Workshop 7: Multiple Regression 2

*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, 25 Dec 2009 11:29:33 -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/25/t1261765889bzm1mji21y09rna.htm/, Retrieved Fri, 25 Dec 2009 19:31:41 +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/25/t1261765889bzm1mji21y09rna.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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Dataseries X:

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25.6 8.1 23.7 7.7 22 7.5 21.3 7.6 20.7 7.8 20.4 7.8 20.3 7.8 20.4 7.5 19.8 7.5 19.5 7.1 23.1 7.5 23.5 7.5 23.5 7.6 22.9 7.7 21.9 7.7 21.5 7.9 20.5 8.1 20.2 8.2 19.4 8.2 19.2 8.2 18.8 7.9 18.8 7.3 22.6 6.9 23.3 6.6 23 6.7 21.4 6.9 19.9 7 18.8 7.1 18.6 7.2 18.4 7.1 18.6 6.9 19.9 7 19.2 6.8 18.4 6.4 21.1 6.7 20.5 6.6 19.1 6.4 18.1 6.3 17 6.2 17.1 6.5 17.4 6.8 16.8 6.8 15.3 6.4 14.3 6.1 13.4 5.8 15.3 6.1 22.1 7.2 23.7 7.3 22.2 6.9 19.5 6.1 16.6 5.8 17.3 6.2 19.8 7.1 21.2 7.7 21.5 7.9 20.6 7.7 19.1 7.4 19.6 7.5 23.5 8 24 8.1

