paper - time-series analysis (3)

*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 17:46:19 +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/t129200308008qb5qw33cz0ep7.htm/, Retrieved Fri, 10 Dec 2010 18:44:50 +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/t129200308008qb5qw33cz0ep7.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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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 6 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Gemiddelde_prijs_vliegticket_in$[t] = + 292.821678002806 + 0.157012705702092`Gemiddelde_olieprijs_in$`[t] + 4.90706693997384Q1[t] + 3.5409335155381Q2[t] -1.63119990889761Q3[t] + 0.413466757769053t + 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) 292.821678002806 6.018055 48.6572 0 0 `Gemiddelde_olieprijs_in$` 0.157012705702092 0.180053 0.872 0.38705 0.193525 Q1 4.90706693997384 6.291533 0.7799 0.438827 0.219413 Q2 3.5409335155381 6.265309 0.5652 0.574302 0.287151 Q3 -1.63119990889761 6.249521 -0.261 0.795076 0.397538 t 0.413466757769053 0.256631 1.6111 0.112981 0.056491 Multiple Linear Regression - Regression Statistics Multiple R 0.558044424545972 R-squared 0.311413579766845 Adjusted R-squared 0.247655577893405 F-TEST (value) 4.88430582227156 F-TEST (DF numerator) 5 F-TEST (DF denominator) 54 p-value 0.000929059806845878 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 17.1005825241527 Sum Squared Residuals 15791.2158239292 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 296.95 300.842830238625 -3.89283023862503 2 296.84 299.890163571959 -3.05016357195858 3 287.54 295.131496905292 -7.59149690529192 4 287.81 297.176163571959 -9.36616357195859 5 283.99 303.03525085026 -19.0452508502597 6 275.79 302.082584183593 -26.292584183593 7 269.52 297.323917516926 -27.8039175169263 8 278.35 299.368584183593 -21.0185841835930 9 283.43 304.483431236866 -21.0534312368661 10 289.46 303.530764570199 -14.0707645701995 11 282.3 298.772097903533 -16.4720979035328 12 293.55 300.816764570199 -7.26676457019944 13 304.78 305.143407840848 -0.363407840848122 14 300.99 304.190741174181 -3.2007411741814 15 315.29 299.432074507515 15.8579254924853 16 316.21 301.476741174181 14.7332588258186 17 331.79 307.604320179233 24.1856798207670 18 329.38 306.651653512566 22.7283464874336 19 317.27 301.8929868459 15.3770131541003 20 317.98 303.937653512566 14.0423464874336 21 340.28 310.865997316699 29.4140026833013 22 339.21 309.913330650032 29.296669349968 23 336.71 305.154663983365 31.5553360166346 24 340.11 307.199330650032 32.910669349968 25 347.72 311.888673270853 35.8313267291475 26 328.68 310.936006604186 17.7439933958142 27 303.05 306.177339937519 -3.12733993751914 28 299.83 308.222006604186 -8.39200660418582 29 320.04 313.643028433578 6.39697156642198 30 317.94 312.690361766911 5.24963823308864 31 303.31 307.931695100245 -4.62169510024469 32 308.85 309.976361766911 -1.12636176691133 33 319.19 315.901394381607 3.28860561839269 34 314.52 314.948727714941 -0.428727714940638 35 312.39 310.190061048274 2.19993895172603 36 315.77 312.234727714941 3.53527228505936 37 320.23 318.97465627223 1.25534372776959 38 309.45 318.021989605564 -8.57198960556375 39 296.54 313.263322938897 -16.7233229388971 40 297.28 315.307989605564 -18.0279896055638 41 301.39 323.165848627452 -21.7758486274525 42 306.68 322.213181960786 -15.5331819607858 43 305.91 317.454515294119 -11.5445152941191 44 314.76 319.499181960786 -4.73918196078577 45 323.34 326.529584023624 -3.18958402362447 46 341.58 325.576917356958 16.0030826430422 47 