Workshop 10; PLC: 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: Mon, 13 Dec 2010 21:38:20 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/13/t1292276252bh2esqkcqhd4zt8.htm/, Retrieved Mon, 13 Dec 2010 22:37:35 +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/2010/Dec/13/t1292276252bh2esqkcqhd4zt8.htm/}, year = {2010}, } @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 = {2010}, 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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107.11 236.67 8.92 1 122.23 258.1 9.32 2 134.69 241.52 8.9 3 128.79 190.71 8.53 4 126.16 200.32 8.51 5 119.98 223.41 9.03 6 108.45 201.38 9.6 7 108.43 211.83 9.88 8 98.17 224.41 10.81 9 106.09 211.57 11.61 10 108.81 194.77 11.81 11 103.03 201.86 13.93 12 124.36 225 16.19 13 118.52 278.9 18.05 14 112.2 259.74 17.08 15 114.71 230.45 17.46 16 107.96 238.26 16.9 17 101.21 250.14 15.69 18 102.77 263.81 15.86 19 112.13 247.22 12.98 20 109.36 229.81 12.31 21 110.91 224.27 11.51 22 123.57 213.23 11.73 23 129.95 239.57 11.7 24 124.46 249.7 10.9 25 122.34 212.5 10.57 26 116.61 203.27 10.37 27 114.59 192.05 9.59 28 112.52 190.04 9.09 29 118.67 202.05 9.26 30 116.8 211.91 9.9 31 123.63 210.39 9.61 32 128.04 231.25 9.85 33 134.57 224.3 9.99 34 130.33 209.64 9.9 35 136.47 206.05 10.45 36 139.05 229.7 11.66 37 158.21 264.67 13.61 38 148.07 246.29 12.88 39 137.74 260.91 12.52 40 139.74 265.14 10.93 41 144.08 284.52 12.07 42 145.35 287.48 13.21 43 145.77 321.9 13.68 44 140.56 321.59 etc...

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 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation Coffee[t] = + 81.5751802572647 + 0.18546007848443Tea[t] -1.11064247481463Sugar[t] + 0.416890792681842Month[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) 81.5751802572647 8.381259 9.733 0 0 Tea 0.18546007848443 0.056268 3.296 0.001706 0.000853 Sugar -1.11064247481463 0.560667 -1.9809 0.052518 0.026259 Month 0.416890792681842 0.103765 4.0176 0.000177 8.9e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.768876611917088 R-squared 0.5911712443531 Adjusted R-squared 0.569269703872017 F-TEST (value) 26.9922220705748 F-TEST (DF numerator) 3 F-TEST (DF denominator) 56 p-value 6.25013374389027e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 10.0744787349589 Sum Squared Residuals 5683.72681974376 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 107.11 115.97797694951 -8.86797694951014 2 122.23 119.925020234187 2.30497976581255 3 134.69 117.73345276502 16.9565472349804 4 128.79 109.138054685589 19.6519453144111 5 126.16 111.359429682002 14.8005703179976 6 119.98 115.481059599986 4.49894040001384 7 108.45 111.179198653012 -2.72919865301167 8 108.43 113.223167372908 -4.79316737290771 9 98.17 114.940248451346 -16.7702484513461 10 106.09 112.087317856436 -5.99731785643614 11 108.81 109.166350835617 -0.356350835616631 12 103.03 108.543591538146 -5.51359153814608 13 124.36 110.741976553877 13.6180234461234 14 118.52 119.089370573714 -0.569370573713992 15 112.2 117.030169463204 -4.83016946320434 16 114.71 111.592890416648 3.11710958335233 17 107.96 114.080184208189 -6.1201842081891 18 101.21 118.044218127792 -16.8342181277917 19 102.77 120.807538972637 -18.0375389726372 20 112.13 121.346297390728 -9.21629739072846 21 109.36 119.278458675122 -9.91845867512217 22 110.91 119.556414612852 -8.64641461285197 23 123.57 117.681484794606 5.88851520539351 24 129.95 123.016713328813 6.93328667118734 25 124.46 126.200828696393 -1.74082869639347 26 122.34 120.085116586143 2.25488341385666 27 116.61 119.012339349377 -2.40233934937682 28 114.59 118.214669191819 -3.62466919181876 29 112.52 118.814106464154 -6.29410646415421 30 118.67 121.269563578716 -2.59956357871557 31 116.8 122.804279561373 -6.00427956137254 32 123.63 123.261357352454 0.368642647545714 33 128.04 127.280391188366 0.759608811634165 34 134.57 126.252844489107 8.31715551089315 35 130.33 124.05084835394 6.27915164605977 36 136.47 123.191084103715 13.2789158962851 37 139.05 126.650228358028 12.3997716419722 38 