Multiple regression

*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 12:20:52 -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/t1258658546xum0adogwxt3cj1.htm/, Retrieved Thu, 19 Nov 2009 20:22:38 +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/t1258658546xum0adogwxt3cj1.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:

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Dataseries X:

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19 613 18 611 19 594 19 595 22 591 23 589 20 584 14 573 14 567 14 569 15 621 11 629 17 628 16 612 20 595 24 597 23 593 20 590 21 580 19 574 23 573 23 573 23 620 23 626 27 620 26 588 17 566 24 557 26 561 24 549 27 532 27 526 26 511 24 499 23 555 23 565 24 542 17 527 21 510 19 514 22 517 22 508 18 493 16 490 14 469 12 478 14 528 16 534 8 518 3 506 0 502 5 516 1 528 1 533 3 536 6 537 7 524 8 536 14 587 14 597 13 581

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 ICONS[t] = + 52.0164275074887 -0.0414585275913714WLH[t] -1.08025189735549M1[t] -5.10599152463905M2[t] -6.0625749461278M3[t] -2.88119657649014M4[t] -1.90810991237075M5[t] -2.60035782483614M6[t] -2.88331496422184M7[t] -4.20872969876033M8[t] -3.99118730436532M9[t] -4.21810064024594M10[t] -0.213546124149344M11[t] -0.28187790341837t + 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) 52.0164275074887 23.892195 2.1771 0.034527 0.017263 WLH -0.0414585275913714 0.036427 -1.1381 0.26083 0.130415 M1 -1.08025189735549 4.120547 -0.2622 0.794342 0.397171 M2 -5.10599152463905 4.502715 -1.134 0.262554 0.131277 M3 -6.0625749461278 4.675074 -1.2968 0.201035 0.100518 M4 -2.88119657649014 4.613657 -0.6245 0.535324 0.267662 M5 -1.90810991237075 4.559214 -0.4185 0.677475 0.338738 M6 -2.60035782483614 4.589188 -0.5666 0.573664 0.286832 M7 -2.88331496422184 4.68833 -0.615 0.541522 0.270761 M8 -4.20872969876033 4.738206 -0.8883 0.378928 0.189464 M9 -3.99118730436532 4.898516 -0.8148 0.419311 0.209655 M10 -4.21810064024594 4.830029 -0.8733 0.386936 0.193468 M11 -0.213546124149344 4.274209 -0.05 0.960365 0.480182 t -0.28187790341837 0.07738 -3.6428 0.000671 0.000336 Multiple Linear Regression - Regression Statistics Multiple R 0.55791164972303 R-squared 0.311265408896673 Adjusted R-squared 0.120764351782986 F-TEST (value) 1.63393008738485 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 0.109387943459274 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 6.73486969870926 Sum Squared Residuals 2131.84808335383 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 19 25.2402202932042 -6.24022029320423 2 18 21.015519817685 -3.015519817685 3 19 20.4818534618312 -1.48185346183120 4 19 23.3398954004591 -4.33989540045913 5 22 24.1969382715256 -2.19693827152562 6 23 23.3057295108246 -0.305729510824605 7 20 22.9481871059774 -2.9481871059774 8 14 21.7969382715256 -7.79693827152563 9 14 21.9813539280505 -7.98135392805049 10 14 21.3896456335688 -7.38964563356877 11 15 22.9564788114957 -7.95647881149567 12 11 22.5564788114957 -11.5564788114957 13 17 21.2358075383132 -4.23580753831319 14 16 17.5915264490732 -1.5915264490732 15 20 17.0578600932194 2.94213990678061 16 24 19.8744435042559 4.12555649574406 17 23 20.7314863753224 2.26851362467756 18 20 19.8817361422128 0.118263857787203 19 21 19.7314863753224 1.26851362467756 20 19 18.3729449029138 0.62705509708619 21 23 18.3500679214818 4.64993207851818 22 23 17.8412766821828 5.15872331781716 23 23 19.6154024980666 