paper Regression Analysis of Time Series eonia

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Title produced by software: Multiple Regression

Date of computation: Sun, 26 Dec 2010 17:37:56 +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/t1293384954rhtgz4cf8pthdpi.htm/, Retrieved Sun, 26 Dec 2010 18:36:03 +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/t1293384954rhtgz4cf8pthdpi.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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2,08 2,09 2,07 2,04 2,35 2,33 2,37 2,59 2,62 2,6 2,83 2,78 3,01 3,06 3,33 3,32 3,6 3,57 3,57 3,83 3,84 3,8 4,07 4,05 4,272 3,858 4,067 3,964 3,782 4,114 4,009 4,025 4,082 4,044 3,916 4,289 4,296 4,193 3,48 2,934 2,221 1,211 1,28 0,96 0,5 0,687 0,344 0,346 0,334 0,34 0,328 0,344 0,341 0,32 0,314 0,325 0,339 0,329 0,48 0,399 0,37

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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation eonia[t] = + 4.15592941176471 -0.226790196078433M1[t] -0.159913725490198M2[t] -0.163582352941180M3[t] -0.24865098039216M4[t] -0.260719607843140M5[t] -0.36098823529412M6[t] -0.311856862745100M7[t] -0.22492549019608M8[t] -0.245194117647061M9[t] -0.179862745098042M10[t] -0.0943313725490219M11[t] -0.0495313725490196t + 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) 4.15592941176471 0.704125 5.9023 0 0 M1 -0.226790196078433 0.821175 -0.2762 0.783597 0.391799 M2 -0.159913725490198 0.861911 -0.1855 0.853592 0.426796 M3 -0.163582352941180 0.86081 -0.19 0.850085 0.425042 M4 -0.24865098039216 0.859824 -0.2892 0.773683 0.386841 M5 -0.260719607843140 0.858953 -0.3035 0.762796 0.381398 M6 -0.36098823529412 0.858197 -0.4206 0.675899 0.337949 M7 -0.311856862745100 0.857558 -0.3637 0.717711 0.358855 M8 -0.22492549019608 0.857034 -0.2624 0.7941 0.39705 M9 -0.245194117647061 0.856626 -0.2862 0.775932 0.387966 M10 -0.179862745098042 0.856335 -0.21 0.834527 0.417264 M11 -0.0943313725490219 0.85616 -0.1102 0.912726 0.456363 t -0.0495313725490196 0.00999 -4.9582 9e-06 5e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.586785378864402 R-squared 0.344317080849039 Adjusted R-squared 0.180396351061299 F-TEST (value) 2.10050968718168 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 0.0348169075442155 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.35361562785521 Sum Squared Residuals 87.9492128627451 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2.08 3.87960784313725 -1.79960784313725 2 2.09 3.89695294117647 -1.80695294117647 3 2.07 3.84375294117647 -1.77375294117647 4 2.04 3.70915294117647 -1.66915294117647 5 2.35 3.64755294117647 -1.29755294117647 6 2.33 3.49775294117647 -1.16775294117647 7 2.37 3.49735294117647 -1.12735294117647 8 2.59 3.53475294117647 -0.94475294117647 9 2.62 3.46495294117647 -0.84495294117647 10 2.6 3.48075294117647 -0.880752941176471 11 2.83 3.51675294117647 -0.686752941176471 12 2.78 3.56155294117647 -0.781552941176473 13 3.01 3.28523137254902 -0.275231372549022 14 3.06 3.30257647058824 -0.242576470588235 15 3.33 3.24937647058823 0.080623529411766 16 3.32 3.11477647058823 0.205223529411765 17 3.6 3.05317647058824 0.546823529411764 18 3.57 2.90337647058824 0.666623529411765 19 3.57 2.90297647058824 0.667023529411764 20 3.83 2.94037647058824 0.889623529411764 21 3.84 2.87057647058824 0.969423529411764 22 3.8 2.88637647058824 0.913623529411764 23 4.07 2.92237647058824 1.14762352941176 24 4.05 2.96717647058824 1.08282352941176 25 4.272 2.69085490196078 1.58114509803922 26 3.858 2.7082 