WS10 Multiple Regression - Priems

*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 11:41:40 +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/t12922405220febs9i3o34ar9q.htm/, Retrieved Mon, 13 Dec 2010 12:42:06 +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/t12922405220febs9i3o34ar9q.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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0.504208603 0.397232704 0.457969746 0.382767296 0.509923035 0.396037736 0.606622221 0.441761006 0.626210885 0.445220126 0.626631316 0.438490566 0.676731276 0.467484277 0.613117455 0.465786164 0.486215861 0.402075472 0.452529881 0.376163522 0.467150592 0.37591195 0.494624486 0.392955975 0.444567428 0.34490566 0.478862605 0.368553459 0.544458459 0.390880503 0.628201498 0.424842767 0.672578445 0.426855346 0.652706633 0.442327044 0.645430599 0.474842767 0.576334011 0.447610063 0.618334234 0.480754717 0.639896351 0.516037736 0.72850438 0.580628931 0.694655375 0.573522013 0.689773225 0.578867925 0.712244845 0.593584906 0.760337031 0.645974843 0.837816503 0.690503145 0.90688735 0.782201258 0.976018259 0.839056604 0.962066806 0.847484277 0.837593417 0.726855346 0.767638807 0.635534591 0.580006349 0.470943396 0.387740568 0.346163522 0.331274078 0.272327044 0.345251272 0.286792453 0.380172806 0.27672956 0.399838692 0.297421384 0.425742404 0.321698113 0.524183377 0.365597484 0.59 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 RuweOlie(NSB)[t] = -0.0541370250440063 + 0.879652544184292LoodvrijeBenzine[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) -0.0541370250440063 0.02054 -2.6357 0.010798 0.005399 LoodvrijeBenzine 0.879652544184292 0.033264 26.4442 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.961578230829939 R-squared 0.924632694006035 Adjusted R-squared 0.923310460567544 F-TEST (value) 699.296105430171 F-TEST (DF numerator) 1 F-TEST (DF denominator) 57 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0344242198136768 Sum Squared Residuals 0.0675465338574791 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 0.397232704 0.389391355384552 0.0078413486154484 2 0.382767296 0.348717227184328 0.0340500688156722 3 0.396037736 0.39441807003192 0.0016196659680803 4 0.441761006 0.47947975501737 -0.0377187490173697 5 0.445220126 0.496710973142141 -0.051490847142141 6 0.438490566 0.497080806340945 -0.0585902403409449 7 0.467484277 0.541151363618476 -0.0736670866184763 8 0.465786164 0.485193304130542 -0.0194071401305421 9 0.402075472 0.3735639941074 0.0285114778926 10 0.376163522 0.343932036097059 0.0322314859029412 11 0.37591195 0.356793181725992 0.019118768274008 12 0.392955975 0.380960662481742 0.0119953125182585 13 0.34490566 0.336927844057661 0.00797781594233911 14 0.368553459 0.367095683758962 0.0014577752410385 15 0.390880503 0.424797243618003 -0.0339167406180029 16 0.424842767 0.498462020932077 -0.0736192539320774 17 0.426855346 0.537498315263759 -0.110642969263759 18 0.442327044 0.520018025280407 -0.0776909812804069 19 0.474842767 0.513617643460735 -0.0387748764607355 20 0.447610063 0.452836654032082 -0.00522659103208166 21 0.480754717 0.489782257050339 -0.00902754005033926 22 0.516037736 0.508749428127389 0.00728830787261139 23 0.580628931 0.586693706272394 -0.00606477527239416 24 0.573522013 0.556918342906037 0.0166036700939627 25 0.578867925 0.552623747237448 0.0262441777625519 26 0.593584906 0.57239096494239 0.0211939410576094 27 0.645974843 0.614695378712675 0.0312794642873251 28 0.690503145 0.68285039337953 0.00765275162046949 29 0.782201258 0.743608739672044 0.0385925183279555 30 0.839056604 0.804419919655667 0.0346366843443327 31 0.847484277 0.79214748852915 0.0553367884708502 32 0.726855346 0.682654155212059 0.0442011907879414 33 0.635534591 0.621118404548139 0.0144161864518613 34 0.470943396 0.456067035496886 0.0148763605031137 35 0.346163522 0.286939952080656 0.0592235699193437 36 0.272327044 0.237269060490999 0.0350579835090006 37 0.286792453 0.249564134753657 0.0372283182463432 38 0.27672956 0.280282950983575 -0.00355339098357509 39 0.297421384 0.297582097637113 -0.00016071363711334 40 0.321698113 0.320368363801731 0.00132974919826947 41 0.365597484 0.406962216153158 -0.0413647321531577 42 0.435220126 