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, 17 Dec 2009 09:21:45 -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/17/t1261066979txy2mlq3slgmqrh.htm/, Retrieved Thu, 17 Dec 2009 17:23:11 +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/17/t1261066979txy2mlq3slgmqrh.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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611 19 594 18 595 19 591 19 589 22 584 23 573 20 567 14 569 14 621 14 629 15 628 11 612 17 595 16 597 20 593 24 590 23 580 20 574 21 573 19 573 23 620 23 626 23 620 23 588 27 566 26 557 17 561 24 549 26 532 24 526 27 511 27 499 26 555 24 565 23 542 23 527 24 510 17 514 21 517 19 508 22 493 22 490 18 469 16 478 14 528 12 534 14 518 16 506 8 502 3 516 0 528 5 533 1 536 1 537 3 524 6 536 7 587 8 597 14 581 14

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 WHL[t] = + 537.42712288551 + 1.13414112306967ICONS[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) 537.42712288551 13.960787 38.4955 0 0 ICONS 1.13414112306967 0.745013 1.5223 0.133365 0.066682 Multiple Linear Regression - Regression Statistics Multiple R 0.196011750165139 R-squared 0.0384206062028008 Adjusted R-squared 0.0218416511373318 F-TEST (value) 2.31743231410429 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.133364918003988 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 41.3262635143747 Sum Squared Residuals 99055.883251453 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 611 558.975804223834 52.0241957761664 2 594 557.841663100764 36.1583368992357 3 595 558.975804223834 36.0241957761660 4 591 558.975804223834 32.0241957761660 5 589 562.378227593043 26.621772406957 6 584 563.512368716113 20.4876312838873 7 573 560.109945346904 12.8900546530964 8 567 553.305098608486 13.6949013915144 9 569 553.305098608486 15.6949013915144 10 621 553.305098608486 67.6949013915144 11 629 554.439239731555 74.5607602684447 12 628 549.902675239277 78.0973247607234 13 612 556.707521977695 55.2924780223054 14 595 555.573380854625 39.4266191453751 15 597 560.109945346904 36.8900546530964 16 593 564.646509839182 28.3534901608177 17 590 563.512368716113 26.4876312838874 18 580 560.109945346904 19.8900546530964 19 574 561.244086469973 12.7559135300267 20 573 558.975804223834 14.0241957761660 21 573 563.512368716113 9.48763128388735 22 620 563.512368716113 56.4876312838874 23 626 563.512368716113 62.4876312838874 24 620 563.512368716113 56.4876312838874 25 588 568.048933208391 19.9510667916087 26 566 566.914792085322 -0.914792085321672 27 557 556.707521977695 0.292478022305391 28 561 564.646509839182 -3.64650983918232 29 549 566.914792085322 -17.9147920853217 30 532 564.646509839182 -32.6465098391823 31 526 568.048933208391 -42.0489332083913 32 511 568.048933208391 -57.0489332083913 33 499 566.914792085322 -67.9147920853217 34 555 564.646509839182 -9.64650983918232 35 565 563.512368716113 1.48763128388735 36 542 563.512368716113 -21.5123687161126 37 527 564.646509839182 -37.6465098391823 38 510 556.707521977695 -46.7075219776946 39 514 561.244086469973 -47.2440864699733 40 517 558.975804223834 -41.975804223834 41 508 562.378227593043 -54.378227593043 42 493 562.378227593043 -69.378227593043 43 490 557.841663100764 -67.8416631007643 44 469 555.573380854625 -86.573380854625 45 478 553.305098608486 -75.3050986084856 46 528 551.036816362346 -23.0368163623462 47 534 553.305098608486 -19.3050986084856 48 518 555.573380854625 -37.5733808546249 49 506 546.500251870068 -40.5002518700675 50 502 540.829546254719 -38.8295462547192 51 516 537.42712288551 -21.4271228855102 52 528 543.097828500859 -15.0978285008585 53 533 