Workshop 7: type of equation lineair trend

*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: Thu, 19 Nov 2009 11:35:38 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Nov/19/t12586558015ktn98f13cd8b4s.htm/, Retrieved Thu, 19 Nov 2009 19:36:52 +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/2009/Nov/19/t12586558015ktn98f13cd8b4s.htm/}, year = {2009}, } @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 = {2009}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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Dataseries X:

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0,7461 0,5270 0,7775 0,4720 0,7790 0,0000 0,7744 0,0520 0,7905 0,3130 0,7719 0,3640 0,7811 0,3630 0,7557 -0,1550 0,7637 0,0520 0,7595 0,5680 0,7471 0,6680 0,7615 1,3780 0,7487 0,2520 0,7389 -0,4020 0,7337 -0,0500 0,7510 0,5550 0,7382 0,0500 0,7159 0,1500 0,7542 0,4500 0,7636 0,2990 0,7433 0,1990 0,7658 0,4960 0,7627 0,4440 0,7480 -0,3930 0,7692 -0,4440 0,7850 0,1980 0,7913 0,4940 0,7720 0,1330 0,7880 0,3880 0,8070 0,4840 0,8268 0,2780 0,8244 0,3690 0,8487 0,1650 0,8572 0,1550 0,8214 0,0870 0,8827 0,4140 0,9216 0,3600 0,8865 0,9750 0,8816 0,2700 0,8884 0,3590 0,9466 0,1690 0,9180 0,3810 0,9337 0,1540 0,9559 0,4860 0,9626 0,9250 0,9434 0,7280 0,8639 -0,0140 0,7996 0,0460 0,6680 -0,8190 0,6572 -1,6740 0,6928 -0,7880 0,6438 0,2790 0,6454 0,3960 0,6873 -0,1410 0,7265 -0,0190 0,7912 0,0990 0,8114 0,7420 0,8281 0,0050 0,8393 0,4480

