Multiple Regression model 1

*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: Fri, 20 Nov 2009 16:43:08 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Nov/21/t1258761845naxgi4u0n23q7sh.htm/, Retrieved Sat, 21 Nov 2009 01:04:16 +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/Nov/21/t1258761845naxgi4u0n23q7sh.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}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

8.4 99 8.4 98.6 8.4 98.6 8.6 98.5 8.9 98.9 8.8 99.4 8.3 99.8 7.5 99.9 7.2 100 7.4 100.1 8.8 100.1 9.3 100.2 9.3 100.3 8.7 100 8.2 99.9 8.3 99.4 8.5 99.8 8.6 99.6 8.5 100 8.2 99.9 8.1 100.3 7.9 100.6 8.6 100.7 8.7 100.8 8.7 100.8 8.5 100.6 8.4 101.1 8.5 101.1 8.7 100.9 8.7 101.1 8.6 101.2 8.5 101.4 8.3 101.9 8 102.1 8.2 102.1 8.1 103 8.1 103.4 8 103.2 7.9 103.1 7.9 103 8 103.7 8 103.4 7.9 103.5 8 103.8 7.7 104 7.2 104.2 7.5 104.4 7.3 104.4 7 104.9 7 105.3 7 105.2 7.2 105.4 7.3 105.4 7.1 105.5 6.8 105.7 6.4 105.6 6.1 105.8 6.5 105.4 7.7 105.5 7.9 105.8 7.5 106.1 6.9 106 6.6 105.5 6.9 105.4 7.7 106 8 106.1 8 106.4 7.7 106 7.3 106 7.4 106 8.1 106 8.3 106.1 8.2 106.1

