WS8 Multiple Regression 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: Mon, 29 Nov 2010 11:10:34 +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/Nov/29/t1291029003fefs5bxgldj3fwd.htm/, Retrieved Mon, 29 Nov 2010 12:10:22 +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/Nov/29/t1291029003fefs5bxgldj3fwd.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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37 0 30 1 47 0 35 0 30 1 43 0 82 0 40 0 47 0 19 1 52 0 136 0 80 0 42 0 54 0 66 0 81 0 63 0 137 0 72 0 107 0 58 0 36 0 52 0 79 0 77 0 54 0 84 0 48 0 96 0 83 0 66 0 61 0 53 0 30 1 74 0 69 0 59 0 42 0 65 0 70 0 100 0 63 0 105 0 82 0 81 0 75 0 102 0 121 0 98 0 76 0 77 0 63 0 37 0 35 0 23 1 40 0 29 1 37 0 51 0 20 1 28 1 13 1 22 1 25 1 13 1 16 1 13 1 16 1 17 1 9 1 17 1 25 1 14 1 8 1 7 1 10 1 7 1 10 1 3 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 14 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Soldiers[t] = + 82.19825815463 -48.8422479664777Dummy[t] -4.8590042294134M1[t] -9.68968586530826M2[t] -24.3324383487682M3[t] -15.4262982655884M4[t] -12.5426941871975M5[t] -14.8997323849432M6[t] -5.27930658747765M7[t] -13.1099882233725M8[t] -13.3136583447774M9[t] -12.9839117965813M10[t] -24.075289231624M11[t] -0.0489972260369107t + 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) 82.19825815463 10.234328 8.0316 0 0 Dummy -48.8422479664777 6.778625 -7.2053 0 0 M1 -4.8590042294134 12.205622 -0.3981 0.691845 0.345922 M2 -9.68968586530826 12.32179 -0.7864 0.434456 0.217228 M3 -24.3324383487682 12.182607 -1.9973 0.049917 0.024959 M4 -15.4262982655884 12.173329 -1.2672 0.209529 0.104765 M5 -12.5426941871975 12.274483 -1.0219 0.310582 0.155291 M6 -14.8997323849432 12.159246 -1.2254 0.224786 0.112393 M7 -5.27930658747765 12.154448 -0.4344 0.66545 0.332725 M8 -13.1099882233725 12.240395 -1.071 0.288051 0.144026 M9 -13.3136583447774 12.596936 -1.0569 0.294412 0.147206 M10 -12.9839117965813 12.823909 -1.0125 0.315008 0.157504 M11 -24.075289231624 12.649201 -1.9033 0.061363 0.030682 t -0.0489972260369107 0.134976 -0.363 0.71776 0.35888 Multiple Linear Regression - Regression Statistics Multiple R 0.781049808513339 R-squared 0.610038803378723 Adjusted R-squared 0.53322826465029 F-TEST (value) 7.94212374340375 F-TEST (DF numerator) 13 F-TEST (DF denominator) 66 p-value 3.04937552986217e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 21.8072579105914 Sum Squared Residuals 31386.7288402174 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 37 77.2902566991794 -40.2902566991794 2 30 23.5683298707702 6.43167012922982 3 47 57.7188281277509 -10.7188281277509 4 35 66.5759709848939 -31.5759709848939 5 30 20.5683298707701 9.4316701292299 6 43 67.0045424134652 -24.0045424134652 7 82 76.5759709848938 5.42402901510618 8 40 68.6962921229621 -28.6962921229621 9 47 68.4436247755203 -21.4436247755203 10 19 19.8821261312018 -0.88212613120179 11 52 57.5839994365999 -5.58399943659992 12 136 81.6102914421869 54.3897085578131 13 80 76.7022899867367 3.29771001326335 14 42 71.8226111248048 -29.8226111248048 15 54 57.130861415308 -3.13086141530805 16 66 65.9880042724509 0.0119957275490812 17 81 68.8226111248049 12.1773888751951 18 63 66.4165757010223 -3.41657570102233 19 137 75.9880042724509 61.0119957275491 20 72 68.1083254105192 3.89167458948085 21 107 67.8556580630773 39.1443419369227 22 58 68.1364073852366 -10.1364073852366 23 36 56.996032724157 -20.9960327241570 24 52 81.022324729744 -29.022324729744 25 79 76.1143232742937 2.88567672570628 26 77 71.2346444123619 5.76535558763807 27 54 56.5428947028651 -2.54289470286513 28 84 65.400037560008 18.599962439992 29 48 68.234644412362 -20.2346444123619 30 96 65.8286089885794 30.1713910114206 31 83 75.400037560008 7.59996243999201 32 66 67.5203586980762 -1.52035869807623 33 61 67.2676913506344 -6.2676913506344 34 53 67.5484406727937 -14.5484406727937 35 30 7.56581804523631 22.4341819547637 36 74 80.434358017301 -6.43435801730108 37 69 75.5263565618508 -6.52635656185079 38 59 70.646677699919 -11.646677699919 39 42 55.9549279904222 -13.9549279904222 40 65 64.8120708475651 0.187929152434931 41 70 67.646677699919 2.35332230008098 42 100 65.2406422761365 34.7593577238635 43 63 74.812070847565 -11.8120708475651 44 105 66.9323919856333 38.0676080143667 45 82 66.6797246381915 15.3202753618085 46 81 66.9604739603507 14.0395260396493 47 75 55.8200992992711 19.1799007007289 48 102 79.8463913048581 22.1536086951418 49 121 74.9383898494079 