WS 7 Multiple regression 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: Thu, 19 Nov 2009 11:54:58 -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/19/t1258656958yi43b1qk0hpzbzr.htm/, Retrieved Thu, 19 Nov 2009 19:56:10 +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/19/t1258656958yi43b1qk0hpzbzr.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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108,01 102,9 101,21 97,4 119,93 111,4 94,76 87,4 95,26 96,8 117,96 114,1 115,86 110,3 111,44 103,9 108,16 101,6 108,77 94,6 109,45 95,9 124,83 104,7 115,31 102,8 109,49 98,1 124,24 113,9 92,85 80,9 98,42 95,7 120,88 113,2 111,72 105,9 116,1 108,8 109,37 102,3 111,65 99 114,29 100,7 133,68 115,5 114,27 100,7 126,49 109,9 131 114,6 104 85,4 108,88 100,5 128,48 114,8 132,44 116,5 128,04 112,9 116,35 102 120,93 106 118,59 105,3 133,1 118,8 121,05 106,1 127,62 109,3 135,44 117,2 114,88 92,5 114,34 104,2 128,85 112,5 138,9 122,4 129,44 113,3 114,96 100 127,98 110,7 127,03 112,8 128,75 109,8 137,91 117,3 128,37 109,1 135,9 115,9 122,19 96 113,08 99,8 136,2 116,8 138 115,7 115,24 99,4 110,95 94,3 99,23 91 102,39 93,2 112,67 103,1

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 Y(omzet)[t] = -2.97734126813718 + 1.15356403007296`X(prod)`[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) -2.97734126813718 8.183472 -0.3638 0.717313 0.358656 `X(prod)` 1.15356403007296 0.077558 14.8737 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.890102564482754 R-squared 0.792282575298776 Adjusted R-squared 0.788701240390134 F-TEST (value) 221.225491474419 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.43221752345103 Sum Squared Residuals 1711.52125888113 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 108.01 115.724397426371 -7.71439742637102 2 101.21 109.379795260970 -8.16979526096953 3 119.93 125.529691681991 -5.59969168199106 4 94.76 97.8441549602399 -3.08415496023991 5 95.26 108.687656842926 -13.4276568429258 6 117.96 128.644314563188 -10.6843145631881 7 115.86 124.260771248911 -8.4007712489108 8 111.44 116.877961456444 -5.43796145644384 9 108.16 114.224764187276 -6.064764187276 10 108.77 106.149815976765 2.62018402323475 11 109.45 107.649449215860 1.80055078413989 12 124.83 117.800812680502 7.0291873194978 13 115.31 115.609041023364 -0.299041023363559 14 109.49 110.187290082021 -0.697290082020628 15 124.24 128.413601757173 -4.17360175717349 16 92.85 90.3459887647657 2.50401123523435 17 98.42 107.418736409846 -8.99873640984552 18 120.88 127.606106936122 -6.72610693612241 19 111.72 119.185089516590 -7.46508951658976 20 116.1 122.530425203801 -6.43042520380136 21 109.37 115.032259008327 -5.66225900832707 22 111.65 111.225497709086 0.424502290913707 23 114.29 113.186556560210 1.10344343978966 24 133.68 130.259304205290 3.42069579470979 25 114.27 113.186556560210 1.08344343978965 26 126.49 123.799345636882 2.69065436311837 27 131 129.221096578225 1.77890342177546 28 104 95.537026900094 8.46297309990602 29 108.88 112.955843754196 -4.07584375419575 30 128.48 129.451809384239 -0.971809384239152 31 132.44 131.412868235363 1.02713176463681 32 128.04 127.260037727101 0.779962272899475 33 116.35 114.686189799305 1.66381020069480 34 120.93 119.300445919597 1.62955408040295 35 118.59 118.492951098546 0.0970489014540293 36 133.1 134.066065504531 -0.966065504531005 37 121.05 119.415802322604 1.63419767739565 38 127.62 123.107207218838 4.51279278116217 39 135.44 132.220363056414 3.21963694358573 40 114.88 103.727331513612 11.1526684863880 41 114.34 117.224030665466 -2.88403066546572 42 128.85 126.798612115071 2.05138788492867 43 138.9 138.218896012794 0.681103987206322 44 129.44 127.721463339130 1.71853666087030 45 114.96 112.379061739159 2.58093826084073 46 127.98 124.72219686094 