Meervoudig regressiemodel II Paper

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Title produced by software: Multiple Regression

Date of computation: Tue, 21 Dec 2010 17:01:49 +0000

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/21/t1292950784nb44umwfatutaxn.htm/, Retrieved Tue, 21 Dec 2010 17:59:55 +0100

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@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/t1292950784nb44umwfatutaxn.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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8.8 8.1 0 8.5 9.9 0 8.6 11.5 0 8.7 23.4 0 9.1 25.4 0 8.8 27.9 0 6.3 26.1 0 2.5 18.8 0 -2.7 14.1 0 -4.5 11.5 0 -7 15.8 0 -9.3 12.4 0 -12.2 4.5 0 -13.2 -2.2 1 -13.7 -4.2 1 -15 -9.4 1 -16.9 -14.5 1 -16.3 -17.9 1 -16.7 -15.1 1 -16 -15.2 1 -14.5 -15.7 1 -12.2 -18 1 -7.5 -18.1 1 -4.4 -13.5 1 -1.1 -9.9 1 1.3 -4.8 1 -0.1 -1.7 0 0.4 -0.1 0 2.4 2.2 0 1 10.2 0 3.3 7.6 0 1.8 10.8 0 3.2 3.8 0 1.3 11 0 1.5 10.8 0 1.3 20.1 0 2 14.9 0 3 13 0 4.4 10.9 0 3.1 9.6 0 2.6 4 0 2.7 -1.1 0 4 -7.7 0 4.1 -8.9 0 3 -8 0 2.7 -7.1 0 4 -5.3 0 4.8 -2.5 0 6 -2.4 0 4.6 -2.9 0 4.4 -4.8 0 6.6 -7.2 0 4.7 1.7 0 7.6 2.2 0 5.3 13.4 0 6.6 12.3 0 4 13.7 0 3.8 4.4 0 1.2 -2.5 0

