Workshop 7: 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 10:54:57 -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/t125865336431m54o45s0gr34w.htm/, Retrieved Thu, 19 Nov 2009 18:56:16 +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/t125865336431m54o45s0gr34w.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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0,7461 0,5270 0,7775 0,4720 0,7790 0,0000 0,7744 0,0520 0,7905 0,3130 0,7719 0,3640 0,7811 0,3630 0,7557 -0,1550 0,7637 0,0520 0,7595 0,5680 0,7471 0,6680 0,7615 1,3780 0,7487 0,2520 0,7389 -0,4020 0,7337 -0,0500 0,7510 0,5550 0,7382 0,0500 0,7159 0,1500 0,7542 0,4500 0,7636 0,2990 0,7433 0,1990 0,7658 0,4960 0,7627 0,4440 0,7480 -0,3930 0,7692 -0,4440 0,7850 0,1980 0,7913 0,4940 0,7720 0,1330 0,7880 0,3880 0,8070 0,4840 0,8268 0,2780 0,8244 0,3690 0,8487 0,1650 0,8572 0,1550 0,8214 0,0870 0,8827 0,4140 0,9216 0,3600 0,8865 0,9750 0,8816 0,2700 0,8884 0,3590 0,9466 0,1690 0,9180 0,3810 0,9337 0,1540 0,9559 0,4860 0,9626 0,9250 0,9434 0,7280 0,8639 -0,0140 0,7996 0,0460 0,6680 -0,8190 0,6572 -1,6740 0,6928 -0,7880 0,6438 0,2790 0,6454 0,3960 0,6873 -0,1410 0,7265 -0,0190 0,7912 0,0990 0,8114 0,7420 0,8281 0,0050 0,8393 0,4480

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.772647651071932 + 0.0700411042991505X[t] -0.000190631685312515M1[t] + 0.00240989211865488M2[t] + 0.00406895727169555M3[t] -0.0260309794167778M4[t] -0.00934246972346833M5[t] -0.00996982849640152M6[t] + 0.0146382701539164M7[t] + 0.0301313224239747M8[t] + 0.0241132248770421M9[t] + 0.0308083018096797M10[t] + 0.0113569242639656M11[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) 0.772647651071932 0.039191 19.7151 0 0 X 0.0700411042991505 0.023839 2.9381 0.005147 0.002573 M1 -0.000190631685312515 0.052114 -0.0037 0.997097 0.498549 M2 0.00240989211865488 0.052392 0.046 0.963512 0.481756 M3 0.00406895727169555 0.052072 0.0781 0.938055 0.469028 M4 -0.0260309794167778 0.051335 -0.5071 0.614523 0.307262 M5 -0.00934246972346833 0.051348 -0.1819 0.856426 0.428213 M6 -0.00996982849640152 0.051366 -0.1941 0.846958 0.423479 M7 0.0146382701539164 0.051369 0.285 0.776953 0.388476 M8 0.0301313224239747 0.051406 0.5861 0.560642 0.280321 M9 0.0241132248770421 0.051312 0.4699 0.640621 0.32031 M10 0.0308083018096797 0.051299 0.6006 0.551082 0.275541 M11 0.0113569242639656 0.051301 0.2214 0.825779 0.41289 Multiple Linear Regression - Regression Statistics Multiple R 0.478529492155792 R-squared 0.228990474862880 Adjusted R-squared 0.0278575552618919 F-TEST (value) 1.13850321129508 F-TEST (DF numerator) 12 F-TEST (DF denominator) 46 p-value 0.354332486664713 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0764656629349064 Sum Squared Residuals 0.268961889971437 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 0.7461 0.80936868135227 -0.0632686813522695 2 0.7775 0.808116944419786 -0.0306169444197858 3 0.779 0.776716608343627 0.00228339165637259 4 0.7744 0.75025880907871 0.02414119092129 5 0.7905 0.785228046994098 0.00527195300590231 6 0.7719 0.788172784540421 -0.0162727845404211 7 0.7811 0.81271084208644 -0.0316108420864399 8 0.7557 0.791922602329538 -0.0362226023295382 9 0.7637 0.80040301337253 -0.0367030133725297 10 0.7595 0.84323930012353 -0.0837393001235291 11 0.7471 0.83079203300773 -0.08369203300773 12 0.7615 0.869164292796161 -0.107664292796161 13 0.7487 0.790107377670005 -0.0414073776700053 14 0.7389 0.746901019262328 -0.00800101926232827 15 0.7337 0.77321455312867 -0.0395145531286699 16 0.751 0.785489484541183 -0.0344894845411826 17 0.7382 0.766807236563421 -0.0286072365634211 18 0.7159 0.773183988220403 -0.057283988220403 19 0.7542 0.818804418160466 -0.064604418160466 20 0.7636 0.823721263681353 -0.0601212636813526 21 0.7433 0.810699055704505 -0.0673990557045049 22 0.7658 