Multi lineair 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: Thu, 19 Nov 2009 12:35:22 -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/19/t1258659420khj3lb0e5nes0j8.htm/, Retrieved Thu, 19 Nov 2009 20:37:12 +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/19/t1258659420khj3lb0e5nes0j8.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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8,9 6,3 8,2 6,2 7,6 6,1 7,7 6,3 8,1 6,5 8,3 6,6 8,3 6,5 7,9 6,2 7,8 6,2 8 5,9 8,5 6,1 8,6 6,1 8,5 6,1 8 6,1 7,8 6,1 8 6,4 8,2 6,7 8,3 6,9 8,2 7 8,1 7 8 6,8 7,8 6,4 7,8 5,9 7,7 5,5 7,6 5,5 7,6 5,6 7,6 5,8 7,8 5,9 8 6,1 8 6,1 7,9 6 7,7 6 7,4 5,9 6,9 5,5 6,7 5,6 6,5 5,4 6,4 5,2 6,7 5,2 6,8 5,2 6,9 5,5 6,9 5,8 6,7 5,8 6,4 5,5 6,2 5,3 5,9 5,1 6,1 5,2 6,7 5,8 6,8 5,8 6,6 5,5 6,4 5 6,4 4,9 6,7 5,3 7,1 6,1 7,1 6,5 6,9 6,8 6,4 6,6 6 6,4 6 6,4

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation wv[t] = + 2.12751659053694 + 0.88013875721796wm[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) 2.12751659053694 0.960147 2.2158 0.030785 0.015393 wm 0.88013875721796 0.160237 5.4927 1e-06 1e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.591710957513192 R-squared 0.350121857241178 Adjusted R-squared 0.338516890406199 F-TEST (value) 30.1700006746993 F-TEST (DF numerator) 1 F-TEST (DF denominator) 56 p-value 1.00029949146041e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.641031880131442 Sum Squared Residuals 23.0116247953117 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.9 7.67239076101003 1.22760923898997 2 8.2 7.58437688528829 0.615623114711712 3 7.6 7.4963630095665 0.103636990433508 4 7.7 7.67239076101008 0.0276092389899161 5 8.1 7.84841851245368 0.251581487546323 6 8.3 7.93643238817547 0.363567611824529 7 8.3 7.84841851245368 0.451581487546325 8 7.9 7.58437688528829 0.315623114711712 9 7.8 7.58437688528829 0.215623114711711 10 8 7.3203352581229 0.679664741877099 11 8.5 7.4963630095665 1.00363699043351 12 8.6 7.4963630095665 1.10363699043351 13 8.5 7.4963630095665 1.00363699043351 14 8 7.4963630095665 0.503636990433508 15 7.8 7.4963630095665 0.303636990433508 16 8 7.76040463673188 0.239595363268119 17 8.2 8.02444626389727 0.175553736102731 18 8.3 8.20047401534086 0.0995259846591403 19 8.2 8.28848789106266 -0.0884878910626567 20 8.1 8.28848789106266 -0.188487891062656 21 8 8.11246013961906 -0.112460139619064 22 7.8 7.76040463673188 0.0395953632681193 23 7.8 7.3203352581229 0.479664741877099 24 7.7 6.96827975523572 0.731720244764283 25 7.6 6.96827975523572 0.631720244764283 26 7.6 7.05629363095751 0.543706369042487 27 7.6 7.2323213824011 0.367678617598895 28 7.8 7.3203352581229 0.479664741877099 29 8 7.4963630095665 0.503636990433508 30 8 7.4963630095665 0.503636990433508 31 7.9 7.4083491338447 0.491650866155304 32 7.7 7.4083491338447 0.291650866155304 33 7.4 7.3203352581229 0.0796647418770996 34 6.9 6.96827975523572 -0.0682797552357164 35 6.7 7.05629363095751 -0.356293630957512 36 6.5 6.88026587951392 -0.380265879513921 37 6.4 6.70423812807033 -0.304238128070329 38 6.7 6.70423812807033 -0.00423812807032881 39 6.8 6.70423812807033 0.0957618719296709 40 6.9 6.96827975523572 -0.0682797552357164 41 6.9 7.2323213824011 -0.332321382401104 42 6.7 7.2323213824011 -0.532321382401104 43 6.4 6.96827975523572 -0.568279755235716 44 6.2 6.79225200379212 -0.592252003792125 45 5.9 6.61622425234853 -0.716224252348532 46 6.1 6.70423812807033 -0.604238128070329 47 6.7 7.2323213824011 -0.532321382401104 48 6.8 7.2323213824011 -0.432321382401104 49 6.6 6.96827975523572 -0.368279755235717 