Multiple Regression analysis 1

*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, 15 Dec 2009 12:14:24 -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/Dec/15/t126090455178m7g8p9cboywoe.htm/, Retrieved Tue, 15 Dec 2009 20:16:05 +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/Dec/15/t126090455178m7g8p9cboywoe.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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103.34 98.60 96.33 102.60 96.90 96.33 100.69 95.10 95.05 105.67 97.00 96.84 123.61 112.70 96.92 113.08 102.90 97.44 106.46 97.40 97.78 123.38 111.40 97.69 109.87 87.40 96.67 95.74 96.80 98.29 123.06 114.10 98.20 123.39 110.30 98.71 120.28 103.90 98.54 115.33 101.60 98.20 110.40 94.60 100.80 114.49 95.90 101.33 132.03 104.70 101.88 123.16 102.80 101.85 118.82 98.10 102.04 128.32 113.90 102.22 112.24 80.90 102.63 104.53 95.70 102.65 132.57 113.20 102.54 122.52 105.90 102.37 131.80 108.80 102.68 124.55 102.30 102.76 120.96 99.00 102.82 122.60 100.70 103.31 145.52 115.50 103.23 118.57 100.70 103.60 134.25 109.90 103.95 136.70 114.60 103.93 121.37 85.40 104.25 111.63 100.50 104.38 134.42 114.80 104.36 137.65 116.50 104.32 137.86 112.90 104.58 119.77 102.00 104.68 130.69 106.00 104.92 128.28 105.30 105.46 147.45 118.80 105.23 128.42 106.10 105.58 136.90 109.30 105.34 143.95 117.20 105.28 135.64 92.50 105.70 122.48 104.20 105.67 136.83 112.50 105.71 153.04 122.40 106.19 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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Uitvoer[t] = -153.26923086961 + 1.12814174357858TIP[t] + 1.55877189719339cons[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) -153.26923086961 23.230546 -6.5977 0 0 TIP 1.12814174357858 0.101805 11.0814 0 0 cons 1.55877189719339 0.201259 7.7451 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.859237237376913 R-squared 0.73828863009511 Adjusted R-squared 0.730476350396457 F-TEST (value) 94.5036095190456 F-TEST (DF numerator) 2 F-TEST (DF denominator) 67 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 8.00601229043713 Sum Squared Residuals 4294.44759724024 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 103.34 108.122041903877 -4.78204190387686 2 102.6 106.204200939793 -3.60420093979347 3 100.69 102.178317772944 -1.48831777294448 4 105.67 107.11198878172 -1.44198878171999 5 123.61 124.948515907679 -1.33851590767922 6 113.08 114.703288207150 -1.62328820714967 7 106.46 109.028491062513 -2.56849106251322 8 123.38 124.682186001866 -1.30218600186597 9 109.87 96.0168368208427 13.8531631791573 10 95.74 109.146579683935 -13.4065796839347 11 123.06 128.523142377097 -5.46314237709676 12 123.39 125.031177419067 -1.64117741906676 13 120.28 117.546079037641 2.73392096235902 14 115.33 114.421370582364 0.908629417635523 15 110.4 110.577185310017 -0.177185310017179 16 114.49 112.869918682182 1.62008131781814 17 132.03 123.654890569130 8.37510943087026 18 123.16 121.464658099415 1.69534190058537 19 118.82 116.458558565062 2.36144143493795 20 128.32 134.563777055098 -6.24377705509848 21 112.24 97.9741959948545 14.2658040051455 22 104.53 114.701869237761 -10.1718692377614 23 132.57 134.272884841695 -1.70288484169536 24 122.52 125.772458891049 -3.25245889104882 25 131.8 129.527289235557 2.27271076444336 26 124.55 122.319069654071 2.23093034592867 27 120.96 118.689728214094 2.2702717859064 28 122.6 121.371367407802 1.22863259219803 29 145.52 137.943163460990 7.57683653901048 30 118.57 121.823411257988 -3.25341125798804 31 134.25 132.747885462929 1.5021145370713 32 136.7 138.018976219804 -1.31897621980418 33 121.37 105.576044314411 15.7939556855886 34 111.63 122.813624989083 -11.1836249890832 35 134.42 138.914876484313 -4.49487648431305 36 137.65 140.770366572509 -3.12036657250888 37 137.86 137.114336988896 0.745663011103736 38 119.77 124.973469173609 -5.20346917360907 39 130.69 129.860141403250 0.829858596750194 40 128.28 129.912179007229 -1.63217900722921 41 147.45 144.783575009186 2.66642499081437 42 128.42 131.001745029755 -2.58174502975530 43 136.9 134.237693353880 2.66230664611966 44 143.95 143.056486814320 0.893513185680429 45 135.64 115.846069944750 19.7939300552502 46 122.48 128.998565187703 -6.51856518770339 47 136.83 138.424492535293 -1.59449253529335 48 153.04 150.341306307374 2.69869369262582 49 142.71 141.228707644732 1.48129235526783 50 123.46 127.019396122706 -3.55939612270564 51 144.37 139.729609256846 4.64039074315425 52 146.15 143.439250749947 2.71074925005292 53 147.61 141.005676376499 6.60432362350071 54 158.51 149.731730675862 8.77826932413844 55 