Workshop 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: Sat, 20 Nov 2010 09:52:53 +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/20/t1290246724k9sxz97n5xmfgr5.htm/, Retrieved Sat, 20 Nov 2010 10:52:06 +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/20/t1290246724k9sxz97n5xmfgr5.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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Dataseries X:

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16198.9 16896.2 16554.2 16698 19554.2 19691.6 15903.8 15930.7 18003.8 17444.6 18329.6 17699.4 16260.7 15189.8 14851.9 15672.7 18174.1 17180.8 18406.6 17664.9 18466.5 17862.9 16016.5 16162.3 17428.5 17463.6 17167.2 16772.1 19630 19106.9 17183.6 16721.3 18344.7 18161.3 19301.4 18509.9 18147.5 17802.7 16192.9 16409.9 18374.4 17967.7 20515.2 20286.6 18957.2 19537.3 16471.5 18021.9 18746.8 20194.3 19009.5 19049.6 19211.2 20244.7 20547.7 21473.3 19325.8 19673.6 20605.5 21053.2 20056.9 20159.5 16141.4 18203.6 20359.8 21289.5 19711.6 20432.3 15638.6 17180.4 14384.5 15816.8 13855.6 15071.8 14308.3 14521.1 15290.6 15668.8 14423.8 14346.9 13779.7 13881 15686.3 15465.9 14733.8 14238.2 12522.5 13557.7 16189.4 16127.6 16059.1 16793.9 16007.1 16014 15806.8 16867.9 15160 16014.6 15692.1 15878.6 18908.9 18664.9 16969.9 17962.5 16997.5 17332.7 19858.9 19542.1 17681.2 17203.6

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 7 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation uitvoer[t] = + 748.072152423624 + 0.892597476409893invoer[t] + 241.389011880087M1[t] + 995.458410491878M2[t] + 1101.31078858059M3[t] + 827.408828727138M4[t] + 1101.50543825314M5[t] + 1536.18475816057M6[t] + 1526.30536759173M7[t] -67.6231584650186M8[t] + 1333.38498903394M9[t] + 1149.19652200137M10[t] + 766.137395534915M11[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) 748.072152423624 789.350269 0.9477 0.3487 0.17435 invoer 0.892597476409893 0.043991 20.2907 0 0 M1 241.389011880087 385.223723 0.6266 0.534299 0.26715 M2 995.458410491878 384.844169 2.5867 0.013247 0.006623 M3 1101.31078858059 394.32318 2.7929 0.007833 0.003916 M4 827.408828727138 385.614746 2.1457 0.037724 0.018862 M5 1101.50543825314 385.648539 2.8562 0.006637 0.003319 M6 1536.18475816057 392.311749 3.9157 0.000325 0.000163 M7 1526.30536759173 384.901578 3.9654 0.00028 0.00014 M8 -67.6231584650186 406.976247 -0.1662 0.868828 0.434414 M9 1333.38498903394 410.424128 3.2488 0.002284 0.001142 M10 1149.19652200137 415.779622 2.764 0.008443 0.004222 M11 766.137395534915 407.67852 1.8793 0.067158 0.033579 Multiple Linear Regression - Regression Statistics Multiple R 0.968101963445662 R-squared 0.937221411627346 Adjusted R-squared 0.919284672092302 F-TEST (value) 52.2514925188187 F-TEST (DF numerator) 12 F-TEST (DF denominator) 42 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 573.625168812105 Sum Squared Residuals 13819925.0403781 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 16198.9 16070.9666452205 127.933354779463 2 16554.2 16648.1232240079 -93.9232240078967 3 19554.2 19426.0554074773 128.14459252274 4 15903.8 15795.1835985938 108.616401406153 5 18003.8 17420.5835276568 583.216472343212 6 18329.6 18082.6966845535 246.903315446544 7 16260.7 15832.7546671864 427.94533281365 8 14851.9 14669.8614624879 182.038537512062 9 18174.1 17416.9958641607 757.104135839344 10 18406.6 17664.9138354581 741.686164541883 11 18466.5 17458.5890093208 1007.91099067918 12 16016.5 15174.5003454032 841.999654596761 13 17428.5 16577.4264533355 