Multiple Linear Regression Model 4

*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: Sat, 21 Nov 2009 06:16:06 -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/21/t1258809454l4a5z8yhpn0xcmq.htm/, Retrieved Sat, 21 Nov 2009 14:17:46 +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/21/t1258809454l4a5z8yhpn0xcmq.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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Dataseries X:

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8.5 104.1 8.6 90.2 8.5 99.2 8.2 116.5 8.1 98.4 7.9 90.6 8.6 130.5 8.7 107.4 8.7 106 8.5 196.5 8.4 107.8 8.5 90.5 8.7 123.8 8.7 114.7 8.6 115.3 8.5 197 8.3 88.4 8 93.8 8.2 111.3 8.1 105.9 8.1 123.6 8 171 7.9 97 7.9 99.2 8 126.6 8 103.4 7.9 121.3 8 129.6 7.7 110.8 7.2 98.9 7.5 122.8 7.3 120.9 7 133.1 7 203.1 7 110.2 7.2 119.5 7.3 135.1 7.1 113.9 6.8 137.4 6.4 157.1 6.1 126.4 6.5 112.2 7.7 128.8 7.9 136.8 7.5 156.5 6.9 215.2 6.6 146.7 6.9 130.8 7.7 133.1 8 153.4 8 159.9 7.7 174.6 7.3 145 7.4 112.9 8.1 137.8 8.3 150.6

