paper: multiple regression (verleden)

*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, 16 Dec 2010 07:22:43 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/16/t1292484069ic8oxmgqpyrxic0.htm/, Retrieved Thu, 16 Dec 2010 08:21:12 +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/2010/Dec/16/t1292484069ic8oxmgqpyrxic0.htm/}, year = {2010}, } @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 = {2010}, 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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567 0 573 584 589 591 569 0 567 573 584 589 621 0 569 567 573 584 629 0 621 569 567 573 628 0 629 621 569 567 612 0 628 629 621 569 595 0 612 628 629 621 597 0 595 612 628 629 593 0 597 595 612 628 590 0 593 597 595 612 580 0 590 593 597 595 574 0 580 590 593 597 573 0 574 580 590 593 573 0 573 574 580 590 620 0 573 573 574 580 626 0 620 573 573 574 620 0 626 620 573 573 588 0 620 626 620 573 566 0 588 620 626 620 557 0 566 588 620 626 561 0 557 566 588 620 549 0 561 557 566 588 532 0 549 561 557 566 526 0 532 549 561 557 511 0 526 532 549 561 499 0 511 526 532 549 555 1 499 511 526 532 565 1 555 499 511 526 542 1 565 555 499 511 527 1 542 565 555 499 510 1 527 542 565 555 514 1 510 527 542 565 517 1 514 510 527 542 508 1 517 514 510 527 493 1 508 517 514 510 490 1 493 508 517 514 469 1 490 493 508 517 478 1 469 490 493 508 528 1 478 469 490 493 534 1 528 478 469 490 518 1 534 528 478 469 506 1 518 534 528 478 502 1 506 518 534 528 516 1 502 506 518 534 528 1 516 50 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 6 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk R Framework error message The field 'Names of X columns' contains a hard return which cannot be interpreted. Please, resubmit your request without hard returns in the 'Names of X columns'. Multiple Linear Regression - Estimated Regression Equation Totaal[t] = + 32.2475215807633 + 2.471430053128Crisis[t] + 1.24026784409011`t-1`[t] -0.575426130628861`t-2`[t] + 0.110311812189182`t-3`[t] + 0.164208502243871`t-4 `[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) 32.2475215807633 63.634431 0.5068 0.614549 0.307275 Crisis 2.471430053128 8.193368 0.3016 0.764179 0.38209 `t-1` 1.24026784409011 0.14149 8.7657 0 0 `t-2` -0.575426130628861 0.225687 -2.5497 0.013895 0.006948 `t-3` 0.110311812189182 0.226094 0.4879 0.627752 0.313876 `t-4 ` 0.164208502243871 0.157658 1.0416 0.302631 0.151315 Multiple Linear Regression - Regression Statistics Multiple R 0.913098051345784 R-squared 0.833748051371467 Adjusted R-squared 0.817122856508614 F-TEST (value) 50.1496709211128 F-TEST (DF numerator) 5 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 17.6529714939156 Sum Squared Residuals 15581.3701282499 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 567 568.893018162697 -1.89301816269679 2 569 566.90112246964 2.0988775303598 3 621 570.799742496293 50.2002575037068 4 629 631.674653729904 -2.67465372990351 5 628 610.910010300839 17.0899896991613 6 612 611.130964650043 0.869035349957042 7 595 601.283441889425 -6.28344188942481 8 597 590.609062835716 6.3909371642835 9 593 600.942645247317 -7.94264524731659 10 590 590.32808476658 -0.328084766580404 11 580 586.338064843058 -6.33806484305808 12 574 575.548834549775 -1.54883454977458 13 573 572.87371934598 0.126280654020495 14 573 573.490264657039 -0.490264657039133 15 620 571.761734892094 48.2382651079058 16 626 628.958760738677 -2.95876073867692 17 620 609.191131161417 10.8088688385828 18 588 603.481622485995 -15.481622485995 19 566 575.625278737482 -9.62527873748165 20 557 567.076402487951 -10.0764024879509 21 561 564.058137761458 -3.05813776145785 22 549 566.516512373512 -17.5165123735122 23 532 544.726200362848 -12.7262003628476 24 526 529.510131309424 -3.51013130942398 