Workshop 7: Multiple Regression Analysis

*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, 20 Nov 2009 03:56:15 -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/20/t1258715142g91xm14skmhwvun.htm/, Retrieved Fri, 20 Nov 2009 12:05:54 +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/20/t1258715142g91xm14skmhwvun.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:

ETSHWW7(1)

Dataseries X:

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1,43 0,51 1,43 0,51 1,43 0,51 1,43 0,51 1,43 0,52 1,43 0,52 1,44 0,52 1,48 0,53 1,48 0,53 1,48 0,52 1,48 0,52 1,48 0,52 1,48 0,52 1,48 0,52 1,48 0,52 1,48 0,52 1,48 0,52 1,48 0,52 1,48 0,52 1,48 0,53 1,48 0,53 1,48 0,53 1,48 0,54 1,48 0,54 1,48 0,54 1,48 0,54 1,48 0,54 1,48 0,54 1,48 0,54 1,48 0,54 1,48 0,54 1,48 0,54 1,48 0,53 1,48 0,53 1,48 0,53 1,48 0,53 1,48 0,53 1,57 0,54 1,58 0,55 1,58 0,55 1,58 0,55 1,58 0,55 1,59 0,55 1,6 0,55 1,6 0,55 1,61 0,55 1,61 0,56 1,61 0,56 1,62 0,56 1,63 0,56 1,63 0,56 1,64 0,55 1,64 0,56 1,64 0,55 1,64 0,55 1,64 0,56 1,65 0,55 1,65 0,55 1,65 0,55 1,65 0,55