0.471116991522979 -0.0358968655229794 43 0.412893082 0.42218576633093 -0.00929268433093034 44 0.458679245 0.48688397081054 -0.0282047258105404 45 0.428427673 0.427846482632646 0.000581190367353617 46 0.463522013 0.451259201273477 0.0122628117265226 47 0.487169811 0.476679587861307 0.0104902231386931 48 0.473584906 0.453855582843239 0.0197293231567609 49 0.491886792 0.486144304412933 0.00574248758706745 50 0.474842767 0.460263512961392 0.0145792540386078 51 0.502327044 0.497865759010833 0.00446128498916735 52 0.539371069 0.54126457753987 -0.00189350853987037 53 0.484402516 0.514115786298244 -0.0297132702982444 54 0.474654088 0.470188634376729 0.00446545362327131 55 0.473522013 0.46130508490869 0.0122169280913104 56 0.48754717 0.480355279352164 0.00719189064783576 57 0.493333333 0.47230196794664 0.0210313650533602 58 0.525157233 0.507700308761262 0.0174569242387378 59 0.542704403 0.516961238846934 0.0257431641530657 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0015945272252874 0.0031890544505748 0.998405472774713 6 0.000642352009735446 0.00128470401947089 0.999357647990265 7 0.000139159194790768 0.000278318389581537 0.99986084080521 8 0.00439265244093759 0.00878530488187518 0.995607347559062 9 0.0018604207074985 0.003720841414997 0.998139579292502 10 0.00060778723526109 0.00121557447052218 0.999392212764739 11 0.000248597573499633 0.000497195146999267 0.9997514024265 12 6.82521151268307e-05 0.000136504230253661 0.999931747884873 13 0.000350072366554159 0.000700144733108319 0.999649927633446 14 0.000225900674042518 0.000451801348085037 0.999774099325958 15 0.000294599651775342 0.000589199303550685 0.999705400348225 16 0.000742853132310857 0.00148570626462171 0.99925714686769 17 0.0177375654125664 0.0354751308251329 0.982262434587434 18 0.044672402060383 0.089344804120766 0.955327597939617 19 0.101653526993022 0.203307053986043 0.898346473006978 20 0.12237662021294 0.24475324042588 0.87762337978706 21 0.236896919421018 0.473793838842036 0.763103080578982 22 0.561685534355 0.87662893129 0.438314465645 23 0.861414818788528 0.277170362422945 0.138585181211472 24 0.948033856997288 0.103932286005423 0.0519661430027117 25 0.97822171292094 0.043556574158119 0.0217782870790595 26 0.986331937410138 0.0273361251797241 0.0136680625898621 27 0.993104464175976 0.0137910716480488 0.00689553582402439 28 0.992794237873755 0.0144115242524898 0.00720576212624491 29 0.995192226175485 0.00961554764903 0.004807773824515 30 0.99480290327592 0.0103941934481585 0.00519709672407925 31 0.997949089383925 0.00410182123215091 0.00205091061607546 32 0.999242002578003 0.00151599484399346 0.000757997421996728 33 0.998970677077536 0.00205864584492848 0.00102932292246424 34 0.998273335267447 0.00345332946510537 0.00172666473255268 35 0.999579288705709 0.000841422588582955 0.000420711294291477 36 0.999547137980543 0.000905724038913812 0.000452862019456906 37 0.999749975531723 0.000500048936553561 0.000250024468276781 38 0.999437356497241 0.00112528700551772 0.00056264350275886 39 0.998820539627554 0.00235892074489281 0.0011794603724464 40 0.997854732990786 0.00429053401842744 0.00214526700921372 41 0.99912868886791 0.00174262226418179 0.000871311132090894 42 0.999775117237146 0.000449765525707808 0.000224882762853904 43 0.99968517099125 0.000629658017498352 0.000314829008749176 44 0.99992709949412 0.000145801011761551 7.29005058807757e-05 45 0.9998834327027 0.000233134594599899 0.00011656729729995 46 0.999654089032257 0.000691821935485236 0.000345910967742618 47 0.99898701080627 0.00202597838746044 0.00101298919373022 48 0.99735289504334 0.00529420991332083 0.00264710495666042 49 0.993039191486698 0.0139216170266034 0.00696080851330172 50 0.98272150974519 0.0345569805096197 0.0172784902548098 51 0.959549234357649 0.0809015312847026 0.0404507656423513 52 0.911542296102803 0.176915407794395 0.0884577038971973 53 0.99698110308385 0.00603779383229985 0.00301889691614993 54 0.990426636265006 0.0191467274699877 0.00957336373499386 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 32 0.64 NOK 5% type I error level 41 0.82 NOK 10% type I error level 43 0.86 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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