JJ Workshop 7, Multiple Regression (Monthly Dummies & Linear 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: Fri, 20 Nov 2009 12:16:41 -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/20/t1258744734dwwy5rx8wptjl2j.htm/, Retrieved Fri, 20 Nov 2009 20:19:06 +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/20/t1258744734dwwy5rx8wptjl2j.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}, }

Original text written by user:

IsPrivate?

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User-defined keywords:

Dataseries X:

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95.1 93.8 97 93.8 112.7 107.6 102.9 101 97.4 95.4 111.4 96.5 87.4 89.2 96.8 87.1 114.1 110.5 110.3 110.8 103.9 104.2 101.6 88.9 94.6 89.8 95.9 90 104.7 93.9 102.8 91.3 98.1 87.8 113.9 99.7 80.9 73.5 95.7 79.2 113.2 96.9 105.9 95.2 108.8 95.6 102.3 89.7 99 92.8 100.7 88 115.5 101.1 100.7 92.7 109.9 95.8 114.6 103.8 85.4 81.8 100.5 87.1 114.8 105.9 116.5 108.1 112.9 102.6 102 93.7 106 103.5 105.3 100.6 118.8 113.3 106.1 102.4 109.3 102.1 117.2 106.9 92.5 87.3 104.2 93.1 112.5 109.1 122.4 120.3 113.3 104.9 100 92.6 110.7 109.8 112.8 111.4 109.8 117.9 117.3 121.6 109.1 117.8 115.9 124.2 96 106.8 99.8 102.7 116.8 116.8 115.7 113.6 99.4 96.1 94.3 85

