*The author of this computation has been verified*

R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Mon, 20 Dec 2010 20:15:14 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9.htm/, Retrieved Mon, 20 Dec 2010 21:18:43 +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/2010/Dec/20/t1292876313no81pmkl0dnlaw9.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}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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313737 312276 309391 302950 300316 304035 333476 337698 335932 323931 313927 314485 313218 309664 302963 298989 298423 301631 329765 335083 327616 309119 295916 291413 291542 284678 276475 272566 264981 263290 296806 303598 286994 276427 266424 267153 268381 262522 255542 253158 243803 250741 280445 285257 270976 261076 255603 260376 263903 264291 263276 262572 256167 264221 293860 300713 287224 275902 271115 277509 279681

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 HPC[t] = + 320655.670588235 + 880.29019607827M1[t] -6186.68627450981M2[t] -10274.9176470588M3[t] -12688.7490196078M4[t] -16929.1803921569M5[t] -11815.0117647059M6[t] + 19340.3568627451M7[t] + 26008.3254901961M8[t] + 16355.4941176471M9[t] + 4966.66274509804M10[t] -2658.76862745098M11[t] -1068.56862745098t + 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) 320655.670588235 6373.95704 50.3072 0 0 M1 880.29019607827 7433.533414 0.1184 0.906228 0.453114 M2 -6186.68627450981 7802.279801 -0.7929 0.431719 0.215859 M3 -10274.9176470588 7792.316204 -1.3186 0.193561 0.096781 M4 -12688.7490196078 7783.390596 -1.6302 0.109598 0.054799 M5 -16929.1803921569 7775.50655 -2.1772 0.03441 0.017205 M6 -11815.0117647059 7768.667239 -1.5209 0.134857 0.067429 M7 19340.3568627451 7762.875422 2.4914 0.016234 0.008117 M8 26008.3254901961 7758.133448 3.3524 0.00157 0.000785 M9 16355.4941176471 7754.44324 2.1092 0.040171 0.020086 M10 4966.66274509804 7751.806302 0.6407 0.524758 0.262379 M11 -2658.76862745098 7750.223709 -0.3431 0.733054 0.366527 t -1068.56862745098 90.431209 -11.8164 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.896894324127192 R-squared 0.804419428651573 Adjusted R-squared 0.755524285814467 F-TEST (value) 16.4519292096453 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 4.12336831345783e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 12253.3454343963 Sum Squared Residuals 7206934768.06276 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 313737 320467.392156864 -6730.39215686355 2 312276 312331.847058824 -55.8470588235415 3 309391 307175.047058824 2215.95294117650 4 302950 303692.647058823 -742.64705882347 5 300316 298383.647058823 1932.35294117654 6 304035 302429.247058823 1605.75294117655 7 333476 332516.047058823 959.952941176511 8 337698 338115.447058823 -417.447058823474 9 335932 327394.047058823 8537.95294117654 10 323931 314936.647058823 8994.3529411765 11 313927 306242.647058823 7684.35294117651 12 314485 307832.847058824 6652.15294117651 13 313218 307644.568627451 5573.43137254917 14 309664 299509.023529412 10154.9764705883 15 302963 294352.223529412 8610.77647058825 16 298989 290869.823529412 8119.17647058825 17 298423 285560.823529412 12862.1764705882 18 301631 289606.423529412 12024.5764705882 19 329765 319693.223529412 10071.7764705883 20 335083 325292.623529412 9790.37647058825 21 327616 314571.223529412 13044.7764705882 22 309119 302113.823529412 7005.17647058825 23 295916 293419.823529412 2496.17647058826 24 291413 295010.023529412 -3597.02352941175 25 291542 294821.745098039 -3279.74509803902 26 284678 286686.2 -2008.19999999999 27 276475 281529.4 -5054.40000000000 28 272566 278047 -5481.00000000002 29 264981 272738 -7757.00000000001 30 263290 276783.6 -13493.6 31 296806 306870.4 -10064.4 32 303598 312469.8 -8871.8 33 286994 301748.4 -14754.4 34 276427 289291 -12864 35 266424 280597 -14173 36 267153 282187.2 -15034.2 37 268381 281998.921568627 -13617.9215686273 38 262522 273863.376470588 -11341.3764705882 39 255542 268706.576470588 -13164.5764705883 40 253158 265224.176470588 -12066.1764705883 41 243803 259915.176470588 -16112.1764705883 42 250741 263960.776470588 -13219.7764705883 43 280445 294047.576470588 -13602.5764705883 44 285257 299646.976470588 -14389.9764705883 45 270976 288925.576470588 -17949.5764705883 46 