3.25780313906001 47 127.03 127.144681324093 -0.114681324093209 48 128.75 123.683989233874 5.06601076612568 49 137.91 132.335719459422 5.57428054057844 50 128.37 122.876494412823 5.49350558717677 51 135.9 130.720729817319 5.17927018268059 52 122.19 107.764805618867 14.4251943811326 53 113.08 112.148348933145 0.931651066855332 54 136.2 131.758937444385 4.44106255561492 55 138 130.490017011305 7.50998298869519 56 115.24 111.686923321115 3.55307667888450 57 110.95 105.803746767743 5.14625323225664 58 99.23 101.996985468503 -2.76698546850257 59 102.39 104.534826334663 -2.14482633466311 60 112.67 115.955110232385 -3.28511023238545 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.485985771177463 0.971971542354927 0.514014228822536 6 0.377578698406419 0.755157396812838 0.622421301593581 7 0.270396268695209 0.540792537390418 0.729603731304791 8 0.217503996674798 0.435007993349597 0.782496003325202 9 0.159714485110923 0.319428970221846 0.840285514889077 10 0.430499976119415 0.86099995223883 0.569500023880585 11 0.514243459977627 0.971513080044745 0.485756540022373 12 0.88272440394581 0.23455119210838 0.11727559605419 13 0.860071678969571 0.279856642060858 0.139928321030429 14 0.81822451274085 0.3635509745183 0.18177548725915 15 0.78323984358287 0.433520312834260 0.216760156417130 16 0.722993685985398 0.554012628029205 0.277006314014602 17 0.820301657979089 0.359396684041822 0.179698342020911 18 0.818843748305606 0.362312503388788 0.181156251694394 19 0.85134730928771 0.297305381424580 0.148652690712290 20 0.870164277501592 0.259671444996816 0.129835722498408 21 0.891620482484321 0.216759035031358 0.108379517515679 22 0.878420817846523 0.243158364306953 0.121579182153477 23 0.869760609562831 0.260478780874338 0.130239390437169 24 0.92051033569975 0.158979328600499 0.0794896643002496 25 0.906847512014291 0.186304975971418 0.093152487985709 26 0.909986706697002 0.180026586605996 0.0900132933029982 27 0.902415198453317 0.195169603093366 0.0975848015466832 28 0.939326274995042 0.121347450009916 0.0606737250049579 29 0.946944979919352 0.106110040161296 0.0530550200806481 30 0.936967909460527 0.126064181078947 0.0630320905394735 31 0.92483456543291 0.150330869134181 0.0751654345670905 32 0.906746890570845 0.186506218858309 0.0932531094291547 33 0.881502094345205 0.236995811309591 0.118497905654795 34 0.852452593427764 0.295094813144471 0.147547406572235 35 0.820645086357496 0.358709827285007 0.179354913642504 36 0.796543703783883 0.406912592432235 0.203456296216117 37 0.75384616793346 0.492307664133079 0.246153832066539 38 0.731645702031237 0.536708595937525 0.268354297968763 39 0.689099330698226 0.621801338603547 0.310900669301774 40 0.861478297421781 0.277043405156437 0.138521702578219 41 0.866728994678978 0.266542010642044 0.133271005321022 42 0.826477002221042 0.347045995557915 0.173522997778958 43 0.803121219186886 0.393757561626227 0.196878780813114 44 0.757134256438812 0.485731487122376 0.242865743561188 45 0.687257641589552 0.625484716820896 0.312742358410448 46 0.61584484058893 0.76831031882214 0.38415515941107 47 0.593689412645424 0.812621174709153 0.406310587354576 48 0.520146767195745 0.95970646560851 0.479853232804255 49 0.444200857936661 0.888401715873322 0.555799142063339 50 0.365709338617382 0.731418677234765 0.634290661382618 51 0.281372768206726 0.562745536413452 0.718627231793274 52 0.875835400226306 0.248329199547388 0.124164599773694 53 0.783558197403234 0.432883605193531 0.216441802596766 54 0.666578020264437 0.666843959471126 0.333421979735563 55 0.577128905821282 0.845742188357436 0.422871094178718 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 0 0 OK 10% type I error level 0 0 OK

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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