*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: Fri, 20 Nov 2009 00:36:00 -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/20/t1258702732d1k3gyx7m6hzi5f.htm/, Retrieved Fri, 20 Nov 2009 08:39:05 +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/20/t1258702732d1k3gyx7m6hzi5f.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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100.03 2 100.25 1.8 99.6 2.7 100.16 2.3 100.49 1.9 99.72 2 100.14 2.3 98.48 2.8 100.38 2.4 101.45 2.3 98.42 2.7 98.6 2.7 100.06 2.9 98.62 3 100.84 2.2 100.02 2.3 97.95 2.8 98.32 2.8 98.27 2.8 97.22 2.2 99.28 2.6 100.38 2.8 99.02 2.5 100.32 2.4 99.81 2.3 100.6 1.9 101.19 1.7 100.47 2 101.77 2.1 102.32 1.7 102.39 1.8 101.16 1.8 100.63 1.8 101.48 1.3 101.44 1.3 100.09 1.3 100.7 1.2 100.78 1.4 99.81 2.2 98.45 2.9 98.49 3.1 97.48 3.5 97.91 3.6 96.94 4.4 98.53 4.1 96.82 5.1 95.76 5.8 95.27 5.9 97.32 5.4 96.68 5.5 97.87 4.8 97.42 3.2 97.94 2.7 99.52 2.1 100.99 1.9 99.92 0.6 101.97 0.7 101.58 -0.2 99.54 -1 100.83 -1.7

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] = + 101.656351098474 -0.878501125566274X[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) 101.656351098474 0.279527 363.673 0 0 X -0.878501125566274 0.098678 -8.9027 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.75989485080661 R-squared 0.5774401842824 Adjusted R-squared 0.570154670218304 F-TEST (value) 79.2586740211045 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 1.91957560957690e-12 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.07801618410123 Sum Squared Residuals 67.402895804682 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 100.03 99.899348847342 0.130651152658071 2 100.25 100.075049072455 0.174950927544810 3 99.6 99.2843980594455 0.315601940554456 4 100.16 99.635798509672 0.524201490327948 5 100.49 99.9871989598986 0.502801040101436 6 99.72 99.899348847342 -0.179348847341932 7 100.14 99.635798509672 0.504201490327952 8 98.48 99.196547946889 -0.716547946888908 9 100.38 99.5479483971154 0.832051602884574 10 101.45 99.635798509672 1.81420149032795 11 98.42 99.2843980594455 -0.864398059445537 12 98.6 99.2843980594455 -0.684398059445544 13 100.06 99.1086978343323 0.951302165667718 14 98.62 99.0208477217757 -0.400847721775652 15 100.84 99.7236486222287 1.11635137777133 16 100.02 99.635798509672 0.384201490327947 17 97.95 99.196547946889 -1.24654794688891 18 98.32 99.196547946889 -0.876547946888918 19 98.27 99.196547946889 -0.926547946888916 20 97.22 99.7236486222287 -2.50364862222868 21 99.28 99.3722481720022 -0.092248172002165 22 100.38 99.196547946889 1.18345205311108 23 99.02 99.4600982845588 -0.440098284558798 24 100.32 99.5479483971154 0.772051602884572 25 99.81 99.635798509672 0.174201490327954 26 100.6 99.9871989598986 0.612801040101436 27 101.19 100.162899185012 1.02710081498818 28 100.47 99.899348847342 0.570651152658068 29 101.77 99.8114987347853 1.95850126521469 30 102.32 100.162899185012 2.15710081498818 31 102.39 100.075049072455 2.31495092754481 32 101.16 100.075049072455 1.08495092754481 33 100.63 100.075049072455 0.55495092754481 34 101.48 100.514299635238 0.96570036476168 35 101.44 100.514299635238 0.925700364761675 36 100.09 100.514299635238 -0.42429963523832 37 100.7 100.602149747795 0.0978502522050524 38 100.78 100.426449522682 0.353550477318306 39 99.81 99.7236486222287 0.0863513777713264 40 98.45 99.1086978343323 -0.658697834332281 41 98.49 98.932997609219 -0.442997609219034 42 97.48 98.5815971589925 -1.10159715899252 43 97.91 98.493747046436 -0.583747046435895 44 96.94 97.7909461459829 -0.850946145982874 45 98.53 98.0544964836528 0.475503516347246 46 96.82 97.1759953580865 -0.355995358086487 47 95.76 96.56104457019 -0.801044570190083 48 95.27 96.4731944576335 -1.20319445763346 49 97.32 