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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Sat, 04 Dec 2010 10:35:01 +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/04/t12914590576ss5iel3lxx4p46.htm/, Retrieved Sat, 04 Dec 2010 11:37:51 +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/04/t12914590576ss5iel3lxx4p46.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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14731798.37 0 16471559.62 0 15213975.95 0 17637387.4 0 17972385.83 0 16896235.55 0 16697955.94 0 19691579.52 0 15930700.75 0 17444615.98 0 17699369.88 0 15189796.81 0 15672722.75 0 17180794.3 0 17664893.45 0 17862884.98 0 16162288.88 0 17463628.82 0 16772112.17 0 19106861.48 0 16721314.25 0 18161267.85 0 18509941.2 0 17802737.97 0 16409869.75 0 17967742.04 0 20286602.27 0 19537280.81 0 18021889.62 0 20194317.23 0 19049596.62 0 20244720.94 0 21473302.24 0 19673603.19 0 21053177.29 0 20159479.84 0 18203628.31 0 21289464.94 0 20432335.71 1 17180395.07 1 15816786.32 1 15071819.75 1 14521120.61 1 15668789.39 1 14346884.11 1 13881008.13 1 15465943.69 1 14238232.92 1 13557713.21 1 16127590.29 1 16793894.2 1 16014007.43 1 16867867.15 1 16014583.21 1 15878594.85 1 18664899.14 1 17962530.06 1 17332692.2 1 19542066.35 1 17203555.19 1

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 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 15832703.2737956 -5049990.48431708X[t] + 111445.020763520t + 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) 15832703.2737956 437948.932719 36.1519 0 0 X -5049990.48431708 687682.616591 -7.3435 0 0 t 111445.020763520 19135.492057 5.824 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.69826695104146 R-squared 0.487576734916737 Adjusted R-squared 0.469596971229605 F-TEST (value) 27.1180836078336 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 5.3017826795454e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1413308.63293638 Sum Squared Residuals 113854153640153 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 14731798.37 15944148.2945591 -1212349.92455909 2 16471559.62 16055593.3153226 415966.304677383 3 15213975.95 16167038.3360861 -953062.38608614 4 17637387.4 16278483.3568497 1358904.04315034 5 17972385.83 16389928.3776132 1582457.45238682 6 16896235.55 16501373.3983767 394862.151623307 7 16697955.94 16612818.4191402 85137.520859786 8 19691579.52 16724263.4399037 2967316.08009627 9 15930700.75 16835708.4606673 -905007.710667253 10 17444615.98 16947153.4814308 497462.498569228 11 17699369.88 17058598.5021943 640771.377805706 12 15189796.81 17170043.5229578 -1980246.71295781 13 15672722.75 17281488.5437213 -1608765.79372133 14 17180794.3 17392933.5644849 -212139.264484851 15 17664893.45 17504378.5852484 160514.864751628 16 17862884.98 17615823.6060119 247061.373988109 17 16162288.88 17727268.6267754 -1564979.74677541 18 17463628.82 17838713.6475389 -375084.82753893 19 16772112.17 17950158.6683025 -1178046.49830245 20 19106861.48 18061603.6890660 1045257.79093403 21 16721314.25 18173048.7098295 -1451734.45982949 22 18161267.85 18284493.730593 -123225.880593008 23 18509941.2 18395938.7513565 114002.44864347 24 17802737.97 18507383.7721200 -704645.80212005 25 16409869.75 18618828.7928836 -2208959.04288357 26 17967742.04 18730273.8136471 -762531.773647089 27 20286602.27 18841718.8344106 1444883.43558939 28 19537280.81 18953163.8551741 584116.954825871 29 18021889.62 19064608.8759376 -1042719.25593765 30 20194317.23 19176053.8967012 1018263.33329883 31 19049596.62 19287498.9174647 -237902.297464686 32 20244720.94 19398943.9382282 845777.001771794 33 21473302.24 19510388.9589917 1962913.28100827 34 19673603.19 19621833.9797552 51769.2102447548 35 21053177.29 19733279.0005188 1319898.28948123 36 20159479.84 19844724.0212823 314755.818717714 37 18203628.31 19956169.0420458 -1752540.73204581 38 21289464.94 20067614.0628093 1221850.87719068 39 20432335.71 15129068.5992558 5303267.11074423 40 17180395.07 15240513.6200193 1939881.44998071 41 15816786.32 15351958.6407828 