WS 7 Multiple regression

*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: Tue, 23 Nov 2010 20:21:47 +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/23/t129054363421iezygol543p7y.htm/, Retrieved Tue, 23 Nov 2010 21:20:44 +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/23/t129054363421iezygol543p7y.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:

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No (this computation is public)

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Dataseries X:

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102,89 167,16 100,70 106,88 97,69 102,64 179,84 99,62 107,45 101,69 103,33 174,44 99,83 107,65 102,72 103,56 180,35 100,74 107,72 101,85 103,60 193,17 100,84 108,10 114,94 104,24 195,16 100,85 108,38 106,20 105,31 202,43 99,71 108,62 106,76 105,40 189,91 100,80 108,79 107,24 105,89 195,98 100,06 109,03 106,50 105,89 212,09 100,57 109,34 106,77 105,54 205,81 99,79 109,73 108,24 106,15 204,31 99,90 109,76 104,43 106,14 196,07 100,12 109,96 100,90 105,85 199,98 100,40 110,49 103,91 106,27 199,10 100,51 111,37 103,81 106,51 198,31 100,70 111,56 104,59 106,82 195,72 100,62 111,90 104,94 106,53 223,04 99,70 111,96 111,64 107,14 238,41 99,48 112,25 111,27 107,39 259,73 99,36 112,39 106,82 107,33 326,54 99,39 112,30 106,07 107,53 335,15 99,45 112,49 111,35 107,42 321,81 99,28 112,77 112,59 108,25 368,62 99,40 113,15 108,59 108,26 369,59 99,10 113,15 106,83 108,93 425,00 99,48 113,28 112,51 109,43 439,72 99,74 113,83 113,61 109,61 362,23 100,42 114,49 114,96 109,74 328,76 100,80 114,76 118, etc...