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 7 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Industriële_productie[t] = + 1.52135612764115 + 0.0240283609530079registratie_personenwagens[t] -14.0818924662892crisis[t] + 1.92197614831955M1[t] + 4.88892712039673M2[t] + 1.95879600098666M3[t] + 1.97668990890989M4[t] + 1.58467572843338M5[t] + 1.95266154795688M6[t] + 1.61824453138508M7[t] + 1.00948140062399M8[t] -0.142942444689056M9[t] -0.493627844326387M10[t] -0.268341604916727M11[t] + 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) 1.52135612764115 2.744599 0.5543 0.582112 0.291056 registratie_personenwagens 0.0240283609530079 0.078581 0.3058 0.761184 0.380592 crisis -14.0818924662892 2.264548 -6.2184 0 0 M1 1.92197614831955 3.518097 0.5463 0.587551 0.293775 M2 4.88892712039673 3.523943 1.3873 0.172169 0.086085 M3 1.95879600098666 3.521063 0.5563 0.580756 0.290378 M4 1.97668990890989 3.517342 0.562 0.576916 0.288458 M5 1.58467572843338 3.515942 0.4507 0.654362 0.327181 M6 1.95266154795688 3.514981 0.5555 0.581287 0.290643 M7 1.61824453138508 3.514406 0.4605 0.647403 0.323702 M8 1.00948140062399 3.51645 0.2871 0.775373 0.387687 M9 -0.142942444689056 3.525252 -0.0405 0.967835 0.483918 M10 -0.493627844326387 3.534077 -0.1397 0.889539 0.444769 M11 -0.268341604916727 3.535943 -0.0759 0.939843 0.469922 Multiple Linear Regression - Regression Statistics Multiple R 0.798836521394294 R-squared 0.638139787913337 Adjusted R-squared 0.533602393310523 F-TEST (value) 6.1044164180476 F-TEST (DF numerator) 13 F-TEST (DF denominator) 45 p-value 2.23704555324389e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.23742733164919 Sum Squared Residuals 1234.37902744377 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.8 3.63796199968007 5.16203800031993 2 8.5 6.64816402147265 1.85183597852735 3 8.6 3.75647827958739 4.84352172041261 4 8.7 4.06030968285142 4.63969031714858 5 9.1 3.71635222428094 5.38364777571906 6 8.8 4.14440894618695 4.65559105381305 7 6.3 3.76674087989973 2.53325912010027 8 2.5 2.98257071418168 -0.482570714181684 9 -2.7 1.71721357238951 -4.41721357238951 10 -4.5 1.30405443427436 -5.80405443427436 11 -7 1.63266262578194 -8.63266262578194 12 -9.3 1.81930780345845 -11.1193078034584 13 -12.2 3.55145990024923 -15.7514599002492 14 -13.2 -7.72447161234797 -5.47552838765203 15 -13.7 -10.7026594536640 -2.99734054633595 16 -15 -10.8097130226965 -4.19028697730354 17 -16.9 -11.3242718440333 -5.5757281559667 18 -16.3 -11.0379824517500 -5.26201754824997 19 -16.7 -11.3051200576534 -5.39487994234659 20 -16 -11.9162860245098 -4.08371397549019 21 -14.5 -13.0807240502994 -1.41927594970065 22 -12.2 -13.4866746801286 1.28667468012861 23 -7.5 -13.2637912768142 5.76379127681424 24 -4.4 -12.8849192115137 8.48491921151368 25 -1.1 -10.8764409637633 9.7764409637633 26 1.3 -7.78694535082579 9.08694535082579 27 -0.1 3.4393039150077 -3.5393039150077 28 0.4 3.49564320045574 -3.09564320045574 29 2.4 3.15889425017115 -0.75889425017115 30 1 3.71910695731871 -2.71910695731871 31 3.3 3.32221620226909 -0.0222162022690867 32 1.8 2.79034382655762 -0.990343826557624 33 3.2 1.46972145457352 1.73027854542648 34 1.3 1.29204025379785 0.00795974620215367 35 1.5 1.51252082101691 -0.0125208210169078 36 1.3 2.00432618279661 -0.704326182796609 37 2 3.80135485416052 -1.80135485416052 38 3 6.72265194042697 -3.72265194042697 39 4.4 3.7420612630156 0.657938736984404 40 3.1 3.72871830169991 -0.628718301699914 41 2.6 3.20214529988656 -0.602145299886565 42 2.7 3.44758647854972 -0.747586478549722 43 4 2.95458227968806 1.04541772031194 44 4.1 2.31698511578337 1.78301488421663 45 3 1.18618679532803 1.81381320467197 46 2.7 0.857126920548406 1.84287307945159 47 4 1.12566420967348 2.87433579032652 48 4.8 1.46128522525863 3.33871477474137 49 6 3.38566420967348 2.61433579032652 50 4.6 6.34060100127415 -1.74060100127415 51 4.4 3.36481599605337 1.03518400394663 52 6.6 3.32504183768938 3.27495816231062 53 4.7 3.14688006969465 1.55311993030535 54 7.6 3.52688006969465 4.07311993030535 55 5.3 3.46158069579653 1.83841930420347 56 6.6 2.82638636798714 3.77361363201286 57 4 1.7076022280083 2.2923977719917 58 3.8 1.13345307150799 2.66654692849201 59 1.2 1.19294362034190 0.00705637965809747 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.993496996049189 0.0130060079016221 0.00650300395081106 18 0.989982661912636 0.0200346761747287 0.0100173380873643 19 0.99078188906804 0.0184362218639201 0.00921811093196005 20 0.99614551623783 0.007708967524339 0.0038544837621695 21 0.999615372486777 0.000769255026446414 0.000384627513223207 22 0.999985395732026 2.92085359483558e-05 1.46042679741779e-05 23 0.99999937987436 1.24025128069398e-06 6.20125640346991e-07 24 0.99999993592856 1.28142880803519e-07 6.40714404017597e-08 25 0.999999909916435 1.80167131110397e-07 9.00835655551983e-08 26 0.99999974173303 5.16533939763787e-07 2.58266969881894e-07 27 0.99999978121243 4.37575140058389e-07 2.18787570029194e-07 28 0.999999823688391 3.52623217563783e-07 1.76311608781891e-07 29 0.999999402806577 1.19438684577047e-06 5.97193422885234e-07 30 0.999999082471374 1.8350572517212e-06 9.175286258606e-07 31 0.99999682871697 6.34256605939192e-06 3.17128302969596e-06 32 0.999994320185446 1.13596291074909e-05 5.67981455374543e-06 33 0.999980993062814 3.80138743719335e-05 1.90069371859668e-05 34 0.999937723907047 0.000124552185905375 6.22760929526877e-05 35 0.999776902656956 0.000446194686088698 0.000223097343044349 36 0.999493471027276 0.00101305794544871 0.000506528972724356 37 0.999269146851522 0.00146170629695584 0.00073085314847792 38 0.997958929152329 0.0040821416953431 0.00204107084767155 39 0.993197446671999 0.0136051066560028 0.00680255332800138 40 0.99184142024545 0.0163171595091011 0.00815857975455053 41 0.97871809414878 0.0425638117024389 0.0212819058512195 42 0.985807465709703 0.0283850685805943 0.0141925342902971 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 19 0.730769230769231 NOK 5% type I error level 26 1 NOK 10% type I error level 26 1 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No 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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