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: Thu, 19 Nov 2009 13:47:35 -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/19/t1258663721n00kwiceoelxh7s.htm/, Retrieved Thu, 19 Nov 2009 21:48:53 +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/19/t1258663721n00kwiceoelxh7s.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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3.75 99.9 100.1 100.7 101.1 101.2 4.11 99.7 99.9 100.1 100.7 101.1 4.25 99.5 99.7 99.9 100.1 100.7 4.25 99.2 99.5 99.7 99.9 100.1 4.5 99 99.2 99.5 99.7 99.9 4.7 99 99 99.2 99.5 99.7 4.75 99.3 99 99 99.2 99.5 4.75 99.5 99.3 99 99 99.2 4.75 99.7 99.5 99.3 99 99 4.75 100 99.7 99.5 99.3 99 4.75 100.4 100 99.7 99.5 99.3 4.75 100.6 100.4 100 99.7 99.5 4.58 100.7 100.6 100.4 100 99.7 4.5 100.7 100.7 100.6 100.4 100 4.5 100.6 100.7 100.7 100.6 100.4 4.49 100.5 100.6 100.7 100.7 100.6 4.03 100.6 100.5 100.6 100.7 100.7 3.75 100.5 100.6 100.5 100.6 100.7 3.39 100.4 100.5 100.6 100.5 100.6 3.25 100.3 100.4 100.5 100.6 100.5 3.25 100.4 100.3 100.4 100.5 100.6 3.25 100.4 100.4 100.3 100.4 100.5 3.25 100.4 100.4 100.4 100.3 100.4 3.25 100.4 100.4 100.4 100.4 100.3 3.25 100.4 100.4 100.4 100.4 100.4 3.25 100.5 100.4 100.4 100.4 100.4 3.25 100.6 100.5 100.4 100.4 100.4 3.25 100.6 100.6 100.5 100.4 100.4 3.25 100.5 100.6 100.6 100.5 100.4 3.25 100.5 100.5 100.6 100.6 100.5 3.25 100.7 100.5 100.5 etc...

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 Tprod[t] = + 9.03287437636819 + 0.0100535761591053Rente[t] + 2.06088898976321y1[t] -1.65997795678992y2[t] + 0.66397755103948y3[t] -0.156268465277844y4[t] + 0.00673278043645512t + 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) 9.03287437636819 4.691774 1.9253 0.060012 0.030006 Rente 0.0100535761591053 0.071091 0.1414 0.88812 0.44406 y1 2.06088898976321 0.14162 14.5523 0 0 y2 -1.65997795678992 0.316673 -5.2419 3e-06 2e-06 y3 0.66397755103948 0.322999 2.0557 0.045164 0.022582 y4 -0.156268465277844 0.152604 -1.024 0.310858 0.155429 t 0.00673278043645512 0.003976 1.6935 0.096715 0.048358 Multiple Linear Regression - Regression Statistics Multiple R 0.993522928795038 R-squared 0.98708781004147 Adjusted R-squared 0.985506725556751 F-TEST (value) 624.310604260666 F-TEST (DF numerator) 6 F-TEST (DF denominator) 49 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.147201856015383 Sum Squared Residuals 1.06175093430430 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 99.9 99.5262774179268 0.373722582073162 2 99.7 99.870474288014 -0.170474288013908 3 99.5 99.4625532180054 0.0374467819945941 4 99.2 99.350069360806 -0.150069360806019 5 99 98.971502612559 0.0284973874410457 6 99 98.9645198801592 0.0354801198407734 7 99.3 99.1358113585054 0.164188641494644 8 99.5 99.6748958652462 -0.174895865246220 9 99.7 99.627066749654 0.0729332503460831 10 100 99.9131750019969 0.0868249980031247 11 100.4 100.292093858629 0.107906141371163 12 100.6 100.726730665086 -0.126730665085955 13 100.7 100.647880525068 0.052119474931733 14 100.7 100.746612807863 -0.0466128078628094 15 100.6 100.657635916717 -0.0576359167170165 16 100.5 100.493323324464 0.00667667553607671 17 100.6 100.439713510042 0.160286489957892 18 100.5 100.749320228705 -0.249320228705347 19 100.4 100.329576118493 0.0704238815069325 20 100.3 100.376834896602 -0.0768348966016582 21 100.4 100.261451972109 0.138548027890979 22 100.4 100.589500538625 -0.189500538624661 23 100.4 100.379464614806 0.0205353851940609 24 100.4 100.468221996874 -0.0682219968741338 25 100.4 100.459327930783 -0.0593279307828032 26 100.5 100.466060711219 0.0339392887807360 27 100.6 100.678882390632 -0.078882390632034 28 100.6 