Meervoudig regressiemodel III Paper

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Title produced by software: Multiple Regression

Date of computation: Tue, 21 Dec 2010 17:51:19 +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/t12929537759o71azla0lwlb3z.htm/, Retrieved Tue, 21 Dec 2010 18:49:38 +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/t12929537759o71azla0lwlb3z.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 5 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation Industriële_productie[t] = -3.01523184590184 + 0.138339800743402registratie_personenwagens[t] -10.5035070224672crisis[t] + 2.75333281992525M1[t] + 4.9492218392974M2[t] + 2.65880688554883M3[t] + 2.46585237141649M4[t] + 1.91100057359639M5[t] + 2.11614877577629M6[t] + 1.60746299788184M7[t] + 1.04162284139986M8[t] + 0.00985374942339004M9[t] -0.307053591118063M10[t] -0.16230073240292M11[t] + 0.105681897448404t + 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) -3.01523184590184 3.442132 -0.876 0.385799 0.1929 registratie_personenwagens 0.138339800743402 0.093919 1.473 0.147877 0.073939 crisis -10.5035070224672 2.789857 -3.7649 0.00049 0.000245 M1 2.75333281992525 3.420774 0.8049 0.425214 0.212607 M2 4.9492218392974 3.402776 1.4545 0.152914 0.076457 M3 2.65880688554883 3.416724 0.7782 0.440631 0.220315 M4 2.46585237141649 3.404527 0.7243 0.472724 0.236362 M5 1.91100057359639 3.398601 0.5623 0.576772 0.288386 M6 2.11614877577629 3.39492 0.6233 0.536287 0.268143 M7 1.60746299788184 3.393447 0.4737 0.638059 0.319029 M8 1.04162284139986 3.395451 0.3068 0.760466 0.380233 M9 0.00985374942339004 3.404719 0.0029 0.997704 0.498852 M10 -0.307053591118063 3.413632 -0.0899 0.928736 0.464368 M11 -0.16230073240292 3.414624 -0.0475 0.962305 0.481153 t 0.105681897448404 0.051171 2.0653 0.044819 0.02241 Multiple Linear Regression - Regression Statistics Multiple R 0.81860760226646 R-squared 0.670118406488443 Adjusted R-squared 0.565156081280221 F-TEST (value) 6.38437082218856 F-TEST (DF numerator) 14 F-TEST (DF denominator) 44 p-value 9.87998990509276e-07 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.05715836863012 Sum Squared Residuals 1125.29343367785 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.8 0.964335257493373 7.83566474250663 2 8.5 3.51491781565203 4.98508218434797 3 8.6 1.55152844054132 7.04847155945868 4 8.7 3.11049945270386 5.58950054729614 5 9.1 2.93800915381897 6.16199084618103 6 8.8 3.59468875530577 5.20531124469423 7 6.3 2.9426732335216 3.3573267664784 8 2.5 1.4726344290612 1.0273655709388 9 -2.7 -0.103649828960856 -2.59635017103914 10 -4.5 -0.674558753986753 -3.82544124601325 11 -7 0.170737145373423 -7.17073714537342 12 -9.3 -0.0316355473028178 -9.26836445269718 13 -12.2 1.73449474419796 -13.934494744198 14 -13.2 -7.39431802642951 -5.80568197357049 15 -13.7 -9.85573068421648 -3.84426931578352 16 -15 -10.6623702647661 -4.33762973523391 17 -16.9 -11.8170731489291 -5.08292685107086 18 -16.3 -11.9765983718284 -4.32340162817159 19 -16.7 -11.9922508101929 -4.70774918980707 20 -16 -12.4662430493008 -3.53375695069916 21 -14.5 -13.4615001442006 -1.03849985579939 22 -12.2 -13.9909071290035 1.79090712900348 23 -7.5 -13.7543063529143 6.25430635291428 24 -4.4 -12.8499606396433 