851.07354666448 14 17167.2 16714.2646970099 452.935302990131 15 19630 18904.1536630204 725.846336979601 16 17183.6 16500.8711634435 682.728836556491 17 18344.7 18060.3081389998 284.391861000242 18 19301.4 18806.1469391837 495.253060816329 19 18147.5 18165.0226132978 -17.5226132977622 20 16192.9 15327.8843220973 865.015677902687 21 18374.4 18119.3808183476 255.0191816524 22 20515.2 20005.0366393619 510.16336063807 23 18957.2 18953.1542238215 4.04577617845619 24 16471.5 16834.3746125351 -362.874612535079 25 18746.8 19014.842382168 -268.042382168018 26 19009.5 18747.1554495334 262.344550466598 27 19211.2 19919.7510716796 -708.551071679574 28 20547.7 20742.4943713433 -194.79437134332 29 19325.8 19410.1833025744 -84.3833025744407 30 20605.5 21076.290100937 -470.790100936955 31 20056.9 20268.6963457006 -211.796345700597 32 16141.4 16928.9364155337 -787.536415533736 33 20359.8 21084.411115486 -724.611115485986 34 19711.6 20135.0880916749 -423.488091674855 35 15638.6 16849.3912316711 -1210.79123167107 36 14384.5 14866.1079173036 -481.607917303621 37 13855.6 14442.5118092583 -586.911809258338 38 14308.3 14705.0277776112 -396.727777611202 39 15290.6 15835.3142793755 -544.714279375542 40 14423.8 14381.4877154559 42.3122845441422 41 13779.7 14239.7231607225 -460.023160722492 42 15686.3 16089.080220992 -402.780220991957 43 14733.8 14983.3589086347 -249.558908634698 44 12522.5 12782.017799881 -259.517799881014 45 16189.4 16476.9122020058 -287.512202005757 46 16059.1 16887.4614335051 -828.361433505098 47 16007.1 15808.2655351866 198.834464813433 48 15806.8 15804.3171247581 2.48287524193745 49 15160 15284.0527100176 -124.052710017586 50 15692.1 15916.7288518376 -224.628851837631 51 18908.9 18509.6255784472 399.274421552776 52 16969.9 17608.7631511635 -638.863151163466 53 16997.5 17320.7018700465 -323.201870046522 54 19858.9 19727.486054334 131.413945666039 55 17681.2 17630.2674651806 50.9325348194077 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.526849618858244 0.946300762283512 0.473150381141756 17 0.4500495750939 0.9000991501878 0.5499504249061 18 0.33128934338609 0.66257868677218 0.66871065661391 19 0.278063006951384 0.556126013902768 0.721936993048616 20 0.417214911669489 0.834429823338978 0.582785088330511 21 0.408796061128732 0.817592122257463 0.591203938871268 22 0.439449735936874 0.878899471873748 0.560550264063126 23 0.524452584219057 0.951094831561887 0.475547415780943 24 0.56624950960972 0.86750098078056 0.43375049039028 25 0.459049058346652 0.918098116693304 0.540950941653348 26 0.478391393319817 0.956782786639635 0.521608606680183 27 0.625332686395399 0.749334627209203 0.374667313604601 28 0.543588323458132 0.912823353083736 0.456411676541868 29 0.464429975360673 0.928859950721347 0.535570024639327 30 0.392113969508405 0.78422793901681 0.607886030491595 31 0.301919182497791 0.603838364995581 0.698080817502209 32 0.359003840931022 0.718007681862045 0.640996159068978 33 0.345541602215006 0.691083204430012 0.654458397784994 34 0.301862209367581 0.603724418735161 0.698137790632419 35 0.882129696486649 0.235740607026702 0.117870303513351 36 0.855645456549728 0.288709086900544 0.144354543450272 37 0.838831392586348 0.322337214827304 0.161168607413652 38 0.744962096951866 0.510075806096268 0.255037903048134 39 0.788903496238349 0.422193007523303 0.211096503761651 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 0 0 OK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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