50 6.4 6.52821037662674 -0.128210376626736 51 6.4 6.44019650090494 -0.0401965009049408 52 6.7 6.79225200379212 -0.0922520037921245 53 7.1 7.4963630095665 -0.396363009566492 54 7.1 7.84841851245368 -0.748418512453677 55 6.9 8.11246013961906 -1.21246013961906 56 6.4 7.93643238817547 -1.53643238817547 57 6 7.76040463673188 -1.76040463673188 58 6 7.76040463673188 -1.76040463673188 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.493060316957685 0.98612063391537 0.506939683042315 6 0.327297767039338 0.654595534078677 0.672702232960662 7 0.202049500139588 0.404099000279176 0.797950499860412 8 0.118308533109868 0.236617066219737 0.881691466890131 9 0.0685967619311509 0.137193523862302 0.93140323806885 10 0.0449877828350931 0.0899755656701861 0.955012217164907 11 0.0541852123694689 0.108370424738938 0.945814787630531 12 0.0723553434282005 0.144710686856401 0.9276446565718 13 0.0735527308363664 0.147105461672733 0.926447269163634 14 0.0514351923245028 0.102870384649006 0.948564807675497 15 0.0404060241655675 0.080812048331135 0.959593975834433 16 0.0273380009137264 0.0546760018274529 0.972661999086273 17 0.0170151282343855 0.034030256468771 0.982984871765614 18 0.0104655627134075 0.0209311254268149 0.989534437286593 19 0.0062307058793782 0.0124614117587564 0.993769294120622 20 0.00377652694642208 0.00755305389284417 0.996223473053578 21 0.00246768522776551 0.00493537045553102 0.997532314772234 22 0.00211300219390693 0.00422600438781387 0.997886997806093 23 0.00191678320513856 0.00383356641027713 0.998083216794861 24 0.00190293864972263 0.00380587729944525 0.998097061350277 25 0.00193637306161227 0.00387274612322453 0.998063626938388 26 0.00195589375509827 0.00391178751019655 0.998044106244902 27 0.00203635615049501 0.00407271230099001 0.997963643849505 28 0.00234942055861552 0.00469884111723105 0.997650579441384 29 0.00410506329896785 0.0082101265979357 0.995894936701032 30 0.0104399952315375 0.0208799904630751 0.989560004768463 31 0.0341748654294997 0.0683497308589994 0.9658251345705 32 0.104075949089818 0.208151898179636 0.895924050910182 33 0.242306434717984 0.484612869435968 0.757693565282016 34 0.388405833720209 0.776811667440418 0.611594166279791 35 0.545593057702556 0.908813884594889 0.454406942297444 36 0.637257010996194 0.725485978007612 0.362742989003806 37 0.657468402457932 0.685063195084136 0.342531597542068 38 0.617805631240488 0.764388737519024 0.382194368759512 39 0.590607653405224 0.818784693189553 0.409392346594776 40 0.600498851101461 0.799002297797078 0.399501148898539 41 0.634531881582804 0.730936236834391 0.365468118417196 42 0.64810371426884 0.703792571462321 0.351896285731160 43 0.629611591943864 0.740776816112273 0.370388408056136 44 0.610155932566109 0.779688134867782 0.389844067433891 45 0.682439135157757 0.635121729684486 0.317560864842243 46 0.693775889908687 0.612448220182626 0.306224110091313 47 0.635024108084508 0.729951783830984 0.364975891915492 48 0.575951363017819 0.848097273964362 0.424048636982181 49 0.472440182948298 0.944880365896596 0.527559817051702 50 0.358488831584643 0.716977663169285 0.641511168415357 51 0.262737551809351 0.525475103618701 0.73726244819065 52 0.162510009731890 0.325020019463781 0.83748999026811 53 0.386405828590746 0.772811657181492 0.613594171409254 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 10 0.204081632653061 NOK 5% type I error level 14 0.285714285714286 NOK 10% type I error level 18 0.36734693877551 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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