158.21 131.386905269422 26.8230947305783 39 148.07 129.205808826174 18.8641911738256 40 137.74 132.733957257232 5.00604274276814 41 139.74 135.701265716858 4.03873428314191 42 144.08 138.44624040928 5.6337595907205 43 145.35 138.145960612987 7.20403938701339 44 145.77 144.42438534394 1.34561465606036 45 140.56 144.406165070854 -3.84616507085434 46 121.41 140.129711328516 -18.7197113285165 47 120.44 132.726025951002 -12.2860259510018 48 116.97 131.387384223214 -14.4173842232141 49 128.03 136.923067137686 -8.89306713768594 50 128.51 137.71128974237 -9.2012897423702 51 127.76 137.226039715103 -9.46603971510276 52 134.58 139.741627713333 -5.16162771333316 53 147.64 139.901255787536 7.73874421246412 54 144.46 138.925837765677 5.53416223432315 55 137.6 147.355333303333 -9.75533330333343 56 146.87 141.699290667191 5.17070933280874 57 145.67 148.731497391082 -3.06149739108214 58 151.95 143.973437775182 7.9765622248183 59 150.23 147.842769210768 2.38723078923222 60 155.86 148.336500765734 7.52349923426561 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.65370391318947 0.69259217362106 0.34629608681053 8 0.488346895888548 0.976693791777096 0.511653104111452 9 0.375721070666069 0.751442141332139 0.624278929333931 10 0.60386887667399 0.792262246652019 0.39613112332601 11 0.592390946134514 0.815218107730971 0.407609053865486 12 0.545239233869372 0.909521532261256 0.454760766130628 13 0.783347972845307 0.433304054309387 0.216652027154693 14 0.72529018420328 0.54941963159344 0.27470981579672 15 0.639636173612031 0.720727652775938 0.360363826387969 16 0.559375814109528 0.881248371780945 0.440624185890472 17 0.469636487469033 0.939272974938066 0.530363512530967 18 0.433839357821971 0.867678715643942 0.566160642178029 19 0.461116021554101 0.922232043108201 0.538883978445899 20 0.567512668264598 0.864974663470805 0.432487331735402 21 0.59870354899022 0.80259290201956 0.40129645100978 22 0.638361714335635 0.72327657132873 0.361638285664365 23 0.707692095161882 0.584615809676235 0.292307904838118 24 0.7549358463591 0.490128307281799 0.2450641536409 25 0.733575991078241 0.532848017843518 0.266424008921759 26 0.688653868498352 0.622692263003295 0.311346131501648 27 0.670008127075593 0.659983745848814 0.329991872924407 28 0.667517591431091 0.664964817137817 0.332482408568909 29 0.714972869035652 0.570054261928696 0.285027130964348 30 0.728461984305578 0.543076031388844 0.271538015694422 31 0.844850399443219 0.310299201113562 0.155149600556781 32 0.882604220533922 0.234791558932155 0.117395779466078 33 0.91329212132412 0.17341575735176 0.0867078786758799 34 0.917219183508256 0.165561632983488 0.082780816491744 35 0.920636212916922 0.158727574166156 0.079363787083078 36 0.920588042555395 0.15882391488921 0.079411957444605 37 0.924344340111655 0.151311319776689 0.0756556598883447 38 0.97301008124754 0.0539798375049199 0.02698991875246 39 0.973584807313748 0.052830385372505 0.0264151926862525 40 0.957376100416444 0.0852477991671121 0.0426238995835561 41 0.939112395996322 0.121775208007357 0.0608876040036784 42 0.927644238636874 0.144711522726252 0.0723557613631259 43 0.93807976771121 0.123840464577581 0.0619202322887903 44 0.95604903864347 0.0879019227130585 0.0439509613565293 45 0.993338774043398 0.013322451913205 0.0066612259566025 46 0.994677701789067 0.0106445964218662 0.00532229821093311 47 0.98995730695727 0.0200853860854595 0.0100426930427298 48 0.996017277479552 0.00796544504089602 0.00398272252044801 49 0.989917319809213 0.0201653603815734 0.0100826801907867 50 0.975696300619862 0.0486073987602753 0.0243036993801377 51 0.986045775902909 0.0279084481941823 0.0139542240970912 52 0.975611890751469 0.0487762184970625 0.0243881092485312 53 0.995049825888176 0.00990034822364847 0.00495017411182423 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 2 0.0425531914893617 NOK 5% type I error level 9 0.191489361702128 NOK 10% type I error level 13 0.276595744680851 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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