*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 10:30:56 -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/t1258738527l5se3b3c6jzz9oc.htm/, Retrieved Fri, 20 Nov 2009 18:35:39 +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/t1258738527l5se3b3c6jzz9oc.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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12,6 18 15,7 16 13,2 19 20,3 18 12,8 23 8 20 0,9 20 3,6 15 14,1 17 21,7 16 24,5 15 18,9 10 13,9 13 11 10 5,8 19 15,5 21 22,4 17 31,7 16 30,3 17 31,4 14 20,2 18 19,7 17 10,8 14 13,2 15 15,1 16 15,6 11 15,5 15 12,7 13 10,9 17 10 16 9,1 9 10,3 17 16,9 15 22 12 27,6 12 28,9 12 31 12 32,9 4 38,1 7 28,8 4 29 3 21,8 3 28,8 0 25,6 5 28,2 3 20,2 4 17,9 3 16,3 10 13,2 4 8,1 1 4,5 1 -0,1 8 0 5 2,3 4 2,8 0 2,9 2 0,1 7 3,5 6 8,6 9 13,8 10

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[t] = + 16.3381031035382 -0.0223985047378945X[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) 16.3381031035382 2.625954 6.2218 0 0 X -0.0223985047378945 0.203793 -0.1099 0.912862 0.456431 Multiple Linear Regression - Regression Statistics Multiple R 0.0144301357461699 R-squared 0.000208228817652892 Adjusted R-squared -0.0170295603406634 F-TEST (value) 0.0120797867835872 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.91286168172439 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 9.77484659466084 Sum Squared Residuals 5541.76230505086 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 12.6 15.9349300182561 -3.33493001825614 2 15.7 15.9797270277319 -0.27972702773189 3 13.2 15.9125315135182 -2.71253151351821 4 20.3 15.9349300182561 4.36506998174390 5 12.8 15.8229374945666 -3.02293749456663 6 8 15.8901330087803 -7.89013300878032 7 0.9 15.8901330087803 -14.9901330087803 8 3.6 16.0021255324698 -12.4021255324698 9 14.1 15.957328522994 -1.857328522994 10 21.7 15.9797270277319 5.7202729722681 11 24.5 16.0021255324698 8.49787446753021 12 18.9 16.1141180561593 2.78588194384074 13 13.9 16.0469225419456 -2.14692254194558 14 11 16.1141180561593 -5.11411805615926 15 5.8 15.9125315135182 -10.1125315135182 16 15.5 15.8677345040424 -0.367734504042422 17 22.4 15.957328522994 6.442671477006 18 31.7 15.9797270277319 15.7202729722681 19 30.3 15.957328522994 14.342671477006 20 31.4 16.0245240372077 15.3754759627923 21 20.2 15.9349300182561 4.26506998174389 22 19.7 15.957328522994 3.742671477006 23 10.8 16.0245240372077 -5.22452403720768 24 13.2 16.0021255324698 -2.80212553246979 25 15.1 15.9797270277319 -0.879727027731895 26 15.6 16.0917195514214 -0.491719551421368 27 15.5 16.0021255324698 -0.502125532469789 28 12.7 16.0469225419456 -3.34692254194558 29 10.9 15.957328522994 -5.057328522994 30 10 15.9797270277319 -5.9797270277319 31 9.1 16.1365165608972 -7.03651656089716 32 10.3 15.957328522994 -5.657328522994 33 16.9 16.0021255324698 0.897874467530209 34 22 16.0693210466835 5.93067895331653 35 27.6 16.0693210466835 11.5306789533165 36 28.9 16.0693210466835 12.8306789533165 37 31 16.0693210466835 14.9306789533165 38 32.9 16.2485090845866 16.6514909154134 39 38.1 16.1813135703729 21.9186864296271 40 28.8 16.2485090845866 12.5514909154134 41 29 16.2709075893245 12.7290924106755 42 21.8 16.2709075893245 5.52909241067548 43 28.8 16.3381031035382 12.4618968964618 44 25.6 16.2261105798487 9.37388942015127 45 28.2 16.2709075893245 11.9290924106755 46 20.2 16.2485090845866 3.95149091541337 47 17.9 16.2709075893245 1.62909241067547 48 16.3 16.1141180561593 0.185881943840738 49 13.2 16.2485090845866 -3.04850908458663 50 8.1 16.3157045988003 -8.21570459880031 51 4.5 16.3157045988003 -11.8157045988003 52 -0.1 16.1589150656350 -16.2589150656351 53 0 16.2261105798487 -16.2261105798487 54 2.3 16.2485090845866 -13.9485090845866 55 2.8 16.3381031035382 -13.5381031035382 56 2.9 16.2933060940624 -13.3933060940624 57 0.1 16.1813135703729 -16.0813135703729 58 3.5 16.2037120751108 -12.7037120751108 59 8.6 16.1365165608972 -7.53651656089716 60 13.8 16.1141180561593 -2.31411805615926 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0479969243628798 0.0959938487257597 0.95200307563712 6 0.0425829529105371 0.0851659058210742 0.957417047089463 7 0.136165740584832 0.272331481169665 0.863834259415168 8 0.179109584571582 0.358219169143164 0.820890415428418 9 0.111895721287303 0.223791442574606 0.888104278712697 10 0.120569295625676 0.241138591251352 0.879430704374324 11 0.126991441958739 0.253982883917478 0.873008558041261 12 0.0795711852848176 0.159142370569635 0.920428814715182 13 0.0507390852357371 0.101478170471474 0.949260914764263 14 0.0423179586592831 0.0846359173185663 0.957682041340717 15 0.0372781151676596 0.0745562303353192 0.96272188483234 16 0.0252087320213632 0.0504174640427265 0.974791267978637 17 0.0257522748187763 0.0515045496375527 0.974247725181224 18 0.086977246541168 0.173954493082336 0.913022753458832 19 0.152964513520119 0.305929027040238 0.847035486479881 20 0.218477246901837 0.436954493803675 0.781522753098163 21 0.173325412388386 0.346650824776772 0.826674587611614 22 0.131119401336548 0.262238802673097 0.868880598663452 23 0.111773711634645 0.223547423269291 0.888226288365355 24 0.0826879777859096 0.165375955571819 0.91731202221409 25 0.0566229398369335 0.113245879673867 0.943377060163066 26 0.0388791628474526 0.0777583256949052 0.961120837152547 27 0.0250601048982752 0.0501202097965504 0.974939895101725 28 0.0174524351866797 0.0349048703733595 0.98254756481332 29 0.0124238428855349 0.0248476857710698 0.987576157114465 30 0.0096639673065342 0.0193279346130684 0.990336032693466 31 0.00883831973226955 0.0176766394645391 0.99116168026773 32 0.00720438611631419 0.0144087722326284 0.992795613883686 33 0.00443836947233736 0.00887673894467473 0.995561630527663 34 0.0028583516466127 0.0057167032932254 0.997141648353387 35 0.00290495034277539 0.00580990068555078 0.997095049657225 36 0.00341726715843991 0.00683453431687982 0.99658273284156 37 0.00611363862089939 0.0122272772417988 0.9938863613791 38 0.00925927312818902 0.0185185462563780 0.99074072687181 39 0.0452521109546697 0.0905042219093394 0.95474788904533 40 0.0558856422620579 0.111771284524116 0.944114357737942 41 0.0763956939943117 0.152791387988623 0.923604306005688 42 0.0766120217054494 0.153224043410899 0.92338797829455 43 0.133766036592175 0.26753207318435 0.866233963407825 44 0.212748538300310 0.425497076600619 0.78725146169969 45 0.532059577922575 0.93588084415485 0.467940422077425 46 0.706437071955735 0.58712585608853 0.293562928044265 47 0.869323139941292 0.261353720117415 0.130676860058708 48 0.907545753564205 0.18490849287159 0.092454246435795 49 0.956482795743348 0.0870344085133033 0.0435172042566517 50 0.968548427222066 0.0629031455558683 0.0314515727779342 51 0.96331665177452 0.0733666964509591 0.0366833482254796 52 0.973393300572818 0.0532133988543636 0.0266066994271818 53 0.967301239018985 0.0653975219620293 0.0326987609810146 54 0.93029552024689 0.139408959506221 0.0697044797531104 55 0.874453894506665 0.251092210986670 0.125546105493335 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.0784313725490196 NOK 5% type I error level 11 0.215686274509804 NOK 10% type I error level 25 0.490196078431373 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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