25 511 531.183860728279 -20.1838607282793 26 499 512.186597016558 -13.1865970165583 27 555 504.952789488757 50.047210511243 28 565 578.672974129048 -13.6729741290485 29 542 555.064919974805 -13.0649199748051 30 527 524.991457710112 2.00854228988831 31 510 529.921035300772 -19.9210353007724 32 514 516.572787252761 -2.57278725276102 33 517 525.884630115365 -8.88463011536535 34 508 522.965300784246 -14.9653007842461 35 493 507.726314506159 -14.7263145061594 36 490 495.28890146601 -5.28890146601057 37 469 499.699309090202 -30.6993090902021 38 478 472.247409053164 5.75259094683614 39 528 492.699705422955 35.3002945770447 40 534 546.725088889097 -12.7250888890966 41 518 522.939817184776 -4.93981718477555 42 506 506.636442025215 -0.636442025214569 43 502 509.832341971524 -7.83234197152367 44 516 510.996646181146 5.00335381885412 45 528 526.711022738751 1.28897726124926 46 533 531.126521763345 1.87347823665518 47 536 531.310278777922 4.68972122207791 48 537 535.776612434732 1.22338756526751 49 524 537.812662974808 -13.8126629748084 50 536 522.265732818795 13.734267181205 51 587 545.232423964972 41.7675760350277 52 597 600.311125389806 -3.31112538980605 53 581 582.556102385735 -1.55610238573508 54 564 564.553960022579 -0.553960022579444 55 558 562.153976499439 -4.15397649943859 56 575 564.371709683 10.6282903169996 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 9 0.843380330089453 0.313239339821094 0.156619669910547 10 0.742510584132716 0.514978831734569 0.257489415867284 11 0.670026958897208 0.659946082205585 0.329973041102792 12 0.58205050513569 0.83589898972862 0.41794949486431 13 0.480519160663438 0.961038321326876 0.519480839336562 14 0.395260364816793 0.790520729633585 0.604739635183207 15 0.764030387903213 0.471939224193574 0.235969612096787 16 0.683768982113144 0.632462035773712 0.316231017886856 17 0.661134283120269 0.677731433759462 0.338865716879731 18 0.598728111997252 0.802543776005496 0.401271888002748 19 0.514736974996103 0.970526050007795 0.485263025003897 20 0.430662058950461 0.861324117900921 0.569337941049539 21 0.379656641943258 0.759313283886515 0.620343358056742 22 0.508384962650747 0.983230074698507 0.491615037349253 23 0.569186461556741 0.861627076886519 0.430813538443259 24 0.504218229977749 0.991563540044503 0.495781770022251 25 0.490881301408404 0.981762602816808 0.509118698591596 26 0.426904552914399 0.853809105828798 0.573095447085601 27 0.652630134027086 0.694739731945828 0.347369865972914 28 0.815235580465246 0.369528839069507 0.184764419534754 29 0.85336295717379 0.29327408565242 0.14663704282621 30 0.79889772564121 0.40220454871758 0.20110227435879 31 0.829391505245878 0.341216989508244 0.170608494754122 32 0.767775198227583 0.464449603544834 0.232224801772417 33 0.717896126558512 0.564207746882975 0.282103873441488 34 0.680622815845007 0.638754368309986 0.319377184154993 35 0.644593100341592 0.710813799316815 0.355406899658408 36 0.567922931594012 0.864154136811977 0.432077068405988 37 0.808665879554268 0.382668240891464 0.191334120445732 38 0.74149869819205 0.517002603615902 0.258501301807951 39 0.866310067568606 0.267379864862788 0.133689932431394 40 0.853471386035212 0.293057227929577 0.146528613964788 41 0.780816202366163 0.438367595267675 0.219183797633837 42 0.685897236830184 0.628205526339631 0.314102763169816 43 0.652613985725667 0.694772028548666 0.347386014274333 44 0.53838588534046 0.92322822931908 0.46161411465954 45 0.435491360428084 0.870982720856167 0.564508639571916 46 0.30720964894744 0.614419297894881 0.69279035105256 47 0.185040696864322 0.370081393728645 0.814959303135678 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 0 0 OK

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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