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 TIA[t] = + 65.1843542853472 + 0.370552244553412IAidM[t] -1.44718191533665M1[t] + 0.208032101299208M2[t] + 6.42047202382795M3[t] + 1.87637352487574M4[t] + 1.38285142693650M5[t] + 8.79445734007539M6[t] -10.5523637676236M7[t] -0.419972158013988M8[t] + 7.74804980808746M9[t] + 6.93384022573633M10[t] + 3.69712861521563M11[t] + 0.0420376319371333t + 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) 65.1843542853472 6.186548 10.5365 0 0 IAidM 0.370552244553412 0.074382 4.9818 9e-06 5e-06 M1 -1.44718191533665 2.527046 -0.5727 0.569652 0.284826 M2 0.208032101299208 2.488868 0.0836 0.933749 0.466875 M3 6.42047202382795 2.784987 2.3054 0.025706 0.012853 M4 1.87637352487574 2.5993 0.7219 0.474024 0.237012 M5 1.38285142693650 2.531074 0.5463 0.587466 0.293733 M6 8.79445734007539 2.723066 3.2296 0.00229 0.001145 M7 -10.5523637676236 2.366246 -4.4595 5.3e-05 2.6e-05 M8 -0.419972158013988 2.365497 -0.1775 0.859863 0.429932 M9 7.74804980808746 2.745065 2.8225 0.007015 0.003508 M10 6.93384022573633 2.800864 2.4756 0.017044 0.008522 M11 3.69712861521563 2.499458 1.4792 0.14591 0.072955 t 0.0420376319371333 0.036484 1.1522 0.255178 0.127589 Multiple Linear Regression - Regression Statistics Multiple R 0.9304892192588 R-squared 0.86581018715685 Adjusted R-squared 0.82788697917944 F-TEST (value) 22.8306156924424 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 8.88178419700125e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.73365659029078 Sum Squared Residuals 641.248810574203 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 95.1 98.5370105410575 -3.4370105410575 2 97 100.234262189631 -3.23426218963068 3 112.7 111.602360718934 1.09763928106633 4 102.9 104.654655037866 -1.75465503786607 5 97.4 102.128078002365 -4.72807800236485 6 111.4 109.989329016450 1.41067098355037 7 87.4 87.9795141554479 -0.579514155447851 8 96.8 97.3757836834324 -0.57578368343245 9 114.1 114.256765804021 -0.156765804020882 10 110.3 113.595759526973 -3.29575952697290 11 103.9 107.955440734337 -4.05544073433680 12 101.6 98.6309004093911 2.96909959060888 13 94.6 97.5592531460897 -2.95925314608968 14 95.9 99.3306152435733 -3.43061524357334 15 104.7 107.030246551798 -2.33024655179752 16 102.8 101.564749848944 1.23525015105642 17 98.1 99.8163325270045 -1.71633252700453 18 113.9 111.679547782266 2.22045221773385 19 80.9 82.6662954992049 -1.76629549920488 20 95.7 94.9528725347061 0.747127465293902 21 113.2 109.72170686134 3.47829313865993 22 105.9 108.319596095185 -2.41959609518527 23 108.8 105.273143014423 3.52685698557693 24 102.3 99.4317937882794 2.86820621172055 25 99 99.1753614629955 -0.175361462995504 26 100.7 99.0939623377121 1.60603766228788 27 115.5 110.202674295828 5.29732570417232 28 100.7 102.587974574564 -1.88797457456395 29 109.9 103.285202066677 6.61479793332259 30 114.6 113.703263568181 0.89673643181925 31 85.4 86.2463307122438 -0.846330712243796 32 100.5 98.3846868499237 2.11531315007635 33 114.8 113.561128645566 1.23887135443362 34 116.5 113.60417163317 2.89582836683012 35 112.9 108.371460309543 4.52853969045745 36 102 101.418454349739 0.581545650261309 37 106 103.644722062963 2.35527793703739 38 105.3 104.267372202331 1.03262779766929 39 118.8 115.227863262625 3.57213673737509 40 106.1 106.686782929978 -0.586782929977654 41 109.3 106.124132790610 3.17586720939048 42 117.2 115.356427109542 1.84357289045808 43 92.5 88.7888196405332 3.71118035946683 44 104.2 101.112451900490 3.08754809951028 45 112.5 115.251347411383 -2.75134741138289 46 122.4 118.629360599967 3.77063940003289 47 113.3 109.728182055261 3.57181794473899 48 100 101.515298463976 -1.51529846397553 49 110.7 106.483652786895 4.2163472131053 50 112.8 108.773788026753 4.02621197324684 51 109.8 117.436855170816 -7.63685517081621 52 117.3 114.305837608649 2.99416239135124 53 109.1 112.446254613344 -3.34625461334369 54 115.9 122.271432523562 -6.37143252356154 55 96 96.5190399925703 -0.519039992570301 56 99.8 105.174205031448 -5.37420503144808 57 116.8 118.609051277690 -1.80905127768977 58 115.7 116.651112144705 -0.951112144704847 59 99.4 106.971773886437 -7.57177388643657 60 94.3 99.2035529886152 -4.90355298861521 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.224211242531592 0.448422485063184 0.775788757468408 18 0.116183487750958 0.232366975501916 0.883816512249042 19 0.0565283374134823 0.113056674826965 0.943471662586518 20 0.0255255383285234 0.0510510766570468 0.974474461671477 21 0.0280107804121861 0.0560215608243722 0.971989219587814 22 0.0176978510607724 0.0353957021215447 0.982302148939228 23 0.075174654112461 0.150349308224922 0.924825345887539 24 0.0467506426477133 0.0935012852954266 0.953249357352287 25 0.036097069275655 0.07219413855131 0.963902930724345 26 0.0285603657120726 0.0571207314241452 0.971439634287927 27 0.0255791975982945 0.0511583951965891 0.974420802401705 28 0.0554206358997861 0.110841271799572 0.944579364100214 29 0.106530034244611 0.213060068489223 0.893469965755389 30 0.129282709774856 0.258565419549712 0.870717290225144 31 0.149146557746801 0.298293115493602 0.850853442253199 32 0.101538825682446 0.203077651364893 0.898461174317554 33 0.0808335943994812 0.161667188798962 0.919166405600519 34 0.0728765485654204 0.145753097130841 0.92712345143458 35 0.0461884604046193 0.0923769208092387 0.95381153959538 36 0.0593088644541373 0.118617728908275 0.940691135545863 37 0.0619569811460214 0.123913962292043 0.938043018853979 38 0.0919086117977765 0.183817223595553 0.908091388202223 39 0.107441985500982 0.214883971001965 0.892558014499018 40 0.267543810663302 0.535087621326604 0.732456189336698 41 0.175603307795952 0.351206615591903 0.824396692204048 42 0.159416224599055 0.318832449198109 0.840583775400945 43 0.0905113834670389 0.181022766934078 0.909488616532961 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 1 0.0370370370370370 OK 10% type I error level 8 0.296296296296296 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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