*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: Tue, 21 Dec 2010 12:14:46 +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/21/t129293357243h3hth1dq8roeq.htm/, Retrieved Tue, 21 Dec 2010 13:13:03 +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/21/t129293357243h3hth1dq8roeq.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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User-defined keywords:

Dataseries X:

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112.3 112.9 88.7 105.1 117.3 130.5 94.6 114.9 111.1 137.9 98.7 106.4 102.2 115 84.2 104.5 104.3 116.8 87.7 121.6 122.9 140.9 103.3 141.4 107.6 120.7 88.2 99 121.3 134.2 93.4 126.7 131.5 147.3 106.3 134.1 89 112.4 73.1 81.3 104.4 107.1 78.6 88.6 128.9 128.4 101.6 132.7 135.9 137.7 101.4 132.9 133.3 135 98.5 134.4 121.3 151 99 103.7 120.5 137.4 89.5 119.7 120.4 132.4 83.5 115 137.9 161.3 97.4 132.9 126.1 139.8 87.8 108.5 133.2 146 90.4 113.9 151.1 166.5 101.6 142 105 143.3 80 97.7 119 121 81.7 92.2 140.4 152.6 96.4 128.8 156.6 154.4 110.2 134.9 137.1 154.6 101.1 128.2 122.7 158 89.3 114.8 125.8 142.6 90 117.9 139.3 153.4 95.4 119.1 134.9 163.4 100.3 120.7 149.2 167.3 99.5 129.1 132.3 154.8 93.9 117.6 149 165.7 100.6 129.2 117.2 144.7 84.7 100 119.6 120.9 81.6 87 152 152.8 109 128 149.4 160.2 99 127.7 127.3 128.3 81.1 93.4 114.1 150.5 81.8 84.1 102.1 117 66.5 71.7 107.7 116 66.4 83.2 104.4 133.3 86.3 89.1 102.1 116.4 73.6 79.6 96 104 71.5 62.8 109.3 126.6 87.2 95.1 90 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 6 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation X4[t] = -53.4834533180354 + 0.450126904808100X1[t] + 0.063803933235003X2[t] + 1.09234278871000X3[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) -53.4834533180354 9.682963 -5.5235 1e-06 0 X1 0.450126904808100 0.147473 3.0523 0.00347 0.001735 X2 0.063803933235003 0.119003 0.5362 0.593977 0.296988 X3 1.09234278871000 0.182007 6.0016 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.918689589195073 R-squared 0.843990561295412 Adjusted R-squared 0.83563291279338 F-TEST (value) 100.984213572785 F-TEST (DF numerator) 3 F-TEST (DF denominator) 56 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 9.16960909056846 Sum Squared Residuals 4708.5769289348 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 105.1 101.160067512723 3.93993248727661 2 114.9 110.978473715089 3.921526284911 3 106.4 113.138441444929 -6.73844144492877 4 104.5 91.8322314847601 12.6677685152399 5 121.6 96.7155448251651 24.8844551748349 6 141.4 123.666127549435 17.7338724505646 7 99 98.9959703450033 0.00402965499666902 8 126.7 111.704244540839 14.9957554591611 9 134.1 131.222592469619 2.8774075303809 10 81.3 73.5996611602011 7.70033883979893 11 88.6 86.2013399860053 2.39866001399467 12 132.7 123.712357072039 8.98764292796053 13 132.9 127.238153427040 5.66184657296031 14 134.4 122.727758767545 11.6722412324549 15 103.7 118.893270235963 -15.1932702359629 16 119.7 107.288178727375 12.4118212726246 17 115 100.370089638460 14.6299103615405 18 132.9 125.274808906162 7.62519109383806 19 108.5 108.105036093258 0.394963906742258 20 113.9 114.536612754098 -0.636612754098283 21 142 136.136104215033 5.86389578496711 22 97.7 90.3103984161913 7.3896015838087 23 92.2 97.0463301131712 -4.84633011317115 24 128.8 124.752689160328 4.04731083967234 25 134.9 147.23392258224 -12.3339225822400 26 128.2 128.528889347868 -0.328889347867957 27 114.8 109.374350384852 5.42564961514774 28 117.9 110.551803170035 7.34819682996467 29 119.1 123.216249922917 -4.11624992291676 30 120.7 127.226210538790 -6.52621053879015 31 129.1 133.037986386194 -3.93798638619451 32 117.6 118.516172912724 -0.916172912724057 33 129.2 134.047451779638 -4.84745177963789 34 100 101.025283268316 -1.02528326831615 35 87 97.2007915838615 -10.2007915838615 36 128 143.750441180495 -15.7504411804947 37 127.7 132.128832446833 -4.42883244683259 38 93.4 100.592746462468 -7.19274646246789 39 84.1 96.832158588915 -12.7321585889150 40 71.7 