*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: Thu, 19 Nov 2009 10:07:50 -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/19/t1258650970f9u7lkiutefyldd.htm/, Retrieved Thu, 19 Nov 2009 18:16:22 +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/19/t1258650970f9u7lkiutefyldd.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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29 27 24 25 22 24 26 28 29 24 25 22 26 25 26 29 24 25 21 19 26 26 29 24 23 19 21 26 26 29 22 19 23 21 26 26 21 20 22 23 21 26 16 16 21 22 23 21 19 22 16 21 22 23 16 21 19 16 21 22 25 25 16 19 16 21 27 29 25 16 19 16 23 28 27 25 16 19 22 25 23 27 25 16 23 26 22 23 27 25 20 24 23 22 23 27 24 28 20 23 22 23 23 28 24 20 23 22 20 28 23 24 20 23 21 28 20 23 24 20 22 32 21 20 23 24 17 31 22 21 20 23 21 22 17 22 21 20 19 29 21 17 22 21 23 31 19 21 17 22 22 29 23 19 21 17 15 32 22 23 19 21 23 32 15 22 23 19 21 31 23 15 22 23 18 29 21 23 15 22 18 28 18 21 23 15 18 28 18 18 21 23 18 29 18 18 18 21 10 22 18 18 18 18 13 26 10 18 18 18 10 24 13 10 18 18 9 27 10 13 10 18 9 27 9 10 13 10 6 23 9 9 10 13 11 21 6 9 9 10 9 19 11 6 9 9 10 17 9 11 6 9 9 19 10 9 11 6 16 21 9 10 9 11 10 13 16 9 10 9 7 8 10 16 9 10 7 5 7 10 16 9 14 10 7 7 10 16 11 6 14 7 7 10 10 6 11 14 7 7 6 8 10 11 14 7 8 11 6 10 11 14 13 12 8 6 10 11 12 13 13 8 6 10 15 19 12 13 8 6 16 19 15 12 13 8 16 18 16 1 etc...

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 s[t] = + 8.53757859793634 + 0.116192357805682consv[t] + 0.431844748244927`y(t-1)`[t] + 0.314619658899698`y(t-2)`[t] -0.104294325773217`y(t-3)`[t] -0.0303437661333572`y(t-4)`[t] -2.05606014705970M1[t] -3.08835545424920M2[t] -4.95092737842675M3[t] -1.76783608517104M4[t] -0.238952069021539M5[t] -2.44084765933435M6[t] -2.88724301052567M7[t] -1.28389824915988M8[t] -1.93079924912808M9[t] -6.85809143702282M10[t] -0.412018727564027M11[t] -0.0810169313766484t + 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) 8.53757859793634 4.84348 1.7627 0.085788 0.042894 consv 0.116192357805682 0.074015 1.5699 0.124528 0.062264 `y(t-1)` 0.431844748244927 0.156282 2.7632 0.008689 0.004344 `y(t-2)` 0.314619658899698 0.171318 1.8365 0.07392 0.03696 `y(t-3)` -0.104294325773217 0.17345 -0.6013 0.551124 0.275562 `y(t-4)` -0.0303437661333572 0.167302 -0.1814 0.857016 0.428508 M1 -2.05606014705970 2.379737 -0.864 0.392878 0.196439 M2 -3.08835545424920 2.42868 -1.2716 0.21104 0.10552 M3 -4.95092737842675 2.285318 -2.1664 0.036453 0.018226 M4 -1.76783608517104 2.288861 -0.7724 0.444554 0.222277 M5 -0.238952069021539 2.121875 -0.1126 0.910915 0.455457 M6 -2.44084765933435 2.246256 -1.0866 0.283871 0.141936 M7 -2.88724301052567 2.359874 -1.2235 0.228492 0.114246 M8 -1.28389824915988 2.245427 -0.5718 0.57075 0.285375 M9 -1.93079924912808 2.200794 -0.8773 0.385686 0.192843 M10 -6.85809143702282 2.358847 -2.9074 0.005984 0.002992 M11 -0.412018727564027 2.528009 -0.163 0.871375 0.435687 t -0.0810169313766484 0.0603 -1.3436 0.186857 0.093428 Multiple Linear Regression - Regression Statistics Multiple R 0.907368783699279 R-squared 0.82331810963191 Adjusted R-squared 0.746302926650947 F-TEST (value) 10.6903350451745 F-TEST (DF numerator) 17 F-TEST (DF denominator) 39 p-value 7.56620321951118e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.09778928349134 Sum Squared Residuals 374.255639351637 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 29 24.7447350564127 4.25526494358728 2 26 