46.0616101505921 50 98 70.058710987476 27.9412890125239 51 76 55.3669612779793 20.6330387220207 52 77 64.2241041351221 12.7758958648779 53 63 67.0587109874761 -4.0587109874761 54 37 64.6526755636936 -27.6526755636936 55 35 74.2241041351221 -39.2241041351221 56 23 17.5021773067127 5.49782269328732 57 40 66.0917579257486 -26.0917579257485 58 29 17.5302592814301 11.4697407185699 59 37 55.2321325868282 -18.2321325868282 60 51 79.2584245924152 -28.2584245924152 61 20 25.5081751704872 -5.50817517048724 62 28 20.6284963085554 7.37150369144455 63 13 5.93674659905864 7.06325340094136 64 22 14.7938894562015 7.20611054379849 65 25 17.6284963085555 7.37150369144453 66 13 15.2224608847729 -2.22246088477292 67 16 24.7938894562015 -8.7938894562015 68 13 16.9142105942697 -3.91421059426974 69 16 16.6615432468279 -0.661543246827923 70 17 16.9422925689872 0.0577074310128269 71 9 5.80191790790755 3.19808209209245 72 17 29.8282099134946 -12.8282099134946 73 25 24.9202084580443 0.0797915419556903 74 14 20.0405295961125 -6.04052959611253 75 8 5.34877988661572 2.65122011338428 76 7 14.2059227437586 -7.20592274375858 77 10 17.0405295961125 -7.04052959611254 78 7 14.63449417233 -7.63449417232999 79 10 24.2059227437586 -14.2059227437586 80 3 16.3262438818268 -13.3262438818268 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.380935654138568 0.761871308277136 0.619064345861432 18 0.234770629908143 0.469541259816286 0.765229370091857 19 0.33733360782745 0.6746672156549 0.66266639217255 20 0.219077011543955 0.438154023087909 0.780922988456045 21 0.277719847770330 0.555439695540661 0.72228015222967 22 0.196856494022704 0.393712988045408 0.803143505977296 23 0.492070044286015 0.98414008857203 0.507929955713985 24 0.989491825224566 0.0210163495508683 0.0105081747754342 25 0.982600828593776 0.0347983428124480 0.0173991714062240 26 0.971640177264651 0.0567196454706976 0.0283598227353488 27 0.963924059189447 0.0721518816211069 0.0360759408105534 28 0.94620289996544 0.107594200069118 0.0537971000345592 29 0.965323643544084 0.0693527129118324 0.0346763564559162 30 0.959707811493314 0.0805843770133722 0.0402921885066861 31 0.97224733808446 0.0555053238310796 0.0277526619155398 32 0.961694501403296 0.0766109971934071 0.0383054985967036 33 0.958181060892384 0.083637878215232 0.041818939107616 34 0.95806743670524 0.0838651265895205 0.0419325632947603 35 0.93706631146336 0.125867377073281 0.0629336885366406 36 0.93246450206734 0.135070995865319 0.0675354979326596 37 0.934953616584808 0.130092766830383 0.0650463834151915 38 0.946836015652895 0.10632796869421 0.053163984347105 39 0.972028658007482 0.0559426839850369 0.0279713419925185 40 0.970262128337427 0.0594757433251466 0.0297378716625733 41 0.963145259447497 0.0737094811050069 0.0368547405525034 42 0.968583700693919 0.0628325986121623 0.0314162993060811 43 0.975620366120315 0.0487592677593706 0.0243796338796853 44 0.984937100144645 0.0301257997107097 0.0150628998553548 45 0.97881620960464 0.0423675807907183 0.0211837903953591 46 0.968531830301299 0.0629363393974025 0.0314681696987013 47 0.95883567260855 0.0823286547828999 0.0411643273914499 48 0.970471569336744 0.0590568613265124 0.0295284306632562 49 0.9984399740002 0.00312005199959997 0.00156002599979999 50 0.999790059211256 0.000419881577488941 0.000209940788744471 51 0.999958190954459 8.36180910824826e-05 4.18090455412413e-05 52 0.99999928666368 1.42667264019151e-06 7.13336320095755e-07 53 0.999999830479782 3.39040435264190e-07 1.69520217632095e-07 54 0.999999604303005 7.91393990156685e-07 3.95696995078342e-07 55 0.999999778857754 4.42284492457833e-07 2.21142246228917e-07 56 0.99999884568695 2.30862610173089e-06 1.15431305086545e-06 57 0.999996407408563 7.18518287461198e-06 3.59259143730599e-06 58 0.999981759712432 3.6480575135959e-05 1.82402875679795e-05 59 0.999921120252994 0.000157759494012743 7.88797470063716e-05 60 0.99964006022929 0.000719879541421295 0.000359939770710647 61 0.999832473543108 0.000335052913784384 0.000167526456892192 62 0.99906264284939 0.00187471430122092 0.000937357150610462 63 0.995346527242309 0.00930694551538237 0.00465347275769118 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 15 0.319148936170213 NOK 5% type I error level 20 0.425531914893617 NOK 10% type I error level 35 0.74468085106383 NOK

Charts produced by software:

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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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