workshop 7,3

*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 06:18:27 -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/t1258723168cgy9qyls7ugs7m1.htm/, Retrieved Fri, 20 Nov 2009 14:19:43 +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/t1258723168cgy9qyls7ugs7m1.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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8,9 11,1 8,9 10,9 8,6 10 8,3 9,2 8,3 9,2 8,3 9,5 8,4 9,6 8,5 9,5 8,4 9,1 8,6 8,9 8,5 9 8,5 10,1 8,4 10,3 8,5 10,2 8,5 9,6 8,5 9,2 8,5 9,3 8,5 9,4 8,5 9,4 8,5 9,2 8,5 9 8,6 9 8,4 9 8,1 9,8 8,0 10 8,0 9,8 8,0 9,3 8,0 9 7,9 9 7,8 9,1 7,8 9,1 7,9 9,1 8,1 9,2 8,0 8,8 7,6 8,3 7,3 8,4 7,0 8,1 6,8 7,7 7,0 7,9 7,1 7,9 7,2 8 7,1 7,9 6,9 7,6 6,7 7,1 6,7 6,8 6,6 6,5 6,9 6,9 7,3 8,2 7,5 8,7 7,3 8,3 7,1 7,9 6,9 7,5 7,1 7,8 7,5 8,3 7,7 8,4 7,8 8,2 7,8 7,7 7,7 7,2 7,8 7,3 7,8 8,1 7,9 8,5

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 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 1.34710753037021 + 0.710518726794275X[t] -0.210593615947514M1[t] -0.194875995818465M2[t] + 0.0545559821203266M3[t] + 0.241356836451462M4[t] + 0.207108701921344M5[t] + 0.116019069247684M6[t] + 0.147033181932879M7[t] + 0.305940665441044M8[t] + 0.507479272556865M9[t] + 0.703228254208571M10[t] + 0.625821617821996M11[t] + 0.00319626185069029t + 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) 1.34710753037021 0.735547 1.8314 0.073378 0.036689 X 0.710518726794275 0.068971 10.3017 0 0 M1 -0.210593615947514 0.152691 -1.3792 0.174357 0.087179 M2 -0.194875995818465 0.158685 -1.2281 0.225535 0.112768 M3 0.0545559821203266 0.160565 0.3398 0.73554 0.36777 M4 0.241356836451462 0.165707 1.4565 0.151895 0.075948 M5 0.207108701921344 0.162893 1.2714 0.209828 0.104914 M6 0.116019069247684 0.159755 0.7262 0.471298 0.235649 M7 0.147033181932879 0.159403 0.9224 0.361032 0.180516 M8 0.305940665441044 0.161149 1.8985 0.063781 0.03189 M9 0.507479272556865 0.164758 3.0801 0.003452 0.001726 M10 0.703228254208571 0.170139 4.1332 0.000146 7.3e-05 M11 0.625821617821996 0.168503 3.714 0.000541 0.00027 t 0.00319626185069029 0.003538 0.9034 0.370931 0.185466 Multiple Linear Regression - Regression Statistics Multiple R 0.937247230548335 R-squared 0.878432371170524 Adjusted R-squared 0.84480728234535 F-TEST (value) 26.1243137746855 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.249200386026404 Sum Squared Residuals 2.91873912259832 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.9 9.02646804368984 -0.126468043689842 2 8.9 8.90327818031073 -0.00327818031073305 3 8.6 8.51643956598537 0.0835604340146348 4 8.3 8.13802170073177 0.161978299268232 5 8.3 8.10696982805234 0.193030171947659 6 8.3 8.23223207526765 0.067767924732346 7 8.4 8.33749432248297 0.0625056775170324 8 8.5 8.4285461951624 0.0714538048376043 9 8.4 8.3490735734112 0.0509264265888036 10 8.6 8.40591507155474 0.194084928445262 11 8.5 8.40275656969828 0.0972434303017193 12 8.5 8.56170181320068 -0.0617018132006776 13 8.4 8.49640820446271 -0.09640820446271 14 8.5 8.44427021376302 0.0557297862369788 15 8.5 8.27058721747594 0.229412782524062 16 8.5 8.17637684294005 0.323623157059947 17 8.5 8.21637684294005 0.283623157059947 18 8.5 8.19953534479651 0.300464655203489 19 8.5 8.2337457193324 0.266254280667603 20 8.5 8.2537457193324 0.246254280667604 21 8.5 8.31637684294005 0.183623157059947 22 8.6 8.51532208644245 0.0846779135575502 23 8.4 8.44111171190657 -0.0411117119065639 24 8.1 8.38690133737068 -0.286901337370680 25 8 8.32160772863271 -0.321607728632711 26 8 8.1984178652536 -0.198417865253596 27 8 8.09578674164594 -0.0957867416459392 28 8 8.07262823978948 -0.0726282397894817 29 7.9 8.04157636711005 -0.141576367110053 30 7.8 8.02473486896651 -0.224734868966511 31 7.8 8.0589452435024 -0.258945243502397 32 7.9 8.22104898886125 -0.321048988861252 33 8.1 8.4968357305072 -0.396835730507191 34 8 8.41157348329188 -0.411573483291879 35 7.6 7.98210374535886 -0.382103745358856 36 7.3 7.43053026206698 -0.130530262066977 37 7 7.00997728993187 -0.00997728993187062 38 6.8 6.7446836811939 0.0553163188060994 39 7 7.13941566634224 -0.139415666342237 40 7.1 7.32941278252406 -0.229412782524063 41 7.2 7.36941278252406 -0.169412782524062 42 7.1 7.21046753902167 -0.110467539021665 43 6.9 7.03152229551927 -0.131522295519267 44 6.7 6.83836667748098 -0.138366677480985 45 6.7 6.82994592840921 -0.129945928409214 46 6.6 6.81573555387333 -0.215735553873329 47 6.9 7.02573267005515 -0.125732670055153 48 7.3 7.3267816589164 -0.0267816589164048 49 7.5 7.47464366821672 0.0253563317832808 50 7.3 7.20935005947875 0.0906499405212503 51 7.1 7.17777080855052 -0.0777708085505209 52 6.9 7.08356043401464 -0.183560434014635 53 7.1 7.26566417937349 -0.165664179373491 54 7.5 7.53303017194766 -0.0330301719476587 55 7.7 7.63829241916297 0.0617075808370284 56 7.8 7.65829241916297 0.141707580837029 57 7.8 7.50776792473235 0.292232075267654 58 7.7 7.35145380483760 0.348546195162396 59 7.8 7.34829530298115 0.451704697018853 60 7.8 7.29408492844526 0.505915071554739 61 7.9 7.37089506506615 0.529104934933852 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.00313608424441361 0.00627216848882722 0.996863915755586 18 0.00238361833414975 0.00476723666829949 0.99761638166585 19 0.000928337645312247 0.00185667529062449 0.999071662354688 20 0.000357322829808654 0.000714645659617307 0.999642677170191 21 0.000152201673941521 0.000304403347883041 0.999847798326059 22 0.00895181493975177 0.0179036298795035 0.991048185060248 23 0.0277916927657106 0.0555833855314212 0.97220830723429 24 0.068766573718415 0.13753314743683 0.931233426281585 25 0.113452238083703 0.226904476167407 0.886547761916297 26 0.109171976131787 0.218343952263575 0.890828023868213 27 0.141403533701404 0.282807067402809 0.858596466298596 28 0.320241774566180 0.640483549132361 0.67975822543382 29 0.536718620308047 0.926562759383907 0.463281379691953 30 0.629572103244496 0.740855793511008 0.370427896755504 31 0.671865386895622 0.656269226208756 0.328134613104378 32 0.690276415420623 0.619447169158754 0.309723584579377 33 0.661473491226986 0.677053017546028 0.338526508773014 34 0.638180878575963 0.723638242848075 0.361819121424037 35 0.634546355272165 0.730907289455669 0.365453644727835 36 0.54082227162493 0.91835545675014 0.45917772837507 37 0.46136436819478 0.92272873638956 0.53863563180522 38 0.457885583369924 0.915771166739848 0.542114416630076 39 0.426776374298019 0.853552748596039 0.57322362570198 40 0.429119209843677 0.858238419687355 0.570880790156323 41 0.654415249221468 0.691169501557064 0.345584750778532 42 0.876124530933486 0.247750938133027 0.123875469066514 43 0.933196380962913 0.133607238074175 0.0668036190370875 44 0.958843509703552 0.0823129805928954 0.0411564902964477 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 5 0.178571428571429 NOK 5% type I error level 6 0.214285714285714 NOK 10% type I error level 8 0.285714285714286 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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