*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: Thu, 19 Nov 2009 01:34:59 -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/19/t1258620251mtdfp2jitw02lhg.htm/, Retrieved Thu, 19 Nov 2009 09:44:23 +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/19/t1258620251mtdfp2jitw02lhg.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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2.057 0 2.058 2.077 2.053 2.085 2.076 0 2.057 2.058 2.077 2.053 2.07 0 2.076 2.057 2.058 2.077 2.062 0 2.07 2.076 2.057 2.058 2.073 0 2.062 2.07 2.076 2.057 2.061 0 2.073 2.062 2.07 2.076 2.094 0 2.061 2.073 2.062 2.07 2.067 0 2.094 2.061 2.073 2.062 2.086 0 2.067 2.094 2.061 2.073 2.276 0 2.086 2.067 2.094 2.061 2.326 0 2.276 2.086 2.067 2.094 2.349 0 2.326 2.276 2.086 2.067 2.52 0 2.349 2.326 2.276 2.086 2.628 0 2.52 2.349 2.326 2.276 2.577 0 2.628 2.52 2.349 2.326 2.698 0 2.577 2.628 2.52 2.349 2.814 0 2.698 2.577 2.628 2.52 2.968 0 2.814 2.698 2.577 2.628 3.041 0 2.968 2.814 2.698 2.577 3.278 0 3.041 2.968 2.814 2.698 3.328 0 3.278 3.041 2.968 2.814 3.5 0 3.328 3.278 3.041 2.968 3.563 0 3.5 3.328 3.278 3.041 3.569 0 3.563 3.5 3.328 3.278 3.69 0 3.569 3.563 3.5 3.328 3.819 0 3.69 3.569 3.563 3.5 3.79 0 3.819 3.69 3.569 3.563 3.956 0 3.79 3.819 3.69 3.569 4.063 0 3.956 3.79 3.819 3.69 4.047 0 4.063 3.956 3.79 3.819 4.029 0 4.047 4.063 3.956 3.79 3.941 0 4.029 4.047 4.063 3.956 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 9 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation intb[t] = + 0.210659035938168 -0.25603237044698X[t] + 1.39418813246313`Yt-1`[t] -0.279777174124115`Yt-2`[t] -0.344875777009029`Yt-3`[t] + 0.169566453810560`Yt-4`[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) 0.210659035938168 0.069803 3.0179 0.003966 0.001983 X -0.25603237044698 0.080161 -3.194 0.002407 0.001203 `Yt-1` 1.39418813246313 0.150028 9.2929 0 0 `Yt-2` -0.279777174124115 0.244782 -1.143 0.25839 0.129195 `Yt-3` -0.344875777009029 0.242766 -1.4206 0.161516 0.080758 `Yt-4` 0.169566453810560 0.134184 1.2637 0.212087 0.106044 Multiple Linear Regression - Regression Statistics Multiple R 0.995246444713004 R-squared 0.990515485713875 Adjusted R-squared 0.989585631372098 F-TEST (value) 1065.23725406385 F-TEST (DF numerator) 5 F-TEST (DF denominator) 51 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.126170925631754 Sum Squared Residuals 0.811874226213457 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2.057 2.14431710788699 -0.0873171078869878 2 2.076 2.13453554089273 -0.0585355408927277 3 2.07 2.17192712723827 -0.101927127238274 4 2.062 2.15536934528975 -0.0933693452897483 5 2.073 2.13917229705781 -0.0661722970578056 6 2.061 2.16203760119235 -0.101037601192348 7 2.094 2.14397140218063 -0.049971402180634 8 2.067 2.18818677146382 -0.121186771463823 9 2.086 2.14731478545725 -0.0613147854572474 10 2.276 2.16794264558837 0.108057354411627 11 2.326 2.442429963403 -0.116429963403002 12 2.349 2.44785077292652 -0.0988507729265203 13 2.52 2.40362360625765 0.116376393742348 14 2.628 2.65056873927755 -0.0225687392775473 15 2.577 2.75384534062766 -0.176845340627663 16 2.698 2.59745208163574 0.100547918364263 17 2.814 2.77216676122874 0.0418332387712635 18 2.968 2.93594138816444 0.0320586118355568 19 3.041 3.06781435020294 -0.0268143502029370 20 3.278 3.10701634983566 0.170983650164338 21 3.328 3.383574042501 -0.055574042500999 22 3.5 3.38791356102191 0.112086438978093 23 3.563 3.54436785307639 0.0186321469236089 24 3.569 3.60702349217487 -0.0380234921748724 25 3.69 3.54692234804481 0.143077651955193 26 3.819 3.72137870513195 0.0976212948680517 27 3.79 3.87598936807869 -0.0859893680786856 28 3.956 3.75875408648001 0.197245913519985 29 4.063 3.97433142019541 0.0886685798045928 30 4.047 4.10894200953918 -0.0619420095391831 31 4.029 3.99453203564449 0.0344679643555126 32 3.941 