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 Yt-4[t] = + 52.0823136100608 + 3.85640117510045X[t] + 17.6916980839663M1[t] + 7.16702112263573M2[t] + 18.1793683493212M3[t] + 46.3402276700147M4[t] + 5.23247106121416M5[t] -7.4523097356024M6[t] + 13.7663006215087M7[t] + 10.7416236601781M8[t] + 21.8799825079597M9[t] + 88.4472518580307M10[t] + 6.95388109059166M11[t] + 0.950420914326608t + 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) 52.0823136100608 35.535106 1.4657 0.150188 0.075094 X 3.85640117510045 4.123602 0.9352 0.35503 0.177515 M1 17.6916980839663 9.253217 1.912 0.06272 0.03136 M2 7.16702112263573 9.285444 0.7719 0.444521 0.222261 M3 18.1793683493212 9.22849 1.9699 0.055465 0.027733 M4 46.3402276700147 9.170715 5.0531 9e-06 4e-06 M5 5.23247106121416 9.183559 0.5698 0.571872 0.285936 M6 -7.4523097356024 9.208474 -0.8093 0.422911 0.211455 M7 13.7663006215087 9.325626 1.4762 0.147354 0.073677 M8 10.7416236601781 9.38255 1.1449 0.258753 0.129377 M9 21.8799825079597 9.676069 2.2612 0.028983 0.014492 M10 88.4472518580307 9.66519 9.1511 0 0 M11 6.95388109059166 9.685206 0.718 0.476738 0.238369 t 0.950420914326608 0.157422 6.0374 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.915004088656497 R-squared 0.837232482258107 Adjusted R-squared 0.786852060099902 F-TEST (value) 16.6182109317985 F-TEST (DF numerator) 13 F-TEST (DF denominator) 42 p-value 1.55486734598753e-12 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 13.6573269030295 Sum Squared Residuals 7833.94828172099 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 104.1 103.503842596708 0.596157403292294 2 90.2 94.3152266672137 -4.11522666721365 3 99.2 105.892354690716 -6.69235469071565 4 116.5 133.846714573206 -17.3467145732056 5 98.4 93.3037387612217 5.09626123877832 6 90.6 80.7980986437116 9.80190135628835 7 130.5 105.666610737720 24.8333892622803 8 107.4 103.977994808226 3.4220051917743 9 106 116.066774570334 -10.0667745703340 10 196.5 182.813184599711 13.6868154002885 11 107.8 101.884594629089 5.91540537091103 12 90.5 96.266774570334 -5.76677457033391 13 123.8 115.680173803647 8.11982619635302 14 114.7 106.105917756643 8.59408224335702 15 115.3 117.683045780145 -2.383045780145 16 197 146.408685897655 50.591314102345 17 88.4 105.480069968161 -17.0800699681611 18 93.8 92.588789733141 1.21121026685903 19 111.3 115.529101239599 -4.22910123959879 20 105.9 113.069205075085 -7.16920507508472 21 123.6 125.157984837193 -1.55798483719297 22 171 192.290034984081 -21.2900349840805 23 97 111.361445013458 -14.3614450134581 24 99.2 105.357984837193 -6.15798483719299 25 126.6 124.385743952996 2.21425604700404 26 103.4 114.811487905992 -11.4114879059920 27 121.3 126.388615929494 -5.08861592949398 28 129.6 155.885536282024 -26.2855362820241 29 110.8 114.57128023502 -3.7712802350201 30 98.9 100.90871976498 -2.0087197649799 31 122.8 124.234671388948 -1.43467138894778 32 120.9 121.389135106924 -0.489135106923657 33 133.1 132.320994516502 0.77900548349823 34 203.1 199.838684780899 3.26131521910062 35 110.2 119.295734927787 -9.09573492778693 36 119.5 114.063554986542 5.43644501345803 37 135.1 133.091314102345 2.00868589765506 38 113.9 122.745777820321 -8.84577782032084 39 137.4 133.551625608803 3.84837439119724 40 157.1 161.120345373783 -4.02034537378268 41 126.4 119.806089326779 6.59391067322133 42 112.2 109.614289914329 2.58571008567113 43 128.8 136.411002595887 -7.61100259588715 44 136.8 135.108026783903 1.69197321609678 45 156.5 145.654246075971 10.8457539240287 46 215.2 210.858095635309 4.34190436469139 47 146.7 129.158225429666 17.5417745703339 48 130.8 124.311685605931 6.48831439406889 49 133.1 146.038925544304 -12.9389255443044 50 153.4 137.621589849831 15.7784101501695 51 159.9 149.584357990843 10.3156420091574 52 174.6 177.538717873333 -2.93871787333257 53 145 135.838821708819 9.1611782911815 54 112.9 124.490101943839 -11.5901019438386 55 137.8 149.358614037847 -11.5586140378466 56 150.6 148.055638225863 2.54436177413731 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.997653314461693 0.00469337107661466 0.00234668553830733 18 0.99637670405127 0.0072465918974607 0.00362329594873035 19 0.9985138805747 0.00297223885060064 0.00148611942530032 20 0.998637540054493 0.00272491989101306 0.00136245994550653 21 0.998712919833262 0.00257416033347562 0.00128708016673781 22 0.9986307237505 0.00273855249899947 0.00136927624949973 23 0.997057072759163 0.00588585448167466 0.00294292724083733 24 0.995044625286528 0.009910749426943 0.0049553747134715 25 0.995477915233546 0.00904416953290819 0.00452208476645410 26 0.990839374539447 0.0183212509211052 0.00916062546055262 27 0.98367828897318 0.0326434220536392 0.0163217110268196 28 0.993824116018194 0.0123517679636124 0.00617588398180618 29 0.98942471741707 0.0211505651658602 0.0105752825829301 30 0.980651960738655 0.0386960785226896 0.0193480392613448 31 0.975342754267022 0.0493144914659558 0.0246572457329779 32 0.961337827063382 0.0773243458732355 0.0386621729366177 33 0.946202307651526 0.107595384696947 0.0537976923484736 34 0.912346182919296 0.175307634161407 0.0876538170807036 35 0.948572594317654 0.102854811364692 0.0514274056823459 36 0.920209995811805 0.159580008376389 0.0797900041881946 37 0.885579712285123 0.228840575429755 0.114420287714877 38 0.97129214626313 0.0574157074737401 0.0287078537368700 39 0.93830155239731 0.123396895205378 0.061698447602689 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 9 0.391304347826087 NOK 5% type I error level 15 0.652173913043478 NOK 10% type I error level 17 0.739130434782609 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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