TG 6

*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: Wed, 18 Nov 2009 11:08:13 -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/18/t1258567717sw6dj3orzi3vxnr.htm/, Retrieved Wed, 18 Nov 2009 19:08:49 +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/18/t1258567717sw6dj3orzi3vxnr.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?

No (this computation is public)

User-defined keywords:

Dataseries X:

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94 0 106.3 101.3 102.8 1 94 106.3 102 1 102.8 94 105.1 1 102 102.8 92.4 0 105.1 102 81.4 0 92.4 105.1 105.8 1 81.4 92.4 120.3 1 105.8 81.4 100.7 1 120.3 105.8 88.8 0 100.7 120.3 94.3 0 88.8 100.7 99.9 0 94.3 88.8 103.4 1 99.9 94.3 103.3 1 103.4 99.9 98.8 0 103.3 103.4 104.2 1 98.8 103.3 91.2 0 104.2 98.8 74.7 0 91.2 104.2 108.5 1 74.7 91.2 114.5 1 108.5 74.7 96.9 0 114.5 108.5 89.6 0 96.9 114.5 97.1 0 89.6 96.9 100.3 1 97.1 89.6 122.6 1 100.3 97.1 115.4 1 122.6 100.3 109 1 115.4 122.6 129.1 1 109 115.4 102.8 1 129.1 109 96.2 0 102.8 129.1 127.7 1 96.2 102.8 128.9 1 127.7 96.2 126.5 1 128.9 127.7 119.8 1 126.5 128.9 113.2 1 119.8 126.5 114.1 1 113.2 119.8 134.1 1 114.1 113.2 130 1 134.1 114.1 121.8 1 130 134.1 132.1 1 121.8 130 105.3 1 132.1 121.8 103 1 105.3 132.1 117.1 1 103 105.3 126.3 1 117.1 103 138.1 1 126.3 117.1 119.5 1 138.1 126.3 138 1 119.5 138.1 135.5 1 138 119.5 178.6 1 135.5 138 162.2 1 178.6 135.5 176.9 1 162.2 178.6 204.9 1 176.9 162.2 132.2 1 etc...

