Workshop 7: Multiple Regression Analysis

*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 05:08:59 -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/t1258719407zfsv4nkj1s7szx0.htm/, Retrieved Fri, 20 Nov 2009 13:17:02 +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/t1258719407zfsv4nkj1s7szx0.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:

ETSHWW7(3)

Dataseries X:

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1.43 0.51 1.43 0.51 1.43 0.51 1.43 0.51 1.43 0.52 1.43 0.52 1.44 0.52 1.48 0.53 1.48 0.53 1.48 0.52 1.48 0.52 1.48 0.52 1.48 0.52 1.48 0.52 1.48 0.52 1.48 0.52 1.48 0.52 1.48 0.52 1.48 0.52 1.48 0.53 1.48 0.53 1.48 0.53 1.48 0.54 1.48 0.54 1.48 0.54 1.48 0.54 1.48 0.54 1.48 0.54 1.48 0.54 1.48 0.54 1.48 0.54 1.48 0.54 1.48 0.53 1.48 0.53 1.48 0.53 1.48 0.53 1.48 0.53 1.57 0.54 1.58 0.55 1.58 0.55 1.58 0.55 1.58 0.55 1.59 0.55 1.6 0.55 1.6 0.55 1.61 0.55 1.61 0.56 1.61 0.56 1.62 0.56 1.63 0.56 1.63 0.56 1.64 0.55 1.64 0.56 1.64 0.55 1.64 0.55 1.64 0.56 1.65 0.55 1.65 0.55 1.65 0.55 1.65 0.55