25.3400238129243 0.659976187075723 3 26 23.3386849610899 2.66131503891013 4 21 24.308618336703 -3.30861833670302 5 23 23.7584258269041 -0.758425826904108 6 22 20.8571358056061 1.14286419439391 7 21 21.1647820792644 -0.164782079264363 8 16 21.4190062500065 -5.4190062500065 9 19 18.9580058588779 0.041994141122085 10 16 13.6905784239437 2.30942157605629 11 25 20.7205437602123 4.27945623978767 12 27 24.2978945984748 2.70210540152518 13 23 25.9617432677395 -2.96174326773950 14 22 22.5540966470171 -0.554096647017124 15 23 18.5546942186782 4.44530578132176 16 20 21.8980987251173 -1.89809872511731 17 24 23.0554900455845 0.94450995441554 18 23 21.4821469805358 1.51785301946424 19 20 22.0639077965079 -2.06390779650795 20 21 21.6499357181698 -0.649935718169819 21 22 20.8576922508333 1.14230774916668 22 17 16.8228819243539 0.177118075646127 23 21 20.2843393724868 0.715660627513198 24 19 21.4483302798886 -2.4483302798886 25 23 20.4295549189053 2.57044508109473 26 22 19.916539167482 2.08346083251800 27 15 19.2353748597117 -4.23537485971172 28 23 18.6434265541504 4.35657344584956 29 21 21.2104409160189 -0.210440916018926 30 18 21.1088154999717 -3.10881549997171 31 18 17.9184890538116 0.0815109461883505 32 18 18.4627964295813 -0.462796429581278 33 18 18.2246413656285 -0.224641365628473 34 10 12.4940170401174 -2.49401704011738 35 13 15.8690842634628 -2.86908426346284 36 10 14.7462783175761 -4.74627831757605 37 9 13.4404576507068 -4.44045765070679 38 9 10.8813088389438 -1.88130883894384 39 6 8.3801825721868 -2.38018257218679 40 11 10.149663597893 0.850336402107004 41 9 12.6108544977134 -3.61085449771338 42 10 11.1178490357408 -1.11784903574085 43 9 10.1949865687638 -1.19498656876376 44 16 11.8893438458987 4.11065615410131 45 10 12.8965738374167 -2.89657383741667 46 7 6.99252261158504 0.00747738841495471 47 7 9.12603260383803 -2.12603260383803 48 14 9.50749680406052 4.49250319593948 49 11 10.4235091062357 0.57649089376428 50 10 10.3080315336328 -0.308031533632758 51 6 6.49106338833338 -0.49106338833338 52 8 8.00019278613623 -0.000192786136232415 53 13 9.36478871377912 3.63521128622088 54 12 10.4340526781456 1.56594732185440 55 15 11.6578345016523 3.34216549834772 56 16 13.5789177563437 2.42108224365629 57 16 14.0630866872436 1.93691331275638 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.0528715942837869 0.105743188567574 0.947128405716213 22 0.0242391212564965 0.0484782425129931 0.975760878743503 23 0.00989240445240087 0.0197848089048017 0.990107595547599 24 0.0518647264654328 0.103729452930866 0.948135273534567 25 0.0411052985087476 0.0822105970174951 0.958894701491252 26 0.0539078459012571 0.107815691802514 0.946092154098743 27 0.251690294505052 0.503380589010105 0.748309705494948 28 0.34610356978254 0.69220713956508 0.65389643021746 29 0.312378969773004 0.624757939546009 0.687621030226996 30 0.230791753193729 0.461583506387458 0.769208246806271 31 0.21140493209738 0.42280986419476 0.78859506790262 32 0.161491625445853 0.322983250891706 0.838508374554147 33 0.247843083074216 0.495686166148433 0.752156916925784 34 0.331374981523709 0.662749963047417 0.668625018476291 35 0.761414159499419 0.477171681001161 0.238585840500581 36 0.946465190038663 0.107069619922674 0.053534809961337 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 2 0.125 NOK 10% type I error level 3 0.1875 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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