Ws7.1 multiple regression

*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 08:14:28 -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/t1258730888gt147eu3qn0uodm.htm/, Retrieved Fri, 20 Nov 2009 16:28:20 +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/t1258730888gt147eu3qn0uodm.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:

ShwWs7.1

Dataseries X:

» Textbox « » Textfile « » CSV «

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Yt[t] = + 365.825961771422 -2.39632617023324Xt[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) 365.825961771422 194.027118 1.8854 0.064383 0.032191 Xt -2.39632617023324 1.775122 -1.35 0.182277 0.091139 Multiple Linear Regression - Regression Statistics Multiple R 0.174536352936846 R-squared 0.0304629384964952 Adjusted R-squared 0.0137467822636762 F-TEST (value) 1.82236502651769 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.182277044430929 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 25.4697443187088 Sum Squared Residuals 37625.0567883032 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 151.7 113.732448662885 37.9675513371147 2 121.3 113.732448662885 7.56755133711488 3 133 112.773918194792 20.2260818052082 4 119.6 112.773918194792 6.82608180520817 5 122.2 111.336122492652 10.8638775073481 6 117.4 111.096489875629 6.30351012437146 7 106.7 110.856857258605 -4.1568572586052 8 87.5 109.658694173489 -22.1586941734886 9 81 108.939796322419 -27.9397963224186 10 110.3 108.700163705395 1.59983629460468 11 87 108.700163705395 -21.7001637053953 12 55.7 108.460531088372 -52.760531088372 13 146 108.101082162837 37.898917837163 14 137.5 107.334257788362 30.1657422116376 15 138.5 106.136094703246 32.3639052967543 16 135.6 106.112131441543 29.4878685584566 17 107.3 107.214441479851 0.085558520149303 18 99 106.950845601125 -7.95084560112504 19 91.4 106.687249722399 -15.2872497223994 20 68.4 106.303837535162 -37.9038375351621 21 82.6 105.489086637283 -22.8890866372828 22 98.4 105.441160113878 -7.0411601138781 23 71.3 104.530556169189 -33.2305561691895 24 47.6 104.554519430892 -56.9545194308918 25 130.8 104.554519430892 26.2454805691082 26 113.6 103.332393084073 10.2676069159272 27 125.7 102.829164588324 22.8708354116761 28 113.6 102.637458494705 10.9625415052948 29 97.1 103.068797205347 -5.96879720534719 30 104.4 102.661421756408 1.73857824359244 31 91.8 102.349899354277 -10.5498993542772 32 75.1 101.942523905338 -26.8425239053376 33 89.2 101.870634120231 -12.6706341202306 34 110.2 101.750817811719 8.4491821882811 35 78.4 102.254046307468 -23.8540463074679 36 68.4 101.894597381933 -33.4945973819329 37 122.8 101.894597381933 20.9054026180671 38 129.7 100.289058847877 29.4109411521234 39 159.1 99.9056466606393 59.1943533393607 40 139 99.8577201372347 39.1422798627653 41 102.2 102.877091111729 -0.677091111728526 42 113.6 102.685385018110 10.9146149818901 43 81.5 102.182156522361 -20.6821565223609 44 77.4 101.798744335124 -24.3987443351236 45 87.6 101.726854550017 -14.1268545500166 46 101.2 101.631001503207 -0.431001503207252 47 87.2 101.367405624482 -14.1674056244816 48 64.9 101.007956698947 -36.1079566989466 49 133.1 100.696434296816 32.4035657031837 50 118 99.953573184044 18.0464268159560 51 135.9 99.4024181648903 36.4975818351097 52 125.7 99.378454903188 26.321545096812 53 108 98.1802918180714 9.8197081819286 54 128.3 98.156328556369 30.1436714436310 55 84.7 97.8448061542387 -13.1448061542387 56 86.4 97.9885857244527 -11.5885857244527 57 92.2 98.084438771262 -5.88443877126205 58 95.8 97.4134674435967 -1.61346744359675 59 92.3 97.7729163691317 -5.47291636913175 60 54.3 97.3415776584897 -43.0415776584898 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.171100588575401 0.342201177150802 0.8288994114246 6 0.0734769637408745 0.146953927481749 0.926523036259125 7 0.0348935994022638 0.0697871988045277 0.965106400597736 8 0.0207043254936354 0.0414086509872708 0.979295674506365 9 0.00893283742056554 0.0178656748411311 0.991067162579435 10 0.0178256647300455 0.0356513294600910 0.982174335269955 11 0.00814155488689397 0.0162831097737879 0.991858445113106 12 0.028464636945616 0.056929273891232 0.971535363054384 13 0.386808980071187 0.773617960142373 0.613191019928813 14 0.578222635484976 0.843554729030048 0.421777364515024 15 0.695382826939092 0.609234346121817 0.304617173060908 16 0.732788765873204 0.534422468253592 0.267211234126796 17 0.668857461697494 0.662285076605011 0.331142538302506 18 0.603762285896497 0.792475428207005 0.396237714103503 19 0.546885769378923 0.906228461242153 0.453114230621077 20 0.59327439828207 0.813451203435861 0.406725601717931 21 0.538659595603253 0.922680808793494 0.461340404396747 22 0.458786733157614 0.917573466315228 0.541213266842386 23 0.44434489542246 0.88868979084492 0.55565510457754 24 0.628596083418815 0.74280783316237 0.371403916581185 25 0.705638263394662 0.588723473210675 0.294361736605338 26 0.681884511935282 0.636230976129436 0.318115488064718 27 0.710055893833342 0.579888212333317 0.289944106166658 28 0.672036340915984 0.655927318168032 0.327963659084016 29 0.600194082004376 0.799611835991248 0.399805917995624 30 0.530671788602366 0.938656422795268 0.469328211397634 31 0.457995878858921 0.915991757717842 0.542004121141079 32 0.438387727190830 0.876775454381661 0.56161227280917 33 0.373645422674904 0.747290845349807 0.626354577325096 34 0.321875919367459 0.643751838734918 0.678124080632541 35 0.296991449989887 0.593982899979775 0.703008550010113 36 0.332040847216316 0.664081694432632 0.667959152783684 37 0.321647036767394 0.643294073534788 0.678352963232606 38 0.356066464908364 0.712132929816729 0.643933535091636 39 0.688376509419476 0.623246981161048 0.311623490580524 40 0.782321633474313 0.435356733051375 0.217678366525687 41 0.715754355383872 0.568491289232256 0.284245644616128 42 0.665456523628025 0.66908695274395 0.334543476371975 43 0.622454116164237 0.755091767671525 0.377545883835763 44 0.612826686231173 0.774346627537653 0.387173313768827 45 0.565619319934384 0.868761360131233 0.434380680065616 46 0.480480804118231 0.960961608236463 0.519519195881769 47 0.475408062274529 0.950816124549058 0.524591937725471 48 0.940428382975215 0.119143234049569 0.0595716170247847 49 0.925951592988213 0.148096814023574 0.0740484070117868 50 0.940842398052434 0.118315203895133 0.0591576019475664 51 0.90181314500877 0.196373709982458 0.098186854991229 52 0.885004227995472 0.229991544009057 0.114995772004528 53 0.796074174790923 0.407851650418154 0.203925825209077 54 0.862796489493782 0.274407021012436 0.137203510506218 55 0.733258260294772 0.533483479410456 0.266741739705228 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 4 0.0784313725490196 NOK 10% type I error level 6 0.117647058823529 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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