*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: Fri, 20 Nov 2009 10:16:53 -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/20/t125873747982jumf7o8n0wwx1.htm/, Retrieved Fri, 20 Nov 2009 18:18:11 +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/20/t125873747982jumf7o8n0wwx1.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:

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Dataseries X:

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8 5560 8.1 3922 7.7 3759 7.5 4138 7.6 4634 7.8 3996 7.8 4308 7.8 4143 7.5 4429 7.5 5219 7.1 4929 7.5 5755 7.5 5592 7.6 4163 7.7 4962 7.7 5208 7.9 4755 8.1 4491 8.2 5732 8.2 5731 8.2 5040 7.9 6102 7.3 4904 6.9 5369 6.7 5578 6.7 4619 6.9 4731 7 5011 7.1 5299 7.2 4146 7.1 4625 6.9 4736 7 4219 6.8 5116 6.4 4205 6.7 4121 6.6 5103 6.4 4300 6.3 4578 6.2 3809 6.5 5526 6.8 4247 6.8 3830 6.4 4394 6.1 4826 5.8 4409 6.1 4569 7.2 4106 7.3 4794 6.9 3914 6.1 3793 5.8 4405 6.2 4022 7.1 4100 7.7 4788 7.9 3163 7.7 3585 7.4 3903 7.5 4178 8 3863 8.1 4187