147.4 141.587696425524 5.81230357447552 56 165.05 149.91374447868 15.1362555213199 57 154.64 128.398986919782 26.2410130802177 58 126.2 132.155943100335 -5.95594310033515 59 157.36 151.755221153413 5.60477884658671 60 154.15 150.732493301084 3.41750669891606 61 123.21 132.032028501314 -8.82202850131436 62 113.07 126.512321393643 -13.4423213936426 63 110.45 123.116795738244 -12.6667957382439 64 113.57 126.066339143275 -12.4963391432748 65 122.44 136.174977514611 -13.7349775146112 66 114.93 126.177579012123 -11.2475790121233 67 111.85 123.255510903482 -11.4055109034820 68 126.04 134.835220835780 -8.79522083577957 69 121.34 112.128396351906 9.21160364809383 70 124.36 119.741896626662 4.61810337333823 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.0129658699645737 0.0259317399291474 0.987034130035426 7 0.00223684556616547 0.00447369113233093 0.997763154433835 8 0.000350271337076383 0.000700542674152767 0.999649728662924 9 0.12766912257823 0.25533824515646 0.87233087742177 10 0.332837336604296 0.665674673208593 0.667162663395704 11 0.240760894092339 0.481521788184679 0.75923910590766 12 0.184184977630580 0.368369955261161 0.81581502236942 13 0.156817856786286 0.313635713572572 0.843182143213714 14 0.107954986977502 0.215909973955005 0.892045013022498 15 0.0679034366383258 0.135806873276652 0.932096563361674 16 0.0420593108001276 0.0841186216002551 0.957940689199873 17 0.0498073923983978 0.0996147847967957 0.950192607601602 18 0.0300244044362671 0.0600488088725342 0.969975595563733 19 0.0175396853088895 0.0350793706177790 0.98246031469111 20 0.0143851698281678 0.0287703396563356 0.985614830171832 21 0.0156738790464374 0.0313477580928748 0.984326120953563 22 0.0536555230570198 0.107311046114040 0.94634447694298 23 0.0357753023609180 0.0715506047218359 0.964224697639082 24 0.0244177839252718 0.0488355678505437 0.975582216074728 25 0.0168176640728641 0.0336353281457281 0.983182335927136 26 0.0102803279580920 0.0205606559161841 0.989719672041908 27 0.00601911456184617 0.0120382291236923 0.993980885438154 28 0.00340927041699307 0.00681854083398614 0.996590729583007 29 0.00480646489361353 0.00961292978722706 0.995193535106387 30 0.00368382520140429 0.00736765040280857 0.996316174798596 31 0.00209392537351895 0.0041878507470379 0.99790607462648 32 0.00116827337029879 0.00233654674059758 0.9988317266297 33 0.00322672288808389 0.00645344577616778 0.996773277111916 34 0.0103884293504766 0.0207768587009533 0.989611570649523 35 0.00738007442476398 0.0147601488495280 0.992619925575236 36 0.0048888931442997 0.0097777862885994 0.9951111068557 37 0.00296899893040774 0.00593799786081549 0.997031001069592 38 0.00267884528725848 0.00535769057451697 0.997321154712742 39 0.00152093640479060 0.00304187280958121 0.99847906359521 40 0.000910542114826722 0.00182108422965344 0.999089457885173 41 0.000629038224146714 0.00125807644829343 0.999370961775853 42 0.000402107088564968 0.000804214177129936 0.999597892911435 43 0.000221875892829267 0.000443751785658535 0.99977812410717 44 0.000131525884246863 0.000263051768493726 0.999868474115753 45 0.00211163137845242 0.00422326275690483 0.997888368621548 46 0.00209358215029974 0.00418716430059947 0.9979064178497 47 0.00128356033659012 0.00256712067318024 0.99871643966341 48 0.000949865220300104 0.00189973044060021 0.9990501347797 49 0.00056544274298929 0.00113088548597858 0.99943455725701 50 0.000567352507194702 0.00113470501438940 0.999432647492805 51 0.000366341197972551 0.000732682395945103 0.999633658802027 52 0.000287202358291658 0.000574404716583316 0.999712797641708 53 0.000171988548562168 0.000343977097124337 0.999828011451438 54 0.000141062061682101 0.000282124123364202 0.999858937938318 55 6.87510380037295e-05 0.000137502076007459 0.999931248961996 56 0.000125713550720363 0.000251427101440726 0.99987428644928 57 0.149569015881361 0.299138031762721 0.85043098411864 58 0.153266611596004 0.306533223192008 0.846733388403996 59 0.192528125853573 0.385056251707146 0.807471874146427 60 0.92534700929302 0.149305981413959 0.0746529907069797 61 0.927068389444075 0.145863221111851 0.0729316105559255 62 0.908348359562462 0.183303280875075 0.0916516404375377 63 0.862744937162433 0.274510125675135 0.137255062837567 64 0.823361052375293 0.353277895249413 0.176638947624706 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 29 0.491525423728814 NOK 5% type I error level 39 0.661016949152542 NOK 10% type I error level 43 0.728813559322034 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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