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: Fri, 18 Dec 2009 09:12:21 -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/Dec/18/t1261152904qsqvom8oms5uqjc.htm/, Retrieved Fri, 18 Dec 2009 17:15:16 +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/Dec/18/t1261152904qsqvom8oms5uqjc.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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113 14,3 110 14,2 107 15,9 103 15,3 98 15,5 98 15,1 137 15 148 12,1 147 15,8 139 16,9 130 15,1 128 13,7 127 14,8 123 14,7 118 16 114 15,4 108 15 111 15,5 151 15,1 159 11,7 158 16,3 148 16,7 138 15 137 14,9 136 14,6 133 15,3 126 17,9 120 16,4 114 15,4 116 17,9 153 15,9 162 13,9 161 17,8 149 17,9 139 17,4 135 16,7 130 16 127 16,6 122 19,1 117 17,8 112 17,2 113 18,6 149 16,3 157 15,1 157 19,2 147 17,7 137 19,1 132 18 125 17,5 123 17,8 117 21,1 114 17,2 111 19,4 112 19,8 144 17,6 150 16,2 149 19,5 134 19,9 123 20 116 17,3

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 WK<25j[t] = + 144.986686095178 -0.89516816235565ExpBe[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) 144.986686095178 19.355018 7.4909 0 0 ExpBe -0.89516816235565 1.163893 -0.7691 0.444945 0.222473 Multiple Linear Regression - Regression Statistics Multiple R 0.100478759942464 R-squared 0.0100959811995754 Adjusted R-squared -0.00697132946939738 F-TEST (value) 0.591539076975329 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.444945412800618 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 17.3107196211353 Sum Squared Residuals 17380.3388004903 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 113 132.185781373492 -19.1857813734922 2 110 132.275298189728 -22.2752981897278 3 107 130.753512313723 -23.7535123137232 4 103 131.290613211137 -28.2906132111366 5 98 131.111579578665 -33.1115795786655 6 98 131.469646843608 -33.4696468436078 7 137 131.559163659843 5.44083634015667 8 148 134.155151330675 13.8448486693253 9 147 130.843029129959 16.1569708700412 10 139 129.858344151368 9.1416558486324 11 130 131.469646843608 -1.46964684360776 12 128 132.722882270906 -4.72288227090567 13 127 131.738197292314 -4.73819729231446 14 123 131.82771410855 -8.82771410855002 15 118 130.663995497488 -12.6639954974877 16 114 131.201096394901 -17.2010963949011 17 108 131.559163659843 -23.5591636598433 18 111 131.111579578665 -20.1115795786655 19 151 131.469646843608 19.5303531563922 20 159 134.513218595617 24.486781404383 21 158 130.395445048781 27.604554951219 22 148 130.037377783839 17.9626222161613 23 138 131.559163659843 6.44083634015667 24 137 131.648680476079 5.35131952392111 25 136 131.917230924786 4.08276907521441 26 133 131.290613211137 1.70938678886337 27 126 128.963175989012 -2.96317598901195 28 120 130.305928232545 -10.3059282325454 29 114 131.201096394901 -17.2010963949011 30 116 128.963175989012 -12.9631759890119 31 153 130.753512313723 22.2464876862768 32 162 132.543848638435 29.4561513615655 33 161 129.052692805248 31.9473071947525 34 149 128.963175989012 20.0368240109880 35 139 129.410760070190 9.58923992981023 36 135 130.037377783839 4.96262221616127 37 130 130.663995497488 -0.66399549748768 38 127 130.126894600074 -3.12689460007429 39 122 127.888974194185 -5.88897419418517 40 117 129.052692805248 -12.0526928052475 41 112 129.589793702661 -17.5897937026609 42 113 128.336558275363 -15.336558275363 43 149 130.395445048781 18.6045549512190 44 157 131.469646843608 25.5303531563922 45 157 127.799457377950 29.2005426220504 46 147 129.142209621483 17.8577903785169 47 137 127.888974194185 9.11102580581483 48 132 128.873659172776 3.12634082722362 49 125 129.321243253954 -4.32124325395421 50 123 129.052692805248 -6.05269280524751 51 117 126.098637869474 -9.09863786947387 52 114 129.589793702661 -15.5897937026609 53 111 127.620423745478 -16.6204237454785 54 112 127.262356480536 -15.2623564805362 55 144 129.231726437719 14.7682735622814 56 150 130.484961865017 19.5150381349834 57 149 127.530906929243 21.4690930707571 58 134 127.172839664301 6.82716033569935 59 123 127.083322848065 -4.08332284806508 60 116 129.500276886425 -13.5002768864253 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0381795061196366 0.0763590122392732 0.961820493880363 6 0.0310077537428022 0.0620155074856045 0.968992246257198 7 0.469908876153145 0.93981775230629 0.530091123846855 8 0.383061490479678 0.766122980959357 0.616938509520322 9 0.844265560051986 0.311468879896029 0.155734439948014 10 0.904907577176669 0.190184845646663 0.0950924228233313 11 0.867444969635386 0.265110060729227 0.132555030364614 12 0.814458691339773 0.371082617320454 0.185541308660227 13 0.754121059636899 0.491757880726202 0.245878940363101 14 0.690277336499956 0.619445327000088 0.309722663500044 15 0.628706635638769 0.742586728722462 0.371293364361231 16 0.597302108681462 0.805395782637076 0.402697891318538 17 0.643935670924609 0.712128658150782 0.356064329075391 18 0.660061625837718 0.679876748324564 0.339938374162282 19 0.769831213693297 0.460337572613406 0.230168786306703 20 0.774391857754078 0.451216284491843 0.225608142245922 21 0.917556912662216 0.164886174675569 0.0824430873377845 22 0.936648056527826 0.126703886944348 0.0633519434721742 23 0.915697365549755 0.168605268900491 0.0843026344502455 24 0.888221867239413 0.223556265521174 0.111778132760587 25 0.853842282324067 0.292315435351867 0.146157717675933 26 0.813167690261477 0.373664619477046 0.186832309738523 27 0.763326110178082 0.473347779643836 0.236673889821918 28 0.736493722894809 0.527012554210383 0.263506277105191 29 0.78767046801986 0.424659063960279 0.212329531980140 30 0.766522535869982 0.466954928260036 0.233477464130018 31 0.791263613406888 0.417472773186225 0.208736386593112 32 0.826766793772205 0.346466412455589 0.173233206227795 33 0.926911652520695 0.146176694958609 0.0730883474793046 34 0.934868610766016 0.130262778467968 0.0651313892339841 35 0.912484756268426 0.175030487463149 0.0875152437315743 36 0.877255009534209 0.245489980931582 0.122744990465791 37 0.837651717414504 0.324696565170993 0.162348282585496 38 0.794822173339261 0.410355653321478 0.205177826660739 39 0.738662018183318 0.522675963633364 0.261337981816682 40 0.718408183192274 0.563183633615453 0.281591816807726 41 0.7683475792426 0.463304841514799 0.231652420757399 42 0.771598093961841 0.456803812076319 0.228401906038159 43 0.737380743197964 0.525238513604072 0.262619256802036 44 0.744332600907101 0.511334798185798 0.255667399092899 45 0.874919994457929 0.250160011084141 0.125080005542071 46 0.87680709357556 0.246385812848881 0.123192906424441 47 0.845209735675864 0.309580528648272 0.154790264324136 48 0.777923502419209 0.444152995161582 0.222076497580791 49 0.697591800893872 0.604816398212255 0.302408199106128 50 0.610585256695129 0.778829486609742 0.389414743304871 51 0.504172696600286 0.99165460679943 0.495827303399715 52 0.535797344694117 0.928405310611767 0.464202655305883 53 0.529408777123264 0.941182445753473 0.470591222876736 54 0.54644532721174 0.90710934557652 0.45355467278826 55 0.409422200078793 0.818844400157586 0.590577799921207 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 2 0.0392156862745098 OK

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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