Paper TSA Multiple Regression

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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 12:51:38 +0000

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/18/t129267669048cyiv61njc9c89.htm/, Retrieved Sat, 18 Dec 2010 13:51:33 +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/t129267669048cyiv61njc9c89.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}, }

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Dataseries X:

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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 Multiple Linear Regression - Estimated Regression Equation Werklozen[t] = + 99575.037037037 + 322135.056712963Oliecrisis[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) 99575.037037037 20048.986324 4.9666 7e-06 3e-06 Oliecrisis 322135.056712963 27223.467188 11.833 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.843023613462403 R-squared 0.710688812855207 Adjusted R-squared 0.705613177993017 F-TEST (value) 140.019688600818 F-TEST (DF numerator) 1 F-TEST (DF denominator) 57 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 104177.58885821 Sum Squared Residuals 618619291157.682 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 36700 99575.0370370368 -62875.0370370368 2 35600 99575.037037037 -63975.037037037 3 80900 99575.037037037 -18675.0370370371 4 174000 99575.037037037 74424.962962963 5 169422 99575.037037037 69846.962962963 6 153452 99575.037037037 53876.962962963 7 173570 99575.037037037 73994.962962963 8 193036 99575.037037037 93460.962962963 9 174652 99575.037037037 75076.962962963 10 105367 99575.037037037 5791.96296296294 11 95963 99575.037037037 -3612.03703703706 12 82896 99575.037037037 -16679.0370370371 13 121747 99575.037037037 22171.962962963 14 120196 99575.037037037 20620.962962963 15 103983 99575.037037037 4407.96296296294 16 81103 99575.037037037 -18472.0370370371 17 70944 99575.037037037 -28631.0370370371 18 57248 99575.037037037 -42327.0370370371 19 47830 99575.037037037 -51745.037037037 20 60095 99575.037037037 -39480.0370370371 21 60931 99575.037037037 -38644.0370370371 22 82955 99575.037037037 -16620.0370370371 23 99559 99575.037037037 -16.0370370370585 24 77911 99575.037037037 -21664.0370370371 25 70753 99575.037037037 -28822.0370370371 26 69287 99575.037037037 -30288.0370370371 27 88426 99575.037037037 -11149.0370370371 28 91756 421710.09375 -329954.09375 29 96933 421710.09375 -324777.09375 30 174484 421710.09375 -247226.09375 31 232595 421710.09375 -189115.09375 32 266197 421710.09375 -155513.09375 33 290435 421710.09375 -131275.09375 34 304296 421710.09375 -117414.09375 35 322310 421710.09375 -99400.09375 36 415555 421710.09375 -6155.09375000001 37 490042 421710.09375 68331.90625 38 545109 421710.09375 123398.90625 39 545720 421710.09375 124009.90625 40 505944 421710.09375 84233.90625 41 477930 421710.09375 56219.90625 42 466106 421710.09375 44395.90625 43 424476 421710.09375 2765.90624999999 44 383018 421710.09375 -38692.09375 45 364696 421710.09375 -57014.09375 46 391116 421710.09375 -30594.09375 47 435721 421710.09375 14010.90625 48 511435 421710.09375 89724.90625 49 553997 421710.09375 132286.90625 50 555252 421710.09375 133541.90625 51 544897 421710.09375 123186.90625 52 540562 421710.09375 118851.90625 53 505282 421710.09375 83571.90625 54 507626 421710.09375 85915.90625 55 474427 421710.09375 52716.90625 56 469740 421710.09375 48029.90625 57 491480 421710.09375 69769.90625 58 538974 421710.09375 117263.90625 59 576612 421710.09375 154901.90625 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.345394145541218 0.690788291082436 0.654605854458782 6 0.231049685681111 0.462099371362222 0.768950314318889 7 0.168513538469144 0.337027076938287 0.831486461530856 8 0.138174213391807 0.276348426783615 0.861825786608193 9 0.0916538345402725 0.183307669080545 0.908346165459728 10 0.0517280007178397 0.103456001435679 0.94827199928216 11 0.0290283474186472 0.0580566948372944 0.970971652581353 12 0.0170346250914096 0.0340692501828192 0.98296537490859 13 0.0081411461609397 0.0162822923218794 0.99185885383906 14 0.00372044957343088 0.00744089914686175 0.99627955042657 15 0.00168869869129113 0.00337739738258225 0.998311301308709 16 0.000880334164898474 0.00176066832979695 0.999119665835101 17 0.000497866569542909 0.000995733139085817 0.999502133430457 18 0.000329658348074333 0.000659316696148667 0.999670341651926 19 0.000240955158948821 0.000481910317897641 0.999759044841051 20 0.000135637684934836 0.000271275369869671 0.999864362315065 21 7.2320726006622e-05 0.000144641452013244 0.999927679273993 22 2.98652381114838e-05 5.97304762229676e-05 0.999970134761889 23 1.12212260009763e-05 2.24424520019525e-05 0.999988778774 24 4.49136513112213e-06 8.98273026224426e-06 0.999995508634869 25 1.86000407651311e-06 3.72000815302621e-06 0.999998139995923 26 7.57223685234961e-07 1.51444737046992e-06 0.999999242776315 27 2.56069280100798e-07 5.12138560201595e-07 0.99999974393072 28 8.47799445077914e-07 1.69559889015583e-06 0.999999152200555 29 6.40985003667144e-06 1.28197000733429e-05 0.999993590149963 30 6.41358874729609e-05 0.000128271774945922 0.999935864112527 31 0.00075537731264705 0.0015107546252941 0.999244622687353 32 0.00644013583695842 0.0128802716739168 0.993559864163042 33 0.0366337513129193 0.0732675026258386 0.96336624868708 34 0.145008259132 0.290016518264001 0.854991740868 35 0.39776729792165 0.7955345958433 0.60223270207835 36 0.644171951285697 0.711656097428605 0.355828048714303 37 0.82712289061729 0.34575421876542 0.17287710938271 38 0.935938696370949 0.128122607258103 0.0640613036290515 39 0.967281297013303 0.0654374059733945 0.0327187029866973 40 0.967275096897031 0.0654498062059373 0.0327249031029687 41 0.95837962440428 0.0832407511914386 0.0416203755957193 42 0.94415673553291 0.111686528934181 0.0558432644670906 43 0.93351176420466 0.13297647159068 0.0664882357953402 44 0.95164163684471 0.09671672631058 0.04835836315529 45 0.98533848991997 0.029323020160061 0.0146615100800305 46 0.997112142944034 0.00577571411193148 0.00288785705596574 47 0.998936018207952 0.00212796358409564 0.00106398179204782 48 0.9976561608544 0.00468767829120133 0.00234383914560066 49 0.996294527752127 0.0074109444957467 0.00370547224787335 50 0.994206339145546 0.0115873217089072 0.00579366085445359 51 0.98872677700437 0.0225464459912582 0.0112732229956291 52 0.976749131005889 0.0465017379882226 0.0232508689941113 53 0.939200000237655 0.121599999524689 0.0607999997623445 54 0.852756086119893 0.294487827760215 0.147243913880107 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 22 0.44 NOK 5% type I error level 29 0.58 NOK 10% type I error level 35 0.7 NOK

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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