72.5803593005822 -0.880359300582165 41 83.2 74.9280317554015 8.27196824459846 42 89.1 96.2840425098294 -7.18404250982944 43 79.6 80.2977107404822 -0.697710740482198 44 62.8 74.4668479927477 -11.6668479927477 45 95.1 99.0452865005536 -3.94528650055362 46 63.6 64.2853376149886 -0.685337614988576 47 61.4 65.7524951868977 -4.35249518689767 48 98.2 108.446580011185 -10.2465800111855 49 95.3 99.457839809406 -4.15783980940607 50 81.5 89.1902259939419 -7.69022599394189 51 85.5 92.6913828324376 -7.19138283243755 52 71.1 67.2515462030641 3.84845379693591 53 78.1 78.3506905368065 -0.250690536806491 54 103 108.274729504575 -5.27472950457533 55 86 88.1542204329596 -2.15422043295956 56 86.2 80.9603317501495 5.23966824985051 57 105.7 114.151812885361 -8.4518128853611 58 57.2 76.8567102108885 -19.6567102108884 59 73.7 80.6678494263517 -6.96784942635167 60 120.5 107.236195684811 13.2638043151894 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.965227706648518 0.0695445867029647 0.0347722933514823 8 0.96046836699205 0.0790632660158988 0.0395316330079494 9 0.933499418969532 0.133001162060937 0.0665005810304684 10 0.895760568863957 0.208478862272087 0.104239431136043 11 0.88020238712994 0.239595225740119 0.119797612870060 12 0.848182551845218 0.303634896309565 0.151817448154783 13 0.802626367692368 0.394747264615265 0.197373632307632 14 0.805454412833903 0.389091174332195 0.194545587166097 15 0.89876110603778 0.202477787924439 0.101238893962220 16 0.932527804235666 0.134944391528668 0.0674721957643341 17 0.95217940020073 0.0956411995985407 0.0478205997992703 18 0.943692765816802 0.112614468366396 0.0563072341831981 19 0.935679835492254 0.128640329015493 0.0643201645077463 20 0.921085477787687 0.157829044424626 0.078914522212313 21 0.910771991723522 0.178456016552957 0.0892280082764785 22 0.901358430441244 0.197283139117512 0.0986415695587562 23 0.924674871926313 0.150650256147374 0.0753251280736869 24 0.911521609286172 0.176956781427656 0.0884783907138278 25 0.938881880626295 0.12223623874741 0.0611181193737050 26 0.918608221988502 0.162783556022996 0.0813917780114978 27 0.909146019387538 0.181707961224924 0.090853980612462 28 0.922051515773656 0.155896968452689 0.0779484842263445 29 0.89917058620714 0.201658827585719 0.100829413792859 30 0.874028579704083 0.251942840591835 0.125971420295917 31 0.83221932434701 0.335561351305979 0.167780675652989 32 0.794720320020112 0.410559359959777 0.205279679979888 33 0.742056476946423 0.515887046107154 0.257943523053577 34 0.698328533024038 0.603342933951924 0.301671466975962 35 0.75739608805241 0.485207823895179 0.242603911947590 36 0.810571640173878 0.378856719652244 0.189428359826122 37 0.749137645143707 0.501724709712585 0.250862354856293 38 0.742594942580089 0.514810114839821 0.257405057419911 39 0.7986118510328 0.402776297934399 0.201388148967200 40 0.74099630058747 0.518007398825060 0.259003699412530 41 0.707314990928214 0.585370018143572 0.292685009071786 42 0.67778265302365 0.644434693952699 0.322217346976350 43 0.599951429313564 0.800097141372871 0.400048570686436 44 0.687140535625979 0.625718928748042 0.312859464374021 45 0.615365001657926 0.769269996684148 0.384634998342074 46 0.525103596451175 0.94979280709765 0.474896403548825 47 0.445335809419974 0.890671618839948 0.554664190580026 48 0.411508410643275 0.82301682128655 0.588491589356725 49 0.342491250314523 0.684982500629047 0.657508749685477 50 0.363642184988615 0.72728436997723 0.636357815011385 51 0.274788760133635 0.549577520267269 0.725211239866365 52 0.257065184233762 0.514130368467523 0.742934815766238 53 0.273143099086323 0.546286198172646 0.726856900913677 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 3 0.0638297872340425 OK

Charts produced by software:

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Parameters (Session):

par1 = 4 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 4 ; 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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