Multiple Regression analysis 1a

*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, 18 Dec 2009 08:33:30 -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/18/t1261150458sxqn3rzvuoiqxwq.htm/, Retrieved Fri, 18 Dec 2009 16:34:30 +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/18/t1261150458sxqn3rzvuoiqxwq.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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100,5 98,60 96,33 106,29 96,90 96,33 101,09 95,10 95,05 104,53 97,00 96,84 122,74 112,70 96,92 109,84 102,90 97,44 101,99 97,40 97,78 125,12 111,40 97,69 103,5 87,40 96,67 102,8 96,80 98,29 118,72 114,10 98,20 119,01 110,30 98,71 118,61 103,90 98,54 120,43 101,60 98,20 111,83 94,60 100,80 116,79 95,90 101,33 131,71 104,70 101,88 120,57 102,80 101,85 117,83 98,10 102,04 130,8 113,90 102,22 107,46 80,90 102,63 112,09 95,70 102,65 129,47 113,20 102,54 119,72 105,90 102,37 134,81 108,80 102,68 135,8 102,30 102,76 129,27 99,00 102,82 126,94 100,70 103,31 153,45 115,50 103,23 121,86 100,70 103,60 133,47 109,90 103,95 135,34 114,60 103,93 117,1 85,40 104,25 120,65 100,50 104,38 132,49 114,80 104,36 137,6 116,50 104,32 138,69 112,90 104,58 125,53 102,00 104,68 133,09 106,00 104,92 129,08 105,30 105,46 145,94 118,80 105,23 129,07 106,10 105,58 139,69 109,30 105,34 142,09 117,20 105,28 137,29 92,50 105,70 127,03 104,20 105,67 137,25 112,50 105,71 156,87 122,40 106,19 150, 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 8 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Invoer[t] = -227.016946529721 + 1.15710544765022TIP[t] + 2.26822918015897CONS[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) -227.016946529721 24.040757 -9.443 0 0 TIP 1.15710544765022 0.105356 10.9829 0 0 CONS 2.26822918015897 0.208278 10.8904 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.887253325178803 R-squared 0.787218463040842 Adjusted R-squared 0.78086677537042 F-TEST (value) 123.938471771309 F-TEST (DF numerator) 2 F-TEST (DF denominator) 67 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 8.2852376560788 Sum Squared Residuals 4599.22592218631 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 100.5 105.572167533305 -5.07216753330467 2 106.29 103.605088272299 2.68491172770106 3 101.09 98.618965115925 2.47103488407495 4 104.53 104.877595698945 -0.347595698945075 5 122.74 123.225609561466 -0.485609561466289 6 109.84 113.065455348177 -3.22545534817676 7 101.99 107.472573307355 -5.4825733073546 8 125.12 123.467908948243 1.65209105175661 9 103.5 93.383784440876 10.1162155591241 10 102.8 107.935106920646 -5.13510692064554 11 118.72 127.74889053878 -9.02889053878007 12 119.01 124.508686719590 -5.49868671959027 13 118.61 116.717612894002 1.89238710599813 14 120.43 113.285072443152 7.14492755684772 15 111.83 111.082730178014 0.747269821985967 16 116.79 113.789128725444 3.00087127455642 17 131.71 125.219182713853 6.49081728614704 18 120.57 122.952635487913 -2.38263548791278 19 117.83 117.945203428187 -0.115203428186952 20 130.8 136.635750753489 -5.83575075348906 21 107.46 99.381244944897 8.07875505510309 22 112.09 116.551770153723 -4.46177015372339 23 129.47 136.551610277785 -7.0816102777848 24 119.72 127.719141549311 -7.99914154931116 25 134.81 131.777898393346 3.03210160665393 26 135.8 124.438171318032 11.3618286819677 27 129.27 120.755817091596 8.5141829084039 28 126.94 123.834328650879 3.10567134912058 29 153.45 140.77803094169 12.67196905831 30 121.86 124.492115113126 -2.63211511312550 31 133.47 135.931365444563 -2.46136544456321 32 135.34 141.324396464916 -5.98439646491607 33 117.1 108.262750731180 8.83724926881955 34 120.65 126.029912784119 -5.37991278411944 35 132.49 142.531156101914 -10.0411561019145 36 137.6 144.407506195713 -6.80750619571347 37 138.69 140.831666171014 -2.14166617101402 38 125.53 128.446039709642 -2.9160397096425 39 133.09 133.618836503482 -0.528836503481527 40 129.08 134.033706447412 -4.95370644741218 41 145.94 149.132937279254 -3.19293727925366 42 129.07 135.231578307151 -6.16157830715146 43 139.69 138.389940736394 1.30005926360597 44 142.09 147.394980022021 -5.30498002202125 45 137.29 119.767131720728 17.5228682792725 46 127.03 133.237218582830 -6.20721858283036 47 137.25 142.931922965534 -5.68192296553354 48 156.87 155.476016903747 1.39398309625294 49 150.89 146.624846923448 4.2651530765523 50 139.14 132.392141351581 6.7478586484192 51 158.3 145.703143605303 12.5968563946967 52 149 150.083742140306 -1.08374214030551 53 158.36 147.996045597252 10.3639544027482 54 168.06 157.059935415255 11.0000645847445 55 153.38 149.182113462437 4.19788653756343 