261076 276468.176470588 -15392.1764705882 47 255603 267774.176470588 -12171.1764705882 48 260376 269364.376470588 -8988.37647058826 49 263903 269176.098039216 -5273.09803921553 50 264291 261040.552941176 3250.4470588235 51 263276 255883.752941176 7392.2470588235 52 262572 252401.352941177 10170.6470588235 53 256167 247092.352941177 9074.64705882349 54 264221 251137.952941177 13083.0470588235 55 293860 281224.752941177 12635.2470588235 56 300713 286824.152941176 13888.8470588235 57 287224 276102.752941177 11121.2470588235 58 275902 263645.352941177 12256.6470588235 59 271115 254951.352941177 16163.6470588235 60 277509 256541.552941177 20967.4470588235 61 279681 256353.274509804 23327.7254901962 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.00285132437314696 0.00570264874629391 0.997148675626853 17 0.000398367638622928 0.000796735277245855 0.999601632361377 18 4.97111211197424e-05 9.94222422394848e-05 0.99995028887888 19 6.23242510916713e-06 1.24648502183343e-05 0.99999376757489 20 7.89719101441688e-07 1.57943820288338e-06 0.999999210280899 21 3.36005310066386e-06 6.72010620132772e-06 0.9999966399469 22 0.000196430859064014 0.000392861718128029 0.999803569140936 23 0.0021043205460262 0.0042086410920524 0.997895679453974 24 0.0153549422742379 0.0307098845484757 0.984645057725762 25 0.0242218323299433 0.0484436646598866 0.975778167670057 26 0.0538940603188521 0.107788120637704 0.946105939681148 27 0.106836222911287 0.213672445822575 0.893163777088713 28 0.142509626583667 0.285019253167334 0.857490373416333 29 0.266883342721632 0.533766685443264 0.733116657278368 30 0.391468510306529 0.782937020613058 0.608531489693471 31 0.461708829804253 0.923417659608507 0.538291170195747 32 0.556874129520643 0.886251740958715 0.443125870479357 33 0.764672593364621 0.470654813270758 0.235327406635379 34 0.905112135141536 0.189775729716928 0.0948878648584642 35 0.965724422774129 0.0685511544517418 0.0342755772258709 36 0.987119047246758 0.0257619055064836 0.0128809527532418 37 0.999063246027523 0.00187350794495326 0.00093675397247663 38 0.999957142062187 8.57158756251123e-05 4.28579378125561e-05 39 0.999987664458388 2.46710832243331e-05 1.23355416121666e-05 40 0.999997967231732 4.06553653670542e-06 2.03276826835271e-06 41 0.99999728911876 5.42176247957969e-06 2.71088123978985e-06 42 0.99999165994723 1.66801055386310e-05 8.34005276931552e-06 43 0.999992354679635 1.52906407299399e-05 7.64532036496995e-06 44 0.999902523021714 0.000194953956571862 9.74769782859308e-05 45 0.998863005308418 0.00227398938316320 0.00113699469158160 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 17 0.566666666666667 NOK 5% type I error level 20 0.666666666666667 NOK 10% type I error level 21 0.7 NOK

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/100nxw1292876107.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/100nxw1292876107.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/11cg71292876106.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/11cg71292876106.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/21cg71292876106.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/21cg71292876106.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/34vi51292876107.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/34vi51292876107.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/44vi51292876107.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/44vi51292876107.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/54vi51292876107.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/54vi51292876107.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/6f5h81292876107.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/6f5h81292876107.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/7peyb1292876107.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/7peyb1292876107.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/8peyb1292876107.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/8peyb1292876107.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/9peyb1292876107.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/20/t1292876313no81pmkl0dnlaw9/9peyb1292876107.ps (open in new window)

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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