8.44996063964331 25 -1.1 -9.4929226395934 8.3929226395934 26 1.3 -6.48581873898151 7.78581873898151 27 -0.1 2.2618086094901 -2.3618086094901 28 0.4 2.39587967399562 -1.99587967399562 29 2.4 2.26489131533374 0.135108684666259 30 1 3.68243982090925 -2.68243982090925 31 3.3 2.91975245853037 0.380247541469633 32 1.8 2.90228156187567 -1.10228156187567 33 3.2 1.0078157621438 2.1921842378562 34 1.3 1.79263688440324 -0.492636884403239 35 1.5 2.01540368041811 -0.515403680418106 36 1.3 3.56994645718307 -2.26994645718307 37 2 5.70959421069104 -3.70959421069103 38 3 7.74831950609912 -4.74831950609912 39 4.4 5.27307286823781 -0.873072868237809 40 3.1 5.00595851058746 -1.90595851058746 41 2.6 3.78208572605271 -1.18208572605271 42 2.7 3.38738284188966 -0.687382841889661 43 4 2.07133627653717 1.92866372346283 44 4.1 1.44517025661151 2.65482974338849 45 3 0.643588882752505 2.3564111172475 46 2.7 0.556869260328518 2.14313073967148 47 4 1.05631565783019 2.94368434216981 48 4.8 1.71164972976304 3.08835027023696 49 6 4.58449842721103 1.41550157278897 50 4.6 6.81689944365988 -2.21689944365988 51 4.4 4.36932076594725 0.0306792340527503 52 6.6 3.95003262747916 2.64996737252084 53 4.7 4.73208695372373 -0.0320869537237294 54 7.6 5.11208695372373 2.48791304627627 55 5.3 6.25848884160379 -0.958488841603787 56 6.6 5.64615680175247 0.953843198247532 57 4 4.91374532826516 -0.913745328265165 58 3.8 3.41595973825848 0.384040261741521 59 1.2 2.71184986929256 -1.51184986929256 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.00604980835866925 0.0120996167173385 0.99395019164133 19 0.0110036332071533 0.0220072664143067 0.988996366792847 20 0.227174476967325 0.454348953934649 0.772825523032675 21 0.968697368023666 0.0626052639526677 0.0313026319763339 22 0.999959560115897 8.08797682050906e-05 4.04398841025453e-05 23 0.999999931679984 1.36640032467931e-07 6.83200162339655e-08 24 0.9999999991399 1.72020128116282e-09 8.60100640581409e-10 25 0.999999999837639 3.24722319093639e-10 1.6236115954682e-10 26 0.999999999658287 6.83426557075237e-10 3.41713278537618e-10 27 0.999999999006012 1.98797605897548e-09 9.93988029487742e-10 28 0.999999997277873 5.44425429671974e-09 2.72212714835987e-09 29 0.999999985758426 2.84831477639625e-08 1.42415738819812e-08 30 0.999999949038121 1.01923757159393e-07 5.09618785796964e-08 31 0.9999997503315 4.99336998287403e-07 2.49668499143701e-07 32 0.999998927917032 2.14416593620822e-06 1.07208296810411e-06 33 0.999996832190347 6.33561930591545e-06 3.16780965295772e-06 34 0.999985180923004 2.96381539914205e-05 1.48190769957103e-05 35 0.999955269801143 8.94603977137624e-05 4.47301988568812e-05 36 0.999825178663483 0.000349642673033163 0.000174821336516581 37 0.999512889136464 0.000974221727072787 0.000487110863536393 38 0.998072839054233 0.00385432189153453 0.00192716094576726 39 0.994252034703419 0.0114959305931627 0.00574796529658134 40 0.980908253271881 0.038183493456237 0.0190917467281185 41 0.937387497506792 0.125225004986416 0.062612502493208 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 17 0.708333333333333 NOK 5% type I error level 21 0.875 NOK 10% type I error level 22 0.916666666666667 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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