paper - time-series analysis (1)

*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 17:15:49 +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/t12920013077b97ymaxft4jrfy.htm/, Retrieved Fri, 10 Dec 2010 18:15:16 +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/t12920013077b97ymaxft4jrfy.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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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 9 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Gemiddelde_prijs_vliegticket_in$[t] = + 297.333069801245 + 0.408609281722776`Gemiddelde_olieprijs_in$`[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) 297.333069801245 4.11862 72.1924 0 0 `Gemiddelde_olieprijs_in$` 0.408609281722776 0.089345 4.5734 2.6e-05 1.3e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.514822138164715 R-squared 0.265041833944489 Adjusted R-squared 0.252370141426290 F-TEST (value) 20.9160562856026 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 2.57099771261426e-05 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 17.0469179513113 Sum Squared Residuals 16854.6498750470 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 296.95 304.361149446876 -7.4111494468763 2 296.84 304.361149446877 -7.52114944687654 3 287.54 304.361149446877 -16.8211494468765 4 287.81 304.361149446877 -16.5511494468765 5 283.99 305.762679283186 -21.7726792831857 6 275.79 305.762679283186 -29.9726792831856 7 269.52 305.762679283186 -36.2426792831857 8 278.35 305.762679283186 -27.4126792831856 9 283.43 305.227401124129 -21.7974011241288 10 289.46 305.227401124129 -15.7674011241288 11 282.3 305.227401124129 -22.9274011241288 12 293.55 305.227401124129 -11.6774011241288 13 304.78 302.640904370824 2.13909562917632 14 300.99 302.640904370824 -1.65090437082365 15 315.29 302.640904370824 12.6490956291764 16 316.21 302.640904370824 13.5690956291763 17 331.79 304.741156078879 27.0488439211213 18 329.38 304.741156078879 24.6388439211213 19 317.27 304.741156078879 12.5288439211213 20 317.98 304.741156078879 13.2388439211213 21 340.28 308.92531512372 31.35468487628 22 339.21 308.92531512372 30.28468487628 23 336.71 308.92531512372 27.7846848762800 24 340.11 308.92531512372 31.1846848762801 25 347.72 307.282705811194 40.4372941888056 26 328.68 307.282705811194 21.3972941888056 27 303.05 307.282705811194 -4.23270581119438 28 299.83 307.282705811194 -7.4527058111944 29 320.04 307.544215751497 12.4957842485031 30 317.94 307.544215751497 10.3957842485030 31 303.31 307.544215751497 -4.23421575149697 32 308.85 307.544215751497 1.30578424850305 33 319.19 309.117361486130 10.0726385138703 34 314.52 309.117361486130 5.40263851387033 35 312.39 309.117361486130 3.27263851387033 36 315.77 309.117361486130 6.65263851387033 37 320.23 312.811189392904 7.41881060709647 38 309.45 312.811189392904 -3.36118939290356 39 296.54 312.811189392904 -16.2711893929035 40 297.28 312.811189392904 -15.5311893929036 41 301.39 319.414315385544 -18.0243153855436 42 306.68 319.414315385544 -12.7343153855436 43 305.91 319.414315385544 -13.5043153855436 44 314.76 319.414315385544 -4.65431538554362 45 323.34 323.864070463505 -0.524070463504671 46 341.58 323.864070463505 17.7159295364953 47 330.12 323.864070463505 6.25592953649536 48 318.16 323.864070463505 -5.70407046350462 49 317.84 326.675302321757 -8.83530232175737 50 325.39 326.675302321757 -1.28530232175736 51 327.56 326.675302321757 0.88469767824266 52 329.77 326.675302321757 3.09469767824264 53 333.29 338.091845653092 -4.80184565309168 54 346.1 338.091845653092 8.00815434690832 55 358 338.091845653092 19.9081543469083 