WS 7 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: Tue, 23 Nov 2010 08:46:58 +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/Nov/23/t1290501907mg793g75w3u8m8e.htm/, Retrieved Tue, 23 Nov 2010 09:45:17 +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/Nov/23/t1290501907mg793g75w3u8m8e.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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151.7 105.2 121.3 105.2 133.0 105.6 119.6 105.6 122.2 106.2 117.4 106.3 106.7 106.4 87.5 106.9 81.0 107.2 110.3 107.3 87.0 107.3 55.7 107.4 146.0 107.55 137.5 107.87 138.5 108.37 135.6 108.38 107.3 107.92 99.0 108.03 91.4 108.14 68.4 108.3 82.6 108.64 98.4 108.66 71.3 109.04 47.6 109.03 130.8 109.03 113.6 109.54 125.7 109.75 113.6 109.83 97.1 109.65 104.4 109.82 91.8 109.95 75.1 110.12 89.2 110.15 110.2 110.2 78.4 109.99 68.4 110.14 122.8 110.14 129.7 110.81 159.1 110.97 139.0 110.99 102.2 109.73 113.6 109.81 81.5 110.02 77.4 110.18 87.6 110.21 101.2 110.25 87.2 110.36 64.9 110.51 133.1 110.64 118.0 110.95 135.9 111.18 125.7 111.19 108.0 111.69 128.3 111.7 84.7 111.83 86.4 111.77 92.2 111.73 95.8 112.01 92.3 111.86 54.3 112.04

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 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Yt[t] = + 247.380841890874 -1.77464261498657Xt[t] + 78.1125743122356M1[t] + 65.7367308094882M2[t] + 80.5308594646116M3[t] + 68.6751867579987M4[t] + 48.8929798102284M5[t] + 54.0815320866646M6[t] + 32.8446193529302M7[t] + 20.7564387499451M8[t] + 28.3924274457508M9[t] + 45.068078292647M10[t] + 25.0159548712641M11[t] + 0.158264129372544t + 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) 247.380841890874 262.863954 0.9411 0.351571 0.175785 Xt -1.77464261498657 2.47651 -0.7166 0.47725 0.238625 M1 78.1125743122356 6.167986 12.6642 0 0 M2 65.7367308094882 6.145482 10.6968 0 0 M3 80.5308594646116 6.171861 13.0481 0 0 M4 68.6751867579987 6.145933 11.1741 0 0 M5 48.8929798102284 6.122277 7.9861 0 0 M6 54.0815320866646 6.117187 8.8409 0 0 M7 32.8446193529302 6.110394 5.3752 2e-06 1e-06 M8 20.7564387499451 6.106396 3.3991 0.001406 0.000703 M9 28.3924274457508 6.104948 4.6507 2.8e-05 1.4e-05 M10 45.068078292647 6.102417 7.3853 0 0 M11 25.0159548712641 6.098994 4.1017 0.000166 8.3e-05 t 0.158264129372544 0.264939 0.5974 0.553196 0.276598 Multiple Linear Regression - Regression Statistics Multiple R 0.943285641139972 R-squared 0.889787800780848 Adjusted R-squared 0.858640874914566 F-TEST (value) 28.5674356628588 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 9.64255873554797 Sum Squared Residuals 4277.03119255066 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 151.7 138.959277235895 12.7407227641054 2 121.3 126.741697862520 -5.44169786251965 3 133 140.984233601021 -7.98423360102095 4 119.6 129.286825023781 -9.68682502378063 5 122.2 108.598096636391 13.6019033636092 6 117.4 113.767448780701 3.63255121929910 7 106.7 92.5113359148405 14.1886640851595 8 87.5 79.6940981337347 7.80590186626531 9 81 86.9559581744169 -5.95595817441691 10 110.3 103.612408889187 6.68759111081307 11 87 83.7185495971766 3.28145040282337 12 55.7 58.6833945937864 -2.98339459378642 13 146 136.688036643147 9.31196335685345 14 137.5 123.902571632976 13.597428367024 15 138.5 137.967643109979 0.532356890021326 16 135.6 126.252488106589 9.3475118934115 17 107.3 107.444880891084 -0.144880891084483 18 99 112.596486609245 -13.5964866092447 19 91.4 91.3226273172343 0.0773726827656793 20 68.4 79.108768025224 -10.7087680252240 21 