PAPER - Multiple Regression 1

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Title produced by software: Multiple Regression

Date of computation: Sat, 18 Dec 2010 15:51:20 +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/18/t1292687380rldkxksy1xl9nt0.htm/, Retrieved Sat, 18 Dec 2010 16:49:43 +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/18/t1292687380rldkxksy1xl9nt0.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}, }

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Dataseries X:

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104,37 167.16 101,56 100,93 104,89 179.84 102,13 101,18 105,15 174.44 102,39 101,11 105,72 180.35 102,42 102,42 106,38 193.17 103,87 102,37 106,40 195.16 104,44 101,95 106,47 202.43 104,97 102,20 106,59 189.91 105,17 103,35 106,76 195.98 105,35 103,65 107,35 212.09 104,65 102,06 107,81 205.81 106,62 102,66 108,03 204.31 107,05 102,32 109,08 196.07 112,30 102,21 109,86 199.98 114,70 102,33 110,29 199.1 115,40 104,41 110,34 198.31 115,64 104,33 110,59 195.72 115,66 105,27 110,64 223.04 114,50 105,34 110,83 238.41 115,14 104,88 111,51 259.73 115,41 105,49 113,32 326.54 119,32 105,90 115,89 335.15 124,77 105,39 116,51 321.81 130,96 104,40 117,44 368.62 141,02 106,19 118,25 369.59 150,60 106,54 118,65 425 151,10 108,26 118,52 439.72 157,19 106,95 119,07 362.23 157,28 108,32 119,12 328.76 156,54 108,35 119,28 348.55 159,62 109,29 119,30 328.18 163,77 109,46 119,44 329.34 165,08 109,50 119,57 295.55 164,75 109,84 119,93 237.38 163,93 108,73 120,03 226.85 157,51 109,38 119,66 220.14 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 5 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation Brood[t] = + 27.2751805068209 + 0.00711546804625393Tarwe[t] + 0.145572171967312Meel[t] + 0.618723143738309Water[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) 27.2751805068209 11.325497 2.4083 0.019538 0.009769 Tarwe 0.00711546804625393 0.002911 2.4443 0.017875 0.008937 Meel 0.145572171967312 0.021506 6.7689 0 0 Water 0.618723143738309 0.127504 4.8526 1.1e-05 6e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.98180639943996 R-squared 0.963943805981258 Adjusted R-squared 0.961902889338687 F-TEST (value) 472.309248636078 F-TEST (DF numerator) 3 F-TEST (DF denominator) 53 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.09978300867724 Sum Squared Residuals 64.1047013072841 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 104.37 105.69663882794 -1.32663882794046 2 104.89 106.024519886723 -1.1345198867229 3 105.15 105.980634503923 -0.830634503922941 4 105.72 106.837581403533 -1.11758140353251 5 106.38 107.108945196051 -0.728945196051182 6 106.4 106.946217395115 -0.546217395114492 7 106.47 107.229780884888 -0.759780884888019 8 106.59 107.881341274641 -1.29134127464143 9 106.76 108.136352099758 -1.3763520997578 10 107.35 107.165311971062 0.184688028938067 11 107.81 107.77863789675 0.0313621032499622 12 108.03 107.620194859756 0.409805140244426 13 109.08 108.257757760072 0.82224223992838 14 109.86 108.709199230103 1.15080076989738 15 110.29 110.091782277575 0.198217722425288 16 110.34 110.071600527591 0.268399472408735 17 110.59 110.637682663905 -0.0476826639048221 18 110.64 110.706524151508 -0.0665241515080871 19 110.83 110.624442439318 0.205557560681535 20 111.51 111.192869822176 0.317130177823866 21 113.32 112.491117923671 0.828882076328725 22 115.89 113.030201637465 2.85979836253518 23 116.51 113.223837125905 3.28616287409547 24 117.44 116.128882662432 1.31111733756758 25 118.25 117.746919174193 0.503080825807459 26 118.65 119.278177151849 -0.628177151849014 27 118.52 119.458924050474 -0.938924050473626 28 119.07 119.768298633968 -0.698298633967948 29 119.12 119.440982205516 -0.320982205516155 30 119.28 120.611759362925 -1.33175936292486 31 119.3 121.176124726923 -1.87612472692253 32 119.44 121.399827140883 -1.9598271408829 33 119.57 121.321722527722 -1.75172252772179 34 119.93 120.101663880908 -0.17166388090847 35 120.03 119.494334701781 0.535665298218829 36 119.66 119.207512052332 