*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: Wed, 09 Dec 2009 06:33:07 -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/Dec/09/t12603657027ns9k60zpy4tq9i.htm/, Retrieved Wed, 09 Dec 2009 14:35:14 +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/09/t12603657027ns9k60zpy4tq9i.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:

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11 8.3 8 8.2 6 8 10 7.9 11 7.6 10 7.6 9 8.3 8 8.4 11 8.4 10 8.4 12 8.4 13 8.6 13 8.9 13 8.8 13 8.3 13 7.5 12 7.2 13 7.4 12 8.8 13 9.3 12 9.3 14 8.7 11 8.2 12 8.3 13 8.5 13 8.6 12 8.5 10 8.2 9 8.1 10 7.9 10 8.6 9 8.7 7 8.7 11 8.5 11 8.4 12 8.5 13 8.7 13 8.7 12 8.6 12 8.5 10 8.3 12 8 12 8.2 12 8.1 10 8.1 13 8 13 7.9 11 7.9 13 8 12 8 11 7.9 12 8 12 7.7 11 7.2 10 7.5 9 7.3 10 7 9 7 6 7 7 7.2 5 7.3 8 7.1 5 6.8 5 6.4 5 6.1 1 6.5 3 7.7 5 7.9 7 7.5 2 6.9 3 6.6 2 6.9

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 Y[t] = -14.1035350383858 + 3.01913453757866X[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) -14.1035350383858 3.263874 -4.3211 5e-05 2.5e-05 X 3.01913453757866 0.408932 7.383 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.661656111589322 R-squared 0.437788810003502 Adjusted R-squared 0.429757221574981 F-TEST (value) 54.5083720237515 F-TEST (DF numerator) 1 F-TEST (DF denominator) 70 p-value 2.49030684962293e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 2.40192550633485 Sum Squared Residuals 403.847229658734 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 11 10.9552816235170 0.0447183764829767 2 8 10.6533681697591 -2.65336816975913 3 6 10.0495412622434 -4.04954126224341 4 10 9.74762780848554 0.252372191514458 5 11 8.84188744721194 2.15811255278806 6 10 8.84188744721194 1.15811255278806 7 9 10.955281623517 -1.95528162351700 8 8 11.2571950772749 -3.25719507727487 9 11 11.2571950772749 -0.257195077274869 10 10 11.2571950772749 -1.25719507727487 11 12 11.2571950772749 0.74280492272513 12 13 11.8610219847906 1.13897801520940 13 13 12.7667623460642 0.233237653935804 14 13 12.4648488923063 0.535151107693668 15 13 10.955281623517 2.04471837648299 16 13 8.53997399345408 4.46002600654592 17 12 7.63423363218048 4.36576636781952 18 13 8.23806053969622 4.76193946030378 19 12 12.4648488923063 -0.464848892306332 20 13 13.9744161610957 -0.974416161095659 21 12 13.9744161610957 -1.97441616109566 22 14 12.1629354385485 1.83706456145154 23 11 10.6533681697591 0.346631830240865 24 12 10.955281623517 1.04471837648300 25 13 11.5591085310327 1.44089146896727 26 13 11.8610219847906 1.13897801520940 27 12 11.5591085310327 0.440891468967266 28 10 10.6533681697591 -0.653368169759135 29 9 10.3514547160013 -1.35145471600127 30 10 9.74762780848554 0.252372191514458 31 10 11.8610219847906 -1.86102198479060 32 9 12.1629354385485 -3.16293543854846 33 7 12.1629354385485 -5.16293543854846 34 11 11.5591085310327 -0.559108531032734 35 11 11.2571950772749 -0.257195077274869 36 12 11.5591085310327 0.440891468967266 37 13 12.1629354385485 0.837064561451538 38 13 12.1629354385485 0.837064561451538 39 12 11.8610219847906 0.138978015209402 40 12 11.5591085310327 0.440891468967266 41 10 10.955281623517 -0.955281623517005 42 12 10.0495412622434 1.95045873775659 43 12 10.6533681697591 1.34663183024086 44 12 10.3514547160013 1.64854528399873 45 10 10.3514547160013 -0.351454716001271 46 13 10.0495412622434 2.95045873775659 47 13 9.74762780848554 3.25237219151446 48 11 9.74762780848554 1.25237219151446 49 13 10.0495412622434 2.95045873775659 50 12 10.0495412622434 1.95045873775659 51 11 9.74762780848554 1.25237219151446 52 12 10.0495412622434 1.95045873775659 53 12 9.14380090096981 2.85619909903019 54 11 7.63423363218048 3.36576636781952 55 10 8.53997399345408 1.46002600654592 56 9 7.93614708593835 1.06385291406165 57 10 7.03040672466475 2.96959327533525 58 9 7.03040672466475 1.96959327533525 59 6 7.03040672466475 -1.03040672466475 60 7 7.63423363218048 -0.634233632180484 61 5 7.93614708593835 -2.93614708593835 62 8 7.33232017842262 0.667679821577383 63 5 6.42657981714902 -1.42657981714902 64 5 5.21892600211756 -0.218926002117562 65 5 4.31318564084396 0.686814359156037 66 1 5.52083945587543 -4.52083945587543 67 3 9.14380090096981 -6.14380090096981 68 5 9.74762780848554 -4.74762780848554 69 7 8.53997399345408 -1.53997399345408 70 2 6.72849327090689 -4.72849327090689 71 3 5.82275290963329 -2.82275290963329 72 2 6.72849327090689 -4.72849327090689 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.61789313795859 0.764213724082821 0.382106862041410 6 0.450570539210524 0.901141078421048 0.549429460789476 7 0.310967187487238 0.621934374974477 0.689032812512762 8 0.215235882245318 0.430471764490637 0.784764117754682 9 0.221019534933623 0.442039069867245 0.778980465066377 10 0.153897824244827 0.307795648489654 0.846102175755173 11 0.178055047254458 0.356110094508917 0.821944952745542 12 0.221735708718971 0.443471417437942 0.778264291281029 13 0.18717283393969 0.37434566787938 0.81282716606031 14 0.147813373899723 0.295626747799446 0.852186626100277 15 0.157698468801577 0.315396937603153 0.842301531198423 16 0.288837349004974 0.577674698009949 0.711162650995026 17 0.320140497225507 0.640280994451014 0.679859502774493 18 0.385966057401436 0.771932114802872 0.614033942598564 19 0.320763465299878 0.641526930599755 0.679236534700122 20 0.279995886158283 0.559991772316565 0.720004113841717 21 0.234909266929152 0.469818533858304 0.765090733070848 22 0.240279255242336 0.480558510484672 0.759720744757664 23 0.184539996359595 0.36907999271919 0.815460003640405 24 0.143403529286958 0.286807058573917 0.856596470713042 25 0.121054542101003 0.242109084202006 0.878945457898997 26 0.0980262826131491 0.196052565226298 0.90197371738685 27 0.0704226849202957 0.140845369840591 0.929577315079704 28 0.0540327346616327 0.108065469323265 0.945967265338367 29 0.0496792088185215 0.099358417637043 0.950320791181478 30 0.0352729512731498 0.0705459025462995 0.96472704872685 31 0.0299327900445599 0.0598655800891197 0.97006720995544 32 0.0398498506050720 0.0796997012101439 0.960150149394928 33 0.150255138354333 0.300510276708666 0.849744861645667 34 0.119053890410717 0.238107780821435 0.880946109589283 35 0.0908751948529564 0.181750389705913 0.909124805147044 36 0.0686019371947821 0.137203874389564 0.931398062805218 37 0.0557476171420214 0.111495234284043 0.944252382857979 38 0.0444122093043038 0.0888244186086076 0.955587790695696 39 0.0328186560765056 0.0656373121530113 0.967181343923494 40 0.0234107258641535 0.0468214517283069 0.976589274135847 41 0.0198034604149054 0.0396069208298108 0.980196539585095 42 0.0142292437640672 0.0284584875281344 0.985770756235933 43 0.00947057448017115 0.0189411489603423 0.990529425519829 44 0.00627670254612176 0.0125534050922435 0.993723297453878 45 0.00440451492728646 0.00880902985457292 0.995595485072714 46 0.00403797143288951 0.00807594286577902 0.99596202856711 47 0.00425131436105576 0.00850262872211151 0.995748685638944 48 0.00257526681604992 0.00515053363209983 0.99742473318395 49 0.0026183758182234 0.0052367516364468 0.997381624181777 50 0.00192057264542117 0.00384114529084234 0.998079427354579 51 0.00124395370089391 0.00248790740178783 0.998756046299106 52 0.00110398865092869 0.00220797730185738 0.998896011349071 53 0.00195155044883513 0.00390310089767027 0.998048449551165 54 0.0050514714540065 0.010102942908013 0.994948528545994 55 0.00769990974398844 0.0153998194879769 0.992300090256012 56 0.0115362550862788 0.0230725101725576 0.988463744913721 57 0.0448057751148655 0.089611550229731 0.955194224885134 58 0.124852669261030 0.249705338522059 0.87514733073897 59 0.149011580503970 0.298023161007939 0.85098841949603 60 0.187854296312689 0.375708592625378 0.812145703687311 61 0.195763070286652 0.391526140573303 0.804236929713348 62 0.387662612885085 0.77532522577017 0.612337387114915 63 0.369437666880671 0.738875333761342 0.630562333119329 64 0.370461875869145 0.740923751738291 0.629538124130855 65 0.607095074410661 0.785809851178679 0.392904925589339 66 0.557847988814895 0.88430402237021 0.442152011185105 67 0.617965525152922 0.764068949694155 0.382034474847078 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 9 0.142857142857143 NOK 5% type I error level 17 0.26984126984127 NOK 10% type I error level 24 0.380952380952381 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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