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Broodprijs[t] = + 0.864897321038386 + 1.05488984057064Bakmeelprijs[t] + 0.00313560599612924t + 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) 0.864897321038386 0.300891 2.8744 0.00568 0.00284 Bakmeelprijs 1.05488984057064 0.586451 1.7988 0.077351 0.038675 t 0.00313560599612924 0.000508 6.1734 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.925793510774895 R-squared 0.857093624592906 Adjusted R-squared 0.852079365806692 F-TEST (value) 170.931270430117 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0288897766764698 Sum Squared Residuals 0.0475732941957291 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.43 1.40602674572555 0.0239732542744507 2 1.43 1.40916235172167 0.0208376482783286 3 1.43 1.4122979577178 0.0177020422821997 4 1.43 1.41543356371393 0.0145664362860704 5 1.43 1.42911806811577 0.000881931884234765 6 1.43 1.43225367411189 -0.00225367411189447 7 1.44 1.43538928010802 0.0046107198919763 8 1.48 1.44907378450986 0.0309262154901407 9 1.48 1.45220939050599 0.0277906094940115 10 1.48 1.44479609809641 0.0352039019035886 11 1.48 1.44793170409254 0.0320682959074594 12 1.48 1.45106731008867 0.0289326899113301 13 1.48 1.4542029160848 0.0257970839152009 14 1.48 1.45733852208093 0.0226614779190717 15 1.48 1.46047412807706 0.0195258719229424 16 1.48 1.46360973407319 0.0163902659268132 17 1.48 1.46674534006932 0.0132546599306839 18 1.48 1.46988094606545 0.0101190539345547 19 1.48 1.47301655206157 0.00698344793842546 20 1.48 1.48670105646341 -0.00670105646341016 21 1.48 1.48983666245954 -0.0098366624595394 22 1.48 1.49297226845567 -0.0129722684556686 23 1.48 1.50665677285750 -0.0266567728575043 24 1.48 1.50979237885363 -0.0297923788536335 25 1.48 1.51292798484976 -0.0329279848497627 26 1.48 1.51606359084589 -0.036063590845892 27 1.48 1.51919919684202 -0.0391991968420212 28 1.48 1.52233480283815 -0.0423348028381505 29 1.48 1.52547040883428 -0.0454704088342797 30 1.48 1.52860601483041 -0.0486060148304089 31 1.48 1.53174162082654 -0.0517416208265382 32 1.48 1.53487722682267 -0.0548772268226674 33 1.48 1.52746393441309 -0.0474639344130903 34 1.48 1.53059954040922 -0.0505995404092195 35 1.48 1.53373514640535 -0.0537351464053487 36 1.48 1.53687075240148 -0.056870752401478 37 1.48 1.54000635839761 -0.0600063583976072 38 1.57 1.55369086279944 0.0163091372005572 39 1.58 1.56737536720128 0.0126246327987216 40 1.58 1.57051097319741 0.00948902680259236 41 1.58 1.57364657919354 0.00635342080646312 42 1.58 1.57678218518967 0.00321781481033388 43 1.59 1.57991779118580 0.0100822088142047 44 1.6 1.58305339718192 0.0169466028180754 45 1.6 1.58618900317805 0.0138109968219462 46 1.61 1.58932460917418 0.0206753908258170 47 1.61 1.60300911357602 0.00699088642398134 48 1.61 1.60614471957215 0.0038552804278521 49 1.62 1.60928032556828 0.0107196744317229 50 1.63 1.61241593156441 0.0175840684355934 51 1.63 1.61555153756054 0.0144484624394642 52 1.64 1.60813824515096 0.0318617548490413 53 1.64 1.62182274955279 0.0181772504472057 54 1.64 1.61440945714322 0.0255905428567828 55 1.64 1.61754506313935 0.0224549368606536 56 1.64 1.63122956754118 0.008770432458818 57 1.65 1.62381627513160 0.0261837248683951 58 1.65 1.62695188112773 0.0230481188722659 59 1.65 1.63008748712386 0.0199125128761366 60 1.65 1.63322309311999 0.0167769068800074 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 3.25334993633869e-43 6.50669987267737e-43 1 7 0.000425956702821977 0.000851913405643954 0.999574043297178 8 0.0278302241393856 0.0556604482787712 0.972169775860614 9 0.0190924313193106 0.0381848626386212 0.98090756868069 10 0.0249896512408090 0.0499793024816179 0.975010348759191 11 0.0135815303389378 0.0271630606778756 0.986418469661062 12 0.00723541537675337 0.0144708307535067 0.992764584623247 13 0.00425414365840690 0.00850828731681379 0.995745856341593 14 0.00289348064355410 0.00578696128710819 0.997106519356446 15 0.0023341891101826 0.0046683782203652 0.997665810889817 16 0.00228662463667483 0.00457324927334966 0.997713375363325 17 0.00283342635475085 0.0056668527095017 0.99716657364525 18 0.00483582020175803 0.00967164040351606 0.995164179798242 19 0.0137155385609565 0.027431077121913 0.986284461439044 20 0.0541010147206224 0.108202029441245 0.945898985279378 21 0.141398146712376 0.282796293424752 0.858601853287624 22 0.316334252659234 0.632668505318468 0.683665747340766 23 0.411894412483112 0.823788824966224 0.588105587516888 24 0.445237560409165 0.89047512081833 0.554762439590835 25 0.448441486904224 0.896882973808449 0.551558513095776 26 0.433266150978375 0.86653230195675 0.566733849021625 27 0.406642075520650 0.813284151041299 0.59335792447935 28 0.375062908760712 0.750125817521423 0.624937091239288 29 0.346038674725471 0.692077349450941 0.653961325274529 30 0.328963306286164 0.657926612572327 0.671036693713836 31 0.337681193956894 0.675362387913787 0.662318806043106 32 0.399572723745708 0.799145447491415 0.600427276254292 33 0.360618827627787 0.721237655255573 0.639381172372213 34 0.331233509551164 0.662467019102328 0.668766490448836 35 0.342540568405555 0.685081136811109 0.657459431594446 36 0.504438967038685 0.99112206592263 0.495561032961315 37 0.999981999858044 3.6000283911358e-05 1.8000141955679e-05 38 0.999999482860813 1.03427837411307e-06 5.17139187056537e-07 39 0.999999879249345 2.41501309885836e-07 1.20750654942918e-07 40 0.999999885475569 2.29048861906836e-07 1.14524430953418e-07 41 0.999999859437624 2.81124752655335e-07 1.40562376327668e-07 42 0.999999937496537 1.25006925936110e-07 6.25034629680549e-08 43 0.99999992780395 1.44392098260608e-07 7.21960491303039e-08 44 0.99999981323112 3.73537761345183e-07 1.86768880672591e-07 45 0.99999976155545 4.76889098146584e-07 2.38444549073292e-07 46 0.999999383808759 1.23238248281469e-06 6.16191241407344e-07 47 0.999998586362513 2.82727497441699e-06 1.41363748720849e-06 48 0.999999801403076 3.97193847041453e-07 1.98596923520727e-07 49 0.999999897809436 2.04381128806766e-07 1.02190564403383e-07 50 0.999999054194992 1.89161001556534e-06 9.45805007782672e-07 51 0.999995677448863 8.64510227360306e-06 4.32255113680153e-06 52 0.999966060781594 6.78784368113524e-05 3.39392184056762e-05 53 0.99985925144494 0.000281497110119654 0.000140748555059827 54 0.998472078233791 0.00305584353241759 0.00152792176620880 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 26 0.530612244897959 NOK 5% type I error level 31 0.63265306122449 NOK 10% type I error level 32 0.653061224489796 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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