*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, 09 Dec 2009 04:55:21 -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/09/t1260359980b782bmicrztomk7.htm/, Retrieved Wed, 09 Dec 2009 12:59:52 +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/09/t1260359980b782bmicrztomk7.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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24 33 22 34 25 36 24 36 29 38 26 42 26 35 21 25 23 24 22 22 21 27 16 17 19 30 16 30 25 34 27 37 23 36 22 33 23 33 20 33 24 37 23 40 20 35 21 37 22 43 17 42 21 33 19 39 23 40 22 37 15 44 23 42 21 43 18 40 18 30 18 30 18 31 10 18 13 24 10 22 9 26 9 28 6 23 11 17 9 12 10 9 9 19 16 21 10 18 7 18 7 15 14 24 11 18 10 19 6 30 8 33 13 35 12 36 15 47 16 46 16 43

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 R Framework error message Warning: there are blank lines in the 'Data X' field. Please, use NA for missing data - blank lines are simply deleted and are NOT treated as missing values. Multiple Linear Regression - Estimated Regression Equation S.[t] = + 5.10746822236075 + 0.395127428651407E.S.[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) 5.10746822236075 2.256009 2.2639 0.027269 0.013634 E.S. 0.395127428651407 0.070181 5.6301 1e-06 0 Multiple Linear Regression - Regression Statistics Multiple R 0.591175992644067 R-squared 0.349489054278698 Adjusted R-squared 0.338463445029184 F-TEST (value) 31.6979358119476 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 5.26712907955584e-07 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.038238229328 Sum Squared Residuals 1497.64682287227 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 24 18.1466733678572 5.85332663214276 2 22 18.5418007965086 3.45819920349143 3 25 19.3320556538114 5.66794434618859 4 24 19.3320556538114 4.66794434618859 5 29 20.1223105111142 8.87768948888578 6 26 21.7028202257198 4.29717977428015 7 26 18.93692822516 7.06307177484 8 21 14.9856539386459 6.01434606135406 9 23 14.5905265099945 8.40947349000547 10 22 13.8002716526917 8.19972834730829 11 21 15.7759087959487 5.22409120405125 12 16 11.8246345094347 4.17536549056532 13 19 16.9612910819030 2.03870891809703 14 16 16.9612910819030 -0.961291081902967 15 25 18.5418007965086 6.4581992034914 16 27 19.7271830824628 7.27281691753719 17 23 19.3320556538114 3.66794434618859 18 22 18.1466733678572 3.85332663214281 19 23 18.1466733678572 4.85332663214281 20 20 18.1466733678572 1.85332663214281 21 24 19.7271830824628 4.27281691753719 22 23 20.9125653684170 2.08743463158297 23 20 18.93692822516 1.06307177484 24 21 19.7271830824628 1.27281691753719 25 22 22.0979476543713 -0.097947654371254 26 17 21.7028202257198 -4.70282022571985 27 21 18.1466733678572 2.85332663214281 28 19 20.5174379397656 -1.51743793976563 29 23 20.9125653684170 2.08743463158297 30 22 19.7271830824628 2.27281691753719 31 15 22.4930750830227 -7.49307508302266 32 23 21.7028202257198 1.29717977428015 33 21 22.0979476543713 -1.09794765437125 34 18 20.9125653684170 -2.91256536841703 35 18 16.9612910819030 1.03870891809703 36 18 16.9612910819030 1.03870891809703 37 18 17.3564185105544 0.643581489445626 38 10 12.2197619380861 -2.21976193808609 39 13 14.5905265099945 -1.59052650999453 40 10 13.8002716526917 -3.80027165269171 41 9 15.3807813672973 -6.38078136729734 42 9 16.1710362246002 -7.17103622460015 43 6 14.1953990813431 -8.19539908134312 44 11 11.8246345094347 -0.82463450943468 45 9 9.84899736617765 -0.848997366177649 46 10 8.66361508022343 1.33638491977657 47 9 12.6148893667375 -3.61488936673749 48 16 13.4051442240403 2.59485577595969 49 10 12.2197619380861 -2.21976193808609 50 7 12.2197619380861 -5.21976193808609 51 7 11.0343796521319 -4.03437965213187 52 14 14.5905265099945 -0.590526509994528 53 11 12.2197619380861 -1.21976193808609 54 10 12.6148893667375 -2.61488936673749 55 6 16.9612910819030 -10.9612910819030 56 8 18.1466733678572 -10.1466733678572 57 13 18.93692822516 -5.93692822516 58 12 19.3320556538114 -7.33205565381141 59 15 23.6784573689769 -8.67845736897688 60 16 23.2833299403255 -7.28332994032548 61 16 22.0979476543713 -6.09794765437125 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0557612081923069 0.111522416384614 0.944238791807693 6 0.0546545794530045 0.109309158906009 0.945345420546996 7 0.0283053355045916 0.0566106710091831 0.971694664495408 8 0.0110591414121162 0.0221182828242324 0.988940858587884 9 0.00749246328682713 0.0149849265736543 0.992507536713173 10 0.00372816367146925 0.0074563273429385 0.99627183632853 11 0.00225045234780598 0.00450090469561196 0.997749547652194 12 0.00277509539631452 0.00555019079262904 0.997224904603685 13 0.00538864864423385 0.0107772972884677 0.994611351355766 14 0.0330982161443004 0.0661964322886008 0.9669017838557 15 0.0267693042387095 0.0535386084774189 0.97323069576129 16 0.0279020792688824 0.0558041585377648 0.972097920731118 17 0.0221667641816602 0.0443335283633204 0.97783323581834 18 0.0176452173028569 0.0352904346057137 0.982354782697143 19 0.0148323122081880 0.0296646244163760 0.985167687791812 20 0.0166999054835323 0.0333998109670646 0.983300094516468 21 0.0154312900140408 0.0308625800280816 0.98456870998596 22 0.0162880525122162 0.0325761050244323 0.983711947487784 23 0.0206046652267144 0.0412093304534289 0.979395334773286 24 0.0232183790832516 0.0464367581665033 0.976781620916748 25 0.0276868273846278 0.0553736547692556 0.972313172615372 26 0.0906323757352462 0.181264751470492 0.909367624264754 27 0.095948496503149 0.191896993006298 0.90405150349685 28 0.106165500939976 0.212331001879952 0.893834499060024 29 0.119664759800560 0.239329519601121 0.88033524019944 30 0.148861193631211 0.297722387262423 0.851138806368789 31 0.311495138865389 0.622990277730779 0.68850486113461 32 0.380204651702648 0.760409303405296 0.619795348297352 33 0.416412013581423 0.832824027162846 0.583587986418577 34 0.44999322454458 0.89998644908916 0.55000677545542 35 0.548827207564037 0.902345584871925 0.451172792435962 36 0.673225148250941 0.653549703498119 0.326774851749059 37 0.808762330248808 0.382475339502385 0.191237669751192 38 0.875969153388706 0.248061693222587 0.124030846611294 39 0.891621734913653 0.216756530172694 0.108378265086347 40 0.905100319522335 0.189799360955330 0.0948996804776652 41 0.931849999783479 0.136300000433043 0.0681500002165213 42 0.949943027853645 0.100113944292709 0.0500569721463546 43 0.978876501578052 0.0422469968438954 0.0211234984219477 44 0.967030186288343 0.0659396274233142 0.0329698137116571 45 0.946364325518414 0.107271348963172 0.053635674481586 46 0.9234797448632 0.153040510273599 0.0765202551367997 47 0.891988286388706 0.216023427222588 0.108011713611294 48 0.956055575611605 0.0878888487767904 0.0439444243883952 49 0.931767811690497 0.136464376619005 0.0682321883095025 50 0.907762701577535 0.184474596844930 0.0922372984224652 51 0.865745863998316 0.268508272003367 0.134254136001684 52 0.875461435044739 0.249077129910523 0.124538564955261 53 0.8720215041222 0.2559569917556 0.1279784958778 54 0.960087330967138 0.0798253380657243 0.0399126690328622 55 0.958708136476097 0.0825837270478068 0.0412918635239034 56 0.984896404211323 0.0302071915773544 0.0151035957886772 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 3 0.0576923076923077 NOK 5% type I error level 16 0.307692307692308 NOK 10% type I error level 25 0.480769230769231 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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