464827.679217191 42 15071819.75 15463403.6615463 -391583.911546329 43 14521120.61 15574848.6823098 -1053728.07230985 44 15668789.39 15686293.7030734 -17504.3130733681 45 14346884.11 15797738.7238369 -1450854.61383689 46 13881008.13 15909183.7446004 -2028175.61460041 47 15465943.69 16020628.7653639 -554685.075363928 48 14238232.92 16132073.7861274 -1893840.86612745 49 13557713.21 16243518.8068910 -2685805.59689097 50 16127590.29 16354963.8276545 -227373.537654488 51 16793894.2 16466408.848418 327485.351581992 52 16014007.43 16577853.8691815 -563846.439181527 53 16867867.15 16689298.8899450 178568.260054952 54 16014583.21 16800743.9107086 -786160.700708565 55 15878594.85 16912188.9314721 -1033594.08147209 56 18664899.14 17023633.9522356 1641265.18776439 57 17962530.06 17135078.9729991 827451.087000873 58 17332692.2 17246523.9937626 86168.2062373542 59 19542066.35 17357969.0145262 2184097.33547384 60 17203555.19 17469414.0352897 -265858.845289683 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.395952952820355 0.79190590564071 0.604047047179645 7 0.343639941753300 0.687279883506599 0.6563600582467 8 0.396535050270518 0.793070100541036 0.603464949729482 9 0.64045095906928 0.719098081861441 0.359549040930720 10 0.54568310679667 0.908633786406661 0.454316893203331 11 0.447748425164294 0.895496850328588 0.552251574835706 12 0.638000942284861 0.723998115430277 0.361999057715139 13 0.633184034369504 0.733631931260992 0.366815965630496 14 0.53749484680553 0.92501030638894 0.46250515319447 15 0.45186871925493 0.90373743850986 0.54813128074507 16 0.371761388510067 0.743522777020133 0.628238611489933 17 0.347718392972109 0.695436785944217 0.652281607027891 18 0.269865289664015 0.53973057932803 0.730134710335985 19 0.217895638688138 0.435791277376276 0.782104361311862 20 0.231587117575958 0.463174235151915 0.768412882424042 21 0.203949618849137 0.407899237698274 0.796050381150863 22 0.155344566909174 0.310689133818348 0.844655433090826 23 0.118622831454250 0.237245662908499 0.88137716854575 24 0.0859976480673072 0.171995296134614 0.914002351932693 25 0.114449600796436 0.228899201592872 0.885550399203564 26 0.0893406238846096 0.178681247769219 0.91065937611539 27 0.123560088673160 0.247120177346319 0.87643991132684 28 0.102819128340260 0.205638256680520 0.89718087165974 29 0.0875332915757036 0.175066583151407 0.912466708424296 30 0.081239533025637 0.162479066051274 0.918760466974363 31 0.0589228921471106 0.117845784294221 0.94107710785289 32 0.0473968438170745 0.094793687634149 0.952603156182926 33 0.0639366334797418 0.127873266959484 0.936063366520258 34 0.0431056863435291 0.0862113726870581 0.956894313656471 35 0.0380748359821163 0.0761496719642326 0.961925164017884 36 0.0249247257046326 0.0498494514092652 0.975075274295367 37 0.0355658696979820 0.0711317393959641 0.964434130302018 38 0.0265710969967705 0.053142193993541 0.973428903003229 39 0.356060240736597 0.712120481473193 0.643939759263403 40 0.679746320857043 0.640507358285913 0.320253679142957 41 0.81122583274558 0.377548334508840 0.188774167254420 42 0.851364771485272 0.297270457029455 0.148635228514728 43 0.850970498200076 0.298059003599848 0.149029501799924 44 0.893857909139186 0.212284181721628 0.106142090860814 45 0.874141209334459 0.251717581331083 0.125858790665541 46 0.85199828741185 0.296003425176298 0.148001712588149 47 0.82535330062374 0.349293398752519 0.174646699376260 48 0.778484414254024 0.443031171491953 0.221515585745976 49 0.870471206928291 0.259057586143418 0.129528793071709 50 0.793926304345994 0.412147391308012 0.206073695654006 51 0.727469051864264 0.545061896271471 0.272530948135736 52 0.600744404811698 0.798511190376605 0.399255595188302 53 0.471485429953469 0.942970859906938 0.528514570046531 54 0.337713650920769 0.675427301841539 0.662286349079231 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 1 0.0204081632653061 OK 10% type I error level 6 0.122448979591837 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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