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 Broodprijs[t] = -0.817116564417187 + 4.36503067484664Bakmeelprijs[t] -0.00707975460122839M1[t] + 0.00419018404907976M2[t] -0.00253987730061343M3[t] + 0.00819018404907986M4[t] -0.0092699386503067M5[t] -0.000539877300613433M6[t] + 0.00346012269938658M7[t] -0.0127300613496933M8[t] + 0.0067300613496933M9[t] + 0.0174601226993866M10[t] + 3.07197928500461e-17M11[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) -0.817116564417187 0.190581 -4.2875 8.9e-05 4.5e-05 Bakmeelprijs 4.36503067484664 0.35136 12.4232 0 0 M1 -0.00707975460122839 0.025531 -0.2773 0.782766 0.391383 M2 0.00419018404907976 0.025463 0.1646 0.869998 0.434999 M3 -0.00253987730061343 0.025415 -0.0999 0.920819 0.46041 M4 0.00819018404907986 0.025463 0.3216 0.749147 0.374573 M5 -0.0092699386503067 0.025386 -0.3652 0.716626 0.358313 M6 -0.000539877300613433 0.025415 -0.0212 0.983142 0.491571 M7 0.00346012269938658 0.025415 0.1361 0.892287 0.446144 M8 -0.0127300613496933 0.025386 -0.5015 0.618383 0.309192 M9 0.0067300613496933 0.025386 0.2651 0.792082 0.396041 M10 0.0174601226993866 0.025415 0.687 0.495455 0.247727 M11 3.07197928500461e-17 0.025376 0 1 0.5 Multiple Linear Regression - Regression Statistics Multiple R 0.879042679543587 R-squared 0.77271603245917 Adjusted R-squared 0.71468608329981 F-TEST (value) 13.3158143967549 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 2.39830377779526e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0401228045698134 Sum Squared Residuals 0.0756624539877296 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.43 1.40196932515338 0.0280306748466199 2 1.43 1.41323926380368 0.0167607361963190 3 1.43 1.40650920245399 0.0234907975460126 4 1.43 1.41723926380368 0.0127607361963194 5 1.43 1.44342944785276 -0.0134294478527605 6 1.43 1.45215950920245 -0.0221595092024538 7 1.44 1.45615950920245 -0.0161595092024538 8 1.48 1.48361963190184 -0.00361963190184033 9 1.48 1.50307975460123 -0.0230797546012269 10 1.48 1.47015950920245 0.00984049079754623 11 1.48 1.45269938650307 0.0273006134969328 12 1.48 1.45269938650307 0.0273006134969328 13 1.48 1.44561963190184 0.0343803680981612 14 1.48 1.45688957055215 0.0231104294478531 15 1.48 1.45015950920245 0.0298404907975462 16 1.48 1.46088957055215 0.0191104294478530 17 1.48 1.44342944785276 0.0365705521472395 18 1.48 1.45215950920245 0.0278404907975463 19 1.48 1.45615950920245 0.0238404907975462 20 1.48 1.48361963190184 -0.00361963190184032 21 1.48 1.50307975460123 -0.0230797546012269 22 1.48 1.51380981595092 -0.0338098159509202 23 1.48 1.54 -0.06 24 1.48 1.54 -0.06 25 1.48 1.53292024539877 -0.0529202453987716 26 1.48 1.54419018404908 -0.0641901840490798 27 1.48 1.53746012269939 -0.0574601226993866 28 1.48 1.54819018404908 -0.0681901840490799 29 1.48 1.53073006134969 -0.0507300613496933 30 1.48 1.53946012269939 -0.0594601226993866 31 1.48 1.54346012269939 -0.0634601226993866 32 1.48 1.52726993865031 -0.0472699386503067 33 1.48 1.50307975460123 -0.0230797546012269 34 1.48 1.51380981595092 -0.0338098159509202 35 1.48 1.49634969325153 -0.0163496932515336 36 1.48 1.49634969325153 -0.0163496932515336 37 1.48 1.48926993865031 -0.0092699386503052 38 1.57 1.54419018404908 0.0258098159509203 39 1.58 1.58111042944785 -0.00111042944785288 40 1.58 1.59184049079755 -0.0118404907975461 41 1.58 1.57438036809816 0.0056196319018404 42 1.58 1.58311042944785 -0.00311042944785286 43 1.59 1.58711042944785 0.00288957055214713 44 1.6 1.57092024539877 0.0290797546012270 45 1.6 1.59038036809816 0.0096196319018404 46 1.61 1.60111042944785 0.00888957055214713 47 1.61 1.62730061349693 -0.0173006134969327 48 1.61 1.62730061349693 -0.0173006134969327 49 1.62 1.62022085889570 -0.000220858895704272 50 1.63 1.63149079754601 -0.00149079754601264 51 1.63 1.62476073619632 0.00523926380368055 52 1.64 1.59184049079755 0.0481595092024537 53 1.64 1.61803067484663 0.0219693251533738 54 1.64 1.58311042944785 0.056889570552147 55 1.64 1.58711042944785 0.052889570552147 56 1.64 1.61457055214724 0.0254294478527604 57 1.65 1.59038036809816 0.0596196319018402 58 1.65 1.60111042944785 0.048889570552147 59 1.65 1.58365030674847 0.0663496932515335 60 1.65 1.58365030674847 0.0663496932515335 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 9.09830190722437e-41 1.81966038144487e-40 1 17 0.0529772391907858 0.105954478381572 0.947022760809214 18 0.0922338824274767 0.184467764854953 0.907766117572523 19 0.0999237697105454 0.199847539421091 0.900076230289455 20 0.0536783156836692 0.107356631367338 0.94632168431633 21 0.0252712073270077 0.0505424146540155 0.974728792672992 22 0.0273194909039926 0.0546389818079853 0.972680509096007 23 0.0514336776896875 0.102867355379375 0.948566322310313 24 0.0490441837146739 0.0980883674293477 0.950955816285326 25 0.0302574617926899 0.0605149235853799 0.96974253820731 26 0.0203649048124582 0.0407298096249165 0.979635095187542 27 0.0112551006822571 0.0225102013645143 0.988744899317743 28 0.00877346009045352 0.0175469201809070 0.991226539909546 29 0.00502845269630191 0.0100569053926038 0.994971547303698 30 0.00455428890504904 0.00910857781009808 0.99544571109495 31 0.00554133737437694 0.0110826747487539 0.994458662625623 32 0.00528106890672979 0.0105621378134596 0.99471893109327 33 0.00357076754403276 0.00714153508806552 0.996429232455967 34 0.00321505498096523 0.00643010996193045 0.996784945019035 35 0.00216332986811938 0.00432665973623876 0.99783667013188 36 0.00224143320389850 0.00448286640779699 0.997758566796102 37 0.00216984335181859 0.00433968670363718 0.997830156648181 38 0.0268079444647863 0.0536158889295725 0.973192055535214 39 0.0629778596450452 0.125955719290090 0.937022140354955 40 0.120846704990485 0.241693409980970 0.879153295009515 41 0.213817255574354 0.427634511148708 0.786182744425646 42 0.308491809087187 0.616983618174375 0.691508190912813 43 0.362035707143125 0.724071414286251 0.637964292856875 44 0.578565933736128 0.842868132527745 0.421434066263872 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 7 0.241379310344828 NOK 5% type I error level 13 0.448275862068966 NOK 10% type I error level 18 0.620689655172414 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 = 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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