Paper Multiple Regression Analysis

*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: Tue, 29 Dec 2009 12:11:43 -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/29/t1262113961xxs6ivzaxjmd9z0.htm/, Retrieved Tue, 29 Dec 2009 20:12:53 +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/29/t1262113961xxs6ivzaxjmd9z0.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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10001.60 49.14 10411.75 44.61 10673.38 40.22 10539.51 44.23 10723.78 45.85 10682.06 53.38 10283.19 53.26 10377.18 51.8 10486.64 55.3 10545.38 57.81 10554.27 63.96 10532.54 63.77 10324.31 59.15 10695.25 56.12 10827.81 57.42 10872.48 63.52 10971.19 61.71 11145.65 63.01 11234.68 68.18 11333.88 72.03 10997.97 69.75 11036.89 74.41 11257.35 74.33 11533.59 64.24 11963.12 60.03 12185.15 59.44 12377.62 62.5 12512.89 55.04 12631.48 58.34 12268.53 61.92 12754.80 67.65 13407.75 67.68 13480.21 70.3 13673.28 75.26 13239.71 71.44 13557.69 76.36 13901.28 81.71 13200.58 92.6 13406.97 90.6 12538.12 92.23 12419.57 94.09 12193.88 102.79 12656.63 109.65 12812.48 124.05 12056.67 132.69 11322.38 135.81 11530.75 116.07 11114.08 101.42 9181.73 75.73 8614.55 55.48 8595.56 43.8 8396.20 45.29 7690.50 44.01 7235.47 47.48 7992.12 51.07 8398.37 57.84 8593.01 69.04 8679.75 65.61 9374.63 72.87 9634.97 68.41

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 R Framework error message The field 'Names of X columns' contains a hard return which cannot be interpreted. Please, resubmit your request without hard returns in the 'Names of X columns'. Multiple Linear Regression - Estimated Regression Equation Dow[t] = + 8856.1158450421 + 53.7362021238763`Olie `[t] -169.423877488405M1[t] + 10.3405026825402M2[t] + 357.029369239348M3[t] + 135.121992321534M4[t] + 55.4608754872516M5[t] -346.360093820586M6[t] -250.825807705426M7[t] -178.173008992735M8[t] -531.167461918275M9[t] -685.03164340598M10[t] -390.74917352739M11[t] -44.5322003331388t + 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) 8856.1158450421 927.093959 9.5526 0 0 `Olie ` 53.7362021238763 9.63131 5.5793 1e-06 1e-06 M1 -169.423877488405 893.017442 -0.1897 0.850363 0.425181 M2 10.3405026825402 894.471753 0.0116 0.990826 0.495413 M3 357.029369239348 896.689552 0.3982 0.692351 0.346176 M4 135.121992321534 894.761207 0.151 0.880625 0.440312 M5 55.4608754872516 893.471913 0.0621 0.950773 0.475387 M6 -346.360093820586 888.590454 -0.3898 0.698494 0.349247 M7 -250.825807705426 886.072082 -0.2831 0.778389 0.389194 M8 -178.173008992735 885.26009 -0.2013 0.841378 0.420689 M9 -531.167461918275 886.501646 -0.5992 0.551998 0.275999 M10 -685.03164340598 887.562365 -0.7718 0.444173 0.222086 M11 -390.74917352739 885.658245 -0.4412 0.661138 0.330569 t -44.5322003331388 11.454434 -3.8878 0.000323 0.000162 Multiple Linear Regression - Regression Statistics Multiple R 0.661037995331441 R-squared 0.436971231271811 Adjusted R-squared 0.277854405326888 F-TEST (value) 2.74622893384679 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0.00582681008887731 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1397.88792891172 Sum Squared Residuals 89888170.4426669 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 10001.6 11282.7567395878 -1281.15673958779 2 10411.75 11174.5639238045 -762.81392380448 3 10673.38 11240.8186627043 -567.438662704332 4 10539.51 11189.8612559701 -650.351255970122 5 10723.78 11152.7205862434 -428.940586243381 6 10682.06 11111.0010185952 -428.941018595194 7 10283.19 11155.5547601223 -872.364760122349 8 10377.18 11105.2205034010 -728.040503401041 9 10486.64 10895.7705575759 -409.130557575931 10 10545.38 10832.2520430860 -286.872043086018 11 10554.27 11412.4799556933 -858.209955693306 12 10532.54 11748.4870504840 -1215.94705048402 13 10324.31 11286.2697188502 -961.95971885017 14 10695.25 11258.6812062526 -563.43120625263 15 10827.81 11630.6949352373 -802.884935237338 16 10872.48 11692.0461909420 -819.566190942031 17 10971.19 11470.5903479304 -499.400347930393 18 11145.65 11094.0942410505 51.5557589495432 19 11234.68 11422.9124918129 -188.232491812919 20 11333.88 11657.9174683694 -324.037468369394 21 10997.97 11137.8722742683 -139.902274268278 22 11036.89 11189.8865943447 -152.996594344698 23 11257.35 11435.3379677202 -177.987967720238 24 11533.59 11239.3566614846 294.233338515424 25 11963.12 10799.1711727215 1163.94882727849 26 12185.15 10902.6989933062 1282.45100669377 27 12377.62 11369.2884380290 1008.33156197104 28 12512.89 10701.9767929339 1810.91320706611 29 12631.48 10755.1129427753 1876.36705722473 30 12268.53 10501.1353767378 1767.39462326224 31 12754.8 10860.0459006896 1894.7540993104 32 13407.75 10889.7785851329 2517.97141486713 33 13480.21 10633.0407814387 2847.16921856126 34 13673.28 10701.1759621523 2972.10403784767 35 13239.71 10745.6539395846 2494.05606041543 36 13557.69 11356.2530272283 2201.43697277171 37 13901.28 11429.7856307695 2471.49436923051 38 13200.58 12150.2050517363 1050.37494826369 39 13406.97 12344.8893137122 1062.08068628778 40 12538.12 12166.0397459232 372.080254076813 41 12419.57 12141.7957647062 277.774235293823 42 12193.88 12162.9475535429 30.9324464570743 43 12656.63 12582.5799858947 74.0500141052617 44 12812.48 13384.5018948581 -572.021894858108 45 12056.67 13451.2560279497 -1394.58602794972 46 11322.38 13420.5165967554 -2098.13659675537 47 11530.75 12609.5142363755 -1078.76423637550 48 11114.08 12168.4958484550 -1054.41584845497 49 9181.73 10574.0567380710 -1392.32673807104 50 8614.55 9621.13082490035 -1006.58082490035 51 8595.56 9295.64865031714 -700.088650317144 52 8396.2 9109.27601423077 -713.076014230766 53 7690.5 8916.30035834478 -1225.80035834478 54 7235.47 8656.41181007366 -1420.94181007366 55 7992.12 8900.3268614804 -908.206861480396 56 8398.37 9292.24154823859 -893.87154823859 57 8593.01 9496.56035876733 -903.550358767327 58 8679.75 9113.84880366159 -434.098803661587 59 9374.63 9753.72390062638 -379.093900626382 60 9634.97 9860.27741234814 -225.307412348143 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.000103210639862960 0.000206421279725921 0.999896789360137 18 1.32981749945004e-05 2.65963499890009e-05 0.999986701825005 19 0.000265426116138330 0.000530852232276659 0.999734573883862 20 0.000170210381064908 0.000340420762129816 0.999829789618935 21 3.9102356850154e-05 7.8204713700308e-05 0.99996089764315 22 1.05909748985832e-05 2.11819497971665e-05 0.999989409025101 23 8.14420532975033e-06 1.62884106595007e-05 0.99999185579467 24 1.87473659634520e-05 3.74947319269039e-05 0.999981252634037 25 5.61151960801353e-05 0.000112230392160271 0.99994388480392 26 3.74794913813736e-05 7.49589827627472e-05 0.999962520508619 27 6.04505428764938e-05 0.000120901085752988 0.999939549457123 28 2.56943238716021e-05 5.13886477432041e-05 0.999974305676128 29 9.23875855560041e-06 1.84775171112008e-05 0.999990761241444 30 5.84199276939733e-06 1.16839855387947e-05 0.99999415800723 31 6.95681687706668e-06 1.39136337541334e-05 0.999993043183123 32 4.19903791083549e-05 8.39807582167099e-05 0.999958009620892 33 0.000228194254631044 0.000456388509262088 0.99977180574537 34 0.00093445781717313 0.00186891563434626 0.999065542182827 35 0.000463346088014656 0.000926692176029313 0.999536653911985 36 0.00156927888598039 0.00313855777196078 0.99843072111402 37 0.00634896780308723 0.0126979356061745 0.993651032196913 38 0.00667794280726335 0.0133558856145267 0.993322057192737 39 0.00814558335434178 0.0162911667086836 0.991854416645658 40 0.00715023905311088 0.0143004781062218 0.992849760946889 41 0.0164519205459902 0.0329038410919804 0.98354807945401 42 0.0629972551176956 0.125994510235391 0.937002744882304 43 0.198831121677325 0.397662243354651 0.801168878322675 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 20 0.740740740740741 NOK 5% type I error level 25 0.925925925925926 NOK 10% type I error level 25 0.925925925925926 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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