WS7-1

*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 08:14:30 -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/t1258643741j23z5s3pkh494kc.htm/, Retrieved Thu, 19 Nov 2009 16:15:54 +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/t1258643741j23z5s3pkh494kc.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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10.9 96.8 10 114.1 9.2 110.3 9.2 103.9 9.5 101.6 9.6 94.6 9.5 95.9 9.1 104.7 8.9 102.8 9 98.1 10.1 113.9 10.3 80.9 10.2 95.7 9.6 113.2 9.2 105.9 9.3 108.8 9.4 102.3 9.4 99 9.2 100.7 9 115.5 9 100.7 9 109.9 9.8 114.6 10 85.4 9.8 100.5 9.3 114.8 9 116.5 9 112.9 9.1 102 9.1 106 9.1 105.3 9.2 118.8 8.8 106.1 8.3 109.3 8.4 117.2 8.1 92.5 7.7 104.2 7.9 112.5 7.9 122.4 8 113.3 7.9 100 7.6 110.7 7.1 112.8 6.8 109.8 6.5 117.3 6.9 109.1 8.2 115.9 8.7 96 8.3 99.8 7.9 116.8 7.5 115.7 7.8 99.4 8.3 94.3 8.4 91 8.2 93.2 7.7 103.1 7.2 94.1 7.3 91.8 8.1 102.7 8.5 82.6

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 Y[t] = + 10.8290392669577 -0.0203421112501333X[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) 10.8290392669577 1.357714 7.9759 0 0 X -0.0203421112501333 0.012921 -1.5744 0.120834 0.060417 Multiple Linear Regression - Regression Statistics Multiple R 0.202448207205748 R-squared 0.0409852766008213 Adjusted R-squared 0.0244505399904907 F-TEST (value) 2.47873779708195 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.120834348025313 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.93948083549344 Sum Squared Residuals 51.1922059350481 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 10.9 8.8599228979448 2.04007710205520 2 10 8.50800437331749 1.49199562668251 3 9.2 8.585304396068 0.614695603932001 4 9.2 8.71549390806885 0.484506091931148 5 9.5 8.76228076394416 0.737719236055842 6 9.6 8.9046755426951 0.695324457304909 7 9.5 8.87823079806992 0.621769201930083 8 9.1 8.69922021906874 0.400779780931255 9 8.9 8.737870230444 0.162129769556002 10 9 8.83347815331962 0.166521846680376 11 10.1 8.51207279556752 1.58792720443248 12 10.3 9.18336246682192 1.11663753317808 13 10.2 8.88229922031994 1.31770077968006 14 9.6 8.52631227344261 1.07368772655739 15 9.2 8.67480968556858 0.525190314431415 16 9.3 8.6158175629432 0.684182437056802 17 9.4 8.74804128606906 0.651958713930936 18 9.4 8.8151702531945 0.584829746805496 19 9.2 8.78058866406928 0.419411335930721 20 9 8.4795254175673 0.520474582432695 21 9 8.78058866406928 0.219411335930722 22 9 8.59344124056805 0.406558759431948 23 9.8 8.49783331769243 1.30216668230758 24 10 9.09182296619632 0.908177033803683 25 9.8 8.7846570863193 1.01534291368070 26 9.3 8.4937648954424 0.806235104557602 27 9 8.45918330631717 0.540816693682828 28 9 8.53241490681765 0.467585093182348 29 9.1 8.7541439194441 0.345856080555895 30 9.1 8.67277547444357 0.427224525556428 31 9.1 8.68701495231866 0.412985047681335 32 9.2 8.41239645044187 0.787603549558134 33 8.8 8.67074126331856 0.129258736681442 34 8.3 8.60564650731813 -0.305646507318131 35 8.4 8.44494382844208 -0.0449438284420785 36 8.1 8.94739397632037 -0.847393976320371 37 7.7 8.70939127469381 -1.00939127469381 38 7.9 8.5405517513177 -0.640551751317705 39 7.9 8.33916484994139 -0.439164849941385 40 8 8.5242780623176 -0.524278062317599 41 7.9 8.79482814194437 -0.894828141944371 42 7.6 8.57716755156794 -0.977167551567946 43 7.1 8.53444911794267 -1.43444911794267 44 6.8 8.59547545169307 -1.79547545169307 45 6.5 8.44290961731707 -1.94290961731707 46 6.9 8.60971492956816 -1.70971492956816 47 8.2 8.47138857306725 -0.271388573067253 48 8.7 8.8761965869449 -0.176196586944905 49 8.3 8.7988965641944 -0.498896564194397 50 7.9 8.45308067294213 -0.553080672942132 51 7.5 8.47545699531728 -0.975456995317279 52 7.8 8.80703340869445 -1.00703340869445 53 8.3 8.91077817607013 -0.61077817607013 54 8.4 8.97790714319557 -0.57790714319557 55 8.2 8.93315449844528 -0.733154498445278 56 7.7 8.73176759706896 -1.03176759706896 57 7.2 8.91484659832016 -1.71484659832016 58 7.3 8.96163345419546 -1.66163345419546 59 8.1 8.73990444156901 -0.639904441569011 60 8.5 9.1487808776967 -0.64878087769669 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.43198430105727 0.86396860211454 0.56801569894273 6 0.321057440516254 0.642114881032509 0.678942559483746 7 0.220058190910954 0.440116381821907 0.779941809089046 8 0.165711586529692 0.331423173059384 0.834288413470308 9 0.145678829256078 0.291357658512157 0.854321170743922 10 0.116031976218238 0.232063952436476 0.883968023781762 11 0.116775852771359 0.233551705542718 0.883224147228641 12 0.108543273451754 0.217086546903507 0.891456726548246 13 0.102706645955131 0.205413291910263 0.897293354044869 14 0.0762472830498628 0.152494566099726 0.923752716950137 15 0.0567598478856049 0.113519695771210 0.943240152114395 16 0.0397232298844685 0.079446459768937 0.960276770115531 17 0.0281008462408832 0.0562016924817664 0.971899153759117 18 0.0203083066190012 0.0406166132380023 0.979691693380999 19 0.0156440400901734 0.0312880801803469 0.984355959909827 20 0.0110771852747621 0.0221543705495243 0.988922814725238 21 0.00981713728859065 0.0196342745771813 0.99018286271141 22 0.00724389189430482 0.0144877837886096 0.992756108105695 23 0.0110442450703891 0.0220884901407782 0.988955754929611 24 0.0136154667553219 0.0272309335106438 0.986384533244678 25 0.0197773358023484 0.0395546716046968 0.980222664197652 26 0.0204090173198993 0.0408180346397987 0.9795909826801 27 0.0197925148848056 0.0395850297696113 0.980207485115194 28 0.0210095574073311 0.0420191148146621 0.97899044259267 29 0.0259530070612563 0.0519060141225126 0.974046992938744 30 0.0338710591591558 0.0677421183183117 0.966128940840844 31 0.0490376066152185 0.098075213230437 0.950962393384782 32 0.115141120693453 0.230282241386907 0.884858879306547 33 0.181935065919368 0.363870131838737 0.818064934080632 34 0.276306838315366 0.552613676630732 0.723693161684634 35 0.374157193746958 0.748314387493915 0.625842806253042 36 0.565099194587236 0.869801610825528 0.434900805412764 37 0.706953428782736 0.586093142434529 0.293046571217264 38 0.74230808463829 0.51538383072342 0.25769191536171 39 0.772654047077807 0.454691905844385 0.227345952922193 40 0.794496837198365 0.411006325603271 0.205503162801635 41 0.812621466104443 0.374757067791114 0.187378533895557 42 0.816613772833913 0.366772454332174 0.183386227166087 43 0.85090229611808 0.298195407763841 0.149097703881920 44 0.923722807981864 0.152554384036272 0.0762771920181362 45 0.974916565299462 0.0501668694010753 0.0250834347005377 46 0.99182248707922 0.0163550258415594 0.00817751292077971 47 0.98765403080771 0.0246919383845804 0.0123459691922902 48 0.98992376135341 0.0201524772931792 0.0100762386465896 49 0.985192060116171 0.0296158797676576 0.0148079398838288 50 0.975495460805338 0.0490090783893244 0.0245045391946622 51 0.951960869131766 0.096078261736467 0.0480391308682335 52 0.914162031163427 0.171675937673145 0.0858379688365726 53 0.868272750435534 0.263454499128931 0.131727249564466 54 0.812498209883744 0.375003580232512 0.187501790116256 55 0.709722347300387 0.580555305399227 0.290277652699613 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 16 0.313725490196078 NOK 10% type I error level 23 0.450980392156863 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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