3.38459750193340 24 23 19.2983195532493 3.70168044675065 25 27 18.1849409180237 8.81505908197628 26 26 15.2039962702457 10.7960037297543 27 17 14.8776225523487 2.12237744765128 28 24 18.1502497668904 5.84975023310964 29 26 18.6756244172259 7.32437558277411 30 24 18.1990009324386 5.80099906756142 31 27 18.3389608586878 8.66103914131218 32 27 16.9804193862792 10.0195806137208 33 26 17.5379617911264 8.4620382088736 34 24 17.5266728829239 6.47332711707611 35 23 18.9276719504853 4.0723280495147 36 23 18.4447548953026 4.55524510469744 37 24 18.0361712291303 5.96382877086975 38 17 14.3504316122989 2.64956838770111 39 21 13.8167652564451 7.18323474355492 40 19 16.5504316122989 2.44956838770112 41 22 17.1172647902258 4.88273520977421 42 22 16.5162657226644 5.48373427733563 43 18 16.5733085937309 1.42669140626912 44 16 15.0903915385481 0.90960846145187 45 14 15.8966851089436 -1.89668510894357 46 12 15.0147671213222 -3.01476712132224 47 14 16.6645173544319 -2.66451735443189 48 16 16.3474344096146 -0.347434409614636 49 8 15.6486410503027 -7.64864105030273 50 3 11.8385258506972 -8.83852585069725 51 0 10.7658986361556 -10.7658986361556 52 5 13.0849797160957 -8.0849797160957 53 1 13.2786861457003 -12.2786861457003 54 1 12.0972676918596 -11.0972676918596 55 3 11.4080570662815 -8.40805706628146 56 6 9.75930590073324 -3.75930590073324 57 7 10.2339312503977 -3.2339312503977 58 8 9.22763768000226 -1.22763768000226 59 14 10.8359293855205 3.16407061447946 60 14 10.3530123303378 3.64698766966221 61 13 9.65421897102589 3.34578102897411 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0665494116213896 0.133098823242779 0.93345058837861 18 0.0467317846533482 0.0934635693066963 0.953268215346652 19 0.0171465417843441 0.0342930835686882 0.982853458215656 20 0.0184751518230795 0.0369503036461590 0.98152484817692 21 0.0597116160559135 0.119423232111827 0.940288383944087 22 0.0801144723153368 0.160228944630674 0.919885527684663 23 0.0863441537806037 0.172688307561207 0.913655846219396 24 0.170820359688203 0.341640719376406 0.829179640311797 25 0.145674181163036 0.291348362326072 0.854325818836964 26 0.098608816625283 0.197217633250566 0.901391183374717 27 0.256186637724912 0.512373275449823 0.743813362275088 28 0.202481258016611 0.404962516033222 0.797518741983389 29 0.145133573339623 0.290267146679246 0.854866426660377 30 0.110866809616557 0.221733619233115 0.889133190383443 31 0.0833771979175793 0.166754395835159 0.91662280208242 32 0.080613620116537 0.161227240233074 0.919386379883463 33 0.0534478355271642 0.106895671054328 0.946552164472836 34 0.0315459147490305 0.063091829498061 0.96845408525097 35 0.0317212909458534 0.0634425818917068 0.968278709054147 36 0.0803405641863101 0.160681128372620 0.91965943581369 37 0.154597268819426 0.309194537638852 0.845402731180574 38 0.429658193695983 0.859316387391966 0.570341806304017 39 0.363296029172209 0.726592058344417 0.636703970827791 40 0.808682399888606 0.382635200222788 0.191317600111394 41 0.921349665818876 0.157300668362249 0.0786503341811245 42 0.929591101478566 0.140817797042868 0.0704088985214338 43 0.872115776804874 0.255768446390251 0.127884223195126 44 0.79587877985539 0.408242440289222 0.204121220144611 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 2 0.0714285714285714 NOK 10% type I error level 5 0.178571428571429 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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