*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, 18 Nov 2009 07:38:53 -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/18/t1258555274dzwt8icvflx4y0q.htm/, Retrieved Wed, 18 Nov 2009 15:41:26 +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/18/t1258555274dzwt8icvflx4y0q.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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8,2 25,5 8,3 25,5 8,1 25,5 7,4 20,9 7,3 20,9 7,7 20,9 8 22,3 8 22,3 7,7 22,3 6,9 19,9 6,6 19,9 6,9 19,9 7,5 24,1 7,9 24,1 7,7 24,1 6,5 13,8 6,1 13,8 6,4 13,8 6,8 16,2 7,1 16,2 7,3 16,2 7,2 18,6 7 18,6 7 18,6 7 22,4 7,3 22,4 7,5 22,4 7,2 22,6 7,7 22,6 8 22,6 7,9 20 8 20 8 20 7,9 21,8 7,9 21,8 8 21,8 8,1 28,7 8,1 28,7 8,2 28,7 8 19,5 8,3 19,5 8,5 19,5 8,6 19,4 8,7 19,4 8,7 19,4 8,5 21,7 8,4 21,7 8,5 21,7 8,7 26,2 8,7 26,2 8,6 26,2 7,9 19,1 8,1 19,1 8,2 19,1 8,5 21,3 8,6 21,3 8,5 21,3 8,3 24,1 8,2 24,1 8,7 24,1

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 5.472559642061 + 0.109979305050487X[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) 5.472559642061 0.46384 11.7984 0 0 X 0.109979305050487 0.021413 5.1362 3e-06 2e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.559136904595822 R-squared 0.312634078080997 Adjusted R-squared 0.300782941496187 F-TEST (value) 26.3800923939819 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 3.42438683531565e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.551638998319588 Sum Squared Residuals 17.6497238990882 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.2 8.27703192084841 -0.0770319208484108 2 8.3 8.27703192084841 0.0229680791515907 3 8.1 8.27703192084841 -0.177031920848410 4 7.4 7.77112711761617 -0.37112711761617 5 7.3 7.77112711761617 -0.471127117616171 6 7.7 7.77112711761617 -0.0711271176161704 7 8 7.92509814468685 0.0749018553131476 8 8 7.92509814468685 0.0749018553131476 9 7.7 7.92509814468685 -0.225098144686852 10 6.9 7.66114781256568 -0.761147812565683 11 6.6 7.66114781256568 -1.06114781256568 12 6.9 7.66114781256568 -0.761147812565683 13 7.5 8.12306089377773 -0.623060893777729 14 7.9 8.12306089377773 -0.223060893777728 15 7.7 8.12306089377773 -0.423060893777728 16 6.5 6.99027405175771 -0.490274051757715 17 6.1 6.99027405175771 -0.890274051757715 18 6.4 6.99027405175771 -0.590274051757714 19 6.8 7.25422438387888 -0.454224383878883 20 7.1 7.25422438387888 -0.154224383878883 21 7.3 7.25422438387888 0.0457756161211171 22 7.2 7.51817471600005 -0.318174716000051 23 7 7.51817471600005 -0.518174716000051 24 7 7.51817471600005 -0.518174716000051 25 7 7.9360960751919 -0.9360960751919 26 7.3 7.9360960751919 -0.636096075191901 27 7.5 7.9360960751919 -0.436096075191901 28 7.2 7.958091936202 -0.758091936201998 29 7.7 7.958091936202 -0.258091936201998 30 8 7.958091936202 0.0419080637980015 31 7.9 7.67214574307073 0.227854256929268 32 8 7.67214574307073 0.327854256929267 33 8 7.67214574307073 0.327854256929267 34 7.9 7.8701084921616 0.0298915078383914 35 7.9 7.8701084921616 0.0298915078383914 36 8 7.8701084921616 0.129891507838391 37 8.1 8.62896569700997 -0.528965697009968 38 8.1 8.62896569700997 -0.528965697009968 39 8.2 8.62896569700997 -0.428965697009968 40 8 7.61715609054549 0.382843909454511 41 8.3 7.61715609054549 0.682843909454512 42 8.5 7.61715609054549 0.88284390945451 43 8.6 7.60615816004044 0.99384183995956 44 8.7 7.60615816004044 1.09384183995956 45 8.7 7.60615816004044 1.09384183995956 46 8.5 7.85911056165656 0.64088943834344 47 8.4 7.85911056165656 0.54088943834344 48 8.5 7.85911056165656 0.64088943834344 49 8.7 8.35401743438375 0.345982565616249 50 8.7 8.35401743438375 0.345982565616249 51 8.6 8.35401743438375 0.245982565616249 52 7.9 7.5731643685253 