link2

*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 04:39:19 -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/t1258717224p2lj2zuy40y1u1s.htm/, Retrieved Fri, 20 Nov 2009 12:40:39 +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/t1258717224p2lj2zuy40y1u1s.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:

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1.58 0.55 1.59 0.55 1.6 0.55 1.6 0.55 1.6 0.55 1.6 0.56 1.61 0.56 1.61 0.56 1.62 0.56 1.63 0.56 1.63 0.55 1.63 0.56 1.63 0.55 1.63 0.55 1.64 0.56 1.64 0.55 1.64 0.55 1.65 0.55 1.65 0.55 1.65 0.53 1.65 0.53 1.65 0.53 1.66 0.53 1.67 0.54 1.68 0.54 1.68 0.54 1.68 0.55 1.68 0.55 1.69 0.54 1.7 0.55 1.7 0.56 1.71 0.58 1.73 0.59 1.73 0.6 1.73 0.6 1.74 0.6 1.74 0.59 1.74 0.6 1.75 0.6 1.78 0.62 1.82 0.65 1.83 0.68 1.84 0.73 1.85 0.78 1.86 0.78 1.86 0.82 1.87 0.82 1.87 0.81 1.87 0.83 1.87 0.85 1.87 0.86 1.87 0.85 1.87 0.85 1.88 0.82 1.88 0.8 1.87 0.81 1.87 0.8 1.87 0.8 1.87 0.8 1.87 0.8 1.87 0.79