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] = + 4.13200139227286 + 2.61329620605639X[t] -0.110936303515468M1[t] -1.14827706230421M2[t] -2.52694744169857M3[t] -3.38187260703098M4[t] -4.07039331709015M5[t] -4.38398886181692M6[t] -4.55492516533241M7[t] -4.32906369648451M8[t] -4.57413853115211M9[t] -3.79147928994083M10[t] -0.624531848242254M11[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) 4.13200139227286 1.668938 2.4758 0.016952 0.008476 X 2.61329620605639 0.221591 11.7933 0 0 M1 -0.110936303515468 0.672119 -0.1651 0.869609 0.434804 M2 -1.14827706230421 0.674744 -1.7018 0.0954 0.0477 M3 -2.52694744169857 0.677141 -3.7318 0.000512 0.000256 M4 -3.38187260703098 0.67282 -5.0264 8e-06 4e-06 M5 -4.07039331709015 0.673068 -6.0475 0 0 M6 -4.38398886181692 0.675166 -6.4932 0 0 M7 -4.55492516533241 0.673651 -6.7615 0 0 M8 -4.32906369648451 0.672119 -6.4409 0 0 M9 -4.57413853115211 0.672601 -6.8007 0 0 M10 -3.79147928994083 0.676096 -5.6079 1e-06 1e-06 M11 -0.624531848242254 0.671944 -0.9294 0.35741 0.178705 Multiple Linear Regression - Regression Statistics Multiple R 0.927515115038964 R-squared 0.860284288625743 Adjusted R-squared 0.824612192104656 F-TEST (value) 24.1164487799925 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 4.44089209850063e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.06234362232883 Sum Squared Residuals 53.0429766794292 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 25.6 25.1887643578141 0.41123564218595 2 23.7 23.1061051166029 0.593894883397148 3 22 21.2047754959972 0.795224504002781 4 21.3 20.6111799512705 0.688820048729549 5 20.7 20.4453184824226 0.254681517577443 6 20.4 20.1317229376958 0.268277062304209 7 20.3 19.9607866341803 0.339213365819701 8 20.4 19.4026592412113 0.99734075878872 9 19.8 19.1575844065437 0.642415593456318 10 19.5 18.8949251653324 0.605074834667594 11 23.1 23.1071910894535 -0.00719108945353017 12 23.5 23.7317229376958 -0.231722937695790 13 23.5 23.8821162547860 -0.38211625478596 14 22.9 23.1061051166029 -0.206105116602860 15 21.9 21.7274347372085 0.172565262791502 16 21.5 21.3951688130874 0.104831186912629 17 20.5 21.2293073442395 -0.729307344239474 18 20.2 21.1770414201183 -0.977041420118344 19 19.4 21.0061051166029 -1.60610511660286 20 19.2 21.2319665854507 -2.03196658545075 21 18.8 20.2029028889662 -1.40290288896624 22 18.8 19.4175844065437 -0.617584406543684 23 22.6 21.5392133658197 1.0607866341803 24 23.3 21.3797563522450 1.92024364775496 25 23 21.5301496693352 1.46985033066479 26 21.4 21.0154681517577 0.384531848242252 27 19.9 19.8981273929690 0.00187260703097682 28 18.8 19.3045318482423 -0.504531848242255 29 18.6 18.8773407587887 -0.277340758788722 30 18.4 18.3024155934563 0.0975844065436837 31 18.6 17.6088200487295 0.991179951270452 32 19.9 18.0960111381831 1.80398886181691 33 19.2 17.3282770623042 1.87172293769579 34 18.4 17.0656178210929 1.33438217890707 35 21.1 21.0165541246084 0.0834458753915786 36 20.5 21.3797563522450 -0.879756352245038 37 19.1 20.7461608075183 -1.64616080751829 38 18.1 19.4474904281239 -1.34749042812391 39 17 17.8074904281239 -0.80749042812391 40 17.1 17.7365541246084 -0.63655412460842 41 17.4 17.8320222763662 -0.432022276366167 42 16.8 17.5184267316394 -0.718426731639396 43 15.3 16.3021719457014 -1.00217194570135 44 14.3 15.7440445527323 -1.44404455273233 45 13.4 14.7149808562478 -1.31498085624782 46 15.3 16.281628959276 -0.981628959276014 47 22.1 22.3232022276366 -0.223202227636617 48 23.7 23.2090636964845 0.490936303515487 49 22.2 22.0528089105465 0.147191089453512 50 19.5 18.9248311869126 0.575168813087369 51 16.6 16.7621719457014 -0.162171945701351 52 17.3 16.9525652627915 0.347434737208496 53 19.8 18.6160111381831 1.18398886181692 54 21.2 19.8703933170902 1.32960668290985 55 21.5 20.2221162547859 1.27788374521406 56 20.6 19.9253184824226 0.674681517577445 57 19.1 18.8962547859380 0.203745214061955 58 19.6 19.9402436477550 -0.340243647754962 59 23.5 24.4138391924817 -0.913839192481731 60 24 25.2997006613296 -1.29970066132962 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.115929469366926 0.231858938733851 0.884070530633075 17 0.0813514546702565 0.162702909340513 0.918648545329743 18 0.056778025873242 0.113556051746484 0.943221974126758 19 0.0681688494372356 0.136337698874471 0.931831150562764 20 0.0871004673631051 0.174200934726210 0.912899532636895 21 0.0728194379903764 0.145638875980753 0.927180562009624 22 0.0454766025863408 0.0909532051726816 0.95452339741366 23 0.0340549023923555 0.068109804784711 0.965945097607644 24 0.0378978207920747 0.0757956415841493 0.962102179207925 25 0.0698367473758958 0.139673494751792 0.930163252624104 26 0.0841055446769391 0.168211089353878 0.915894455323061 27 0.100890595250194 0.201781190500388 0.899109404749806 28 0.149158453296846 0.298316906593693 0.850841546703154 29 0.126934418945599 0.253868837891198 0.8730655810544 30 0.0879897726307702 0.175979545261540 0.91201022736923 31 0.063340760135202 0.126681520270404 0.936659239864798 32 0.0944224298310053 0.188844859662011 0.905577570168995 33 0.146990370983203 0.293980741966406 0.853009629016797 34 0.210160366873253 0.420320733746505 0.789839633126747 35 0.231378617998143 0.462757235996285 0.768621382001857 36 0.296237524013120 0.592475048026239 0.70376247598688 37 0.526627165531212 0.946745668937577 0.473372834468788 38 0.688748841845226 0.622502316309548 0.311251158154774 39 0.663469303937421 0.673061392125157 0.336530696062579 40 0.629885272086887 0.740229455826226 0.370114727913113 41 0.590087036821371 0.819825926357258 0.409912963178629 42 0.546664160974829 0.906671678050343 0.453335839025171 43 0.500475478284469 0.999049043431061 0.499524521715531 44 0.494346461200130 0.988692922400261 0.505653538799870 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 3 0.103448275862069 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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