*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: Mon, 21 Dec 2009 04:18:14 -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/21/t1261394443avja5ao25vaax7v.htm/, Retrieved Mon, 21 Dec 2009 12:20:54 +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/21/t1261394443avja5ao25vaax7v.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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User-defined keywords:

Dataseries X:

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2.6 30.5 2.4 28.6 2.5 30 2.7 28.2 3.2 27.6 2.8 24.9 2.8 23.8 3 24.3 3.1 23.6 3.1 24.2 3 28.1 2.4 30.1 2.7 31.1 3 32 2.7 32.4 2.7 34 2 35.1 2.4 37.1 2.6 37.3 2.4 38.1 2.3 39.5 2.4 38.3 2.5 37.3 2.6 38.7 2.6 37.5 2.6 38.7 2.7 37.9 2.8 36.6 2.6 35.5 2.6 37.6 2 38.6 2 40.3 2.1 39 1.9 36.8 2 36.5 2.5 34.1 2.9 34.2 3.3 31.9 3.5 33.7 3.8 33.5 4.6 33.8 4.4 29.9 5.3 32.3 5.8 30.5 5.9 28.5 5.6 29 5.8 23.8 5.5 17.9 4.6 9.9 4.2 3 4 4.2 3.5 0.4 2.3 0 2.2 2.4 1.4 4.2 0.6 8.2 0 9 0.5 13.6 0.1 14 0.1 17.6

