0.756730212908018 0.486539574183963 0.243269787091981 54 0.640440747128255 0.71911850574349 0.359559252871745 55 0.560957352441368 0.878085295117264 0.439042647558632 56 0.623497437016022 0.753005125967956 0.376502562983978 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 0.025 OK 10% type I error level 4 0.1 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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