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 Bier[t] = -17.2101366875923 -0.0019115819018345Tarwe[t] + 0.187982201415514Suiker[t] + 1.00066611432293Mineraalwater[t] -0.0520181143946909Fruit[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) -17.2101366875923 16.756988 -1.027 0.309065 0.154533 Tarwe -0.0019115819018345 0.001873 -1.0205 0.312105 0.156053 Suiker 0.187982201415514 0.143322 1.3116 0.195306 0.097653 Mineraalwater 1.00066611432293 0.033099 30.2326 0 0 Fruit -0.0520181143946909 0.024704 -2.1056 0.039992 0.019996 Multiple Linear Regression - Regression Statistics Multiple R 0.988890264126732 R-squared 0.977903954484638 Adjusted R-squared 0.976236328408007 F-TEST (value) 586.404811119399 F-TEST (DF numerator) 4 F-TEST (DF denominator) 53 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.812567660931656 Sum Squared Residuals 34.9941087903729 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 102.89 103.269675667857 -0.37967566785704 2 102.64 103.404723259398 -0.764723259398334 3 103.33 103.601076629004 -0.271076629003569 4 103.56 103.876145370778 -0.316145370777806 5 103.6 103.569773116954 0.0302268830459472 6 104.24 104.302673722804 -0.0626737228035759 7 105.31 104.28550653614 1.02449346385997 8 105.4 104.659484685619 0.740515314380644 9 105.89 104.787427826517 1.10257217348268 10 105.89 105.148664769354 0.741335230645782 11 105.54 105.327836543019 0.212163456980617 12 106.15 105.579590957301 0.570409042698698 13 106.14 106.020455643162 0.119544356838327 14 105.85 106.439394890585 -0.589394890584986 15 106.27 107.347543116858 -1.07754311685796 16 106.51 107.534322317323 -1.02432231732285 17 106.82 107.846254877167 -1.02625487716703 18 106.53 107.332605434722 -0.802605434721564 19 107.14 107.571308212059 -0.431308212058648 20 107.39 107.879569286803 -0.48956928680326 21 107.33 107.706449601491 -0.376449601491115 22 107.53 107.616740731119 -0.0867407311186347 23 107.42 107.825968309609 -0.405968309609473 24 108.25 108.347370605976 -0.097370605975953 25 108.26 108.380673592441 -0.120673592441168 26 108.93 108.180809780899 0.749190219101455 27 109.43 108.694693104715 0.735306895284975 28 109.61 109.560864664271 0.0491353357289608 29 109.74 109.773991374670 -0.0339913746701849 30 110.12 110.004649851698 0.115350148302389 31 110.16 110.337163143390 -0.177163143390146 32 110.44 110.997057047080 -0.557057047079817 33 111.23 111.561932077062 -0.331932077061866 34 112.86 112.478595607787 0.381404392212776 35 112.77 113.383117591706 -0.61311759170566 36 113.04 113.555280562017 -0.515280562017024 37 112.79 113.692386678622 -0.902386678622353 38 113.87 114.164991484063 -0.294991484062631 39 114.28 114.456821509553 -0.176821509552931 40 115.51 114.661394241797 0.848605758202528 41 116.76 114.494517755164 2.26548224483639 42 116.91 114.836453517077 2.07354648292345 43 116.47 115.722983854469 0.747016145530638 44 116.94 116.090360638561 0.84963936143926 45 117.24 116.389460165153 0.85053983484724 46 116.82 116.526630861703 0.293369138297139 47 117.48 116.440499634468 1.03950036553201 48 117.11 116.843828344924 0.266171655075977 49 117.31 117.355007799778 -0.0450077997784247 50 117.77 117.532550799849 0.237449200150527 51 118.37 118.087531386159 0.282468613841315 52 117.91 118.216637617134 -0.306637617133607 53 118.12 117.559763907109 0.560236092891495 54 118.02 117.612692478639 0.407307521360612 55 117.77 118.483187154614 -0.71318715461382 56 117.85 118.376299915237 -0.526299915237298 57 118.68 120.631437404962 -1.95143740496156 58 118.9 120.725172341612 -1.82517234161151 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.212088462413219 0.424176924826438 0.787911537586781 9 0.108952444256102 0.217904888512204 0.891047555743898 10 0.0640270242196385 0.128054048439277 0.935972975780361 11 0.128407432180850 0.256814864361699 0.87159256781915 12 0.0800168373870307 0.160033674774061 0.919983162612969 13 0.055648806552363 0.111297613104726 0.944351193447637 14 0.0776865495528841 0.155373099105768 0.922313450447116 15 0.0694808245196448 0.138961649039290 0.930519175480355 16 0.045571408413727 0.091142816827454 0.954428591586273 17 0.0320988818928830 0.0641977637857661 0.967901118107117 18 0.0599971634422805 0.119994326884561 0.94000283655772 19 0.0700337964546971 0.140067592909394 0.929966203545303 20 0.096566765370472 0.193133530740944 0.903433234629528 21 0.101016725961956 0.202033451923912 0.898983274038044 22 0.0738000267377063 0.147600053475413 0.926199973262294 23 0.0707399066953263 0.141479813390653 0.929260093304674 24 0.0547111060689607 0.109422212137921 0.94528889393104 25 0.0572992903086442 0.114598580617288 0.942700709691356 26 0.0419516111882869 0.0839032223765737 0.958048388811713 27 0.0286334936738851 0.0572669873477703 0.971366506326115 28 0.0178989380364287 0.0357978760728574 0.982101061963571 29 0.0108573758904101 0.0217147517808202 0.98914262410959 30 0.00651531057335713 0.0130306211467143 0.993484689426643 31 0.00361497359248906 0.00722994718497813 0.99638502640751 32 0.00228402703707015 0.00456805407414031 0.99771597296293 33 0.00163974970018582 0.00327949940037164 0.998360250299814 34 0.00583561802614232 0.0116712360522846 0.994164381973858 35 0.00755289514421145 0.0151057902884229 0.992447104855789 36 0.0134348853912624 0.0268697707825247 0.986565114608738 37 0.101645934534660 0.203291869069319 0.89835406546534 38 0.326870521385151 0.653741042770301 0.67312947861485 39 0.920083604712839 0.159832790574322 0.079916395287161 40 0.997170357242021 0.00565928551595782 0.00282964275797891 41 0.99903192733268 0.00193614533464115 0.000968072667320576 42 0.999669876142896 0.000660247714208514 0.000330123857104257 43 0.999365173678639 0.00126965264272274 0.000634826321361371 44 0.998226810371752 0.00354637925649517 0.00177318962824759 45 0.996103017466803 0.00779396506639374 0.00389698253319687 46 0.996230099360724 0.0075398012785519 0.00376990063927595 47 0.993150536729835 0.0136989265403292 0.00684946327016459 48 0.983774543208196 0.0324509135836074 0.0162254567918037 49 0.973350310501078 0.0532993789978448 0.0266496894989224 50 0.956691452883653 0.086617094232695 0.0433085471163475 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 10 0.232558139534884 NOK 5% type I error level 18 0.418604651162791 NOK 10% type I error level 24 0.558139534883721 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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