paper Regression Analysis of Time Series 2jaar

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Title produced by software: Multiple Regression

Date of computation: Sun, 26 Dec 2010 17:36:00 +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/26/t1293384841fvv1yk1j4z527zo.htm/, Retrieved Sun, 26 Dec 2010 18:34:11 +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/26/t1293384841fvv1yk1j4z527zo.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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61,2 62 65,1 63,2 66,3 61,9 62,1 66,3 72 65,3 67,6 70,5 74,2 77,8 78,5 77,8 81,4 84,5 88 93,9 98,9 96,7 98,9 102,2 105,4 105,1 116,6 112 108,8 106,9 109,5 106,7 118,9 117,5 113,7 119,6 120,6 117,5 120,3 119,8 108 98,8 94,6 84,6 84,4 79,1 73,3 74,3 67,8 64,8 66,5 57,7 53,8 51,8 50,9 49 48,1 42,6 40,9 43,3 43,7

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 2JAAR[t] = + 88.4611764705882 -4.06349673202618M1[t] + 1.65967320261435M2[t] + 5.7997058823529M3[t] + 2.67973856209145M4[t] + 0.419771241830021M5[t] -2.28019607843141M6[t] -1.86016339869285M7[t] -2.60013071895428M8[t] + 1.93990196078429M9[t] -2.10006535947717M10[t] -3.2800326797386M11[t] -0.180032679738562t + 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) 88.4611764705882 13.854364 6.3851 0 0 M1 -4.06349673202618 16.157447 -0.2515 0.802506 0.401253 M2 1.65967320261435 16.95895 0.0979 0.922448 0.461224 M3 5.7997058823529 16.937294 0.3424 0.733528 0.366764 M4 2.67973856209145 16.917893 0.1584 0.874809 0.437404 M5 0.419771241830021 16.900756 0.0248 0.980288 0.490144 M6 -2.28019607843141 16.88589 -0.135 0.893148 0.446574 M7 -1.86016339869285 16.873301 -0.1102 0.912676 0.456338 M8 -2.60013071895428 16.862994 -0.1542 0.878105 0.439053 M9 1.93990196078429 16.854973 0.1151 0.908851 0.454425 M10 -2.10006535947717 16.849242 -0.1246 0.90133 0.450665 M11 -3.2800326797386 16.845802 -0.1947 0.846442 0.423221 t -0.180032679738562 0.19656 -0.9159 0.36429 0.182145 Multiple Linear Regression - Regression Statistics Multiple R 0.184169168221619 R-squared 0.0339182825234429 Adjusted R-squared -0.207602146845696 F-TEST (value) 0.140436494801035 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 0.99963124189441 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 26.6337381254669 Sum Squared Residuals 34049.0883137255 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 61.2 84.2176470588235 -23.0176470588235 2 62 89.7607843137255 -27.7607843137255 3 65.1 93.7207843137255 -28.6207843137255 4 63.2 90.4207843137255 -27.2207843137255 5 66.3 87.9807843137255 -21.6807843137255 6 61.9 85.1007843137255 -23.2007843137255 7 62.1 85.3407843137255 -23.2407843137255 8 66.3 84.4207843137255 -18.1207843137255 9 72 88.7807843137255 -16.7807843137255 10 65.3 84.5607843137255 -19.2607843137255 11 67.6 83.2007843137255 -15.6007843137255 12 70.5 86.3007843137255 -15.8007843137255 13 74.2 82.0572549019608 -7.85725490196077 14 77.8 87.6003921568628 -9.80039215686275 15 78.5 91.5603921568627 -13.0603921568627 16 77.8 88.2603921568627 -10.4603921568627 17 81.4 85.8203921568627 -4.42039215686273 18 84.5 82.9403921568627 1.55960784313726 19 88 83.1803921568627 4.81960784313726 20 93.9 82.2603921568628 11.6396078431372 21 98.9 86.6203921568628 12.2796078431372 22 96.7 82.4003921568628 14.2996078431373 23 98.9 81.0403921568628 17.8596078431373 24 102.2 84.1403921568628 18.0596078431372 25 105.4 79.896862745098 25.503137254902 26 105.1 85.44 19.66 27 116.6 89.4 27.2 28 112 86.1 25.9 29 108.8 83.66 25.14 30 106.9 80.78 26.12 31 109.5 81.02 28.48 32 106.7 80.1 26.6 33 118.9 84.46 34.44 34 117.5 80.24 37.26 