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 Omzet[t] = + 44.1021461554302 + 14.4797682265509Uitvoer[t] + 0.371730114647078`Omzet-1`[t] + 0.0213314976228769`Omzet-2`[t] + 0.59215776931114t + 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) 44.1021461554302 12.581856 3.5052 0.000937 0.000468 Uitvoer 14.4797682265509 5.517833 2.6242 0.011322 0.005661 `Omzet-1` 0.371730114647078 0.13168 2.823 0.006687 0.003344 `Omzet-2` 0.0213314976228769 0.134712 0.1583 0.874784 0.437392 t 0.59215776931114 0.215651 2.7459 0.008223 0.004112 Multiple Linear Regression - Regression Statistics Multiple R 0.843792683885146 R-squared 0.711986093378098 Adjusted R-squared 0.690249194765124 F-TEST (value) 32.7547230198308 F-TEST (DF numerator) 4 F-TEST (DF denominator) 53 p-value 9.39248678832882e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 14.9583866316309 Sum Squared Residuals 11858.9265229317 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 94 86.3700958209234 7.62990417907663 2 102.8 96.9763988947406 5.82360110525942 3 102 100.577404252185 1.42259574781541 4 105.1 101.059895108859 4.04010489114062 5 92.4 88.3075828089273 4.09241719107274 6 81.4 84.2448957648514 -2.84489576485143 7 105.8 94.956880479785 10.8431195202149 8 120.3 104.384606572633 15.9153934273667 9 100.7 110.887339546325 -10.1873395463252 10 88.8 90.0231255575345 -1.22312555753448 11 94.3 85.773597609137 8.526402390863 12 99.9 88.1564261872948 11.7435738127052 13 103.4 105.427364062106 -2.02736406210632 14 103.3 107.440033619370 -4.14003361937035 15 98.8 93.589910392346 5.21008960765403 16 104.2 106.986917722534 -2.78691772253385 17 91.2 95.0106581450854 -3.81065814508538 18 74.7 90.885514511148 -16.1855145111480 19 108.5 99.5465841462359 8.95341585376413 20 114.5 112.351250079841 2.14874992015921 21 96.9 101.415024930137 -4.51502493013675 22 89.6 95.5927216673966 -5.99272166739659 23 97.1 93.0958152416214 4.00418475837858 24 100.3 110.799997164690 -10.4999971646895 25 122.6 112.741677533043 9.8583224669571 26 115.4 121.691677651377 -6.29167765137708 27 109 120.083070992219 -11.0830709922194 28 129.1 118.142569244905 10.9574307550954 29 102.8 126.069980733836 -23.2699807338355 30 96.2 102.834631363597 -6.63463136359747 31 127.7 114.892120215307 12.8078797846929 32 128.9 127.052988711690 1.84701128830976 33 126.5 128.763164793699 -2.2631647936985 34 119.8 128.488768085004 -8.6887680850041 35 113.2 126.539138491885 -13.3391384918849 36 114.1 124.534956470452 -10.4349564704521 37 134.1 125.320883458635 8.7791165413654 38 130 133.366841868748 -3.36684186874788 39 121.8 132.861536120464 -11.0615361204635 40 132.1 130.318047809415 1.78195219058515 41 105.3 134.564107479083 -29.2641074790833 42 103 125.413612601368 -22.4136126013684 43 117.1 124.579106970698 -7.47910697069814 44 126.3 130.363596912000 -4.06359691200046 45 138.1 134.676445852547 3.42355414745271 46 119.5 139.851268752824 -20.3512687528244 47 138 133.78095806165 4.21904193835015 48 135.5 140.853357096146 -5.35335709614643 49 178.6 140.910822284863 37.6891777151369 50 162.2 157.471219251406 4.72878074859387 51 176.9 152.886390688051 24.0136093119488 52 204.9 158.593144581659 46.3068554183408 53 132.2 169.907318576145 -37.7073185761448 54 142.5 144.071978944054 -1.5719789440539 55 164.3 146.942157017047 17.3578429829532 56 174.9 155.85774571118 19.0422542888201 57 175.4 160.855269343929 14.5447306560712 58 143 161.859406045366 -18.8594060453659 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.0612862801563654 0.122572560312731 0.938713719843635 9 0.0205131024601551 0.0410262049203101 0.979486897539845 10 0.0182540536437750 0.0365081072875499 0.981745946356225 11 0.00618388755052202 0.0123677751010440 0.993816112449478 12 0.00199401806434940 0.00398803612869881 0.99800598193565 13 0.000955340331653993 0.00191068066330799 0.999044659668346 14 0.000284968758789604 0.000569937517579208 0.99971503124121 15 0.000139145682773082 0.000278291365546163 0.999860854317227 16 3.81509848738377e-05 7.63019697476754e-05 0.999961849015126 17 1.80120020016546e-05 3.60240040033091e-05 0.999981987997998 18 0.000146527815721147 0.000293055631442293 0.999853472184279 19 7.39335472497422e-05 0.000147867094499484 0.99992606645275 20 2.77911963089686e-05 5.55823926179372e-05 0.999972208803691 21 1.66269834376209e-05 3.32539668752419e-05 0.999983373016562 22 6.02345264521861e-06 1.20469052904372e-05 0.999993976547355 23 2.28294667367221e-06 4.56589334734442e-06 0.999997717053326 24 1.65195510408972e-06 3.30391020817944e-06 0.999998348044896 25 9.96044890369203e-06 1.99208978073841e-05 0.999990039551096 26 3.88494943319547e-06 7.76989886639095e-06 0.999996115050567 27 1.55565184085773e-06 3.11130368171545e-06 0.99999844434816 28 1.86584817473016e-05 3.73169634946033e-05 0.999981341518253 29 2.17797203294129e-05 4.35594406588258e-05 0.99997822027967 30 8.9365884207812e-06 1.78731768415624e-05 0.99999106341158 31 1.75135648060094e-05 3.50271296120188e-05 0.999982486435194 32 1.31449814469865e-05 2.62899628939729e-05 0.999986855018553 33 1.03756209695794e-05 2.07512419391589e-05 0.99998962437903 34 4.20399818477828e-06 8.40799636955656e-06 0.999995796001815 35 1.71299982160223e-06 3.42599964320446e-06 0.999998287000178 36 6.45417241473547e-07 1.29083448294709e-06 0.999999354582759 37 1.1541593582129e-06 2.3083187164258e-06 0.999998845840642 38 5.70348400384082e-07 1.14069680076816e-06 0.9999994296516 39 1.98804677932492e-07 3.97609355864983e-07 0.999999801195322 40 1.88215569477847e-07 3.76431138955694e-07 0.99999981178443 41 5.84329664509224e-07 1.16865932901845e-06 0.999999415670336 42 9.02633579550313e-07 1.80526715910063e-06 0.99999909736642 43 4.20861925435484e-07 8.41723850870969e-07 0.999999579138075 44 1.70497547201686e-07 3.40995094403371e-07 0.999999829502453 45 1.06116091344093e-07 2.12232182688187e-07 0.999999893883909 46 3.10991242624278e-07 6.21982485248556e-07 0.999999689008757 47 5.76645264919166e-07 1.15329052983833e-06 0.999999423354735 48 1.67199390254901e-05 3.34398780509803e-05 0.999983280060974 49 0.000536571172266622 0.00107314234453324 0.999463428827733 50 0.001117045781726 0.002234091563452 0.998882954218274 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 39 0.906976744186046 NOK 5% type I error level 42 0.976744186046512 NOK 10% type I error level 42 0.976744186046512 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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