Multiple Regression

*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, 01 Dec 2010 14:36:47 +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/01/t1291214141cg48t45tzyzbd3o.htm/, Retrieved Wed, 01 Dec 2010 15:35:51 +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/01/t1291214141cg48t45tzyzbd3o.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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2 3 3 2 14 2 5 4 1 18 4 3 2 2 11 3 3 2 2 12 3 4 4 1 16 2 5 4 1 18 4 4 4 2 14 3 4 4 3 14 2 4 3 2 15 2 4 3 2 15 2 4 5 2 17 1 5 4 1 19 2 2 2 4 10 1 4 3 2 16 2 5 5 2 18 3 4 4 3 14 2 4 3 3 14 2 4 4 1 17 3 4 2 1 14 2 5 3 2 16 1 4 4 1 18 3 3 2 3 11 4 3 5 2 14 3 3 3 3 12 2 5 4 2 17 4 2 3 4 9 2 4 4 2 16 4 4 4 2 14 3 4 4 2 15 4 3 2 2 11 2 4 4 2 16 3 3 4 3 13 1 4 4 2 17 2 4 3 2 15 3 4 4 3 14 2 4 4 2 16 4 2 3 4 9 2 4 3 2 15 2 5 4 2 17 2 3 4 4 13 2 4 4 3 15 2 4 4 2 16 2 5 4 3 16 3 3 4 4 12 2 4 2 12 4 3 3 3 11 2 4 4 3 15 2 4 3 2 15 3 5 4 1 17 4 4 3 2 13 2 3 4 1 16 2 3 3 2 14 4 4 2 3 11 2 3 3 4 12 3 4 4 5 12 2 4 4 3 15 2 4 4 2 16 2 3 4 2 15 3 3 3 3 12 4 3 3 2 12 5 3 2 4 8 3 4 3 3 13 5 4 2 2 11 3 4 3 2 14 3 4 4 2 15 4 3 2 3 10