0.83819634061399 -0.0723963406139903 23 0.7627 0.81510282564472 -0.0524028256447202 24 0.748 0.745121497082366 0.00287850291763425 25 0.7692 0.741358769077797 0.0278412309222034 26 0.785 0.788925681841819 -0.00392568184181853 27 0.7913 0.811316913867408 -0.0200169138674078 28 0.772 0.755932138526941 0.0160678614730589 29 0.788 0.790481129816534 -0.00248112981653392 30 0.807 0.79657771705632 0.0104222829436808 31 0.8268 0.806757348221012 0.0200426517789878 32 0.8244 0.828624140982293 -0.00422414098229306 33 0.8487 0.808317658158334 0.0403823418416663 34 0.8572 0.81431232404798 0.04288767595202 35 0.8214 0.790098151409924 0.0313018485900764 36 0.8827 0.80164466825178 0.0810553317482199 37 0.9216 0.797671816934314 0.123928183065686 38 0.8865 0.843347619882258 0.0431523801177415 39 0.8816 0.795627706504398 0.085972293495602 40 0.8884 0.771761428098549 0.116638571901451 41 0.9466 0.77514212797502 0.17145787202498 42 0.918 0.789363483313507 0.128636516686493 43 0.9337 0.798072251287917 0.135627748712083 44 0.9559 0.836818950185294 0.119081049814706 45 0.9626 0.861548897425688 0.101051102574312 46 0.9434 0.854445876811393 0.0889541231886068 47 0.8639 0.78302399987571 0.0808760001242906 48 0.7996 0.775869541869693 0.0237304581303072 49 0.668 0.715093354965615 -0.0470933549656151 50 0.6572 0.657808734593809 -0.00060873459380887 51 0.6928 0.721524218155897 -0.0287242181558969 52 0.6438 0.766158139754617 -0.122358139754617 53 0.6454 0.791041458650927 -0.145641458650927 54 0.6873 0.75280202686935 -0.0655020268693501 55 0.7265 0.785955140244164 -0.0594551402441644 56 0.7912 0.809713042821522 -0.0185130428215224 57 0.8114 0.848731375338944 -0.0373313753389436 58 0.8281 0.803806158403107 0.0242938415968925 59 0.8393 0.815382990061917 0.0239170099380832 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0335973880111620 0.0671947760223240 0.966402611988838 17 0.0183222105214037 0.0366444210428075 0.981677789478596 18 0.0110959883845923 0.0221919767691846 0.988904011615408 19 0.00476679846746194 0.00953359693492388 0.995233201532538 20 0.00151447342514872 0.00302894685029744 0.998485526574851 21 0.000572182650164849 0.00114436530032970 0.999427817349835 22 0.000206083350186539 0.000412166700373079 0.999793916649814 23 8.30191698789731e-05 0.000166038339757946 0.99991698083012 24 2.56769367998686e-05 5.13538735997372e-05 0.9999743230632 25 1.12358654741466e-05 2.24717309482932e-05 0.999988764134526 26 4.21186873817185e-06 8.4237374763437e-06 0.999995788131262 27 2.03887384487026e-06 4.07774768974053e-06 0.999997961126155 28 5.16425638641425e-07 1.03285127728285e-06 0.999999483574361 29 1.52754195348215e-07 3.0550839069643e-07 0.999999847245805 30 2.9254657857638e-07 5.8509315715276e-07 0.999999707453421 31 4.17141620585397e-07 8.34283241170794e-07 0.99999958285838 32 4.94532215011616e-07 9.8906443002323e-07 0.999999505467785 33 2.32009284874381e-06 4.64018569748761e-06 0.999997679907151 34 6.70097181141657e-06 1.34019436228331e-05 0.999993299028189 35 5.17694808816737e-06 1.03538961763347e-05 0.999994823051912 36 2.11031267868055e-05 4.22062535736111e-05 0.999978896873213 37 0.000229366867670294 0.000458733735340589 0.99977063313233 38 0.000357652211756362 0.000715304423512724 0.999642347788244 39 0.000322287981371040 0.000644575962742079 0.99967771201863 40 0.00134317553741778 0.00268635107483556 0.998656824462582 41 0.126793939227939 0.253587878455878 0.873206060772061 42 0.153447792911637 0.306895585823275 0.846552207088363 43 0.333380267350408 0.666760534700815 0.666619732649592 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 22 0.785714285714286 NOK 5% type I error level 24 0.857142857142857 NOK 10% type I error level 25 0.892857142857143 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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