Paper TSA Multiple Regression Model 2

*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, 18 Dec 2010 13:26:55 +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/18/t12926787800u4yv6ev8n1goyd.htm/, Retrieved Sat, 18 Dec 2010 14:26:20 +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/18/t12926787800u4yv6ev8n1goyd.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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36700 0 35600 0 80900 0 174000 0 169422 0 153452 0 173570 0 193036 0 174652 0 105367 0 95963 0 82896 0 121747 0 120196 0 103983 0 81103 0 70944 0 57248 0 47830 0 60095 0 60931 0 82955 0 99559 0 77911 0 70753 0 69287 0 88426 0 91756 1 96933 1 174484 1 232595 1 266197 1 290435 1 304296 1 322310 1 415555 1 490042 1 545109 1 545720 1 505944 1 477930 1 466106 1 424476 1 383018 1 364696 1 391116 1 435721 1 511435 1 553997 1 555252 1 544897 1 540562 1 505282 1 507626 1 474427 1 469740 1 491480 1 538974 1 576612 1

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 5 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Werklozen[t] = + 14396.70103154 + 142652.134415666Oliecrisis[t] + 6084.1668575355t + 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) 14396.70103154 25857.421714 0.5568 0.579901 0.289951 Oliecrisis 142652.134415666 46732.352015 3.0525 0.003467 0.001734 t 6084.1668575355 1367.173085 4.4502 4.1e-05 2.1e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.886720046261259 R-squared 0.786272440441569 Adjusted R-squared 0.778639313314482 F-TEST (value) 103.007905849152 F-TEST (DF numerator) 2 F-TEST (DF denominator) 56 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 90336.9379592833 Sum Squared Residuals 457002692152.126 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 36700 20480.8678890754 16219.1321109246 2 35600 26565.0347466110 9034.96525338904 3 80900 32649.2016041466 48250.7983958534 4 174000 38733.368461682 135266.631538318 5 169422 44817.5353192175 124604.464680782 6 153452 50901.702176753 102550.297823247 7 173570 56985.8690342885 116584.130965711 8 193036 63070.035891824 129965.964108176 9 174652 69154.2027493595 105497.797250640 10 105367 75238.369606895 30128.6303931050 11 95963 81322.5364644305 14640.4635355695 12 82896 87406.703321966 -4510.70332196605 13 121747 93490.8701795015 28256.1298204985 14 120196 99575.037037037 20620.9629629630 15 103983 105659.203894573 -1676.20389457255 16 81103 111743.370752108 -30640.3707521080 17 70944 117827.537609644 -46883.5376096435 18 57248 123911.704467179 -66663.704467179 19 47830 129995.871324715 -82165.8713247146 20 60095 136080.03818225 -75985.03818225 21 60931 142164.205039786 -81233.2050397855 22 82955 148248.371897321 -65293.371897321 23 99559 154332.538754857 -54773.5387548566 24 77911 160416.705612392 -82505.705612392 25 70753 166500.872469928 -95747.8724699276 26 69287 172585.039327463 -103298.039327463 27 88426 178669.206184999 -90243.2061849986 28 91756 327405.5074582 -235649.507458200 29 96933 333489.674315735 -236556.674315735 30 174484 339573.841173271 -165089.841173271 31 232595 345658.008030806 -113063.008030806 32 266197 351742.174888342 -85545.1748883417 33 290435 357826.341745877 -67391.3417458772 34 304296 363910.508603413 -59614.5086034128 35 322310 369994.675460948 -47684.6754609483 36 415555 376078.842318484 39476.1576815162 37 490042 382163.009176019 107878.990823981 38 545109 388247.176033555 156861.823966445 39 545720 394331.34289109 151388.657108910 40 505944 400415.509748626 105528.490251374 41 477930 406499.676606161 71430.3233938387 42 466106 412583.843463697 53522.1565363032 43 424476 418668.010321232 5807.98967876775 44 383018 424752.177178768 -41734.1771787677 45 364696 430836.344036303 -66140.3440363032 46 391116 