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 werkl[t] = + 26.928348083077 -0.184940580085961afzetp[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) 26.928348083077 2.35187 11.4498 0 0 afzetp -0.184940580085961 0.022887 -8.0805 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.6921468536664 R-squared 0.479067267040296 Adjusted R-squared 0.471730186294385 F-TEST (value) 65.2939886626248 F-TEST (DF numerator) 1 F-TEST (DF denominator) 71 p-value 1.18598464382558e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.505111309201547 Sum Squared Residuals 18.1147578625144 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.4 8.61923065456692 -0.219230654566918 2 8.4 8.69320688660125 -0.293206886601253 3 8.4 8.69320688660125 -0.293206886601253 4 8.6 8.71170094460985 -0.111700944609849 5 8.9 8.63772471257546 0.262275287424537 6 8.8 8.54525442253248 0.254745577467518 7 8.3 8.4712781904981 -0.171278190498099 8 7.5 8.4527841324895 -0.952784132489502 9 7.2 8.4342900744809 -1.23429007448091 10 7.4 8.41579601647231 -1.01579601647231 11 8.8 8.41579601647231 0.384203983527689 12 9.3 8.39730195846371 0.902698041536286 13 9.3 8.37880790045512 0.921192099544881 14 8.7 8.4342900744809 0.265709925519092 15 8.2 8.4527841324895 -0.252784132489503 16 8.3 8.54525442253248 -0.245254422532482 17 8.5 8.4712781904981 0.0287218095019002 18 8.6 8.50826630651529 0.0917336934847071 19 8.5 8.4342900744809 0.0657099255190929 20 8.2 8.4527841324895 -0.252784132489503 21 8.1 8.37880790045512 -0.278807900455120 22 7.9 8.32332572642933 -0.423325726429331 23 8.6 8.30483166842073 0.295168331579266 24 8.7 8.28633761041214 0.413662389587861 25 8.7 8.28633761041214 0.413662389587861 26 8.5 8.32332572642933 0.176674273570669 27 8.4 8.23085543638635 0.169144563613649 28 8.5 8.23085543638635 0.269144563613649 29 8.7 8.26784355240354 0.432156447596458 30 8.7 8.23085543638635 0.469144563613648 31 8.6 8.21236137837775 0.387638621622246 32 8.5 8.17537326236056 0.324626737639440 33 8.3 8.08290297231758 0.217097027682421 34 8 8.04591485630039 -0.0459148563003898 35 8.2 8.04591485630039 0.154085143699610 36 8.1 7.87946833422302 0.220531665776976 37 8.1 7.80549210218864 0.294507897811361 38 8 7.84248021820583 0.157519781794169 39 7.9 7.86097427621443 0.0390257237855717 40 7.9 7.87946833422302 0.0205316657769767 41 8 7.75000992816285 0.249990071837150 42 8 7.80549210218864 0.194507897811362 43 7.9 7.78699804418004 0.113001955819957 44 8 7.73151587015426 0.268484129845745 45 7.7 7.69452775413706 0.00547224586293761 46 7.2 7.65753963811987 -0.45753963811987 47 7.5 7.62055152210268 -0.120551522102677 48 7.3 7.62055152210268 -0.320551522102677 49 7 7.5280812320597 -0.528081232059696 50 7 7.45410500002531 -0.454105000025314 51 7 7.47259905803391 -0.472599058033909 52 7.2 7.43561094201672 -0.235610942016716 53 7.3 7.43561094201672 -0.135610942016716 54 7.1 7.41711688400812 -0.317116884008121 55 6.8 7.38012876799093 -0.580128767990928 56 6.4 7.39862282599953 -0.998622825999525 57 6.1 7.36163470998233 -1.26163470998233 58 6.5 7.43561094201672 -0.935610942016716 59 7.7 7.41711688400812 0.282883115991879 60 7.9 7.36163470998233 0.538365290017667 61 7.5 7.30615253595654 0.193847464043455 62 6.9 7.32464659396514 -0.42464659396514 63 6.6 7.41711688400812 -0.817116884008121 64 6.9 7.43561094201672 -0.535610942016716 65 7.7 7.32464659396514 0.37535340603486 66 8 7.30615253595655 0.693847464043455 67 8 7.25067036193076 0.749329638069245 68 7.7 7.32464659396514 0.37535340603486 69 7.3 7.32464659396514 -0.0246465939651404 70 7.4 7.32464659396514 0.0753534060348602 71 8.1 7.32464659396514 0.77535340603486 72 8.3 7.30615253595655 0.993847464043455 73 8.2 7.30615253595655 0.893847464043454 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.120308041159809 0.240616082319617 0.879691958840191 6 0.0459774827973531 0.0919549655947063 0.954022517202647 7 0.0512154894282182 0.102430978856436 0.948784510571782 8 0.243388694474696 0.486777388949392 0.756611305525304 9 0.413457669686282 0.826915339372564 0.586542330313718 10 0.391069535296666 0.782139070593332 0.608930464703334 11 0.716465501539696 0.567068996920608 0.283534498460304 12 0.94407182707037 0.111856345859261 0.0559281729296306 13 0.981481584112281 0.0370368317754370 0.0185184158877185 14 0.97284185621388 0.0543162875722397 0.0271581437861198 15 0.960605253273256 0.0787894934534874 0.0393947467267437 16 0.944107194490241 0.111785611019518 0.0558928055097588 17 0.918694941506865 0.162610116986270 0.0813050584931351 18 0.887659842242933 0.224680315514134 0.112340157757067 19 0.847213681300238 0.305572637399525 0.152786318699763 20 0.813715702080725 0.372568595838551 0.186284297919275 21 0.783238411495859 0.433523177008283 0.216761588504141 22 0.777262116293696 0.445475767412608 0.222737883706304 23 0.736278902691174 0.527442194617652 0.263721097308826 24 0.700433703521774 0.599132592956451 0.299566296478226 25 0.65667667053027 0.68664665893946 0.34332332946973 26 0.590074092460385 0.819851815079231 0.409925907539615 27 0.519174529671352 0.961650940657297 0.480825470328648 28 0.450224516251885 0.90044903250377 0.549775483748115 29 0.399849743796976 0.799699487593952 0.600150256203024 30 0.353632881160214 0.707265762320428 0.646367118839786 31 0.300991518794889 0.601983037589779 0.69900848120511 32 0.249834967865582 0.499669935731164 0.750165032134418 33 0.206028410547535 0.41205682109507 0.793971589452465 34 0.180829200088715 0.361658400177429 0.819170799911285 35 0.144339463187428 0.288678926374855 0.855660536812572 36 0.116776680854567 0.233553361709135 0.883223319145433 37 0.0948304551706452 0.189660910341290 0.905169544829355 38 0.0753901389475923 0.150780277895185 0.924609861052408 39 0.0595656320034297 0.119131264006859 0.94043436799657 40 0.0463542943097668 0.0927085886195335 0.953645705690233 41 0.0375642662284261 0.0751285324568522 0.962435733771574 42 0.0323928888096017 0.0647857776192034 0.967607111190398 43 0.0299498411857484 0.0598996823714969 0.970050158814252 44 0.0381793095177877 0.0763586190355754 0.961820690482212 45 0.0476874643278710 0.0953749286557419 0.95231253567213 46 0.0579701952866474 0.115940390573295 0.942029804713353 47 0.0778909563510546 0.155781912702109 0.922109043648945 48 0.144246344048131 0.288492688096262 0.855753655951869 49 0.172595953799715 0.34519190759943 0.827404046200285 50 0.154285736131711 0.308571472263422 0.845714263868289 51 0.145213220939476 0.290426441878952 0.854786779060524 52 0.125468295258099 0.250936590516197 0.874531704741901 53 0.124040394102029 0.248080788204058 0.875959605897971 54 0.100220672260085 0.200441344520171 0.899779327739915 55 0.0884699184452868 0.176939836890574 0.911530081554713 56 0.129007264489856 0.258014528979712 0.870992735510144 57 0.535771524063957 0.928456951872086 0.464228475936043 58 0.557217426490694 0.885565147018612 0.442782573509306 59 0.654074795486485 0.69185040902703 0.345925204513515 60 0.728049358934722 0.543901282130557 0.271950641065278 61 0.684620953018527 0.630758093962945 0.315379046981473 62 0.860041886038905 0.27991622792219 0.139958113961095 63 0.86293425558666 0.274131488826678 0.137065744413339 64 0.791462665227604 0.417074669544793 0.208537334772396 65 0.704818244154789 0.590363511690423 0.295181755845211 66 0.61672509623942 0.766549807521161 0.383274903760581 67 0.792608226674928 0.414783546650144 0.207391773325072 68 0.649699641451238 0.700600717097525 0.350300358548762 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 1 0.015625 OK 10% type I error level 10 0.15625 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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