96.9124450204166 0.407554979583396 50 96.68 96.82459490786 -0.144594907859964 51 97.87 97.4395456957564 0.430454304243642 52 97.42 98.8451474966624 -1.4251474966624 53 97.94 99.2843980594455 -1.34439805944554 54 99.52 99.8114987347853 -0.291498734785307 55 100.99 99.9871989598986 1.00280104010144 56 99.92 101.129250423135 -1.20925042313471 57 101.97 101.041400310578 0.928599689421911 58 101.58 101.832051323588 -0.252051323587737 59 99.54 102.534852224041 -2.99485222404075 60 100.83 103.149803011937 -2.31980301193715 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0081186983200256 0.0162373966400512 0.991881301679974 6 0.0074241283376767 0.0148482566753534 0.992575871662323 7 0.00205831001983873 0.00411662003967747 0.997941689980161 8 0.00964083925579564 0.0192816785115913 0.990359160744204 9 0.00997928369822485 0.0199585673964497 0.990020716301775 10 0.0728615680988856 0.145723136197771 0.927138431901114 11 0.0816919549974866 0.163383909994973 0.918308045002513 12 0.0611322488081268 0.122264497616254 0.938867751191873 13 0.0719835142720185 0.143967028544037 0.928016485727981 14 0.0455147350638137 0.0910294701276275 0.954485264936186 15 0.0402392339206905 0.0804784678413811 0.95976076607931 16 0.0234717900970272 0.0469435801940544 0.976528209902973 17 0.0333331061845356 0.0666662123690712 0.966666893815464 18 0.0267100321216212 0.0534200642432424 0.973289967878379 19 0.0211890238250689 0.0423780476501377 0.978810976174931 20 0.245150991851617 0.490301983703235 0.754849008148383 21 0.184846622726637 0.369693245453275 0.815153377273363 22 0.224154015899269 0.448308031798539 0.77584598410073 23 0.176268521281538 0.352537042563076 0.823731478718462 24 0.151072827759010 0.302145655518020 0.84892717224099 25 0.109517141843436 0.219034283686871 0.890482858156564 26 0.080997537485542 0.161995074971084 0.919002462514458 27 0.0662003732858558 0.132400746571712 0.933799626714144 28 0.0471965851667197 0.0943931703334395 0.95280341483328 29 0.0958213424364417 0.191642684872883 0.904178657563558 30 0.177658930802060 0.355317861604121 0.82234106919794 31 0.370149362022599 0.740298724045197 0.629850637977401 32 0.38014693378206 0.76029386756412 0.61985306621794 33 0.354691085540845 0.70938217108169 0.645308914459155 34 0.392272967815302 0.784545935630604 0.607727032184698 35 0.445776618976264 0.891553237952529 0.554223381023736 36 0.474723953263663 0.949447906527325 0.525276046736337 37 0.470748038687698 0.941496077375397 0.529251961312302 38 0.472403933014426 0.944807866028852 0.527596066985574 39 0.431126456378574 0.862252912757148 0.568873543621426 40 0.359627811184610 0.719255622369219 0.64037218881539 41 0.287254450254295 0.574508900508589 0.712745549745705 42 0.240568854633188 0.481137709266375 0.759431145366812 43 0.181356251857044 0.362712503714088 0.818643748142956 44 0.143498229531064 0.286996459062127 0.856501770468936 45 0.150335755260523 0.300671510521047 0.849664244739477 46 0.114311456357802 0.228622912715603 0.885688543642199 47 0.0929497019085435 0.185899403817087 0.907050298091457 48 0.113701860321375 0.22740372064275 0.886298139678625 49 0.0904704768298262 0.180940953659652 0.909529523170174 50 0.0668154879655338 0.133630975931068 0.933184512034466 51 0.043290854079309 0.086581708158618 0.95670914592069 52 0.075205441999986 0.150410883999972 0.924794558000014 53 0.242254566076618 0.484509132153237 0.757745433923382 54 0.308794576565860 0.617589153131721 0.69120542343414 55 0.195241287691061 0.390482575382122 0.804758712308939 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.0196078431372549 NOK 5% type I error level 7 0.137254901960784 NOK 10% type I error level 13 0.254901960784314 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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