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 8 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 R Framework error message Warning: there are blank lines in the 'Data X' field. Please, use NA for missing data - blank lines are simply deleted and are NOT treated as missing values. Multiple Linear Regression - Estimated Regression Equation PSS[t] = + 10.1553588186132 -0.520868917539678IDT[t] + 1.95835376594495HPP[t] + 0.0420113088658949TGYW[t] -0.876103192773687POP[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) 10.1553588186132 1.266727 8.017 0 0 IDT -0.520868917539678 0.218734 -2.3813 0.020388 0.010194 HPP 1.95835376594495 0.216052 9.0643 0 0 TGYW 0.0420113088658949 0.19652 0.2138 0.831434 0.415717 POP -0.876103192773687 0.04024 -21.7721 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.978298437003284 R-squared 0.957067831843069 Adjusted R-squared 0.95425260770163 F-TEST (value) 339.961503510753 F-TEST (DF numerator) 4 F-TEST (DF denominator) 61 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.26192173482487 Sum Squared Residuals 97.1392343542274 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 14 13.3625098224190 0.637490177581043 2 18 18.1973318559485 -0.197331855948484 3 11 12.2787606784737 -1.27876067847374 4 12 12.7996295960134 -0.79962959601344 5 16 15.7181091724639 0.281890827536148 6 18 18.1973318559485 -0.197331855948480 7 14 14.3211370621505 -0.321137062150491 8 14 13.9659027869165 0.0340972130835167 9 15 15.3208635883640 -0.320863588363956 10 15 15.3208635883640 -0.320863588363956 11 17 15.4048862060957 1.59511379390425 12 19 18.7182007734882 0.281799226511841 13 10 9.6099383620608 0.390061637939198 14 16 15.8417325059036 0.158267494096365 15 18 17.3632399720407 0.636760027959312 16 14 13.9659027869165 0.0340972130835167 17 14 14.4447603955903 -0.444760395590268 18 17 16.2389780900035 0.761021909996462 19 14 15.6340865547321 -1.63408655473207 20 16 17.2792173543089 -1.27921735430890 21 18 16.7598470075432 1.24015299245678 22 11 11.9235264032398 -0.923526403239752 23 14 12.4047946050714 1.59520539492856 24 12 11.9655377121056 0.0344622878943541 25 17 17.3212286631748 -0.321228663174793 26 9 8.61021183584734 0.389788164152664 27 16 15.3628748972299 0.63712510277015 28 14 14.3211370621505 -0.321137062150491 29 15 14.8420059796902 0.15799402030983 30 11 12.2787606784738 -1.27876067847376 31 16 15.3628748972299 0.63712510277015 32 13 12.0075490209715 0.99245097902846 33 17 15.8837438147695 1.11625618523047 34 15 15.3208635883640 -0.320863588363956 35 14 13.9659027869165 0.0340972130835167 36 16 15.3628748972299 0.63712510277015 37 9 8.61021183584734 0.389788164152664 38 15 15.3208635883640 -0.320863588363956 39 17 17.3212286631748 -0.321228663174793 40 13 11.6523147457375 1.34768525426247 41 15 14.4867717044562 0.513228295543837 42 16 15.3628748972299 0.63712510277015 43 16 16.4451254704011 -0.445125470401106 44 12 11.1314458281979 0.868554171802146 45 4 6.51782035176119 -2.51782035176119 46 2 4.95671216991615 -2.95671216991615 47 2 2.88978424722667 -0.889784247226666 48 3 0.889419172415828 2.11058082758417 49 4 0.532686326407821 3.46731367359218 50 2 2.6416255579632 -0.641625557963202 51 2 2.45052735426087 -0.450527354260868 52 4 2.28639128272919 1.71360871727081 53 2 2.47748948643153 -0.477489486431528 54 3 4.12262028600836 -1.12262028600836 55 2 5.60211644327951 -3.60211644327952 56 2 2.88978424722667 -0.889784247226666 57 2 1.97166974558708 0.0283302544129174 58 3 3.36864185590045 -0.368641855900449 59 4 4.08060897714246 -0.0806089771424625 60 5 4.03859766827657 0.961402331723432 61 3 5.66867929115816 -2.66867929115816 62 5 2.68363686682910 2.31636313317090 63 3 2.43547817756563 0.564521822434367 64 3 1.76552236518951 1.23447763481049 65 4 2.84777293836077 1.15222706163923 66 2 3.87446159674489 -1.87446159674489 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 4.11588828496535e-47 8.2317765699307e-47 1 9 2.26952480416351e-61 4.53904960832702e-61 1 10 9.4213181480394e-74 1.88426362960788e-73 1 11 9.68790836133826e-87 1.93758167226765e-86 1 12 7.47441199972816e-105 1.49488239994563e-104 1 13 2.84465013323496e-119 5.68930026646993e-119 1 14 5.67470699116044e-128 1.13494139823209e-127 1 15 3.75254467215592e-146 7.50508934431184e-146 1 16 1.05897517656618e-168 2.11795035313236e-168 1 17 2.19396382287957e-181 4.38792764575914e-181 1 18 1.8913562405899e-186 3.7827124811798e-186 1 19 2.44495688794345e-199 4.88991377588691e-199 1 20 3.98282786161020e-224 7.96565572322039e-224 1 21 1.58455670913723e-227 3.16911341827446e-227 1 22 2.13000917136951e-247 4.26001834273902e-247 1 23 1.32751170163987e-258 2.65502340327974e-258 1 24 2.87715741018553e-273 5.75431482037105e-273 1 25 6.81670357889293e-294 1.36334071577859e-293 1 26 3.88115644895887e-307 7.76231289791774e-307 1 27 0 0 1 28 0 0 1 29 0 0 1 30 0 0 1 31 0 0 1 32 0 0 1 33 0 0 1 34 0 0 1 35 0 0 1 36 0 0 1 37 0 0 1 38 0 0 1 39 0 0 1 40 0 0 1 41 0 0 1 42 0 0 1 43 0 0 1 44 0 0 1 45 0 0 1 46 2.16775240647444e-14 4.33550481294887e-14 0.999999999999978 47 2.17161799031552e-07 4.34323598063105e-07 0.9999997828382 48 0.00795327242063898 0.0159065448412780 0.99204672757936 49 0.257307345031696 0.514614690063392 0.742692654968304 50 0.244452910896673 0.488905821793345 0.755547089103327 51 0.33534414646283 0.67068829292566 0.66465585353717 52 0.295571000928419 0.591142001856837 0.704428999071581 53 0.259797516979347 0.519595033958695 0.740202483020653 54 0.246673236802684 0.493346473605369 0.753326763197316 55 0.330158003559143 0.660316007118286 0.669841996440857 56 0.309329550128397 0.618659100256794 0.690670449871603 57 0.394726138363535 0.78945227672707 0.605273861636465 58 0.361849254561772 0.723698509123543 0.638150745438228 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 40 0.784313725490196 NOK 5% type I error level 41 0.80392156862745 NOK 10% type I error level 41 0.80392156862745 NOK

Charts produced by software:

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Parameters (Session):

par1 = 5 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 5 ; 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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