WS 7: Multiple Regression

*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, 26 Nov 2009 08:53:55 -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/26/t12592509051pujbnjvo7tdios.htm/, Retrieved Thu, 26 Nov 2009 16:55:18 +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/26/t12592509051pujbnjvo7tdios.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:

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Invloed onderzoeken

Dataseries X:

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8.1 1.3 7.7 1.3 7.5 1.2 7.6 1.1 7.8 1.4 7.8 1.2 7.8 1.5 7.5 1.1 7.5 1.3 7.1 1.5 7.5 1.1 7.5 1.4 7.6 1.3 7.7 1.5 7.7 1.6 7.9 1.7 8.1 1.1 8.2 1.6 8.2 1.3 8.2 1.7 7.9 1.6 7.3 1.7 6.9 1.9 6.6 1.8 6.7 1.9 6.9 1.6 7.0 1.5 7.1 1.6 7.2 1.6 7.1 1.7 6.9 2.0 7.0 2.0 6.8 1.9 6.4 1.7 6.7 1.8 6.6 1.9 6.4 1.7 6.3 2.0 6.2 2.1 6.5 2.4 6.8 2.5 6.8 2.5 6.4 2.6 6.1 2.2 5.8 2.5 6.1 2.8 7.2 2.8 7.3 2.9 6.9 3.0 6.1 3.1 5.8 2.9 6.2 2.7 7.1 2.2 7.7 2.5 7.9 2.3 7.7 2.6 7.4 2.3 7.5 2.2 8.0 1.8 8.1 1.8

