Paper Multiple Regressio met Monthly Dummies

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Title produced by software: Multiple Regression

Date of computation: Thu, 17 Dec 2009 09:52:21 -0700

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/17/t1261068802mzd934m2t2373zm.htm/, Retrieved Thu, 17 Dec 2009 17:53:34 +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/Dec/17/t1261068802mzd934m2t2373zm.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}, }

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Dataseries X:

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7.6 1.62 8.3 1.49 8.4 1.79 8.4 1.8 8.4 1.58 8.4 1.86 8.6 1.74 8.9 1.59 8.8 1.26 8.3 1.13 7.5 1.92 7.2 2.61 7.4 2.26 8.8 2.41 9.3 2.26 9.3 2.03 8.7 2.86 8.2 2.55 8.3 2.27 8.5 2.26 8.6 2.57 8.5 3.07 8.2 2.76 8.1 2.51 7.9 2.87 8.6 3.14 8.7 3.11 8.7 3.16 8.5 2.47 8.4 2.57 8.5 2.89 8.7 2.63 8.7 2.38 8.6 1.69 8.5 1.96 8.3 2.19 8 1.87 8.2 1.6 8.1 1.63 8.1 1.22 8 1.21 7.9 1.49 7.9 1.64 8 1.66 8 1.77 7.9 1.82 8 1.78 7.7 1.28 7.2 1.29 7.5 1.37 7.3 1.12 7 1.51 7 2.24 7 2.94 7.2 3.09 7.3 3.46 7.1 3.64 6.8 4.39 6.4 4.15 6.1 5.21 6.5 5.8 7.7 5.91 7.9 5.39 7.5 5.46 6.9 4.72 6.6 3.14 6.9 2.63 7.7 2.32 8 1.93 8 0.62 7.7 0.6 7.3 -0.37 7.4 -1.1

