Regressieanalyse van tijdsreeksen: BRU Passengers

*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 14:44:12 +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/t129268335499khwnyfwv2n42y.htm/, Retrieved Sat, 18 Dec 2010 15:42:36 +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/t129268335499khwnyfwv2n42y.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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989236 1008380 1207763 1368839 1469798 1498721 1761769 1653214 1599104 1421179 1163995 1037735 1015407 1039210 1258049 1469445 1552346 1549144 1785895 1662335 1629440 1467430 1202209 1076982 1039367 1063449 1335135 1491602 1591972 1641248 1898849 1798580 1762444 1622044 1368955 1262973 1195650 1269530 1479279 1607819 1712466 1721766 1949843 1821326 1757802 1590367 1260647 1149235 1016367 1027885 1262159 1520854 1544144 1564709 1821776 1741365 1623386 1498658 1241822 1136029

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 PassengersBRU[t] = + 1046423.8 -55056.5944444449M1[t] -26964.7222222221M2[t] + 197427.95M3[t] + 378269.222222222M4[t] + 458309.094444444M5[t] + 476887.966666667M6[t] + 723003.238888889M7[t] + 612347.311111111M8[t] + 549024.983333333M9[t] + 392131.855555556M10[t] + 117328.327777778M11[t] + 2393.52777777778t + 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) 1046423.8 39446.619714 26.5276 0 0 M1 -55056.5944444449 47989.020536 -1.1473 0.257075 0.128537 M2 -26964.7222222221 47917.321202 -0.5627 0.57629 0.288145 M3 197427.95 47852.357805 4.1258 0.00015 7.5e-05 M4 378269.222222222 47794.157814 7.9145 0 0 M5 458309.094444444 47742.745961 9.5996 0 0 M6 476887.966666667 47698.144198 9.998 0 0 M7 723003.238888889 47660.371644 15.1699 0 0 M8 612347.311111111 47629.444546 12.8565 0 0 M9 549024.983333333 47605.376245 11.5328 0 0 M10 392131.855555556 47588.177149 8.2401 0 0 M11 117328.327777778 47577.854707 2.466 0.017367 0.008683 t 2393.52777777778 572.231258 4.1828 0.000125 6.2e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.968492495041633 R-squared 0.937977712951967 Adjusted R-squared 0.922142235407788 F-TEST (value) 59.2326761434978 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 75221.7523381548 Sum Squared Residuals 265940665166.667 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 989236 993760.733333335 -4524.7333333352 2 1008380 1024246.13333333 -15866.1333333334 3 1207763 1251032.33333333 -43269.3333333334 4 1368839 1434267.13333333 -65428.1333333333 5 1469798 1516700.53333333 -46902.5333333334 6 1498721 1537672.93333333 -38951.9333333333 7 1761769 1786181.73333333 -24412.7333333333 8 1653214 1677919.33333333 -24705.3333333334 9 1599104 1616990.53333333 -17886.5333333331 10 1421179 1462490.93333333 -41311.9333333332 11 1163995 1190080.93333333 -26085.9333333333 12 1037735 1075146.13333333 -37411.1333333332 13 1015407 1022483.06666667 -7076.06666666607 14 1039210 1052968.46666667 -13758.4666666666 15 1258049 1279754.66666667 -21705.6666666666 16 1469445 1462989.46666667 6455.53333333338 17 1552346 1545422.86666667 6923.13333333342 18 1549144 1566395.26666667 -17251.2666666666 19 1785895 1814904.06666667 -29009.0666666666 20 1662335 1706641.66666667 -44306.6666666666 21 1629440 1645712.86666667 -16272.8666666667 22 1467430 1491213.26666667 -23783.2666666666 23 1202209 1218803.26666667 -16594.2666666666 24 1076982 1103868.46666667 -26886.4666666666 25 1039367 1051205.4 -11838.3999999995 26 1063449 1081690.8 -18241.7999999999 27 