56 173.86 158.003086762125 15.8569132378752 57 162.47 136.337625861981 26.1323741380192 58 145.17 139.963428641798 5.20657135820241 59 168.89 160.246643130494 8.64335686950568 60 166.64 159.291379223301 7.34862077669866 61 140.07 139.976914590571 0.093085409429095 62 128.84 134.415911184579 -5.57591118457857 63 123.41 131.073791335166 -7.66379133516625 64 120.3 134.299892074044 -13.9998920740444 65 129.67 144.212840163274 -14.5428401632735 66 118.1 134.025714052437 -15.9257140524374 67 113.91 130.888043395009 -16.9780433950085 68 131.09 142.457107351523 -11.3671073515228 69 119.15 119.153923270150 -0.00392327015017413 70 122.3 127.083389932305 -4.78338993230523 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.0720885830072275 0.144177166014455 0.927911416992772 7 0.0248628165633069 0.0497256331266137 0.975137183436693 8 0.0145989587971502 0.0291979175943004 0.98540104120285 9 0.0806222012697229 0.161244402539446 0.919377798730277 10 0.0484966612541535 0.096993322508307 0.951503338745846 11 0.0306186558941568 0.0612373117883135 0.969381344105843 12 0.0151867282216094 0.0303734564432188 0.98481327177839 13 0.0138252699446379 0.0276505398892758 0.986174730055362 14 0.0241213318642965 0.0482426637285931 0.975878668135703 15 0.0125739087202752 0.0251478174405503 0.987426091279725 16 0.00699521462670587 0.0139904292534117 0.993004785373294 17 0.00742032308982085 0.0148406461796417 0.99257967691018 18 0.00440889420606703 0.00881778841213406 0.995591105793933 19 0.00226200587873914 0.00452401175747829 0.99773799412126 20 0.00124058497136634 0.00248116994273268 0.998759415028634 21 0.000684294775322816 0.00136858955064563 0.999315705224677 22 0.000637642721767863 0.00127528544353573 0.999362357278232 23 0.000362995528824194 0.000725991057648387 0.999637004471176 24 0.00031026033775595 0.0006205206755119 0.999689739662244 25 0.000309336538368198 0.000618673076736396 0.999690663461632 26 0.00138884735843599 0.00277769471687197 0.998611152641564 27 0.00154072166390922 0.00308144332781843 0.99845927833609 28 0.00086482884412258 0.00172965768824516 0.999135171155877 29 0.00556875234064983 0.0111375046812997 0.99443124765935 30 0.00402256830826633 0.00804513661653267 0.995977431691734 31 0.00241779651913973 0.00483559303827947 0.99758220348086 32 0.00172801841362372 0.00345603682724745 0.998271981586376 33 0.00181598518116457 0.00363197036232915 0.998184014818835 34 0.00162794968993516 0.00325589937987032 0.998372050310065 35 0.00184787198511336 0.00369574397022673 0.998152128014887 36 0.00132645894505269 0.00265291789010538 0.998673541054947 37 0.000742550676816429 0.00148510135363286 0.999257449323184 38 0.000438064680768397 0.000876129361536793 0.999561935319232 39 0.000224832143243908 0.000449664286487816 0.999775167856756 40 0.000143672389071230 0.000287344778142459 0.999856327610929 41 8.8483609989468e-05 0.000176967219978936 0.99991151639001 42 6.76118890980166e-05 0.000135223778196033 0.999932388110902 43 3.63239943391035e-05 7.2647988678207e-05 0.999963676005661 44 3.04839265395739e-05 6.09678530791479e-05 0.99996951607346 45 0.000540511973804903 0.00108102394760981 0.999459488026195 46 0.000420532439840931 0.000841064879681862 0.99957946756016 47 0.000375553216579474 0.000751106433158949 0.99962444678342 48 0.000564931828960435 0.00112986365792087 0.99943506817104 49 0.000509525981549054 0.00101905196309811 0.99949047401845 50 0.000291435503178646 0.000582871006357291 0.999708564496821 51 0.000454059670955084 0.000908119341910168 0.999545940329045 52 0.000421366648142302 0.000842733296284604 0.999578633351858 53 0.000328505097423319 0.000657010194846638 0.999671494902577 54 0.000310677895524263 0.000621355791048525 0.999689322104476 55 0.000149964236165169 0.000299928472330339 0.999850035763835 56 0.000267533935089708 0.000535067870179417 0.99973246606491 57 0.115988440660768 0.231976881321537 0.884011559339232 58 0.130068040866488 0.260136081732976 0.869931959133512 59 0.133793088328675 0.267586176657351 0.866206911671325 60 0.424090167554187 0.848180335108374 0.575909832445813 61 0.757273790731717 0.485452418536566 0.242726209268283 62 0.81630855805683 0.367382883886340 0.183691441943170 63 0.820913022969418 0.358173954061163 0.179086977030582 64 0.845385617756865 0.309228764486271 0.154614382243135 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 38 0.64406779661017 NOK 5% type I error level 47 0.796610169491525 NOK 10% type I error level 49 0.830508474576271 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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