82.6 86.2996423613068 -3.69964236130678 22 98.4 103.098064485276 -4.69806448527572 23 71.3 82.5298409995705 -11.2298409995705 24 47.6 57.6898966838288 -10.0898966838288 25 130.8 135.960735125437 -5.16073512543691 26 113.6 122.838088018419 -9.23808801841893 27 125.7 137.417805853768 -11.7178058537677 28 113.6 125.578425867328 -11.9784258673285 29 97.1 106.273918719628 -9.17391871962822 30 104.4 111.319045880889 -6.91904588088926 31 91.8 90.0096937365792 1.79030626342085 32 75.1 77.778088018419 -2.67808801841894 33 89.2 85.5191015651476 3.68089843485243 34 110.2 102.264284410667 7.93571558933307 35 78.4 82.7431000678038 -4.34310006780376 36 68.4 57.6192129336643 10.7807870663358 37 122.8 135.890051375272 -13.0900513752724 38 129.7 122.483461449857 7.21653855014348 39 159.1 137.151911415955 21.9480885840454 40 139 125.419009986415 13.5809900135854 41 102.2 108.031116862900 -5.83111686289982 42 113.6 113.235961859510 0.364038140490366 43 81.5 91.7846383060006 -10.2846383060006 44 77.4 79.5707790139903 -2.17077901399025 45 87.6 87.311792560719 0.288207439281084 46 101.2 104.074721832388 -2.87472183238813 47 87.2 83.9856518527292 3.21434814727076 48 64.9 58.8617647185897 6.03823528141027 49 133.1 136.901899620250 -3.8018996202496 50 118 124.134181036229 -6.1341810362289 51 135.9 138.678406019278 -2.77840601927798 52 125.7 126.963251015888 -1.26325101588779 53 108 106.451986889997 1.54801311000334 54 128.3 111.781056869656 16.5189431303445 55 84.7 90.4717047253454 -5.77170472534544 56 86.4 78.6482668086321 7.75173319136786 57 92.2 86.5135053384098 5.68649466159017 58 95.8 102.850520382482 -7.05052038248228 59 92.3 83.2228574827199 9.0771425172801 60 54.3 58.0457310701308 -3.74573107013079 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.216429971759887 0.432859943519774 0.783570028240113 18 0.104564506953540 0.209129013907079 0.89543549304646 19 0.0541140938956981 0.108228187791396 0.945885906104302 20 0.0334247402823049 0.0668494805646097 0.966575259717695 21 0.259038010175747 0.518076020351493 0.740961989824253 22 0.186319109823230 0.372638219646459 0.81368089017677 23 0.143237867082294 0.286475734164588 0.856762132917706 24 0.100827552130177 0.201655104260353 0.899172447869823 25 0.068930099092351 0.137860198184702 0.931069900907649 26 0.0566050765816964 0.113210153163393 0.943394923418304 27 0.060780564506705 0.12156112901341 0.939219435493295 28 0.065116785863892 0.130233571727784 0.934883214136108 29 0.0455646093874829 0.0911292187749659 0.954435390612517 30 0.141047999970787 0.282095999941575 0.858952000029213 31 0.117013039181407 0.234026078362814 0.882986960818593 32 0.201073127543708 0.402146255087416 0.798926872456292 33 0.502199229976008 0.995601540047983 0.497800770023992 34 0.584692248936274 0.830615502127451 0.415307751063726 35 0.787962707135546 0.424074585728908 0.212037292864454 36 0.840234919535031 0.319530160929937 0.159765080464969 37 0.957786161255727 0.0844276774885454 0.0422138387442727 38 0.947951486291878 0.104097027416245 0.0520485137081223 39 0.96878761837345 0.0624247632530994 0.0312123816265497 40 0.946224573400444 0.107550853199112 0.0537754265995561 41 0.889695624402324 0.220608751195353 0.110304375597676 42 0.878928973320536 0.242142053358927 0.121071026679464 43 0.787526801025403 0.424946397949195 0.212473198974597 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 4 0.148148148148148 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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