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 TWG[t] = + 7.8669608182082 -0.186281824962715Infl[t] -0.049592609250319M1[t] + 0.81064029069286M2[t] + 0.891391168780046M3[t] + 0.770998865614126M4[t] + 0.517894168531413M5[t] + 0.334772607326373M6[t] + 0.475768985786509M7[t] + 0.748546349038621M8[t] + 0.753725636499254M9[t] + 0.544623317379412M10[t] + 0.258594454251616M11[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) 7.8669608182082 0.296099 26.5687 0 0 Infl -0.186281824962715 0.060129 -3.098 0.002963 0.001481 M1 -0.049592609250319 0.359537 -0.1379 0.890754 0.445377 M2 0.81064029069286 0.373823 2.1685 0.034095 0.017048 M3 0.891391168780046 0.37346 2.3868 0.020163 0.010082 M4 0.770998865614126 0.373402 2.0648 0.043272 0.021636 M5 0.517894168531413 0.373356 1.3871 0.170533 0.085266 M6 0.334772607326373 0.373158 0.8971 0.373234 0.186617 M7 0.475768985786509 0.373082 1.2752 0.207141 0.10357 M8 0.748546349038621 0.373022 2.0067 0.049293 0.024647 M9 0.753725636499254 0.372992 2.0208 0.047774 0.023887 M10 0.544623317379412 0.373057 1.4599 0.149536 0.074768 M11 0.258594454251616 0.372999 0.6933 0.490806 0.245403 Multiple Linear Regression - Regression Statistics Multiple R 0.533413911883252 R-squared 0.284530401390593 Adjusted R-squared 0.141436481668712 F-TEST (value) 1.98841713151481 F-TEST (DF numerator) 12 F-TEST (DF denominator) 60 p-value 0.0411896849717841 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.646036995708778 Sum Squared Residuals 25.0418279894654 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.6 7.51559165251832 0.0844083474816785 2 8.3 8.40004118970663 -0.100041189706625 3 8.4 8.424907520305 -0.0249075203049971 4 8.4 8.30265239888945 0.0973476011105517 5 8.4 8.09052970329853 0.309470296701467 6 8.4 7.85524923110393 0.544750768896065 7 8.6 8.0185994285596 0.581400571440404 8 8.9 8.31931906555612 0.580680934443885 9 8.8 8.38597135525444 0.414028644745557 10 8.3 8.20108567337975 0.0989143266202461 11 7.5 7.76789416853141 -0.267894168531413 12 7.2 7.38076525505552 -0.180765255055524 13 7.4 7.39637128454216 0.00362871545784463 14 8.8 8.22866191074093 0.571338089259073 15 9.3 8.33735506257252 0.96264493742748 16 9.3 8.25980757914802 1.04019242085198 17 8.7 7.85208896734626 0.84791103265374 18 8.2 7.72671477187966 0.473285228120339 19 8.3 7.91987006132936 0.380129938670644 20 8.5 8.1945102428311 0.305489757168904 21 8.6 8.14194216455329 0.458057835446712 22 8.5 7.83969893295209 0.660301067047913 23 8.2 7.61141743556273 0.588582564437267 24 8.1 7.3993934375518 0.700606562448204 25 7.9 7.2827393713149 0.617260628685101 26 8.6 8.09267617851815 0.507323821481854 27 8.7 8.17901551135421 0.520984488645787 28 8.7 8.04930911694016 0.650690883059843 29 8.5 7.92473887908172 0.575261120918283 30 8.4 7.7229891353804 0.677010864619595 31 8.5 7.80437532985247 0.695624670147527 32 8.7 8.1255859675949 0.574414032405108 33 8.7 8.1773357112962 0.522664288703796 34 8.6 8.09676785140064 0.503232148599365 35 8.5 7.7604428955329 0.739557104467096 36 8.3 7.45900362153986 0.840996378460136 37 8 7.46902119627761 0.530978803722386 38 8.2 8.37955018896073 -0.179550188960727 39 8.1 8.45471261229903 -0.354712612299032 40 8.1 8.41069585736782 -0.310695857367824 41 8 8.15945397853474 -0.159453978534739 42 7.9 7.92417350634014 -0.0241735063401384 43 7.9 8.03722761105587 -0.137227611055867 44 8 8.30627933780872 -0.306279337808726 45 8 8.29096762452346 -0.290967624523459 46 7.9 8.07255121415548 -0.172551214155481 47 8 7.7939736240262 0.206026375973806 48 7.7 7.62852008225594 0.0714799177440648 49 7.2 7.57706465475599 -0.377064654755989 50 7.5 8.42239500870215 -0.922395008702151 51 7.3 8.54971634303002 -1.24971634303002 52 7 8.35667412812864 -1.35667412812864 53 7 7.96758369882314 -0.967583698823142 54 7 7.6540648601442 -0.654064860144201 55 7.2 7.76711896485993 -0.56711896485993 56 7.3 7.97097205287584 -0.670972052875838 57 7.1 7.94262061184318 -0.842620611843182 58 6.8 7.5938069240013 -0.793806924001303 59 6.4 7.35248569886456 -0.952485698864558 60 6.1 6.89643251015246 -0.796432510152465 61 6.5 6.73693362417414 -0.236933624174143 62 7.7 7.57667552337142 0.123324476628576 63 7.9 7.75429295043922 0.145707049560779 64 7.5 7.62086091952591 -0.120860919525911 65 6.9 7.5056047729156 -0.605604772915608 66 6.6 7.61680849515166 -1.01680849515166 67 6.9 7.85280860434278 -0.952808604342778 68 7.7 8.18333333333333 -0.483333333333333 69 8 8.26116253252943 -0.261162532529425 70 8 8.29608940411074 -0.29608940411074 71 7.7 8.0137861774822 -0.313786177482197 72 7.3 7.93588509344442 -0.635885093444416 73 7.4 8.02227821641688 -0.622278216416879 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.296154672993926 0.592309345987852 0.703845327006074 17 0.191059252023993 0.382118504047987 0.808940747976007 18 0.130449303832068 0.260898607664136 0.869550696167932 19 0.088839819125364 0.177679638250728 0.911160180874636 20 0.067118393519552 0.134236787039104 0.932881606480448 21 0.0424649560103514 0.0849299120207027 0.957535043989649 22 0.0228808291224942 0.0457616582449884 0.977119170877506 23 0.0198284642062840 0.0396569284125679 0.980171535793716 24 0.0311995682012476 0.0623991364024951 0.968800431798752 25 0.0204404216039552 0.0408808432079104 0.979559578396045 26 0.0124410688241746 0.0248821376483493 0.987558931175825 27 0.00911820082345796 0.0182364016469159 0.990881799176542 28 0.0078324604853171 0.0156649209706342 0.992167539514683 29 0.00592802052811803 0.0118560410562361 0.994071979471882 30 0.00508611444179216 0.0101722288835843 0.994913885558208 31 0.00472348704870728 0.00944697409741456 0.995276512951293 32 0.00400154416471797 0.00800308832943595 0.995998455835282 33 0.00334093321044252 0.00668186642088503 0.996659066789557 34 0.00312052222930809 0.00624104445861617 0.996879477770692 35 0.00851064347398992 0.0170212869479798 0.99148935652601 36 0.0292702595851261 0.0585405191702522 0.970729740414874 37 0.0417028796999313 0.0834057593998626 0.958297120300069 38 0.0321353217700569 0.0642706435401138 0.967864678229943 39 0.0368203691065129 0.0736407382130258 0.963179630893487 40 0.0411145233257457 0.0822290466514914 0.958885476674254 41 0.0435711345832487 0.0871422691664973 0.956428865416751 42 0.0549303439543742 0.109860687908748 0.945069656045626 43 0.0637457089956815 0.127491417991363 0.936254291004318 44 0.0643911626839482 0.128782325367896 0.935608837316052 45 0.06481201098607 0.12962402197214 0.93518798901393 46 0.061044180166856 0.122088360333712 0.938955819833144 47 0.0784747327606746 0.156949465521349 0.921525267239325 48 0.107222213216195 0.214444426432390 0.892777786783805 49 0.0776280930970073 0.155256186194015 0.922371906902993 50 0.117074899959319 0.234149799918638 0.88292510004068 51 0.376067655813818 0.752135311627637 0.623932344186182 52 0.84679426011697 0.30641147976606 0.15320573988303 53 0.882308309973954 0.235383380052092 0.117691690026046 54 0.883450922729672 0.233098154540656 0.116549077270328 55 0.869641671526748 0.260716656946504 0.130358328473252 56 0.806516852533455 0.386966294933091 0.193483147466546 57 0.788031693975442 0.423936612049115 0.211968306024558 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.0952380952380952 NOK 5% type I error level 13 0.309523809523810 NOK 10% type I error level 21 0.5 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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