*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: Wed, 22 Dec 2010 18:39:39 +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/22/t1293043157fsd787p7h50ydb4.htm/, Retrieved Wed, 22 Dec 2010 19:39:28 +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/22/t1293043157fsd787p7h50ydb4.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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-999 -999 38.6 6.654 5.712 645 3 5 3 6.3 2 4.5 1 6.6 42 3 1 3 -999 -999 14 3.385 44.5 60 1 1 1 -999 -999 -999 0.92 5.7 25 5 2 3 2.1 1.8 69 2547 4603 624 3 5 4 0.1 0.7 27 10.55 0.5 180 4 4 4 15.8 3.9 19 0.023 0.3 35 1 1 1 5.2 1 30.4 160 169 392 4 5 4 10.9 3.6 28 3.3 25.6 63 1 2 1 8.3 1.4 50 52.16 440 230 1 1 1 11 1.5 7 0.425 6.4 112 5 4 4 3.2 0.7 30 465 423 281 5 5 5 7.6 2.7 -999 0.55 2.4 -999 2 1 2 -999 -999 40 187.1 419 365 5 5 5 6.3 2.1 3.5 0.075 1.2 42 1 1 1 8.6 0 50 3 25 28 2 2 2 6.6 4.1 6 0.785 3.5 42 2 2 2 9.5 1.2 10.4 0.2 5 120 2 2 2 4.8 1.3 34 1.41 17.5 -999 1 2 1 12 6.1 7 60 81 -999 1 1 1 -999 0.3 28 529 680 400 5 5 5 3.3 0.5 20 27.66 115 148 5 5 5 11 3.4 3.9 0.12 1 16 3 1 2 -999 -999 39.3 207 406 252 1 4 1 4.7 1.5 41 85 325 310 1 3 1 -999 -999 16.2 36.33 119.5 63 1 1 1 10.4 3.4 9 0.101 4 28 5 1 3 7.4 0.8 7.6 1.04 5.5 68 5 3 4 2.1 0.8 46 521 655 336 5 5 5 2.1 -999 22.4 100 157 100 1 1 1 -999 -999 16.3 35 56 33 3 5 4 7.7 1. 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 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation PS[t] = -147.448186216999 + 0.827854459100156SWS[t] + 0.0153005707500430L[t] -0.000146716870009194WB[t] + 0.0270059287704017WBR[t] -0.0209700304582949TG[t] + 3.15985232691063P[t] -9.02103450957327S[t] + 50.7708110913449D[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) -147.448186216999 66.192415 -2.2276 0.030178 0.015089 SWS 0.827854459100156 0.072478 11.4221 0 0 L 0.0153005707500430 0.11626 0.1316 0.895794 0.447897 WB -0.000146716870009194 0.294549 -5e-04 0.999604 0.499802 WBR 0.0270059287704017 0.160936 0.1678 0.867375 0.433688 TG -0.0209700304582949 0.102843 -0.2039 0.83921 0.419605 P 3.15985232691063 50.314252 0.0628 0.95016 0.47508 S -9.02103450957327 33.416919 -0.27 0.788244 0.394122 D 50.7708110913449 65.737628 0.7723 0.443352 0.221676 Multiple Linear Regression - Regression Statistics Multiple R 0.86874491034203 R-squared 0.754717719245183 Adjusted R-squared 0.71769397875389 F-TEST (value) 20.3846966630147 F-TEST (DF numerator) 8 F-TEST (DF denominator) 53 p-value 1.15019105351166e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 211.835697265371 Sum Squared Residuals 2378341.21970301 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 -999 -870.56975915472 -128.430240845279 2 2 9.90445632266566 -7.90445632266566 3 -999 -929.407888592692 -69.5921114073079 4 -999 -840.060887094893 -158.939112905107 5 1.8 133.652979583378 -131.852979583378 6 0.7 28.9235798928042 -28.2235798928042 7 3.9 -89.8935986721811 93.793598672181 8 1 24.2595507696231 -23.2595507696231 9 3.6 -102.737806540719 106.337806540719 10 1.4 -87.8504878586546 89.2504878586546 11 1.5 42.3878169681239 -40.8878169681239 12 0.7 85.6708156845783 -84.9708156845783 13 2.7 -36.5876762175865 39.2876762175865 14 -999 -745.6806511735 -253.3193488265 15 2.1 -98.1178673868504 100.217867386850 16 0 -49.6568042980538 49.6568042980538 17 4.1 -52.8596212463687 56.9596212463687 18 1.2 -51.9865884569005 53.1865884569005 19 1.3 -85.6442136981765 86.9442136981765 20 6.1 -69.3694621578266 75.4694621578266 21 0.3 -739.599824177322 739.899824177322 22 0.5 80.1359485685872 -79.6359485685872 23 3.4 -36.590502451711 39.990502451711 24 -999 -950.377364034415 -48.6226359655849 25 1.5 -113.798640494583 115.298640494583 26 -999 -927.416526357919 -71.5834736420808 27 3.4 19.9107137372525 -16.5107137372525 28 0.8 49.3360415407831 -48.5360415407831 29 0.8 90.1087925663753 -89.3087925663753 30 -999 -98.3290740752834 -900.670925924717 31 -999 -805.952576841016 -193.047423158984 32 1.4 59.3594360243981 -57.9594360243981 33 2 -88.3945003003143 90.3945003003143 34 1.9 -65.9188566341772 67.8188566341772 35 2.4 -108.423669622974 110.823669622974 36 2.8 -3.73362461001548 6.53362461001548 37 1.3 17.9953897927168 -16.6953897927168 38 2 16.8338551552218 -14.8338551552218 39 5.6 -87.9595630249994 93.5595630249993 40 3.1 -89.6661856397193 92.7661856397193 41 1 -745.596140366598 746.596140366598 42 1.8 -47.3792165646299 49.1792165646299 43 0.9 40.3172549998679 -39.4172549998679 44 1.8 -38.6063790181572 40.4063790181572 45 1.9 40.4058428490583 -38.5058428490583 46 0.9 83.2606110449034 -82.3606110449034 47 -999 -884.709023854426 -114.290976145574 48 2.6 13.7808390214966 -11.1808390214966 49 2.4 -95.9924528373733 98.3924528373733 50 1.2 -57.8582903209903 59.0582903209903 51 0.9 -57.1892479544068 58.0892479544068 52 0.5 -2.12888295258638 2.62888295258638 53 -999 -750.183204817024 -248.816795182976 54 0.6 81.6274941903236 -81.0274941903236 55 -999 -886.0111498274 -112.988850172600 56 2.2 -17.7128108932199 19.9128108932199 57 2.3 -40.6207607294625 42.9207607294625 58 0.5 5.63188405688672 -5.13188405688672 59 2.6 -44.4033277375874 47.0033277375874 60 0.6 46.7707886073313 -46.1707886073313 61 6.6 -41.3140782414969 47.9140782414969 62 -999 -923.384904447443 -75.615095552557 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 12 9.46804519936682e-07 1.89360903987336e-06 0.99999905319548 13 7.9819937869379e-08 1.59639875738758e-07 0.999999920180062 14 2.65934814392256e-09 5.31869628784513e-09 0.999999997340652 15 5.38217896052781e-11 1.07643579210556e-10 0.999999999946178 16 8.44315293097584e-13 1.68863058619517e-12 0.999999999999156 17 2.20933975009410e-14 4.41867950018821e-14 0.999999999999978 18 4.1729430221813e-16 8.3458860443626e-16 1 19 1.40035374310464e-17 2.80070748620928e-17 1 20 2.05281127880055e-19 4.10562255760111e-19 1 21 0.538827153600218 0.922345692799564 0.461172846399782 22 0.446730314617958 0.893460629235916 0.553269685382042 23 0.356018735440757 0.712037470881513 0.643981264559243 24 0.275168166151238 0.550336332302476 0.724831833848762 25 0.254180723705874 0.508361447411748 0.745819276294126 26 0.208346242738713 0.416692485477427 0.791653757261287 27 0.151331510044566 0.302663020089132 0.848668489955434 28 0.10660475292595 0.2132095058519 0.89339524707405 29 0.100271256837565 0.200542513675131 0.899728743162435 30 0.999775816598942 0.000448366802116225 0.000224183401058112 31 0.99966234246299 0.000675315074018896 0.000337657537009448 32 0.999248968733381 0.00150206253323725 0.000751031266618623 33 0.998527947483167 0.00294410503366530 0.00147205251683265 34 0.999957973609397 8.40527812056543e-05 4.20263906028271e-05 35 0.999912509274482 0.000174981451036571 8.74907255182856e-05 36 0.99998594446268 2.81110746418541e-05 1.40555373209270e-05 37 0.999960444196716 7.91116065675302e-05 3.95558032837651e-05 38 0.99990903391786 0.000181932164278221 9.09660821391105e-05 39 0.999752138385124 0.000495723229752425 0.000247861614876212 40 0.99935069728785 0.00129860542430221 0.000649302712151103 41 1 3.77052106115249e-22 1.88526053057625e-22 42 1 5.46264601692828e-21 2.73132300846414e-21 43 1 3.21724030047991e-19 1.60862015023995e-19 44 1 2.09716872798975e-17 1.04858436399487e-17 45 1 1.04698409116610e-15 5.23492045583048e-16 46 0.999999999999958 8.4130959256597e-14 4.20654796282985e-14 47 0.999999999995544 8.91270535751416e-12 4.45635267875708e-12 48 0.999999999657233 6.85534503086415e-10 3.42767251543207e-10 49 0.999999966549504 6.69009930125927e-08 3.34504965062963e-08 50 0.999996832623228 6.33475354315818e-06 3.16737677157909e-06 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 30 0.769230769230769 NOK 5% type I error level 30 0.769230769230769 NOK 10% type I error level 30 0.769230769230769 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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