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: Thu, 19 Nov 2009 08:22:56 -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/t1258644239hrohv287gxr2clg.htm/, Retrieved Thu, 19 Nov 2009 16:24:11 +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/t1258644239hrohv287gxr2clg.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:

» Textbox « » Textfile « » CSV «

1.4 1.9 1 1.6 -0.8 0 -2.9 -1.3 -0.7 -0.4 -0.7 -0.3 1.5 1.4 3 2.6 3.2 2.8 3.1 2.6 3.9 3.4 1 1.7 1.3 1.2 0.8 0 1.2 0 2.9 1.6 3.9 2.5 4.5 3.2 4.5 3.4 3.3 2.3 2 1.9 1.5 1.7 1 1.9 2.1 3.3 3 3.8 4 4.4 5.1 4.5 4.5 3.5 4.2 3 3.3 2.8 2.7 2.9 1.8 2.6 1.4 2.1 0.5 1.5 -0.4 1.1 0.8 1.5 0.7 1.7 1.9 2.3 2 2.3 1.1 1.9 0.9 2 0.4 1.6 0.7 1.2 2.1 1.9 2.8 2.1 3.9 2.4 3.5 2.9 2 2.5 2 2.3 1.5 2.5 2.5 2.6 3.1 2.4 2.7 2.5 2.8 2.1 2.5 2.2 3 2.7 3.2 3 2.8 3.2 2.4 3 2 2.7 1.8 2.5 1.1 1.6 -1.5 0.1 -3.7 -1.9

