WS 7 Multiple Regression (model 3)

*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, 21 Nov 2009 00:18:18 -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/21/t12587879523vnjwac9z6zq8c3.htm/, Retrieved Sat, 21 Nov 2009 08:19:24 +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/21/t12587879523vnjwac9z6zq8c3.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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83.4 108.8 113.6 128.4 112.9 121.1 104 119.5 109.9 128.7 99 108.7 106.3 105.5 128.9 119.8 111.1 111.3 102.9 110.6 130 120.1 87 97.5 87.5 107.7 117.6 127.3 103.4 117.2 110.8 119.8 112.6 116.2 102.5 111 112.4 112.4 135.6 130.6 105.1 109.1 127.7 118.8 137 123.9 91 101.6 90.5 112.8 122.4 128 123.3 129.6 124.3 125.8 120 119.5 118.1 115.7 119 113.6 142.7 129.7 123.6 112 129.6 116.8 151.6 127 110.4 112.1 99.2 114.2 130.5 121.1 136.2 131.6 129.7 125 128 120.4 121.6 117.7 135.8 117.5 143.8 120.6 147.5 127.5 136.2 112.3 156.6 124.5 123.3 115.2 104.5 104.7 139.8 130.9 136.5 129.2 112.1 113.5 118.5 125.6 94.4 107.6 102.3 107 111.4 121.6 99.2 110.7 87.8 106.3 115.8 118.6 79.7 104.6

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation inv[t] = -98.6031754376958 + 1.81249154252463cons[t] -10.1515734108020M1[t] -10.2323024500268M2[t] -10.1369413355362M3[t] -7.44036083710648M4[t] -8.40747637998386M5[t] -1.19343749233295M6[t] + 8.4281775125963M7[t] + 1.59241261367585M8[t] + 5.03144811833661M9[t] + 6.55181126262129M10[t] + 9.91851760828455M11[t] + 0.122127045043892t + 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) -98.6031754376958 29.989951 -3.2879 0.001939 0.00097 cons 1.81249154252463 0.281335 6.4425 0 0 M1 -10.1515734108020 6.604696 -1.537 0.131138 0.065569 M2 -10.2323024500268 8.846138 -1.1567 0.253368 0.126684 M3 -10.1369413355362 8.570762 -1.1827 0.242991 0.121496 M4 -7.44036083710648 7.712021 -0.9648 0.339703 0.169851 M5 -8.40747637998386 7.913696 -1.0624 0.293602 0.146801 M6 -1.19343749233295 6.703974 -0.178 0.859489 0.429745 M7 8.4281775125963 6.633268 1.2706 0.210263 0.105131 M8 1.59241261367585 8.284602 0.1922 0.848421 0.42421 M9 5.03144811833661 6.846958 0.7348 0.466164 0.233082 M10 6.55181126262129 6.741149 0.9719 0.336177 0.168088 M11 9.91851760828455 7.980517 1.2428 0.220224 0.110112 t 0.122127045043892 0.078371 1.5583 0.126011 0.063005 Multiple Linear Regression - Regression Statistics Multiple R 0.86276085439231 R-squared 0.744356291871749 Adjusted R-squared 0.67210915696594 F-TEST (value) 10.3029177951787 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 1.11456865958814e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 10.2131534985726 Sum Squared Residuals 4798.19120172865 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 83.4 88.5664580232261 -5.16645802322608 2 113.6 124.132690262528 -10.5326902625276 3 112.9 111.118990161632 1.7810098383677 4 104 111.037711237067 -7.03771123706654 5 109.9 126.867644930460 -16.9676449304596 6 99 97.9539800126619 1.04601998733814 7 106.3 101.897749126556 4.40225087344379 8 128.9 121.102740330782 7.79725966921819 9 111.1 109.257724769027 1.84227523097287 10 102.9 109.631470878588 -6.73147087858845 11 130 130.338973923280 -0.338973923279582 12 87 79.5802744989823 7.41972550101768 13 87.5 88.0382418669755 -0.538241866975468 14 117.6 123.604474106277 -6.00447410627726 15 103.4 105.515797686313 -2.11579768631297 16 110.8 113.046983240351 -2.24698324035063 17 112.6 105.677025189428 6.92297481057151 18 102.5 103.588235100995 -1.08823510099521 19 112.4 115.869465310503 -3.46946531050285 20 135.6 142.143173530575 -6.54317353057452 21 105.1 106.735767916000 -1.63576791599965 22 