1.1498 27 4.067 2.655 1.412 28 3.964 2.5204 1.4436 29 3.782 2.4588 1.3232 30 4.114 2.309 1.805 31 4.009 2.3086 1.7004 32 4.025 2.346 1.679 33 4.082 2.2762 1.8058 34 4.044 2.292 1.752 35 3.916 2.328 1.588 36 4.289 2.3728 1.91620000000000 37 4.296 2.09647843137255 2.19952156862745 38 4.193 2.11382352941176 2.07917647058824 39 3.48 2.06062352941176 1.41937647058824 40 2.934 1.92602352941176 1.00797647058824 41 2.221 1.86442352941176 0.356576470588236 42 1.211 1.71462352941176 -0.503623529411764 43 1.28 1.71422352941176 -0.434223529411764 44 0.96 1.75162352941177 -0.791623529411765 45 0.5 1.68182352941176 -1.18182352941176 46 0.687 1.69762352941176 -1.01062352941176 47 0.344 1.73362352941176 -1.38962352941176 48 0.346 1.77842352941177 -1.43242352941177 49 0.334 1.50210196078431 -1.16810196078431 50 0.34 1.51944705882353 -1.17944705882353 51 0.328 1.46624705882353 -1.13824705882353 52 0.344 1.33164705882353 -0.987647058823528 53 0.341 1.27004705882353 -0.929047058823529 54 0.32 1.12024705882353 -0.800247058823529 55 0.314 1.11984705882353 -0.80584705882353 56 0.325 1.15724705882353 -0.83224705882353 57 0.339 1.08744705882353 -0.74844705882353 58 0.329 1.10324705882353 -0.774247058823529 59 0.48 1.13924705882353 -0.65924705882353 60 0.399 1.18404705882353 -0.785047058823532 61 0.37 0.90772549019608 -0.537725490196079 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.00406742703168054 0.00813485406336109 0.99593257296832 17 0.000681033441420467 0.00136206688284093 0.99931896655858 18 0.000101963455902953 0.000203926911805907 0.999898036544097 19 1.34893749211852e-05 2.69787498423704e-05 0.999986510625079 20 1.80372337198144e-06 3.60744674396288e-06 0.999998196276628 21 2.15393752279348e-07 4.30787504558695e-07 0.999999784606248 22 2.55613153994926e-08 5.11226307989852e-08 0.999999974438685 23 2.99712232682948e-09 5.99424465365895e-09 0.999999997002878 24 4.34921728333755e-10 8.6984345666751e-10 0.999999999565078 25 5.52344309651239e-11 1.10468861930248e-10 0.999999999944766 26 4.43818377017946e-09 8.87636754035892e-09 0.999999995561816 27 3.7694772406998e-09 7.5389544813996e-09 0.999999996230523 28 3.60926102177713e-09 7.21852204355426e-09 0.99999999639074 29 1.43929129983413e-07 2.87858259966825e-07 0.99999985607087 30 6.7606411873192e-08 1.35212823746384e-07 0.999999932393588 31 4.48853660572322e-08 8.97707321144643e-08 0.999999955114634 32 7.5108672807777e-08 1.50217345615554e-07 0.999999924891327 33 7.85290861301129e-08 1.57058172260226e-07 0.999999921470914 34 6.65591828729338e-08 1.33118365745868e-07 0.999999933440817 35 2.59067683543586e-07 5.18135367087172e-07 0.999999740932316 36 3.14176047373146e-07 6.28352094746292e-07 0.999999685823953 37 1.25017550757980e-06 2.50035101515959e-06 0.999998749824492 38 3.97279677059104e-05 7.94559354118208e-05 0.999960272032294 39 0.00963550284288465 0.0192710056857693 0.990364497157115 40 0.369866977418164 0.739733954836327 0.630133022581836 41 0.946401698724054 0.107196602551891 0.0535983012759456 42 0.987696179462225 0.0246076410755509 0.0123038205377754 43 0.99749046102575 0.00501907794850191 0.00250953897425096 44 0.999092651266529 0.00181469746694204 0.000907348733471018 45 0.995943575441432 0.00811284911713694 0.00405642455856847 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 26 0.866666666666667 NOK 5% type I error level 28 0.933333333333333 NOK 10% type I error level 28 0.933333333333333 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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