Multiple Regression

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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Wed, 22 Dec 2010 14:47:15 +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/Dec/22/t1293029479kgm8zalg4kgvr5j.htm/, Retrieved Wed, 22 Dec 2010 15:51:19 +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/Dec/22/t1293029479kgm8zalg4kgvr5j.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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109,99 89 103.88 103.77 112,01 86,4 103.91 103.88 111,96 84,5 103.91 103.91 111,41 82,7 103.92 103.91 112,11 80,8 104.05 103.92 111,67 81,8 104.23 104.05 111,95 81,8 104.30 104.23 112,31 82,9 104.31 104.30 113,26 83,8 104.31 104.31 113,5 86,2 104.34 104.31 114,43 86,1 104.55 104.34 115,02 86,2 104.65 104.55 115,1 88,8 104.73 104.65 117,11 89,6 104.75 104.73 117,52 87,8 104.75 104.75 116,1 88,3 104.76 104.75 116,39 88,6 104.94 104.76 116,01 91 105.29 104.94 116,74 91,5 105.38 105.29 116,68 95,4 105.43 105.38 117,45 98,7 105.43 105.43 117,8 99,9 105.42 105.43 119,37 98,6 105.52 105.42 118,9 100,3 105.69 105.52 119,05 100,2 105.72 105.69 120,46 100,4 105.74 105.72 120,99 101,4 105.74 105.74 119,86 103 105.74 105.74 120,18 109,1 105.95 105.74 119,81 111,4 106.17 105.95 120,15 114,1 106.34 106.17 119,8 121,8 106.37 106.34 120,27 127,6 106.37 106.37 120,71 129,9 106.36 106.37 121,87 128 106.44 106.36 121,87 123,5 106.29 106.44 121,92 124 106.23 106.29 123,72 127,4 106.23 106.23 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 4 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 118.427761851709 -0.0801344199419449X[t] -1.64513563400788Y1[t] + 1.62948208638638Y2[t] + 0.420973600787796t + 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) 118.427761851709 31.98194 3.703 0.000541 0.00027 X -0.0801344199419449 0.017396 -4.6064 2.9e-05 1.5e-05 Y1 -1.64513563400788 1.181023 -1.393 0.169915 0.084958 Y2 1.62948208638638 1.225912 1.3292 0.189938 0.094969 t 0.420973600787796 0.032121 13.1058 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.98962756756525 R-squared 0.979362722485113 Adjusted R-squared 0.977678046769612 F-TEST (value) 581.336047925333 F-TEST (DF numerator) 4 F-TEST (DF denominator) 49 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.730093650301397 Sum Squared Residuals 26.1188001723105 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 109.99 109.911438521240 0.0785614787604981 2 112.01 110.670650574359 1.33934942564122 3 111.96 111.292764035628 0.667235964372123 4 111.41 111.841528235971 -0.431528235971084 5 112.11 112.217184423091 -0.107184423091427 6 111.67 112.473731861046 -0.80373186104607 7 111.95 113.072852743003 -1.12285274300288 8 112.31 113.503290871561 -1.19329087156149 9 113.26 113.868438315265 -0.608438315265402 10 113.5 114.047735239172 -0.547735239172297 11 114.43 114.180128261404 0.249871738595770 12 115.02 114.770766094938 0.249233905061831 13 115.1 115.014727561795 0.085272438205065 14 117.11 115.46904948086 1.64095051914007 15 117.52 116.066854679271 1.45314532072905 16 116.1 116.431309713748 -0.331309713747685 17 116.39 116.548413395295 -0.158413395295359 18 116.01 116.494573691869 -0.484573691869246 19 116.74 117.297736605861 -0.557736605860636 20 116.68 117.470582574949 -0.790582574949178 21 117.45 117.708586694248 -0.258586694247897 22 117.8 118.049850347445 -0.249850347445452 23 119.37 118.394190309893 0.975809690106882 24 118.9 118.562210547637 0.337789452363106 25 119.05 119.218855476084 -0.168855476084341 26 120.46 119.639783942795 0.820216057204807 27 120.99 120.013212765369 0.976787234631236 28 119.86 120.305971294249 -0.445971294249443 29 120.18 119.892646450250 0.287353549750298 30 119.81 120.109572283830 -0.299572283830449 31 120.15 120.392995951999 -0.242995951998651 32 119.8 120.424592404899 -0.624592404898929 33 120.27 120.429670832615 -0.159670832615038 34 120.71 120.682786623876 0.0272133761235505 35 121.87 121.108109950969 0.76189004903057 36 121.87 122.266817353508 -0.396817353508053 37 121.92 122.502009569407 -0.582009569407412 38 123.72 122.552757217209 1.16724278279059 39 124.38 122.957703934009 1.42229606599118 40 123.21 123.133605079102 0.0763949208978032 41 123.17 123.328904886166 -0.158904886165873 42 122.95 123.616329341807 -0.666329341807101 43 123.46 124.059645379171 -0.599645379170719 44 123.24 123.671484141343 -0.431484141343407 45 123.86 123.785511513641 0.0744884863590672 46 124.28 124.311738006466 -0.0317380064660795 47 124.78 124.681349445461 0.098650554539271 48 125.19 125.835836125314 -0.645836125314084 49 125.46 126.327413350951 -0.867413350950698 50 127.6 126.611086690335 0.988913309664724 51 127.8 126.517926586440 1.28207341356045 52 126.63 126.885732265653 -0.255732265652858 53 127.06 127.358116029897 -0.298116029897032 54 126.77 127.309240327633 -0.539240327632887 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.425351786077308 0.850703572154616 0.574648213922692 9 0.533957462090459 0.932085075819082 0.466042537909541 10 0.43239116866759 0.86478233733518 0.56760883133241 11 0.546384540636606 0.907230918726788 0.453615459363394 12 0.552827697318838 0.894344605362324 0.447172302681162 13 0.440338654031651 0.880677308063302 0.559661345968349 14 0.657196917167441 0.685606165665119 0.342803082832559 15 0.77589129041681 0.448217419166379 0.224108709583190 16 0.793492943582606 0.413014112834788 0.206507056417394 17 0.73193407780455 0.536131844390899 0.268065922195450 18 0.70621688651292 0.587566226974159 0.293783113487079 19 0.698172390304744 0.603655219390513 0.301827609695256 20 0.782133266165397 0.435733467669205 0.217866733834603 21 0.779648271303431 0.440703457393138 0.220351728696569 22 0.793838263241177 0.412323473517646 0.206161736758823 23 0.763815897765433 0.472368204469134 0.236184102234567 24 0.694509937565115 0.61098012486977 0.305490062434885 25 0.649636444125778 0.700727111748444 0.350363555874222 26 0.604312072384599 0.791375855230803 0.395687927615401 27 0.593879435838559 0.812241128322883 0.406120564161441 28 0.668822504186514 0.662354991626972 0.331177495813486 29 0.616248951204143 0.767502097591713 0.383751048795857 30 0.631404689325268 0.737190621349463 0.368595310674732 31 0.604510319852827 0.790979360294346 0.395489680147173 32 0.658011208072595 0.683977583854809 0.341988791927405 33 0.623883364375427 0.752233271249145 0.376116635624573 34 0.587119177351061 0.825761645297878 0.412880822648939 35 0.509708922465747 0.980582155068506 0.490291077534253 36 0.460435256928598 0.920870513857196 0.539564743071402 37 0.554534218764589 0.890931562470821 0.445465781235411 38 0.487993485569209 0.975986971138418 0.512006514430791 39 0.609057789359538 0.781884421280925 0.390942210640462 40 0.567944469430287 0.864111061139425 0.432055530569713 41 0.530776400471926 0.938447199056148 0.469223599528074 42 0.468618624623386 0.937237249246771 0.531381375376614 43 0.396719856675648 0.793439713351296 0.603280143324352 44 0.298314707601288 0.596629415202577 0.701685292398712 45 0.36088115244975 0.7217623048995 0.63911884755025 46 0.224588411665873 0.449176823331745 0.775411588334127 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 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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