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] = + 2.63835457876613 + 0.0084265340572524X[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) 2.63835457876613 0.449474 5.8699 0 0 X 0.0084265340572524 0.015094 0.5583 0.578806 0.289403 Multiple Linear Regression - Regression Statistics Multiple R 0.0731087363377837 R-squared 0.00534488732890758 Adjusted R-squared -0.0118043387516285 F-TEST (value) 0.311669302381749 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.578806475583351 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.33847340344487 Sum Squared Residuals 103.907641000298 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2.6 2.89536386751233 -0.295363867512332 2 2.4 2.87935345280355 -0.479353452803550 3 2.5 2.89115060048370 -0.391150600483703 4 2.7 2.87598283918065 -0.175982839180649 5 3.2 2.8709269187463 0.329073081253702 6 2.8 2.84817527679172 -0.0481752767917163 7 2.8 2.83890608932874 -0.0389060893287387 8 3 2.84311935635736 0.156880643642635 9 3.1 2.83722078251729 0.262779217482712 10 3.1 2.84227670295164 0.257723297048361 11 3 2.87514018577492 0.124859814225076 12 2.4 2.89199325388943 -0.491993253889429 13 2.7 2.90041978794668 -0.200419787946681 14 3 2.90800366859821 0.0919963314017917 15 2.7 2.91137428222111 -0.211374282221109 16 2.7 2.92485673671271 -0.224856736712713 17 2 2.93412592417569 -0.93412592417569 18 2.4 2.95097899229020 -0.550978992290196 19 2.6 2.95266429910165 -0.352664299101646 20 2.4 2.95940552634745 -0.559405526347448 21 2.3 2.9712026740276 -0.671202674027601 22 2.4 2.9610908331589 -0.561090833158899 23 2.5 2.95266429910165 -0.452664299101646 24 2.6 2.9644614467818 -0.364461446781799 25 2.6 2.95434960591310 -0.354349605913096 26 2.6 2.9644614467818 -0.364461446781799 27 2.7 2.957720219536 -0.257720219535997 28 2.8 2.94676572526157 -0.146765725261570 29 2.6 2.93749653779859 -0.337496537798592 30 2.6 2.95519225931882 -0.355192259318822 31 2 2.96361879337607 -0.963618793376074 32 2 2.97794390127340 -0.977943901273403 33 2.1 2.96698940699898 -0.866989406998975 34 1.9 2.94845103207302 -1.04845103207302 35 2 2.94592307185584 -0.945923071855844 36 2.5 2.92569939011844 -0.425699390118438 37 2.9 2.92654204352416 -0.0265420435241637 38 3.3 2.90716101519248 0.392838984807517 39 3.5 2.92232877649554 0.577671223504463 40 3.8 2.92064346968409 0.879356530315913 41 4.6 2.92317142990126 1.67682857009874 42 4.4 2.89030794707798 1.50969205292202 43 5.3 2.91053162881538 2.38946837118462 44 5.8 2.89536386751233 2.90463613248767 45 5.9 2.87851079939782 3.02148920060218 46 5.6 2.88272406642645 2.71727593357355 47 5.8 2.83890608932874 2.96109391067126 48 5.5 2.78918953839095 2.71081046160905 49 4.6 2.72177726593293 1.87822273406707 50 4.2 2.66363418093789 1.53636581906211 51 4 2.67374602180659 1.32625397819341 52 3.5 2.64172519238903 0.858274807610968 53 2.3 2.63835457876613 -0.338354578766131 54 2.2 2.65857826050354 -0.458578260503537 55 1.4 2.67374602180659 -1.27374602180659 56 0.6 2.7074521580356 -2.1074521580356 57 0 2.71419338528140 -2.71419338528140 58 0.5 2.75295544194476 -2.25295544194476 59 0.1 2.75632605556767 -2.65632605556767 60 0.1 2.78666157817377 -2.68666157817377 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0114474240933169 0.0228948481866339 0.988552575906683 6 0.00298500936945455 0.0059700187389091 0.997014990630545 7 0.00056896212571626 0.00113792425143252 0.999431037874284 8 9.77474507862184e-05 0.000195494901572437 0.999902252549214 9 1.67431297662078e-05 3.34862595324156e-05 0.999983256870234 10 2.79960135118431e-06 5.59920270236863e-06 0.999997200398649 11 8.02080278311731e-07 1.60416055662346e-06 0.999999197919722 12 1.69215982171278e-07 3.38431964342555e-07 0.999999830784018 13 3.11196997634010e-08 6.22393995268019e-08 0.9999999688803 14 2.6862967816924e-08 5.3725935633848e-08 0.999999973137032 15 4.26364085413442e-09 8.52728170826885e-09 0.99999999573636 16 7.01530338111902e-10 1.40306067622380e-09 0.99999999929847 17 7.42465560293367e-10 1.48493112058673e-09 0.999999999257534 18 1.20607259947445e-10 2.41214519894889e-10 0.999999999879393 19 2.88886878206947e-11 5.77773756413893e-11 0.999999999971111 20 4.38062769055744e-12 8.7612553811149e-12 0.99999999999562 21 6.46425123327324e-13 1.29285024665465e-12 0.999999999999354 22 9.29840533979355e-14 1.85968106795871e-13 0.999999999999907 23 1.37029385672614e-14 2.74058771345227e-14 0.999999999999986 24 3.04697292313371e-15 6.09394584626741e-15 0.999999999999997 25 5.16053193033838e-16 1.03210638606768e-15 1 26 9.56843273447087e-17 1.91368654689417e-16 1 27 2.17251641861236e-17 4.34503283722471e-17 1 28 5.85499864891341e-18 1.17099972978268e-17 1 29 7.54519508467045e-19 1.50903901693409e-18 1 30 1.05711060918354e-19 2.11422121836708e-19 1 31 1.43638526280649e-19 2.87277052561299e-19 1 32 9.90280598568281e-20 1.98056119713656e-19 1 33 4.52827887082971e-20 9.05655774165942e-20 1 34 1.61674138336717e-19 3.23348276673433e-19 1 35 2.63381537289425e-19 5.26763074578849e-19 1 36 7.71703075450152e-20 1.54340615090030e-19 1 37 5.14164488050178e-20 1.02832897610036e-19 1 38 1.69878728152952e-19 3.39757456305903e-19 1 39 3.43260265762977e-18 6.86520531525955e-18 1 40 2.19123970455543e-16 4.38247940911086e-16 1 41 6.47178680749915e-13 1.29435736149983e-12 0.999999999999353 42 7.66672256424672e-12 1.53334451284934e-11 0.999999999992333 43 1.98434346182306e-09 3.96868692364611e-09 0.999999998015657 44 1.64470119507978e-07 3.28940239015956e-07 0.99999983552988 45 2.62573705213292e-06 5.25147410426583e-06 0.999997374262948 46 1.38287057936088e-05 2.76574115872176e-05 0.999986171294206 47 0.000265196804237043 0.000530393608474087 0.999734803195763 48 0.0269936081497211 0.0539872162994422 0.97300639185028 49 0.285018394387286 0.570036788774573 0.714981605612714 50 0.475644239424695 0.95128847884939 0.524355760575305 51 0.866457535425815 0.267084929148371 0.133542464574185 52 0.963472478832376 0.0730550423352479 0.0365275211676239 53 0.942647377292558 0.114705245414884 0.0573526227074419 54 0.96395812900575 0.0720837419885004 0.0360418709942502 55 0.974409513539867 0.0511809729202664 0.0255904864601332 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 42 0.823529411764706 NOK 5% type I error level 43 0.843137254901961 NOK 10% type I error level 47 0.92156862745098 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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