35 113.7 78.88 34.82 36 119.6 81.98 37.62 37 120.6 77.7364705882353 42.8635294117647 38 117.5 83.2796078431373 34.2203921568627 39 120.3 87.2396078431372 33.0603921568628 40 119.8 83.9396078431372 35.8603921568628 41 108 81.4996078431372 26.5003921568628 42 98.8 78.6196078431373 20.1803921568627 43 94.6 78.8596078431373 15.7403921568627 44 84.6 77.9396078431373 6.66039215686274 45 84.4 82.2996078431373 2.10039215686275 46 79.1 78.0796078431372 1.02039215686275 47 73.3 76.7196078431373 -3.41960784313726 48 74.3 79.8196078431373 -5.51960784313729 49 67.8 75.5760784313725 -7.77607843137253 50 64.8 81.1192156862745 -16.3192156862745 51 66.5 85.0792156862745 -18.5792156862745 52 57.7 81.7792156862745 -24.0792156862745 53 53.8 79.3392156862745 -25.5392156862745 54 51.8 76.4592156862745 -24.6592156862745 55 50.9 76.6992156862745 -25.7992156862745 56 49 75.7792156862745 -26.7792156862745 57 48.1 80.1392156862745 -32.0392156862745 58 42.6 75.9192156862745 -33.3192156862745 59 40.9 74.5592156862745 -33.6592156862745 60 43.3 77.6592156862745 -34.3592156862746 61 43.7 73.4156862745098 -29.7156862745098 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.000129183321643540 0.000258366643287081 0.999870816678357 17 7.7892052154236e-06 1.55784104308472e-05 0.999992210794785 18 0.000160430027781579 0.000320860055563157 0.999839569972218 19 0.000369398109612584 0.000738796219225167 0.999630601890387 20 0.000454788353263518 0.000909576706527036 0.999545211646736 21 0.000348516563217316 0.000697033126434631 0.999651483436783 22 0.000592321737571758 0.00118464347514352 0.999407678262428 23 0.000721224494024026 0.00144244898804805 0.999278775505976 24 0.00102191375310585 0.00204382750621170 0.998978086246894 25 0.000968658378527503 0.00193731675705501 0.999031341621472 26 0.00105046972669654 0.00210093945339308 0.998949530273303 27 0.00224131463315082 0.00448262926630165 0.99775868536685 28 0.00375438049024312 0.00750876098048623 0.996245619509757 29 0.00562611226385703 0.0112522245277141 0.994373887736143 30 0.0102797143611653 0.0205594287223307 0.989720285638835 31 0.0176159366598813 0.0352318733197625 0.982384063340119 32 0.0590149304853838 0.118029860970768 0.940985069514616 33 0.046655781394335 0.09331156278867 0.953344218605665 34 0.0308400973232865 0.0616801946465729 0.969159902676713 35 0.0236869713767780 0.0473739427535561 0.976313028623222 36 0.0135677877061681 0.0271355754123362 0.986432212293832 37 0.00921300691635775 0.0184260138327155 0.990786993083642 38 0.0139724051085712 0.0279448102171424 0.986027594891429 39 0.0277207074970955 0.055441414994191 0.972279292502904 40 0.180051036943227 0.360102073886454 0.819948963056773 41 0.656740261642973 0.686519476714054 0.343259738357027 42 0.905758900550627 0.188482198898746 0.0942410994493732 43 0.977434298372745 0.0451314032545097 0.0225657016272548 44 0.976114556435214 0.0477708871295716 0.0238854435647858 45 0.96763849351523 0.0647230129695385 0.0323615064847693 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 13 0.433333333333333 NOK 5% type I error level 22 0.733333333333333 NOK 10% type I error level 26 0.866666666666667 NOK

Charts produced by software:

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Parameters (Session):

par1 = 12 ; par2 = 0.3 ; par3 = 1 ; par4 = 0 ; par5 = 12 ; par6 = 0 ; par7 = 0 ; par8 = 1 ; par9 = 0 ; par10 = FALSE ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ; par4 = 0 ; par5 = 12 ; par6 = 0 ; par7 = 0 ; par8 = 1 ; par9 = 0 ; par10 = FALSE ;

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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