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] = + 8.55056685635983 -0.000128679428252901X[t] + 0.0738198554733517M1[t] -0.371626817583526M2[t] -0.529082118656772M3[t] -0.590577953777158M4[t] -0.308473981755963M5[t] -0.0330163030212729M6[t] + 0.165507164044993M7[t] + 0.0760396380719248M8[t] -0.0464566797393355M9[t] -0.179002860352318M10[t] -0.410294417357079M11[t] -0.0192537224129794t + 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) 8.55056685635983 0.89064 9.6005 0 0 X -0.000128679428252901 0.00016 -0.8049 0.424948 0.212474 M1 0.0738198554733517 0.377536 0.1955 0.84582 0.42291 M2 -0.371626817583526 0.402273 -0.9238 0.3603 0.18015 M3 -0.529082118656772 0.395282 -1.3385 0.187175 0.093588 M4 -0.590577953777158 0.39124 -1.5095 0.137865 0.068932 M5 -0.308473981755963 0.389053 -0.7929 0.431829 0.215915 M6 -0.0330163030212729 0.397889 -0.083 0.934221 0.46711 M7 0.165507164044993 0.388255 0.4263 0.671846 0.335923 M8 0.0760396380719248 0.390305 0.1948 0.846373 0.423186 M9 -0.0464566797393355 0.390087 -0.1191 0.905709 0.452855 M10 -0.179002860352318 0.39007 -0.4589 0.648421 0.324211 M11 -0.410294417357079 0.387826 -1.0579 0.295493 0.147747 t -0.0192537224129794 0.005158 -3.7325 0.000511 0.000256 Multiple Linear Regression - Regression Statistics Multiple R 0.572985276305044 R-squared 0.328312126862368 Adjusted R-squared 0.142526119398767 F-TEST (value) 1.76715206567261 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 0.0776405512042548 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.612630271023394 Sum Squared Residuals 17.6398449017873 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8 7.88967536833406 0.110324631665942 2 8.1 7.63575187634246 0.464248123657535 3 7.7 7.48001759966146 0.219982400338537 4 7.5 7.35049853882025 0.149501461179753 5 7.6 7.54952379201502 0.0504762079849763 6 7.8 7.88782522356209 -0.087825223562086 7 7.8 8.02694698660047 -0.226946986600467 8 7.8 7.93945784387615 -0.139457843876148 9 7.5 7.76090548717158 -0.260905487171578 10 7.5 7.50744883582582 -0.0074488358258244 11 7.1 7.29422059060143 -0.194220590601427 12 7.5 7.57897207780863 -0.0789720778086287 13 7.5 7.65451295767422 -0.154512957674223 14 7.6 7.37369546517776 0.226304534822237 15 7.7 7.09417157851747 0.605828421482531 16 7.7 6.98176688163389 0.71823311836611 17 7.9 7.30290891224067 0.59709108775933 18 8.1 7.59308423762115 0.506915762378853 19 8.2 7.61266281181258 0.587337188187416 20 8.2 7.50407024285479 0.695929757145211 21 8.2 7.4512376875533 0.748762312446696 22 7.9 7.16278023172276 0.73721976827724 23 7.3 7.066392907352 0.233607092648004 24 6.9 7.3975976681585 -0.497597668158496 25 6.7 7.42526980071401 -0.725269800714011 26 6.7 7.08397297693869 -0.383972976938687 27 6.9 6.89285185748814 0.00714814251186351 28 7 6.77607206004396 0.223927939956041 29 7.1 7.00186263431534 0.0981373656846603 30 7.2 7.40643397141265 -0.206433971412645 31 7.1 7.5240662699328 -0.424066269932792 32 6.9 7.40106160501067 -0.501061605010672 33 7 7.32583882919318 -0.325838829193182 34 6.8 7.05861347902437 -0.258613479024368 35 6.4 6.92529515874502 -0.525295158745021 36 6.7 7.32714492566236 -0.627144925662364 37 6.6 7.25534786017839 -0.655347860178387 38 6.4 6.89397704559561 -0.493977045595610 39 6.3 6.68149514105508 -0.381495141055078 40 6.2 6.6997000638482 -0.499700063848193 41 6.5 6.74160773514618 -0.241607735146178 42 6.8 7.16239268020335 -0.362392680203350 43 6.8 7.3953217464381 -0.595321746438095 44 6.4 7.21402530051741 -0.814025300517411 45 6.1 7.01668574728792 -0.91668574728792 46 5.8 6.91854516584342 -1.11854516584342 47 6.1 6.64741117790521 -0.547411177905212 48 7.2 7.0980304481304 0.101969551869596 49 7.3 7.06406513455278 0.235934865447219 50 6.9 6.71260263594548 0.187397364054524 51 6.1 6.55146382327785 -0.451463823277853 52 5.8 6.39196245565371 -0.591962455653711 53 6.2 6.70409692628279 -0.504096926282789 54 7.1 6.95026388720077 0.149736112799227 55 7.7 7.04100218521606 0.658997814783937 56 7.9 7.14138500774098 0.75861499225902 57 7.7 6.94533224879402 0.754667751205984 58 7.4 6.75261228758363 0.647387712416369 59 7.5 6.46668016539634 1.03331983460366 60 8 6.8982548802401 1.10174511975989 61 8.1 6.91112887854654 1.18887112145346 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.079937227776215 0.15987445555243 0.920062772223785 18 0.0450569346603546 0.0901138693207093 0.954943065339645 19 0.0172198156982225 0.034439631396445 0.982780184301777 20 0.00638093653319808 0.0127618730663962 0.993619063466802 21 0.0129874557005699 0.0259749114011398 0.98701254429943 22 0.00958584666748967 0.0191716933349793 0.99041415333251 23 0.00604143075932119 0.0120828615186424 0.993958569240679 24 0.00380986067289928 0.00761972134579856 0.9961901393271 25 0.00861941999895475 0.0172388399979095 0.991380580001045 26 0.0244149665077097 0.0488299330154194 0.97558503349229 27 0.0212239353623586 0.0424478707247173 0.97877606463764 28 0.026986367118293 0.053972734236586 0.973013632881707 29 0.0362145464013921 0.0724290928027842 0.963785453598608 30 0.0273376458397556 0.0546752916795111 0.972662354160244 31 0.0170488829159141 0.0340977658318282 0.982951117084086 32 0.0118875248032109 0.0237750496064217 0.988112475196789 33 0.0106991930169611 0.0213983860339223 0.989300806983039 34 0.0209100789066254 0.0418201578132509 0.979089921093375 35 0.0162855215702096 0.0325710431404192 0.98371447842979 36 0.024833822324723 0.049667644649446 0.975166177675277 37 0.0140573697285998 0.0281147394571997 0.9859426302714 38 0.00820236045698902 0.0164047209139780 0.99179763954301 39 0.0137534187496892 0.0275068374993784 0.98624658125031 40 0.0329435821232518 0.0658871642465036 0.967056417876748 41 0.424658503512361 0.849317007024721 0.57534149648764 42 0.684294871755839 0.631410256488322 0.315705128244161 43 0.588847311689932 0.822305376620137 0.411152688310068 44 0.420971325356986 0.841942650713973 0.579028674643014 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.0357142857142857 NOK 5% type I error level 18 0.642857142857143 NOK 10% type I error level 23 0.821428571428571 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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