127.7 125.959426067817 1.74057393218288 23 137 138.6919663254 -1.6919663253999 24 91 88.47701436386 2.52298563614 25 90.5 98.7474732743778 -8.24747327437777 26 122.4 126.338742726571 -3.93874272657119 27 123.3 129.456217354145 -6.15621735414507 28 124.3 125.387457036025 -1.08745703602511 29 120 113.123771820286 6.87622817971354 30 118.1 113.572469891388 4.52753010861231 31 119 119.509979702059 -0.509979702059099 32 142.7 141.977455682829 0.722544317170941 33 123.6 113.457517929848 10.1424820701522 34 129.6 123.799967523295 5.80003247670542 35 151.6 145.776214647753 5.82378535224705 36 110.4 108.973700100895 1.42629989910470 37 99.2 102.750485974439 -3.55048597443897 38 130.5 115.298075623678 15.2019243763221 39 136.2 134.546724979721 1.65327502027895 40 129.7 125.402988342532 4.29701165746787 41 128 116.220538749085 11.7794612509147 42 121.6 118.662977516964 2.93702248303635 43 135.8 128.044221258432 7.75577874156814 44 143.8 126.949307186382 16.8506928136184 45 147.5 143.016661379506 4.48333862049376 46 136.2 117.109280122460 19.0907198775395 47 156.6 142.710510331968 13.8894896680319 48 123.3 116.057948423248 7.24205157675162 49 104.5 86.9973408609817 17.5026591390183 50 139.8 134.526017280946 5.27398271905397 51 136.5 131.662269818189 4.83773018181137 52 112.1 106.024860144026 6.0751398559744 53 118.5 127.11101931074 -8.6110193107401 54 94.4 101.822337477992 -7.42233747799158 55 102.3 110.47858460245 -8.17858460244998 56 111.4 130.227323269433 -18.8273232694330 57 99.2 114.032328005619 -14.8323280056192 58 87.8 107.699855407839 -19.8998554078394 59 115.8 133.482334771599 -17.6823347715995 60 79.7 98.311062613014 -18.611062613014 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.110598995204015 0.221197990408031 0.889401004795985 18 0.0394088564873929 0.0788177129747858 0.960591143512607 19 0.0130672167230793 0.0261344334461586 0.98693278327692 20 0.00412495730458185 0.0082499146091637 0.995875042695418 21 0.00312190383405896 0.00624380766811792 0.99687809616594 22 0.0171937578253876 0.0343875156507752 0.982806242174612 23 0.00755196325263573 0.0151039265052715 0.992448036747364 24 0.00309372504350047 0.00618745008700095 0.9969062749565 25 0.00197674120717798 0.00395348241435597 0.998023258792822 26 0.00129096028039929 0.00258192056079859 0.9987090397196 27 0.000810087660193326 0.00162017532038665 0.999189912339807 28 0.000796598487466947 0.00159319697493389 0.999203401512533 29 0.000411549687191279 0.000823099374382558 0.999588450312809 30 0.000246841308631414 0.000493682617262828 0.999753158691369 31 0.000132282220033331 0.000264564440066663 0.999867717779967 32 6.61612990178008e-05 0.000132322598035602 0.999933838700982 33 4.02360403236123e-05 8.04720806472245e-05 0.999959763959676 34 2.34080956225199e-05 4.68161912450399e-05 0.999976591904377 35 1.68394153099284e-05 3.36788306198569e-05 0.99998316058469 36 1.20754593279287e-05 2.41509186558574e-05 0.999987924540672 37 0.000257666659385439 0.000515333318770879 0.999742333340615 38 0.000272608650448351 0.000545217300896702 0.999727391349552 39 0.00337892349603056 0.00675784699206112 0.99662107650397 40 0.0292190563471465 0.058438112694293 0.970780943652854 41 0.094070711854913 0.188141423709826 0.905929288145087 42 0.303237790261706 0.606475580523413 0.696762209738294 43 0.589993326310043 0.820013347379914 0.410006673689957 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 18 0.666666666666667 NOK 5% type I error level 21 0.777777777777778 NOK 10% type I error level 23 0.851851851851852 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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