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.728300505019516 + 0.0833205341372568X[t] + 0.0115275508742551M1[t] + 0.0136250999361690M2[t] + 0.0130176820646555M3[t] -0.0222569327829172M4[t] -0.00672208989372865M5[t] -0.00846062129530073M6[t] + 0.0148610162525153M7[t] + 0.0293756901923157M8[t] + 0.0194232132331628M9[t] + 0.0251478794934456M10[t] + 0.00522539783728926M11[t] + 0.00131833173411334t + 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) 0.728300505019516 0.042483 17.1433 0 0 X 0.0833205341372568 0.023619 3.5278 0.000978 0.000489 M1 0.0115275508742551 0.050252 0.2294 0.8196 0.4098 M2 0.0136250999361690 0.050494 0.2698 0.788521 0.394261 M3 0.0130176820646555 0.050098 0.2598 0.79617 0.398085 M4 -0.0222569327829172 0.049261 -0.4518 0.65357 0.326785 M5 -0.00672208989372865 0.049258 -0.1365 0.892061 0.44603 M6 -0.00846062129530073 0.049266 -0.1717 0.864417 0.432209 M7 0.0148610162525153 0.049265 0.3017 0.764303 0.382152 M8 0.0293756901923157 0.049301 0.5958 0.554262 0.277131 M9 0.0194232132331628 0.049254 0.3943 0.695186 0.347593 M10 0.0251478794934456 0.049262 0.5105 0.612205 0.306102 M11 0.00522539783728926 0.049275 0.106 0.916018 0.458009 t 0.00131833173411334 0.000589 2.2393 0.030122 0.015061 Multiple Linear Regression - Regression Statistics Multiple R 0.553438452854356 R-squared 0.306294121097823 Adjusted R-squared 0.105890200526083 F-TEST (value) 1.52838387704185 F-TEST (DF numerator) 13 F-TEST (DF denominator) 45 p-value 0.144426630482259 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0733325746213827 Sum Squared Residuals 0.24199499252703 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 0.7461 0.785056309118216 -0.0389563091182163 2 0.7775 0.783889560536696 -0.00638956053669647 3 0.779 0.745273182286511 0.0337268177134889 4 0.7744 0.715649566948189 0.0587504330518109 5 0.7905 0.754249400981315 0.0362505990186848 6 0.7719 0.758078548554856 0.0138214514451435 7 0.7811 0.782635197302649 -0.00153519730264859 8 0.7557 0.755308166293463 0.000391833706536697 9 0.7637 0.763921371634836 -0.000221371634835889 10 0.7595 0.813957765244056 -0.0544577652440566 11 0.7471 0.80368566873574 -0.0565856687357393 12 0.7615 0.858936181870016 -0.0974361818700156 13 0.7487 0.777963143039833 -0.0292631430398330 14 0.7389 0.726887394510094 0.0120126054899058 15 0.7337 0.756927136389008 -0.0232271363890084 16 0.751 0.77337977642859 -0.0223797764285894 17 0.7382 0.748156081312577 -0.00995608131257673 18 0.7159 0.756067935058844 -0.0401679350588436 19 0.7542 0.80570406458195 -0.0515040645819501 20 0.7636 0.808955669601138 -0.0453556696011381 21 0.7433 0.791989470962373 -0.0486894709623728 22 0.7658 0.823778667595534 -0.0579786675955342 23 0.7627 0.800841849898354 -0.0381418498983538 24 0.748 0.727195496722294 0.0208045032777060 25 0.7692 0.735792032089662 0.0334079679103376 26 0.785 0.792699695801808 -0.00769969580180838 27 0.7913 0.818073487769036 -0.0267734877690363 28 0.772 0.754038491832027 0.0179615081679728 29 0.788 0.79213840266033 -0.00413840266032959 30 0.807 0.799716974270048 0.00728302572995251 31 0.8268 0.807192913519702 0.0196070864802979 32 0.8244 0.830608087800106 -0.00620808780010616 33 0.8487 0.804976553611066 0.0437234463889338 34 0.8572 0.81118634626409 0.0460136537359102 35 0.8214 0.786916400020713 0.0344835999792867 36 0.8827 0.81025514858042 0.0724448514195797 37 0.9216 0.818601722345377 0.102998277654623 38 0.8865 0.873259731635817 0.0132402683641829 39 0.8816 0.815229668931651 0.0663703310683491 40 0.8884 0.788688913356407 0.0997110866435926 41 0.9466 0.78971118649363 0.156888813506369 42 0.918 0.80695494006327 0.111045059936730 43 0.9337 0.812681148096042 0.121018851903958 44 0.9559 0.856176571103525 0.0997234288964746 45 0.9626 0.884120140364742 0.0784798596352585 46 0.9434 0.874748993134098 0.068651006865902 47 0.8639 0.79432100688221 0.0695789931177895 48 0.7996 0.79541317282727 0.00418682717273 49 0.668 0.736186793406911 -0.0681867934069113 50 0.6572 0.668363617515584 -0.0111636175155839 51 0.6928 0.742896524623793 -0.0500965246237933 52 0.6438 0.797843251434787 -0.154043251434787 53 0.6454 0.824444928552148 -0.179044928552148 54 0.6873 0.779281602052982 -0.0919816020529823 55 0.7265 0.814086676499657 -0.087586676499657 56 0.7912 0.839751505201767 -0.048551505201767 57 0.8114 0.884692463426984 -0.0732924634269837 58 0.8281 0.830328227762221 -0.00222822776222153 59 0.8393 0.848635074462983 -0.00933507446298322 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0121420089510593 0.0242840179021186 0.98785799104894 18 0.00323914096999308 0.00647828193998616 0.996760859030007 19 0.000603548102345543 0.00120709620469109 0.999396451897654 20 0.000254145043654589 0.000508290087309177 0.999745854956345 21 4.76750949686066e-05 9.53501899372132e-05 0.999952324905031 22 3.25283980094326e-05 6.50567960188653e-05 0.99996747160199 23 3.58535201936147e-05 7.17070403872295e-05 0.999964146479806 24 1.14127261060168e-05 2.28254522120336e-05 0.999988587273894 25 1.55192060463475e-05 3.10384120926951e-05 0.999984480793954 26 1.41820830104169e-05 2.83641660208339e-05 0.99998581791699 27 1.2216401943403e-05 2.4432803886806e-05 0.999987783598057 28 3.44765160223052e-06 6.89530320446103e-06 0.999996552348398 29 1.32969380578783e-06 2.65938761157567e-06 0.999998670306194 30 2.54207828090322e-06 5.08415656180644e-06 0.99999745792172 31 3.60835167831222e-06 7.21670335662444e-06 0.999996391648322 32 7.27932903332751e-06 1.45586580666550e-05 0.999992720670967 33 2.56437931760776e-05 5.12875863521551e-05 0.999974356206824 34 0.000221865957431791 0.000443731914863582 0.999778134042568 35 0.0132290382284959 0.0264580764569917 0.986770961771504 36 0.0452769430373638 0.0905538860747277 0.954723056962636 37 0.0832293371364838 0.166458674272968 0.916770662863516 38 0.0504177632716821 0.100835526543364 0.949582236728318 39 0.0302525147176625 0.0605050294353249 0.969747485282338 40 0.0267786894012801 0.0535573788025603 0.97322131059872 41 0.327723850406819 0.655447700813637 0.672276149593181 42 0.318630719967677 0.637261439935354 0.681369280032323 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 17 0.653846153846154 NOK 5% type I error level 19 0.730769230769231 NOK 10% type I error level 22 0.846153846153846 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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