436920.510893839 -45804.5108938387 47 435721 443004.677751374 -7283.67775137425 48 511435 449088.84460891 62346.1553910903 49 553997 455173.011466445 98823.9885335548 50 555252 461257.178323981 93994.8216760193 51 544897 467341.345181516 77555.6548184838 52 540562 473425.512039052 67136.4879609483 53 505282 479509.678896587 25772.3211034127 54 507626 485593.845754123 22032.1542458772 55 474427 491678.012611658 -17251.0126116583 56 469740 497762.179469194 -28022.1794691938 57 491480 503846.346326729 -12366.3463267293 58 538974 509930.513184265 29043.4868157353 59 576612 516014.6800418 60597.3199581997 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.102158005583088 0.204316011166175 0.897841994416912 7 0.056111113579937 0.112222227159874 0.943888886420063 8 0.0298784176215857 0.0597568352431713 0.970121582378414 9 0.0296371244122873 0.0592742488245746 0.970362875587713 10 0.115092774605073 0.230185549210146 0.884907225394927 11 0.163983726683 0.327967453366 0.836016273317 12 0.183689043267973 0.367378086535947 0.816310956732026 13 0.144424701244189 0.288849402488379 0.85557529875581 14 0.112861084265813 0.225722168531625 0.887138915734187 15 0.0901929671818347 0.180385934363669 0.909807032818165 16 0.0762046040761087 0.152409208152217 0.923795395923891 17 0.062041549350045 0.12408309870009 0.937958450649955 18 0.0497378453108057 0.0994756906216115 0.950262154689194 19 0.0379447534269879 0.0758895068539758 0.962055246573012 20 0.0243928816117849 0.0487857632235698 0.975607118388215 21 0.0146473050764578 0.0292946101529156 0.985352694923542 22 0.00845786161766636 0.0169157232353327 0.991542138382334 23 0.00524397409882668 0.0104879481976534 0.994756025901173 24 0.00279209861979008 0.00558419723958017 0.99720790138021 25 0.00141753651478611 0.00283507302957221 0.998582463485214 26 0.000693553692669727 0.00138710738533945 0.99930644630733 27 0.000349788920624718 0.000699577841249436 0.999650211079375 28 0.000475299415019561 0.000950598830039121 0.99952470058498 29 0.00127367129239000 0.00254734258478000 0.99872632870761 30 0.00427147888159167 0.00854295776318335 0.995728521118408 31 0.017060917446541 0.034121834893082 0.982939082553459 32 0.0535206387728042 0.107041277545608 0.946479361227196 33 0.130281896219330 0.260563792438660 0.86971810378067 34 0.264655271204782 0.529310542409564 0.735344728795218 35 0.473477133625974 0.946954267251947 0.526522866374026 36 0.6824084901787 0.635183019642599 0.317591509821299 37 0.8559891122285 0.288021775543002 0.144010887771501 38 0.962392483593158 0.0752150328136846 0.0376075164068423 39 0.99049100122838 0.0190179975432387 0.00950899877161933 40 0.994276169251942 0.0114476614961157 0.00572383074805784 41 0.994043562916576 0.0119128741668481 0.00595643708342403 42 0.992452208077784 0.0150955838444329 0.00754779192221647 43 0.985732131983693 0.0285357360326134 0.0142678680163067 44 0.978840404975685 0.042319190048629 0.0211595950243145 45 0.985391890824624 0.0292162183507518 0.0146081091753759 46 0.993603977843916 0.0127920443121674 0.00639602215608368 47 0.997122488624328 0.00575502275134379 0.00287751137567190 48 0.994093797591865 0.0118124048162699 0.00590620240813497 49 0.988147175220908 0.0237056495581834 0.0118528247790917 50 0.979584299406903 0.040831401186194 0.020415700593097 51 0.966475915023025 0.0670481699539508 0.0335240849769754 52 0.962774998340586 0.0744500033188282 0.0372250016594141 53 0.929838152281527 0.140323695436947 0.0701618477184734 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 8 0.166666666666667 NOK 5% type I error level 24 0.5 NOK 10% type I error level 31 0.645833333333333 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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