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 Y[t] = + 8.4683207267475 -0.682770154699912X[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) 8.4683207267475 0.25883 32.7177 0 0 X -0.682770154699912 0.131238 -5.2026 3e-06 1e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.564075245240582 R-squared 0.318180882293222 Adjusted R-squared 0.306425380263795 F-TEST (value) 27.0665499012062 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 2.68574621342665e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.549622647050563 Sum Squared Residuals 17.5209331407503 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.1 7.58071952563763 0.519280474362368 2 7.7 7.58071952563761 0.119280474362386 3 7.5 7.6489965411076 -0.148996541107605 4 7.6 7.7172735565776 -0.117273556577596 5 7.8 7.51244251016762 0.287557489832377 6 7.8 7.6489965411076 0.151003458892395 7 7.8 7.44416549469763 0.355834505302368 8 7.5 7.7172735565776 -0.217273556577596 9 7.5 7.58071952563761 -0.0807195256376138 10 7.1 7.44416549469763 -0.344165494697632 11 7.5 7.7172735565776 -0.217273556577596 12 7.5 7.51244251016762 -0.0124425101676226 13 7.6 7.58071952563761 0.0192804743623859 14 7.7 7.44416549469763 0.255834505302369 15 7.7 7.37588847922764 0.32411152077236 16 7.9 7.30761146375765 0.592388536242351 17 8.1 7.7172735565776 0.382726443422404 18 8.2 7.37588847922764 0.82411152077236 19 8.2 7.58071952563761 0.619280474362386 20 8.2 7.30761146375765 0.89238853624235 21 7.9 7.37588847922764 0.52411152077236 22 7.3 7.30761146375765 -0.00761146375764923 23 6.9 7.17105743281767 -0.271057432817666 24 6.6 7.23933444828766 -0.639334448287658 25 6.7 7.17105743281767 -0.471057432817667 26 6.9 7.37588847922764 -0.47588847922764 27 7 7.44416549469763 -0.444165494697631 28 7.1 7.37588847922764 -0.275888479227641 29 7.2 7.37588847922764 -0.17588847922764 30 7.1 7.30761146375765 -0.207611463757649 31 6.9 7.10278041734768 -0.202780417347675 32 7 7.10278041734768 -0.102780417347675 33 6.8 7.17105743281767 -0.371057432817667 34 6.4 7.30761146375765 -0.907611463757649 35 6.7 7.23933444828766 -0.539334448287658 36 6.6 7.17105743281767 -0.571057432817667 37 6.4 7.30761146375765 -0.907611463757649 38 6.3 7.10278041734768 -0.802780417347676 39 6.2 7.03450340187768 -0.834503401877684 40 6.5 6.82967235546771 -0.329672355467711 41 6.8 6.76139533999772 0.0386046600022803 42 6.8 6.76139533999772 0.0386046600022803 43 6.4 6.69311832452773 -0.293118324527728 44 6.1 6.96622638640769 -0.866226386407693 45 5.8 6.76139533999772 -0.96139533999772 46 6.1 6.55656429358775 -0.456564293587746 47 7.2 6.55656429358775 0.643435706412254 48 7.3 6.48828727811775 0.811712721882245 49 6.9 6.42001026264776 0.479989737352237 50 6.1 6.35173324717777 -0.251733247177773 51 5.8 6.48828727811775 -0.688287278117755 52 6.2 6.62484130905774 -0.424841309057737 53 7.1 6.96622638640769 0.133773613592307 54 7.7 6.76139533999772 0.93860466000228 55 7.9 6.8979493709377 1.00205062906230 56 7.7 6.69311832452773 1.00688167547227 57 7.4 6.8979493709377 0.502050629062298 58 7.5 6.96622638640769 0.533773613592307 59 8 7.23933444828766 0.760665551712342 60 8.1 7.23933444828766 0.860665551712342 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0721586444035165 0.144317288807033 0.927841355596484 6 0.0259612519526049 0.0519225039052098 0.974038748047395 7 0.00972069728902371 0.0194413945780474 0.990279302710976 8 0.00338210861184596 0.00676421722369193 0.996617891388154 9 0.00209684098865681 0.00419368197731363 0.997903159011343 10 0.0162812189862059 0.0325624379724119 0.983718781013794 11 0.00783030794069253 0.0156606158813851 0.992169692059307 12 0.00346036010554652 0.00692072021109304 0.996539639894453 13 0.00134762804835965 0.0026952560967193 0.99865237195164 14 0.000540454264287984 0.00108090852857597 0.999459545735712 15 0.000207438382976996 0.000414876765953991 0.999792561617023 16 0.000118151473315926 0.000236302946631852 0.999881848526684 17 0.000239564906440689 0.000479129812881378 0.99976043509356 18 0.000520765977535002 0.00104153195507000 0.999479234022465 19 0.00102074084154516 0.00204148168309032 0.998979259158455 20 0.00144617910160117 0.00289235820320235 0.9985538208984 21 0.000991662916262005 0.00198332583252401 0.999008337083738 22 0.00160075206831937 0.00320150413663873 0.99839924793168 23 0.00601853913845127 0.0120370782769025 0.993981460861549 24 0.0224167639872003 0.0448335279744007 0.9775832360128 25 0.0275621946745745 0.0551243893491489 0.972437805325426 26 0.0284145739214852 0.0568291478429704 0.971585426078515 27 0.0263921868981144 0.0527843737962288 0.973607813101886 28 0.0192126424444375 0.0384252848888751 0.980787357555562 29 0.0127026817006406 0.0254053634012812 0.98729731829936 30 0.00817326736715269 0.0163465347343054 0.991826732632847 31 0.00488031309285812 0.00976062618571624 0.995119686907142 32 0.00274357651087097 0.00548715302174194 0.99725642348913 33 0.00177247873252579 0.00354495746505159 0.998227521267474 34 0.00415467853070333 0.00830935706140665 0.995845321469297 35 0.00337243435603489 0.00674486871206979 0.996627565643965 36 0.00279372813080746 0.00558745626161491 0.997206271869193 37 0.00675441142566419 0.0135088228513284 0.993245588574336 38 0.0108815346372229 0.0217630692744459 0.989118465362777 39 0.0212378999470492 0.0424757998940983 0.97876210005295 40 0.0183514888210183 0.0367029776420367 0.981648511178982 41 0.0159939191959480 0.0319878383918960 0.984006080804052 42 0.0122308481027064 0.0244616962054127 0.987769151897294 43 0.00894251324748831 0.0178850264949766 0.991057486752512 44 0.0378139875480245 0.0756279750960489 0.962186012451976 45 0.165187024735422 0.330374049470843 0.834812975264578 46 0.195810376170026 0.391620752340053 0.804189623829974 47 0.260151299793239 0.520302599586478 0.739848700206761 48 0.406782029891492 0.813564059782983 0.593217970108508 49 0.447287831982346 0.894575663964692 0.552712168017654 50 0.34383087769825 0.6876617553965 0.65616912230175 51 0.461555633404227 0.923111266808454 0.538444366595773 52 0.843116584634721 0.313766830730557 0.156883415365279 53 0.95538464387337 0.0892307122532596 0.0446153561266298 54 0.916734453719719 0.166531092560563 0.0832655462802815 55 0.877623617645775 0.24475276470845 0.122376382354225 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 19 0.372549019607843 NOK 5% type I error level 34 0.666666666666667 NOK 10% type I error level 40 0.784313725490196 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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