56 344.82 338.091845653092 6.72815434690829 57 313.3 322.360388306765 -9.06038830676482 58 301.26 322.360388306765 -21.1003883067648 59 306.38 322.360388306765 -15.9803883067648 60 319.31 322.360388306765 -3.05038830676482 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0384401133101991 0.0768802266203981 0.96155988668980 6 0.0202692489238833 0.0405384978477665 0.979730751076117 7 0.0211184469777525 0.042236893955505 0.978881553022247 8 0.00910615225072779 0.0182123045014556 0.990893847749272 9 0.00364187274777891 0.00728374549555783 0.99635812725222 10 0.00258322053377302 0.00516644106754603 0.997416779466227 11 0.00126229677589706 0.00252459355179412 0.998737703224103 12 0.00213896220608730 0.00427792441217461 0.997861037793913 13 0.00103485579460784 0.00206971158921569 0.998965144205392 14 0.000585769353399025 0.00117153870679805 0.999414230646601 15 0.00059678242645885 0.0011935648529177 0.999403217573541 16 0.000442180715822176 0.00088436143164435 0.999557819284178 17 0.237115768872504 0.474231537745009 0.762884231127496 18 0.584295732994544 0.831408534010912 0.415704267005456 19 0.631595236326198 0.736809527347603 0.368404763673802 20 0.662671232120835 0.67465753575833 0.337328767879165 21 0.966344370186928 0.0673112596261442 0.0336556298130721 22 0.986460060474128 0.0270798790517445 0.0135399395258722 23 0.991216729906532 0.0175665401869351 0.00878327009346756 24 0.99593360725208 0.00813278549583933 0.00406639274791967 25 0.999808309378595 0.000383381242810559 0.000191690621405279 26 0.999906122951194 0.000187754097611319 9.38770488056596e-05 27 0.999832267126064 0.000335465747871685 0.000167732873935842 28 0.999737327003975 0.000525345992049522 0.000262672996024761 29 0.999703958577531 0.000592082844937284 0.000296041422468642 30 0.99965410103864 0.000691797922717321 0.000345898961358660 31 0.99938321302214 0.00123357395572082 0.00061678697786041 32 0.998940379613102 0.00211924077379542 0.00105962038689771 33 0.998974053442774 0.00205189311445136 0.00102594655722568 34 0.998817608193886 0.00236478361222705 0.00118239180611352 35 0.998647682855899 0.00270463428820256 0.00135231714410128 36 0.999169506945024 0.00166098610995273 0.000830493054976366 37 0.999692277420195 0.000615445159609337 0.000307722579804668 38 0.999716014124686 0.000567971750628706 0.000283985875314353 39 0.99959046169052 0.000819076618959871 0.000409538309479935 40 0.999304615568387 0.00139076886322553 0.000695384431612766 41 0.99917131417012 0.00165737165975942 0.000828685829879711 42 0.998425535964684 0.00314892807063137 0.00157446403531568 43 0.997185478374221 0.00562904325155729 0.00281452162577864 44 0.99427250689964 0.0114549862007210 0.00572749310036049 45 0.989032312525303 0.0219353749493933 0.0109676874746967 46 0.997904561315433 0.00419087736913375 0.00209543868456687 47 0.998335525454909 0.00332894909018174 0.00166447454509087 48 0.995983863534837 0.00803227293032567 0.00401613646516283 49 0.991398927394157 0.0172021452116851 0.00860107260584256 50 0.981339351599984 0.0373212968000315 0.0186606484000157 51 0.965647693421295 0.0687046131574103 0.0343523065787051 52 0.950605819667737 0.0987883606645268 0.0493941803322634 53 0.96408602290369 0.0718279541926185 0.0359139770963093 54 0.921379165522133 0.157241668955734 0.078620834477867 55 0.891109436107422 0.217781127785157 0.108890563892578 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 31 0.607843137254902 NOK 5% type I error level 40 0.784313725490196 NOK 10% type I error level 45 0.88235294117647 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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