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 bbp[t] = -0.635828009397949 + 1.25032818779747dnst[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) -0.635828009397949 0.191327 -3.3233 0.001496 0.000748 dnst 1.25032818779747 0.080561 15.5203 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.891794576951301 R-squared 0.79529756747975 Adjusted R-squared 0.791995915342326 F-TEST (value) 240.878667520800 F-TEST (DF numerator) 1 F-TEST (DF denominator) 62 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.776230952059975 Sum Squared Residuals 37.357138438028 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.4 1.73979554741723 -0.339795547417234 2 1 1.364697091078 -0.364697091077999 3 -0.8 -0.635828009397951 -0.164171990602049 4 -2.9 -2.26125465353467 -0.638745346465333 5 -0.7 -1.13595928451694 0.435959284516941 6 -0.7 -1.01092646573719 0.310926465737194 7 1.5 1.11463145351851 0.38536854648149 8 3 2.61502527887548 0.384974721124522 9 3.2 2.86509091643497 0.334909083565028 10 3.1 2.61502527887548 0.484974721124523 11 3.9 3.61528782911346 0.284712170886544 12 1 1.48972990985775 -0.489729909857752 13 1.3 0.864565815959015 0.435434184040985 14 0.8 -0.635828009397952 1.43582800939795 15 1.2 -0.635828009397952 1.83582800939795 16 2.9 1.36469709107800 1.53530290892200 17 3.9 2.48999246009573 1.41000753990427 18 4.5 3.36522219155396 1.13477780844604 19 4.5 3.61528782911346 0.884712170886545 20 3.3 2.23992682253624 1.06007317746376 21 2 1.73979554741725 0.260204452582754 22 1.5 1.48972990985775 0.0102700901422481 23 1 1.73979554741725 -0.739795547417246 24 2.1 3.49025501033371 -1.39025501033371 25 3 4.11541910423245 -1.11541910423244 26 4 4.86561601691093 -0.865616016910929 27 5.1 4.99064883569068 0.109351164309324 28 4.5 3.7403206478932 0.759679352106797 29 4.2 3.11515655399447 1.08484344600553 30 3.3 2.86509091643497 0.434909083565028 31 2.7 2.99012373521472 -0.290123735214719 32 1.8 2.61502527887548 -0.815025278875477 33 1.4 1.98986118497674 -0.589861184976741 34 0.5 1.23966427229826 -0.739664272298257 35 -0.4 0.739532997179268 -1.13953299717927 36 0.8 1.23966427229826 -0.439664272298257 37 0.7 1.48972990985775 -0.789729909857752 38 1.9 2.23992682253624 -0.339926822536235 39 2 2.23992682253624 -0.239926822536235 40 1.1 1.73979554741725 -0.639795547417246 41 0.9 1.86482836619699 -0.964828366196994 42 0.4 1.36469709107800 -0.964697091078005 43 0.7 0.864565815959015 -0.164565815959015 44 2.1 1.73979554741725 0.360204452582754 45 2.8 1.98986118497674 0.810138815023259 46 3.9 2.36495964131598 1.53504035868402 47 3.5 2.99012373521472 0.509876264785281 48 2 2.48999246009573 -0.48999246009573 49 2 2.23992682253624 -0.239926822536235 50 1.5 2.48999246009573 -0.98999246009573 51 2.5 2.61502527887548 -0.115025278875478 52 3.1 2.36495964131598 0.735040358684017 53 2.7 2.48999246009573 0.21000753990427 54 2.8 1.98986118497674 0.810138815023259 55 2.5 2.11489400375649 0.385105996243511 56 3 2.74005809765522 0.259941902344775 57 3.2 3.11515655399447 0.0848434460055337 58 2.8 3.36522219155396 -0.565222191553962 59 2.4 3.11515655399447 -0.715156553994467 60 2 2.74005809765522 -0.740058097655225 61 1.8 2.48999246009573 -0.68999246009573 62 1.1 1.36469709107800 -0.264697091078005 63 -1.5 -0.510795190618205 -0.989204809381795 64 -3.7 -3.01145156621315 -0.68854843378685 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.198800457351679 0.397600914703359 0.80119954264832 6 0.140045963286010 0.280091926572021 0.85995403671399 7 0.109478539723691 0.218957079447382 0.890521460276309 8 0.0666252653777274 0.133250530755455 0.933374734622273 9 0.0334301773241603 0.0668603546483207 0.96656982267584 10 0.0181467682192202 0.0362935364384405 0.98185323178078 11 0.00795682354129819 0.0159136470825964 0.992043176458702 12 0.00900612879838255 0.0180122575967651 0.990993871201618 13 0.00577514034351986 0.0115502806870397 0.99422485965648 14 0.0675285141867549 0.135057028373510 0.932471485813245 15 0.333329306630556 0.666658613261111 0.666670693369444 16 0.526462799335845 0.94707440132831 0.473537200664155 17 0.642792069074526 0.714415861850947 0.357207930925474 18 0.65809575959695 0.683808480806099 0.341904240403049 19 0.627693101191058 0.744613797617884 0.372306898808942 20 0.65258436324679 0.694831273506421 0.347415636753210 21 0.599917873577701 0.800164252844598 0.400082126422299 22 0.553942850033449 0.892114299933102 0.446057149966551 23 0.631639515979195 0.73672096804161 0.368360484020805 24 0.859670501837415 0.280658996325169 0.140329498162585 25 0.92252692687951 0.154946146240978 0.0774730731204892 26 0.943818887223077 0.112362225553847 0.0561811127769233 27 0.922549394484952 0.154901211030096 0.0774506055150478 28 0.916116648049598 0.167766703900804 0.0838833519504018 29 0.941389266933403 0.117221466133194 0.0586107330665969 30 0.926655080439133 0.146689839121733 0.0733449195608666 31 0.90493379982892 0.190132400342159 0.0950662001710795 32 0.914890260658781 0.170219478682439 0.0851097393412193 33 0.903968093309135 0.192063813381731 0.0960319066908655 34 0.899972444329894 0.200055111340213 0.100027555670106 35 0.926977123153672 0.146045753692656 0.0730228768463281 36 0.905360668728574 0.189278662542851 0.0946393312714256 37 0.901480521203414 0.197038957593173 0.0985194787965864 38 0.8705760786697 0.2588478426606 0.1294239213303 39 0.829035528679767 0.341928942640467 0.170964471320233 40 0.806280299008043 0.387439401983914 0.193719700991957 41 0.828252375144459 0.343495249711082 0.171747624855541 42 0.846294342127688 0.307411315744625 0.153705657872312 43 0.794462528422556 0.411074943154888 0.205537471577444 44 0.751502540995059 0.496994918009882 0.248497459004941 45 0.77281541899082 0.454369162018361 0.227184581009180 46 0.9472475366577 0.105504926684600 0.0527524633423001 47 0.941610792098747 0.116778415802506 0.0583892079012528 48 0.918567100570662 0.162865798858675 0.0814328994293376 49 0.87785140194718 0.244297196105639 0.122148598052819 50 0.903534393649034 0.192931212701932 0.0964656063509662 51 0.852831252577309 0.294337494845382 0.147168747422691 52 0.884027855430858 0.231944289138285 0.115972144569142 53 0.843167625663856 0.313664748672288 0.156832374336144 54 0.939583476667174 0.120833046665652 0.060416523332826 55 0.962897903656644 0.0742041926867123 0.0371020963433562 56 0.980325370577757 0.0393492588444852 0.0196746294222426 57 0.99371653523122 0.0125669295375592 0.00628346476877958 58 0.978567724795464 0.0428645504090718 0.0214322752045359 59 0.933672904190525 0.132654191618950 0.0663270958094748 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 7 0.127272727272727 NOK 10% type I error level 9 0.163636363636364 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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