*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 05:06:30 -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/t1258718869ksc7v5id58k5mo0.htm/, Retrieved Fri, 20 Nov 2009 13:08:01 +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/t1258718869ksc7v5id58k5mo0.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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1,4 2 1,2 2 1 2 1,7 2 2,4 2 2 2 2,1 2 2 2 1,8 2 2,7 2 2,3 2 1,9 2 2 2 2,3 2 2,8 2 2,4 2 2,3 2 2,7 2 2,7 2 2,9 2 3 2 2,2 2 2,3 2 2,8 2,21 2,8 2,25 2,8 2,25 2,2 2,45 2,6 2,5 2,8 2,5 2,5 2,64 2,4 2,75 2,3 2,93 1,9 3 1,7 3,17 2 3,25 2,1 3,39 1,7 3,5 1,8 3,5 1,8 3,65 1,8 3,75 1,3 3,75 1,3 3,9 1,3 4 1,2 4 1,4 4 2,2 4 2,9 4 3,1 4 3,5 4 3,6 4 4,4 4 4,1 4 5,1 4 5,8 4 5,9 4,18 5,4 4,25 5,5 4,25 4,8 3,97 3,2 3,42 2,7 2,75

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.989987872559212 + 0.559260034717772X[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) 0.989987872559212 0.476845 2.0761 0.042325 0.021163 X 0.559260034717772 0.157282 3.5558 0.000758 0.000379 Multiple Linear Regression - Regression Statistics Multiple R 0.423057547683513 R-squared 0.178977688651988 Adjusted R-squared 0.164822131559781 F-TEST (value) 12.6436344035177 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.000758008081633177 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.06648468124056 Sum Squared Residuals 65.9685953686052 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.4 2.10850794199477 -0.708507941994769 2 1.2 2.10850794199476 -0.90850794199476 3 1 2.10850794199476 -1.10850794199476 4 1.7 2.10850794199476 -0.408507941994758 5 2.4 2.10850794199476 0.291492058005242 6 2 2.10850794199476 -0.108507941994758 7 2.1 2.10850794199476 -0.00850794199475767 8 2 2.10850794199476 -0.108507941994758 9 1.8 2.10850794199476 -0.308507941994758 10 2.7 2.10850794199476 0.591492058005242 11 2.3 2.10850794199476 0.191492058005242 12 1.9 2.10850794199476 -0.208507941994758 13 2 2.10850794199476 -0.108507941994758 14 2.3 2.10850794199476 0.191492058005242 15 2.8 2.10850794199476 0.691492058005242 16 2.4 2.10850794199476 0.291492058005242 17 2.3 2.10850794199476 0.191492058005242 18 2.7 2.10850794199476 0.591492058005242 19 2.7 2.10850794199476 0.591492058005242 20 2.9 2.10850794199476 0.791492058005242 21 3 2.10850794199476 0.891492058005242 22 2.2 2.10850794199476 0.0914920580052424 23 2.3 2.10850794199476 0.191492058005242 24 2.8 2.22595254928549 0.57404745071451 25 2.8 2.2483229506742 0.551677049325799 26 2.8 2.2483229506742 0.551677049325799 27 2.2 2.36017495761776 -0.160174957617755 28 2.6 2.38813795935364 0.211862040646356 29 2.8 2.38813795935364 0.411862040646356 30 2.5 2.46643436421413 0.0335656357858682 31 2.4 2.52795296803309 -0.127952968033087 32 2.3 2.62861977428229 -0.328619774282286 33 1.9 2.66776797671253 -0.76776797671253 34 1.7 2.76284218261455 -1.06284218261455 35 2 2.80758298539197 -0.807582985391972 36 2.1 2.88587939025246 -0.78587939025246 37 1.7 2.94739799407142 -1.24739799407142 38 1.8 2.94739799407142 -1.14739799407142 39 1.8 3.03128699927908 -1.23128699927908 40 1.8 3.08721300275086 -1.28721300275086 41 1.3 3.08721300275086 -1.78721300275086 42 1.3 3.17110200795852 -1.87110200795852 43 1.3 3.2270280114303 -1.9270280114303 44 1.2 3.2270280114303 -2.0270280114303 45 1.4 3.2270280114303 -1.8270280114303 46 2.2 3.2270280114303 -1.02702801143030 47 2.9 3.2270280114303 -0.327028011430301 48 3.1 3.2270280114303 -0.127028011430301 49 3.5 3.2270280114303 0.272971988569699 50 3.6 3.2270280114303 0.372971988569699 51 4.4 3.2270280114303 1.1729719885697 52 4.1 3.2270280114303 0.872971988569698 53 5.1 3.2270280114303 1.87297198856970 54 5.8 3.2270280114303 