Workshop 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 07:57:35 -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/t12587291590yogt3yo5c3onfe.htm/, Retrieved Fri, 20 Nov 2009 15:59:32 +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/t12587291590yogt3yo5c3onfe.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:

model 3

Dataseries X:

» Textbox « » Textfile « » CSV «

108.2 108.5 108.8 112.3 110.2 116.6 109.5 115.5 109.5 120.1 116 132.9 111.2 128.1 112.1 129.3 114 132.5 119.1 131 114.1 124.9 115.1 120.8 115.4 122 110.8 122.1 116 127.4 119.2 135.2 126.5 137.3 127.8 135 131.3 136 140.3 138.4 137.3 134.7 143 138.4 134.5 133.9 139.9 133.6 159.3 141.2 170.4 151.8 175 155.4 175.8 156.6 180.9 161.6 180.3 160.7 169.6 156 172.3 159.5 184.8 168.7 177.7 169.9 184.6 169.9 211.4 185.9 215.3 190.8 215.9 195.8 244.7 211.9 259.3 227.1 289 251.3 310.9 256.7 321 251.9 315.1 251.2 333.2 270.3 314.1 267.2 284.7 243 273.9 229.9 216 187.2 196.4 178.2 190.9 175.2 206.4 192.4 196.3 187 199.5 184 198.9 194.1 214.4 212.7 214.2 217.5 187.6 200.5 180.6 205.9 172.2 196.5