100.725706274366 -0.125706274365815 29 100.5 100.632839014227 -0.132839014227226 30 100.5 100.484253804264 0.0157461957364671 31 100.7 100.641357533851 0.0586424661488154 32 101.1 100.989848926673 0.110151073327290 33 101.5 101.503563200568 -0.00356320056830893 34 101.9 101.803455904402 0.0965440955980037 35 102.1 102.202879710156 -0.102879710156175 36 102.1 102.160380061326 -0.0603800613259744 37 102.1 102.038200884709 0.0617991152908789 38 102.4 102.111200358779 0.288799641221327 39 102.8 102.703940785473 0.0960592145273543 40 103.1 103.037035774777 0.0629642252226264 41 103.1 103.197237334739 -0.0972373347386817 42 102.9 102.924687208971 -0.0246872089705846 43 102.4 102.655928070655 -0.255928070655128 44 101.9 101.917331407985 -0.0173314079845912 45 101.3 101.590813161727 -0.290813161726524 46 100.7 100.890266444236 -0.190266444235819 47 100.6 100.402598062008 0.197401937992481 48 101 100.878976419557 0.121023580443197 49 101.5 101.571437140121 -0.0714371401205832 50 101.9 101.971986556785 -0.0719865567854098 51 102.1 102.254303821676 -0.154303821675791 52 102.3 102.278704606757 0.0212953932425151 53 102.5 102.553076381565 -0.0530763815654929 54 102.9 102.710279492693 0.189720507306649 55 103.6 103.310914094829 0.289085905170546 56 104.3 104.197819802536 0.102180197463521 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 10 0.271181533643645 0.542363067287290 0.728818466356355 11 0.617484323797822 0.765031352404357 0.382515676202178 12 0.748682317858286 0.502635364283427 0.251317682141714 13 0.714446407998396 0.571107184003208 0.285553592001604 14 0.63128678058339 0.73742643883322 0.36871321941661 15 0.518963628551646 0.962072742896709 0.481036371448354 16 0.429113147700908 0.858226295401816 0.570886852299092 17 0.433309302455932 0.866618604911864 0.566690697544068 18 0.52274649930562 0.95450700138876 0.47725350069438 19 0.456213793848702 0.912427587697403 0.543786206151298 20 0.382904594504487 0.765809189008973 0.617095405495513 21 0.421654931882504 0.843309863765008 0.578345068117496 22 0.403051616033867 0.806103232067733 0.596948383966133 23 0.340062673122351 0.680125346244703 0.659937326877649 24 0.272373771339463 0.544747542678925 0.727626228660537 25 0.204473000046142 0.408946000092284 0.795526999953858 26 0.159790512233035 0.31958102446607 0.840209487766965 27 0.118118936985378 0.236237873970756 0.881881063014622 28 0.0919210414102472 0.183842082820494 0.908078958589753 29 0.0886529171193103 0.177305834238621 0.91134708288069 30 0.0580341433475005 0.116068286695001 0.9419658566525 31 0.0533490258322642 0.106698051664528 0.946650974167736 32 0.090549438920489 0.181098877840978 0.909450561079511 33 0.0810178521635331 0.162035704327066 0.918982147836467 34 0.0727419075884224 0.145483815176845 0.927258092411578 35 0.0613366679856953 0.122673335971391 0.938663332014305 36 0.0448228584033816 0.089645716806763 0.955177141596618 37 0.0787022207466955 0.157404441493391 0.921297779253305 38 0.107111728595488 0.214223457190975 0.892888271404512 39 0.101966213059102 0.203932426118205 0.898033786940898 40 0.103420104704999 0.206840209409998 0.896579895295001 41 0.0897536164216421 0.179507232843284 0.910246383578358 42 0.185823457638471 0.371646915276942 0.814176542361529 43 0.188133462461684 0.376266924923368 0.811866537538316 44 0.653024817680585 0.693950364638830 0.346975182319415 45 0.673893861864632 0.652212276270736 0.326106138135368 46 0.620291329403054 0.759417341193892 0.379708670596946 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 1 0.0270270270270270 OK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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