0.452487947667925 37 119.46 120.548563937332 -1.08856393733194 38 119.48 119.822199311023 -0.34219931102302 39 119.56 119.687161548791 -0.127161548791163 40 119.43 118.952964837456 0.477035162544173 41 119.57 118.805621456818 0.764378543181783 42 119.59 119.263727554931 0.326272445068886 43 119.5 118.708547485889 0.791452514111197 44 119.54 118.758029025989 0.78197097401068 45 119.56 119.249762623894 0.310237376106077 46 119.61 118.512653656934 1.0973463430656 47 119.64 118.569198652984 1.0708013470155 48 119.6 117.250733429584 2.34926657041626 49 119.71 118.280748065233 1.42925193476732 50 119.72 119.401249596324 0.318750403675717 51 119.66 119.603474916486 0.0565250835136374 52 119.76 119.759980275129 1.9724871011073e-05 53 119.8 119.769361314704 0.0306386852958137 54 119.88 120.438638884186 -0.558638884186446 55 119.78 120.389531932415 -0.609531932415295 56 120.08 120.835605233978 -0.755605233978133 57 120.22 121.003946228585 -0.783946228584525 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.00731760001396593 0.0146352000279319 0.992682399986034 8 0.00445191459263527 0.00890382918527053 0.995548085407365 9 0.00169469847303157 0.00338939694606315 0.998305301526968 10 0.00072746124482725 0.0014549224896545 0.999272538755173 11 0.0009263575406263 0.0018527150812526 0.999073642459374 12 0.000581851216805094 0.00116370243361019 0.999418148783195 13 0.000178662537657191 0.000357325075314383 0.999821337462343 14 5.3286328930436e-05 0.000106572657860872 0.99994671367107 15 1.70026493308216e-05 3.40052986616432e-05 0.99998299735067 16 6.30884262379384e-06 1.26176852475877e-05 0.999993691157376 17 4.79873349175221e-06 9.59746698350443e-06 0.999995201266508 18 1.09352375949204e-05 2.18704751898408e-05 0.999989064762405 19 0.00029842650690346 0.00059685301380692 0.999701573493097 20 0.0157127144019413 0.0314254288038825 0.98428728559806 21 0.706204322679306 0.587591354641388 0.293795677320694 22 0.803778462134748 0.392443075730504 0.196221537865252 23 0.963023015307993 0.0739539693840151 0.0369769846920075 24 0.99999998670935 2.65813005598526e-08 1.32906502799263e-08 25 0.999999999990131 1.97370688609101e-11 9.86853443045504e-12 26 0.999999999999697 6.06330792741829e-13 3.03165396370914e-13 27 0.999999999999952 9.55705546408133e-14 4.77852773204066e-14 28 0.999999999999833 3.33054101238272e-13 1.66527050619136e-13 29 0.999999999999378 1.24387636905123e-12 6.21938184525613e-13 30 0.99999999999801 3.98160243773045e-12 1.99080121886522e-12 31 0.999999999996653 6.69464902705846e-12 3.34732451352923e-12 32 0.999999999993307 1.33854610639422e-11 6.69273053197109e-12 33 0.999999999988927 2.21466824403792e-11 1.10733412201896e-11 34 0.999999999984382 3.12352504036653e-11 1.56176252018327e-11 35 0.999999999998262 3.47644760463728e-12 1.73822380231864e-12 36 0.999999999991876 1.62481846061782e-11 8.12409230308911e-12 37 0.999999999992757 1.44863521523635e-11 7.24317607618173e-12 38 0.999999999986996 2.60078829997037e-11 1.30039414998518e-11 39 0.999999999950718 9.85648796235672e-11 4.92824398117836e-11 40 0.999999999909694 1.80611352307819e-10 9.03056761539094e-11 41 0.99999999954023 9.1954135768042e-10 4.5977067884021e-10 42 0.99999999914522 1.70955901691853e-09 8.54779508459265e-10 43 0.999999998441802 3.11639614380511e-09 1.55819807190256e-09 44 0.999999995682954 8.63409201901848e-09 4.31704600950924e-09 45 0.99999998923093 2.15381424229861e-08 1.0769071211493e-08 46 0.999999907869794 1.84260412097819e-07 9.21302060489093e-08 47 0.999999239017117 1.52196576556385e-06 7.60982882781923e-07 48 0.999991580788395 1.68384232104029e-05 8.41921160520143e-06 49 0.999965290211745 6.94195765107814e-05 3.47097882553907e-05 50 0.99946464703577 0.00107070592845864 0.00053535296422932 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 39 0.886363636363636 NOK 5% type I error level 41 0.931818181818182 NOK 10% type I error level 42 0.954545454545455 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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