330.12 320.818250690291 9.3017493097089 48 318.16 322.862917356958 -4.70291735695774 49 317.84 329.263698469931 -11.4236984699311 50 325.39 328.311031803264 -2.92103180326438 51 327.56 323.552365136598 4.00763486340230 52 329.77 325.597031803264 4.17296819673562 53 333.29 335.304500498324 -2.0145004983237 54 346.1 334.351833831657 11.748166168343 55 358 329.593167164990 28.4068328350096 56 344.82 331.637833831657 13.1821661683430 57 313.3 330.913378359869 -17.6133783598694 58 301.26 329.960711693203 -28.7007116932027 59 306.38 325.202045026536 -18.8220450265360 60 319.31 327.246711693203 -7.93671169320268 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 9 0.0199897812368523 0.0399795624737046 0.980010218763148 10 0.0162348120361732 0.0324696240723463 0.983765187963827 11 0.0100484527893442 0.0200969055786884 0.989951547210656 12 0.0166301168333204 0.0332602336666408 0.98336988316668 13 0.0275343711443689 0.0550687422887377 0.972465628855631 14 0.0250359225929328 0.0500718451858655 0.974964077407067 15 0.05986749298942 0.11973498597884 0.94013250701058 16 0.0407101809817369 0.0814203619634739 0.959289819018263 17 0.511354304218223 0.977291391563554 0.488645695781777 18 0.663943378721438 0.672113242557124 0.336056621278562 19 0.621131621444777 0.757736757110446 0.378868378555223 20 0.547164804434164 0.905670391131672 0.452835195565836 21 0.673317433052486 0.653365133895028 0.326682566947514 22 0.657550107506634 0.684899784986732 0.342449892493366 23 0.63709943377641 0.72580113244718 0.36290056622359 24 0.628664076761227 0.742671846477546 0.371335923238773 25 0.792722560069443 0.414554879861114 0.207277439930557 26 0.82979952243968 0.34040095512064 0.17020047756032 27 0.93361325760029 0.132773484799420 0.0663867423997101 28 0.980942308787635 0.0381153824247307 0.0190576912123654 29 0.987592950367727 0.0248140992645450 0.0124070496322725 30 0.988001616982993 0.0239967660340141 0.0119983830170071 31 0.988560915458366 0.0228781690832674 0.0114390845416337 32 0.986023591487586 0.0279528170248286 0.0139764085124143 33 0.99040239803153 0.019195203936939 0.0095976019684695 34 0.989641101128603 0.0207177977427943 0.0103588988713972 35 0.986980291721988 0.0260394165560247 0.0130197082780124 36 0.988299672826336 0.0234006543473279 0.0117003271736640 37 0.996677773581693 0.00664445283661456 0.00332222641830728 38 0.995929896506646 0.00814020698670848 0.00407010349335424 39 0.993891249724934 0.0122175005501318 0.00610875027506588 40 0.990686892731659 0.0186262145366828 0.00931310726834142 41 0.985573750450029 0.0288524990999422 0.0144262495499711 42 0.981220926407478 0.037558147185043 0.0187790735925215 43 0.986166092259355 0.0276678154812892 0.0138339077406446 44 0.978990140865529 0.0420197182689421 0.0210098591344710 45 0.962157150432397 0.0756856991352067 0.0378428495676033 46 0.98980763807381 0.0203847238523815 0.0101923619261907 47 0.981841185798931 0.0363176284021378 0.0181588142010689 48 0.966486550881642 0.0670268982367152 0.0335134491183576 49 0.926788892600038 0.146422214799923 0.0732111073999615 50 0.853722411660407 0.292555176679185 0.146277588339593 51 0.721113072352448 0.557773855295105 0.278886927647552 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 2 0.0465116279069767 NOK 5% type I error level 23 0.534883720930233 NOK 10% type I error level 28 0.651162790697674 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Quarterly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Quarterly 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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