1335135 1308477 26658 28 1491602 1491711.8 -109.799999999995 29 1591972 1574145.2 17826.8 30 1641248 1595117.6 46130.4 31 1898849 1843626.4 55222.6 32 1798580 1735364 63216 33 1762444 1674435.2 88008.8 34 1622044 1519935.6 102108.4 35 1368955 1247525.6 121429.4 36 1262973 1132590.8 130382.2 37 1195650 1079927.73333333 115722.266666667 38 1269530 1110413.13333333 159116.866666667 39 1479279 1337199.33333333 142079.666666667 40 1607819 1520434.13333333 87384.8666666667 41 1712466 1602867.53333333 109598.466666667 42 1721766 1623839.93333333 97926.0666666667 43 1949843 1872348.73333333 77494.2666666666 44 1821326 1764086.33333333 57239.6666666666 45 1757802 1703157.53333333 54644.4666666666 46 1590367 1548657.93333333 41709.0666666666 47 1260647 1276247.93333333 -15600.9333333333 48 1149235 1161313.13333333 -12078.1333333334 49 1016367 1108650.06666667 -92283.0666666663 50 1027885 1139135.46666667 -111250.466666667 51 1262159 1365921.66666667 -103762.666666667 52 1520854 1549156.46666667 -28302.4666666668 53 1544144 1631589.86666667 -87445.8666666667 54 1564709 1652562.26666667 -87853.2666666667 55 1821776 1901071.06666667 -79295.0666666667 56 1741365 1792808.66666667 -51443.6666666667 57 1623386 1731879.86666667 -108493.866666667 58 1498658 1577380.26666667 -78722.2666666668 59 1241822 1304970.26666667 -63148.2666666668 60 1136029 1190035.46666667 -54006.4666666668 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0306945672352756 0.0613891344705512 0.969305432764724 17 0.0102126958972082 0.0204253917944165 0.989787304102792 18 0.00249280108633591 0.00498560217267182 0.997507198913664 19 0.00100204598152835 0.0020040919630567 0.998997954018472 20 0.000655926307042261 0.00131185261408452 0.999344073692958 21 0.000205023895720114 0.000410047791440227 0.99979497610428 22 6.40178952850278e-05 0.000128035790570056 0.999935982104715 23 2.12106220977289e-05 4.24212441954578e-05 0.999978789377902 24 9.91500346851211e-06 1.98300069370242e-05 0.999990084996532 25 6.5457480787722e-06 1.30914961575444e-05 0.999993454251921 26 4.70389379977967e-06 9.40778759955933e-06 0.9999952961062 27 8.76409790421142e-06 1.75281958084228e-05 0.999991235902096 28 8.76767665632982e-06 1.75353533126596e-05 0.999991232323344 29 9.39211096578133e-06 1.87842219315627e-05 0.999990607889034 30 3.9817474157732e-05 7.9634948315464e-05 0.999960182525842 31 0.000166518347364995 0.000333036694729991 0.999833481652635 32 0.00128667368735134 0.00257334737470269 0.998713326312649 33 0.00379700713029599 0.00759401426059198 0.996202992869704 34 0.0199202742439781 0.0398405484879562 0.980079725756022 35 0.0448416559328392 0.0896833118656784 0.95515834406716 36 0.101550193773449 0.203100387546898 0.898449806226551 37 0.0870586431540711 0.174117286308142 0.912941356845929 38 0.251392068590792 0.502784137181583 0.748607931409208 39 0.461747770129652 0.923495540259304 0.538252229870348 40 0.348746555510781 0.697493111021561 0.65125344448922 41 0.387312281542639 0.774624563085279 0.61268771845736 42 0.447978110211687 0.895956220423375 0.552021889788313 43 0.439064835949506 0.878129671899011 0.560935164050494 44 0.301552693473766 0.603105386947532 0.698447306526234 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 16 0.551724137931034 NOK 5% type I error level 18 0.620689655172414 NOK 10% type I error level 20 0.689655172413793 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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