2.5729719885697 55 5.9 3.3276948176795 2.5723051823205 56 5.4 3.36684302010974 2.03315697989026 57 5.5 3.36684302010974 2.13315697989026 58 4.8 3.21025021038877 1.58974978961123 59 3.2 2.90265719129399 0.297342808706007 60 2.7 2.52795296803309 0.172047031966914 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.200072967767426 0.400145935534852 0.799927032232574 6 0.112556220434799 0.225112440869599 0.8874437795652 7 0.0646251308724237 0.129250261744847 0.935374869127576 8 0.0314141943650823 0.0628283887301646 0.968585805634918 9 0.0129303919768203 0.0258607839536407 0.98706960802318 10 0.0191566888484000 0.0383133776967999 0.9808433111516 11 0.0109683541828002 0.0219367083656004 0.9890316458172 12 0.00470296536408427 0.00940593072816855 0.995297034635916 13 0.00195323651189147 0.00390647302378295 0.998046763488109 14 0.000999212390519086 0.00199842478103817 0.99900078760948 15 0.00128814032288588 0.00257628064577176 0.998711859677114 16 0.000674625381837698 0.00134925076367540 0.999325374618162 17 0.000304496085784720 0.000608992171569441 0.999695503914215 18 0.000239702238881494 0.000479404477762987 0.999760297761119 19 0.000173843275123097 0.000347686550246194 0.999826156724877 20 0.000177061399639548 0.000354122799279095 0.99982293860036 21 0.000203338303489169 0.000406676606978338 0.99979666169651 22 8.4902033518996e-05 0.000169804067037992 0.99991509796648 23 3.54032884184309e-05 7.08065768368618e-05 0.999964596711582 24 1.52146335384930e-05 3.04292670769861e-05 0.999984785366462 25 6.63575890205921e-06 1.32715178041184e-05 0.999993364241098 26 3.00114723492254e-06 6.00229446984507e-06 0.999996998852765 27 2.79922027178866e-06 5.59844054357731e-06 0.999997200779728 28 1.2171555156696e-06 2.4343110313392e-06 0.999998782844484 29 6.20558491794676e-07 1.24111698358935e-06 0.999999379441508 30 3.28208097868096e-07 6.56416195736192e-07 0.999999671791902 31 1.83652697894670e-07 3.67305395789341e-07 0.999999816347302 32 1.00992403915410e-07 2.01984807830820e-07 0.999999899007596 33 7.36703682021473e-08 1.47340736404295e-07 0.999999926329632 34 5.0938801869688e-08 1.01877603739376e-07 0.999999949061198 35 1.83197362199678e-08 3.66394724399356e-08 0.999999981680264 36 5.86017794595607e-09 1.17203558919121e-08 0.999999994139822 37 2.76940340495631e-09 5.53880680991261e-09 0.999999997230597 38 1.05147609657173e-09 2.10295219314346e-09 0.999999998948524 39 4.49973488474852e-10 8.99946976949704e-10 0.999999999550026 40 2.25935512652416e-10 4.51871025304831e-10 0.999999999774065 41 3.94265703448335e-10 7.8853140689667e-10 0.999999999605734 42 1.09970340046032e-09 2.19940680092065e-09 0.999999998900297 43 6.63395393669835e-09 1.32679078733967e-08 0.999999993366046 44 1.68162014191063e-07 3.36324028382127e-07 0.999999831837986 45 1.09723346631841e-05 2.19446693263683e-05 0.999989027665337 46 0.000319193710727251 0.000638387421454502 0.999680806289273 47 0.00501239223657857 0.0100247844731571 0.994987607763421 48 0.0497833902009396 0.0995667804018792 0.95021660979906 49 0.210380131697049 0.420760263394098 0.789619868302951 50 0.559563427171843 0.880873145656315 0.440436572828157 51 0.686497405892803 0.627005188214395 0.313502594107197 52 0.864145325325663 0.271709349348673 0.135854674674337 53 0.845683699729879 0.308632600540242 0.154316300270121 54 0.9222686651026 0.155462669794802 0.0777313348974008 55 0.942503322180677 0.114993355638646 0.0574966778193231 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 35 0.686274509803922 NOK 5% type I error level 39 0.764705882352941 NOK 10% type I error level 41 0.80392156862745 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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