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = -73.1510170709155 + 1.59969233247625X[t] + 11.1606847397861M1[t] + 6.0224322816004M2[t] + 5.10915205278968M3[t] -0.50326670695449M4[t] -3.26028849504525M5[t] -0.0384486529738717M6[t] + 1.08645587982531M7[t] -1.87090434254156M8[t] -5.83979691027234M9[t] -8.29572307978729M10[t] -6.8884307248126M11[t] -0.601101440014372t + 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) -73.1510170709155 10.228513 -7.1517 0 0 X 1.59969233247625 0.07222 22.1502 0 0 M1 11.1606847397861 8.892078 1.2551 0.215772 0.107886 M2 6.0224322816004 8.878778 0.6783 0.500983 0.250491 M3 5.10915205278968 8.868485 0.5761 0.567354 0.283677 M4 -0.50326670695449 8.877189 -0.0567 0.955036 0.477518 M5 -3.26028849504525 8.892682 -0.3666 0.715579 0.35779 M6 -0.0384486529738717 8.88698 -0.0043 0.996567 0.498283 M7 1.08645587982531 8.861671 0.1226 0.902957 0.451478 M8 -1.87090434254156 8.876112 -0.2108 0.83399 0.416995 M9 -5.83979691027234 8.911837 -0.6553 0.515548 0.257774 M10 -8.29572307978729 8.861467 -0.9362 0.354081 0.177041 M11 -6.8884307248126 8.821069 -0.7809 0.438858 0.219429 t -0.601101440014372 0.182858 -3.2873 0.001943 0.000971 Multiple Linear Regression - Regression Statistics Multiple R 0.98098756551415 R-squared 0.96233660369338 Adjusted R-squared 0.951692600389335 F-TEST (value) 90.4111522896355 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 13.9379095736167 Sum Squared Residuals 8936.20487098658 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 108.2 110.975184302529 -2.77518430252921 2 108.8 111.314661267739 -2.51466126773874 3 110.2 116.678956628562 -6.47895662856164 4 109.5 108.705774863079 0.794225136920803 5 109.5 112.706236364365 -3.20623636436480 6 116 135.803036622118 -19.8030366221178 7 111.2 128.648316519017 -17.4483165190166 8 112.1 127.009485655607 -14.9094856556069 9 114 127.558507111786 -13.5585071117857 10 119.1 122.101941003542 -3.00194100354202 11 114.1 113.150008690397 0.94999130960277 12 115.1 112.878599412043 2.22140058795716 13 115.4 125.357813510786 -9.95781351078607 14 110.8 119.778428845834 -8.9784288458336 15 116 126.742416539133 -10.7424165391326 16 119.2 133.006496532689 -13.8064965326888 17 126.5 133.007727202784 -6.50772720278385 18 127.8 131.949173240145 -4.14917324014544 19 131.3 134.072668665406 -2.7726686654065 20 140.3 134.353468600968 5.94653139903174 21 137.3 123.864612963061 13.4353870369390 22 143 126.726446983694 16.2735530163062 23 134.5 120.334022402511 14.165977597489 24 139.9 126.141443987566 13.7585560124337 25 159.3 148.858689014158 10.4413109858424 26 170.4 160.076073840206 10.3239261597943 27 175 164.320584568295 10.6794154317049 28 175.8 160.026695167508 15.7733048324919 29 180.9 164.667033601784 16.2329663982158 30 180.3 165.848048904613 14.4519510953875 31 169.6 158.853298034759 10.746701965241 32 172.3 160.893759536045 11.4062404639554 33 184.8 171.040934987081 13.7590650129191 34 177.7 169.903538176523 7.79646182347684 35 184.6 170.709729091483 13.8902709085165 36 211.4 202.592135695902 8.80786430409833 37 215.3 220.990211424807 -5.69021142480705 38 215.9 223.249319188988 -7.34931918898821 39 244.7 247.489984073031 -2.78998407303071 40 259.3 265.591787326911 -6.29178732691112 41 289 300.946218544731 -11.9462185447312 42 310.9 312.20529554216 -1.30529554215997 43 321 305.050575439059 15.9494245609412 44 315.1 300.372329143944 14.7276708560559 45 333.2 326.356458686495 6.84354131350461 46 314.1 318.340384846290 -4.24038484628962 47 284.7 280.434021315325 4.26597868467522 48 273.9 265.765381044684 8.13461895531582 49 216 208.01810174772 7.98189825227993 50 196.4 187.881516857234 8.51848314276628 51 190.9 181.56805819098 9.33194180902012 52 206.4 202.869246109813 3.53075389018716 53 196.3 190.872784286336 5.42721571366405 54 199.5 188.694445690964 10.8055543090358 55 198.9 205.375141341759 -6.47514134175912 56 214.4 231.570957063436 -17.1709570634361 57 214.2 234.679486251577 -20.4794862515770 58 187.6 204.427688989951 -16.8276889899514 59 180.6 213.872218500284 -33.2722185002835 60 172.2 205.122439859805 -32.922439859805 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0182214542183781 0.0364429084367562 0.981778545781622 18 0.0223383754802711 0.0446767509605421 0.977661624519729 19 0.0340856820240459 0.0681713640480918 0.965914317975954 20 0.0832368997365275 0.166473799473055 0.916763100263473 21 0.0521837515871436 0.104367503174287 0.947816248412856 22 0.0340875532131432 0.0681751064262864 0.965912446786857 23 0.0175753554567922 0.0351507109135844 0.982424644543208 24 0.0136325892755490 0.0272651785510979 0.986367410724451 25 0.05412061301816 0.10824122603632 0.94587938698184 26 0.0790382883989726 0.158076576797945 0.920961711601027 27 0.0698140093499216 0.139628018699843 0.930185990650078 28 0.0538865463029298 0.107773092605860 0.94611345369707 29 0.0361243050540651 0.0722486101081303 0.963875694945935 30 0.0275220948747921 0.0550441897495842 0.972477905125208 31 0.0176591054635552 0.0353182109271103 0.982340894536445 32 0.0096869953941402 0.0193739907882804 0.99031300460586 33 0.00472693247756947 0.00945386495513895 0.99527306752243 34 0.00364272054589821 0.00728544109179642 0.996357279454102 35 0.00187561810696449 0.00375123621392898 0.998124381893035 36 0.000937080700437572 0.00187416140087514 0.999062919299562 37 0.00099217923653893 0.00198435847307786 0.999007820763461 38 0.00335499158042932 0.00670998316085863 0.99664500841957 39 0.0117907893164319 0.0235815786328639 0.988209210683568 40 0.335347773992306 0.670695547984612 0.664652226007694 41 0.764260316454262 0.471479367091476 0.235739683545738 42 0.993606485394936 0.0127870292101271 0.00639351460506355 43 0.9827755588013 0.034448882397402 0.017224441198701 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 6 0.222222222222222 NOK 5% type I error level 15 0.555555555555556 NOK 10% type I error level 19 0.703703703703704 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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