538.56126400858 -5.56126400857983 54 536 538.56126400858 -2.56126400857983 55 537 540.829546254719 -3.82954625471918 56 524 544.231969623928 -20.2319696239282 57 536 545.366110746998 -9.36611074699787 58 587 546.500251870068 40.4997481299325 59 597 553.305098608486 43.6949013915144 60 581 553.305098608486 27.6949013915144 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.015469357775253 0.030938715550506 0.984530642224747 6 0.00299958991673753 0.00599917983347505 0.997000410083262 7 0.00404320064990040 0.00808640129980079 0.9959567993501 8 0.00533862599524203 0.0106772519904841 0.994661374004758 9 0.00195070080707641 0.00390140161415283 0.998049299192924 10 0.0095487669170356 0.0190975338340712 0.990451233082964 11 0.0209497004099759 0.0418994008199517 0.979050299590024 12 0.0224183622760578 0.0448367245521156 0.977581637723942 13 0.0170678320864025 0.034135664172805 0.982932167913597 14 0.0102299322931578 0.0204598645863156 0.989770067706842 15 0.00609501876795978 0.0121900375359196 0.99390498123204 16 0.00356407728770665 0.00712815457541331 0.996435922712293 17 0.0019301316062224 0.0038602632124448 0.998069868393778 18 0.00119240073022551 0.00238480146045103 0.998807599269774 19 0.000783068415301409 0.00156613683060282 0.999216931584699 20 0.000612549452476003 0.00122509890495201 0.999387450547524 21 0.000337399096880385 0.00067479819376077 0.99966260090312 22 0.00137818182786673 0.00275636365573347 0.998621818172133 23 0.00716534209287138 0.0143306841857428 0.992834657907129 24 0.0227241479337893 0.0454482958675787 0.97727585206621 25 0.0244216893057329 0.0488433786114658 0.975578310694267 26 0.0270284068106951 0.0540568136213902 0.972971593189305 27 0.0460424928281265 0.0920849856562531 0.953957507171873 28 0.0539602748021227 0.107920549604245 0.946039725197877 29 0.0655407913476673 0.131081582695335 0.934459208652333 30 0.106660333762251 0.213320667524502 0.893339666237749 31 0.132950672586843 0.265901345173687 0.867049327413157 32 0.186966363176958 0.373932726353916 0.813033636823042 33 0.293696999256691 0.587393998513381 0.70630300074331 34 0.271826015698397 0.543652031396794 0.728173984301603 35 0.289911366395709 0.579822732791418 0.710088633604291 36 0.284312286295847 0.568624572591694 0.715687713704153 37 0.278720905339812 0.557441810679624 0.721279094660188 38 0.417534181700528 0.835068363401056 0.582465818299472 39 0.439648979432437 0.879297958864873 0.560351020567563 40 0.458165863023242 0.916331726046485 0.541834136976758 41 0.456381704703124 0.912763409406248 0.543618295296876 42 0.502935095805529 0.994129808388943 0.497064904194471 43 0.609265469502392 0.781469060995216 0.390734530497608 44 0.856042030204997 0.287915939590006 0.143957969795003 45 0.966725407733795 0.0665491845324102 0.0332745922662051 46 0.956003302226774 0.0879933955464525 0.0439966977732262 47 0.941316114505384 0.117367770989232 0.0586838854946159 48 0.981516287059766 0.036967425880468 0.018483712940234 49 0.994798742602888 0.0104025147942233 0.00520125739711163 50 0.995990057661506 0.00801988467698714 0.00400994233849357 51 0.989460850306451 0.0210782993870975 0.0105391496935487 52 0.980673795602782 0.0386524087944357 0.0193262043972178 53 0.953377067396082 0.0932458652078357 0.0466229326039179 54 0.908426024597958 0.183147950804083 0.0915739754020417 55 0.815779442548782 0.368441114902436 0.184220557451218 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 11 0.215686274509804 NOK 5% type I error level 26 0.509803921568627 NOK 10% type I error level 31 0.607843137254902 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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