paper - time-series analysis (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: Fri, 10 Dec 2010 17:23:42 +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/10/t12920026763d6dovjjsr9p4qy.htm/, Retrieved Fri, 10 Dec 2010 18:37:56 +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/10/t12920026763d6dovjjsr9p4qy.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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296.95 17.20 296.84 17.20 287.54 17.20 287.81 17.20 283.99 20.63 275.79 20.63 269.52 20.63 278.35 20.63 283.43 19.32 289.46 19.32 282.30 19.32 293.55 19.32 304.78 12.99 300.99 12.99 315.29 12.99 316.21 12.99 331.79 18.13 329.38 18.13 317.27 18.13 317.98 18.13 340.28 28.37 339.21 28.37 336.71 28.37 340.11 28.37 347.72 24.35 328.68 24.35 303.05 24.35 299.83 24.35 320.04 24.99 317.94 24.99 303.31 24.99 308.85 24.99 319.19 28.84 314.52 28.84 312.39 28.84 315.77 28.84 320.23 37.88 309.45 37.88 296.54 37.88 297.28 37.88 301.39 54.04 306.68 54.04 305.91 54.04 314.76 54.04 323.34 64.93 341.58 64.93 330.12 64.93 318.16 64.93 317.84 71.81 325.39 71.81 327.56 71.81 329.77 71.81 333.29 99.75 346.10 99.75 358.00 99.75 344.82 99.75 313.30 61.25 301.26 61.25 306.38 61.25 319.31 61.25

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 Gemiddelde_prijs_vliegticket_in$[t] = + 296.249069801245 + 0.408609281722776`Gemiddelde_olieprijs_in$`[t] + 3.66666666666665Q1[t] + 2.71400000000000Q2[t] -2.04466666666666Q3[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) 296.249069801245 5.710603 51.877 0 0 `Gemiddelde_olieprijs_in$` 0.408609281722776 0.090917 4.4943 3.6e-05 1.8e-05 Q1 3.66666666666665 6.334186 0.5789 0.565041 0.28252 Q2 2.71400000000000 6.334186 0.4285 0.669983 0.334992 Q3 -2.04466666666666 6.334186 -0.3228 0.748072 0.374036 Multiple Linear Regression - Regression Statistics Multiple R 0.527554415011112 R-squared 0.278313660797717 Adjusted R-squared 0.225827381583006 F-TEST (value) 5.30259841165704 F-TEST (DF numerator) 4 F-TEST (DF denominator) 55 p-value 0.00110098507590728 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 17.3468832420402 Sum Squared Residuals 16550.2897017137 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 296.95 306.943816113544 -9.99381611354354 2 296.84 305.991149446877 -9.15114944687654 3 287.54 301.23248278021 -13.6924827802099 4 287.81 303.277149446877 -15.4671494468765 5 283.99 308.345345949852 -24.3553459498523 6 275.79 307.392679283186 -31.6026792831856 7 269.52 302.634012616519 -33.114012616519 8 278.35 304.678679283186 -26.3286792831856 9 283.43 307.810067790795 -24.3800677907955 10 289.46 306.857401124129 -17.3974011241288 11 282.3 302.098734457462 -19.7987344574621 12 293.55 304.143401124129 -10.5934011241288 13 304.78 305.22357103749 -0.443571037490312 14 300.99 304.270904370824 -3.28090437082363 15 315.29 299.512237704157 15.7777622958430 16 316.21 301.556904370824 14.6530956291763 17 331.79 307.323822745545 24.4661772544547 18 329.38 306.371156078879 23.0088439211213 19 317.27 301.612489412212 15.6575105877879 20 317.98 303.657156078879 14.3228439211213 21 340.28 311.507981790387 28.7720182096134 22 339.21 310.55531512372 28.6546848762800 23 336.71 305.796648457053 30.9133515429467 24 340.11 307.84131512372 32.2686848762801 25 347.72 309.865372477861 37.854627522139 26 328.68 308.912705811194 19.7672941888056 27 303.05 304.154039144528 -1.10403914452771 28 299.83 306.198705811194 -6.3687058111944 29 320.04 310.126882418164 9.91311758183642 30 317.94 309.174215751497 8.76578424850305 31 303.31 304.41554908483 -1.10554908483029 32 308.85 306.460215751497 2.38978424850307 33 319.19 311.700028152796 7.48997184720371 34 314.52 310.747361486130 3.77263851387034 35 312.39 305.988694819463 6.401305180537 36 315.77 308.033361486130 7.73663851387033 37 320.23 315.39385605957 4.83614394042983 38 309.45 314.441189392904 -4.99118939290356 39 296.54 309.682522726237 -13.1425227262369 40 297.28 311.727189392904 -14.4471893929036 41 301.39 321.99698205221 -20.6069820522103 42 306.68 321.044315385544 -14.3643153855436 43 305.91 316.285648718877 -10.3756487188769 44 314.76 318.330315385544 -3.57031538554362 45 323.34 326.446737130171 -3.10673713017132 46 341.58 325.494070463505 16.0859295364953 47 330.12 320.735403796838 9.38459620316202 48 318.16 322.780070463505 -4.62007046350463 49 317.84 329.257968988424 -11.4179689884240 50 325.39 328.305302321757 -2.91530232175736 51 327.56 323.546635655091 4.01336434490932 52 329.77 325.591302321757 4.17869767824263 53 333.29 340.674512319758 -7.38451231975835 54 346.1 339.721845653092 6.3781543469083 55 358 334.963178986425 23.0368210135749 56 344.82 337.007845653092 7.81215434690827 57 313.3 324.943054973431 -11.6430549734315 58 301.26 323.990388306765 -22.7303883067648 59 306.38 319.231721640098 -12.8517216400982 60 319.31 321.276388306765 -1.96638830676483 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.0167007562518466 0.0334015125036933 0.983299243748153 9 0.00749861439464502 0.0149972287892900 0.992501385605355 10 0.00326602804666645 0.0065320560933329 0.996733971953334 11 0.00183043290401987 0.00366086580803974 0.99816956709598 12 0.00494295435746176 0.00988590871492353 0.995057045642538 13 0.00281155144493026 0.00562310288986052 0.99718844855507 14 0.00158809706450756 0.00317619412901512 0.998411902935492 15 0.00435475730952197 0.00870951461904394 0.995645242690478 16 0.00232885328432736 0.00465770656865471 0.997671146715673 17 0.261124739724768 0.522249479449535 0.738875260275232 18 0.619244866068345 0.76151026786331 0.380755133931655 19 0.714868153261258 0.570263693477483 0.285131846738742 20 0.74480458117528 0.510390837649441 0.255195418824720 21 0.9658022194581 0.0683955610838002 0.0341977805419001 22 0.9853452918087 0.0293094163826009 0.0146547081913005 23 0.992658693036374 0.0146826139272520 0.00734130696362602 24 0.996774304452962 0.00645139109407593 0.00322569554703796 25 0.999760034076726 0.000479931846547108 0.000239965923273554 26 0.999826602299623 0.000346795400754595 0.000173397700377297 27 0.999646668791035 0.00070666241793008 0.00035333120896504 28 0.999410668835545 0.00117866232891093 0.000589331164455463 29 0.999350030411627 0.00129993917674602 0.000649969588373012 30 0.999118825898846 0.00176234820230780 0.000881174101153899 31 0.998300740094233 0.00339851981153384 0.00169925990576692 32 0.996998420399136 0.0060031592017272 0.0030015796008636 33 0.99760166528256 0.00479666943488128 0.00239833471744064 34 0.997024911551198 0.00595017689760334 0.00297508844880167 35 0.996265978623443 0.00746804275311343 0.00373402137655671 36 0.997036279865191 0.00592744026961784 0.00296372013480892 37 0.999424698501814 0.00115060299637110 0.000575301498185552 38 0.999379435763557 0.0012411284728866 0.0006205642364433 39 0.99891094587428 0.00217810825144084 0.00108905412572042 40 0.99803716966654 0.00392566066691818 0.00196283033345909 41 0.997056084857398 0.00588783028520349 0.00294391514260174 42 0.994572761172957 0.0108544776540863 0.00542723882704314 43 0.991067928600926 0.0178641427981476 0.00893207139907382 44 0.982000616064014 0.0359987678719714 0.0179993839359857 45 0.974308777161567 0.0513824456768656 0.0256912228384328 46 0.996649210018409 0.0067015799631826 0.0033507899815913 47 0.995323325172952 0.00935334965409564 0.00467667482704782 48 0.987764949578153 0.0244701008436948 0.0122350504218474 49 0.970660558252086 0.0586788834958284 0.0293394417479142 50 0.949633249176121 0.100733501647757 0.0503667508238785 51 0.888510303314173 0.222979393371653 0.111489696685827 52 0.782219534909733 0.435560930180535 0.217780465090267 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 27 0.6 NOK 5% type I error level 35 0.777777777777778 NOK 10% type I error level 38 0.844444444444444 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Quarterly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Quarterly 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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