0.326835631474706 53 8.1 7.5731643685253 0.526835631474705 54 8.2 7.5731643685253 0.626835631474705 55 8.5 7.81511883963637 0.684881160363635 56 8.6 7.81511883963637 0.784881160363634 57 8.5 7.81511883963637 0.684881160363635 58 8.3 8.12306089377773 0.176939106222272 59 8.2 8.12306089377773 0.0769391062222707 60 8.7 8.12306089377773 0.576939106222271 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0056132409262552 0.0112264818525104 0.994386759073745 6 0.0120077638702899 0.0240155277405798 0.98799223612971 7 0.0105757058475014 0.0211514116950028 0.989424294152499 8 0.00632523420377365 0.0126504684075473 0.993674765796226 9 0.00204847647134887 0.00409695294269775 0.997951523528651 10 0.00492522152982804 0.00985044305965609 0.995074778470172 11 0.0204173168848079 0.0408346337696159 0.979582683115192 12 0.0137482905653106 0.0274965811306213 0.98625170943469 13 0.0206616245659847 0.0413232491319693 0.979338375434015 14 0.0108393137776728 0.0216786275553456 0.989160686222327 15 0.00735600786658001 0.0147120157331600 0.99264399213342 16 0.00837440175361944 0.0167488035072389 0.99162559824638 17 0.00726983925952113 0.0145396785190423 0.992730160740479 18 0.00665544694190448 0.0133108938838090 0.993344553058096 19 0.00570122941195992 0.0114024588239198 0.99429877058804 20 0.00773169895158138 0.0154633979031628 0.992268301048419 21 0.0144185203194598 0.0288370406389196 0.98558147968054 22 0.0121984541169854 0.0243969082339709 0.987801545883014 23 0.0149453775742074 0.0298907551484148 0.985054622425793 24 0.0235194794214056 0.0470389588428111 0.976480520578594 25 0.116160918547651 0.232321837095302 0.883839081452349 26 0.190485198786044 0.380970397572089 0.809514801213956 27 0.230705704661471 0.461411409322942 0.769294295338529 28 0.544341089493678 0.911317821012644 0.455658910506322 29 0.609703902704789 0.780592194590422 0.390296097295211 30 0.634841597192154 0.730316805615692 0.365158402807846 31 0.753854851580228 0.492290296839544 0.246145148419772 32 0.84067182331379 0.31865635337242 0.15932817668621 33 0.894483387942494 0.211033224115013 0.105516612057506 34 0.91798357930066 0.164032841398679 0.0820164206993394 35 0.942611979685097 0.114776040629807 0.0573880203149033 36 0.95648417342912 0.0870316531417612 0.0435158265708806 37 0.960896983742777 0.0782060325144452 0.0391030162572226 38 0.97175643905693 0.056487121886141 0.0282435609430705 39 0.982146378789552 0.0357072424208969 0.0178536212104484 40 0.991912165874663 0.0161756682506749 0.00808783412533747 41 0.994348441825203 0.0113031163495936 0.00565155817479682 42 0.99648382314113 0.00703235371773893 0.00351617685886947 43 0.99816263084728 0.00367473830544029 0.00183736915272014 44 0.999457654779584 0.00108469044083252 0.000542345220416262 45 0.999907022270076 0.000185955459848870 9.29777299244352e-05 46 0.999823912444733 0.000352175110533204 0.000176087555266602 47 0.999561956657129 0.000876086685742321 0.000438043342871161 48 0.999147971470772 0.00170405705845656 0.000852028529228278 49 0.997779349728424 0.00444130054315211 0.00222065027157606 50 0.994485136370502 0.0110297272589957 0.00551486362949783 51 0.986015501721802 0.0279689965563956 0.0139844982781978 52 0.98557515545014 0.0288496890997194 0.0144248445498597 53 0.977130405232985 0.0457391895340305 0.0228695947670152 54 0.973751772499291 0.0524964550014169 0.0262482275007085 55 0.923272094387163 0.153455811225674 0.0767279056128368 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 10 0.196078431372549 NOK 5% type I error level 35 0.686274509803922 NOK 10% type I error level 39 0.764705882352941 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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