3.96515940723872 -0.0241594072387241 33 4.022 3.87116846370608 0.150831536293924 34 3.879 4.01221279448371 -0.133212794483706 35 4.022 3.81747881264563 0.204521187354371 36 4.028 4.01399906561455 0.0140009343854544 37 4.091 4.04540817738052 0.0455918226194789 38 3.987 4.05799812767375 -0.0709981276737537 39 4.01 3.91755534816062 0.092444651839375 40 4.007 3.95800872608748 0.0489912739125212 41 4.191 3.99394105408424 0.197058945915761 42 4.299 4.22574394791232 0.0732560520876786 43 4.273 4.32977192194817 -0.0567719219481741 44 3.82 4.19934125336763 -0.379341253367634 45 3.15 3.31296950902625 -0.162969509026250 46 2.486 2.53288246736795 -0.0468824673679502 47 1.812 1.94641225326160 -0.134412253261603 48 1.257 1.34675466261973 -0.0897546626197302 49 1.062 0.876938056343265 0.185061943656735 50 0.842 0.880201850525712 -0.0382018505257123 51 0.782 0.70515527670972 0.0768447232902804 52 0.698 0.656196361721137 0.041803638278863 53 0.358 0.598678401490608 -0.240678401490608 54 0.347 0.131543645861787 0.215456354138213 55 0.363 0.230127393647017 0.132872606352983 56 0.359 0.358526134744775 0.000473865255225039 57 0.355 0.294613986680445 0.0603860133195546 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 9 0.00366016674867997 0.00732033349735993 0.99633983325132 10 0.0510923140925298 0.102184628185060 0.94890768590747 11 0.0198462823884211 0.0396925647768421 0.980153717611579 12 0.0175675602788059 0.0351351205576118 0.982432439721194 13 0.0279969910171315 0.0559939820342631 0.972003008982868 14 0.0129068312229832 0.0258136624459664 0.987093168777017 15 0.0126125215001469 0.0252250430002939 0.987387478499853 16 0.00650987375425439 0.0130197475085088 0.993490126245746 17 0.00291846839116363 0.00583693678232726 0.997081531608836 18 0.0165622873597578 0.0331245747195156 0.983437712640242 19 0.0144926298144739 0.0289852596289478 0.985507370185526 20 0.0136885094468115 0.0273770188936230 0.986311490553188 21 0.0238407911262578 0.0476815822525156 0.976159208873742 22 0.0132665591540514 0.0265331183081028 0.986733440845949 23 0.0180411556931174 0.0360823113862348 0.981958844306883 24 0.0205701601131259 0.0411403202262519 0.979429839886874 25 0.0117667718274274 0.0235335436548548 0.988233228172573 26 0.00741577782012623 0.0148315556402525 0.992584222179874 27 0.00709729408764504 0.0141945881752901 0.992902705912355 28 0.0054273677164535 0.010854735432907 0.994572632283546 29 0.00339484425447425 0.0067896885089485 0.996605155745526 30 0.00210612792907717 0.00421225585815433 0.997893872070923 31 0.00269263984923867 0.00538527969847733 0.997307360150761 32 0.00297526266567714 0.00595052533135429 0.997024737334323 33 0.00291742985049857 0.00583485970099714 0.997082570149501 34 0.00357278721355358 0.00714557442710717 0.996427212786446 35 0.0064827257794275 0.012965451558855 0.993517274220572 36 0.00345721638470501 0.00691443276941003 0.996542783615295 37 0.00213826578085027 0.00427653156170054 0.99786173421915 38 0.00260215046107167 0.00520430092214334 0.997397849538928 39 0.00145575507167300 0.00291151014334600 0.998544244928327 40 0.000693364567499667 0.00138672913499933 0.9993066354325 41 0.00454864556076925 0.0090972911215385 0.99545135443923 42 0.0123871497835120 0.0247742995670240 0.987612850216488 43 0.0843416646619357 0.168683329323871 0.915658335338064 44 0.307355613954084 0.614711227908168 0.692644386045916 45 0.29464130399277 0.58928260798554 0.70535869600723 46 0.391546989468856 0.783093978937712 0.608453010531144 47 0.384012123163679 0.768024246327358 0.615987876836321 48 0.255133798431794 0.510267596863587 0.744866201568206 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 14 0.35 NOK 5% type I error level 32 0.8 NOK 10% type I error level 33 0.825 NOK

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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