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] = + 1.21294728270632 + 0.820321325217045X[t] -0.0109867997205874M1[t] -0.0179058616904500M2[t] -0.0168277896417523M3[t] -0.0108277896417522M4[t] -0.00410907494262042M5[t] + 0.000609639756511355M6[t] -0.001952930845225M7[t] -0.0097967867478295M8[t] -0.00179678674782949M9[t] -0.00799999999999997M10[t] -0.00235935734956588M11[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) 1.21294728270632 0.035105 34.5522 0 0 X 0.820321325217045 0.045123 18.1798 0 0 M1 -0.0109867997205874 0.024986 -0.4397 0.662108 0.331054 M2 -0.0179058616904500 0.026154 -0.6846 0.496875 0.248438 M3 -0.0168277896417523 0.026135 -0.6439 0.522722 0.261361 M4 -0.0108277896417522 0.026135 -0.4143 0.680501 0.34025 M5 -0.00410907494262042 0.026124 -0.1573 0.875675 0.437838 M6 0.000609639756511355 0.026114 0.0233 0.981472 0.490736 M7 -0.001952930845225 0.026098 -0.0748 0.94066 0.47033 M8 -0.0097967867478295 0.026083 -0.3756 0.708867 0.354433 M9 -0.00179678674782949 0.026083 -0.0689 0.945365 0.472683 M10 -0.00799999999999997 0.026079 -0.3068 0.760352 0.380176 M11 -0.00235935734956588 0.026079 -0.0905 0.928291 0.464145 Multiple Linear Regression - Regression Statistics Multiple R 0.936168716070138 R-squared 0.876411864948412 Adjusted R-squared 0.845514831185514 F-TEST (value) 28.3655664706835 F-TEST (DF numerator) 12 F-TEST (DF denominator) 48 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0412343776253169 Sum Squared Residuals 0.0816131471110674 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.58 1.65313721185511 -0.0731372118551079 2 1.59 1.64621814988524 -0.0562181498852411 3 1.6 1.64729622193394 -0.0472962219339386 4 1.6 1.65329622193394 -0.0532962219339386 5 1.6 1.66001493663307 -0.0600149366330704 6 1.6 1.67293686458437 -0.0729368645843727 7 1.61 1.67037429398264 -0.0603742939826363 8 1.61 1.66253043808003 -0.0525304380800318 9 1.62 1.67053043808003 -0.0505304380800318 10 1.63 1.66432722482786 -0.0343272248278615 11 1.63 1.66176465422613 -0.0317646542261252 12 1.63 1.67232722482786 -0.0423272248278615 13 1.63 1.65313721185510 -0.0231372118551036 14 1.63 1.64621814988524 -0.016218149885241 15 1.64 1.65549943518611 -0.0154994351861092 16 1.64 1.65329622193394 -0.0132962219339388 17 1.64 1.66001493663307 -0.0200149366330706 18 1.65 1.66473365133220 -0.0147336513322024 19 1.65 1.66217108073047 -0.012171080730466 20 1.65 1.63792079832352 0.0120792016764794 21 1.65 1.64592079832352 0.00407920167647939 22 1.65 1.63971758507135 0.0102824149286499 23 1.66 1.64535822772178 0.0146417722782158 24 1.67 1.65592079832352 0.0140792016764795 25 1.68 1.64493399860293 0.0350660013970669 26 1.68 1.63801493663307 0.0419850633669295 27 1.68 1.64729622193394 0.0327037780660613 28 1.68 1.65329622193394 0.0267037780660613 29 1.69 1.6518117233809 0.0381882766190999 30 1.7 1.66473365133220 0.0352663486677977 31 1.7 1.67037429398264 0.0296257060173636 32 1.71 1.67893686458437 0.0310631354156273 33 1.73 1.69514007783654 0.0348599221634568 34 1.73 1.69714007783654 0.0328599221634568 35 1.73 1.70278072048698 0.0272192795130227 36 1.74 1.70514007783654 0.0348599221634569 37 1.74 1.68595006486379 0.0540499351362148 38 1.74 1.68723421614609 0.0527657838539069 39 1.75 1.68831228819479 0.0616877118052091 40 1.78 1.71071871469913 0.0692812853008682 41 1.82 1.74204706915478 0.077952930845225 42 1.83 1.77137542361042 0.0586245763895819 43 1.84 1.80982891926953 0.0301710807304660 44 1.85 1.84300112962778 0.00699887037221826 45 1.86 1.85100112962778 0.00899887037221827 46 1.86 1.87761076938429 -0.017610769384293 47 1.87 1.88325141203473 -0.0132514120347271 48 1.87 1.87740755613212 -0.0074075561321226 49 1.87 1.88282718291588 -0.0128271829158760 50 1.87 1.89231454745035 -0.0223145474503543 51 1.87 1.90159583275122 -0.0315958327512226 52 1.87 1.89939261949905 -0.0293926194990522 53 1.87 1.90611133419818 -0.0361113341981840 54 1.88 1.88622040914080 -0.00622040914080454 55 1.88 1.86725141203473 0.0127485879652727 56 1.87 1.86761076938429 0.00238923061570689 57 1.87 1.86740755613212 0.00259244386787736 58 1.87 1.86120434287995 0.00879565712004784 59 1.87 1.86684498553039 0.00315501446961374 60 1.87 1.86920434287995 0.000795657120047871 61 1.87 1.85001432990719 0.0199856700928058 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.719102643543961 0.561794712912079 0.280897356456039 17 0.751421136690233 0.497157726619534 0.248578863309767 18 0.893831126296353 0.212337747407295 0.106168873703648 19 0.906413552086463 0.187172895827074 0.0935864479135371 20 0.86697279979967 0.26605440040066 0.13302720020033 21 0.848351074427127 0.303297851145746 0.151648925572873 22 0.820640771972725 0.358718456054551 0.179359228027275 23 0.77822748162691 0.44354503674618 0.22177251837309 24 0.756480403917769 0.487039192164463 0.243519596082231 25 0.885327280107935 0.229345439784129 0.114672719892064 26 0.916445372047645 0.16710925590471 0.083554627952355 27 0.933227796498253 0.133544407003493 0.0667722035017466 28 0.961956860254136 0.0760862794917291 0.0380431397458645 29 0.971667284568492 0.0566654308630159 0.0283327154315080 30 0.987179340095743 0.0256413198085133 0.0128206599042567 31 0.996927290430171 0.00614541913965717 0.00307270956982858 32 0.9995700102812 0.000859979437601168 0.000429989718800584 33 0.99982730363452 0.000345392730958800 0.000172696365479400 34 0.99983305738535 0.000333885229299968 0.000166942614649984 35 0.99989581787344 0.000208364253118787 0.000104182126559394 36 0.999941600059113 0.000116799881773352 5.8399940886676e-05 37 0.999982330242287 3.53395154251091e-05 1.76697577125545e-05 38 0.999987989867457 2.40202650868354e-05 1.20101325434177e-05 39 0.999990713812351 1.85723752978944e-05 9.28618764894718e-06 40 0.99997908585113 4.18282977424013e-05 2.09141488712006e-05 41 0.999947115314597 0.000105769370805826 5.28846854029131e-05 42 0.999703956576784 0.000592086846431685 0.000296043423215842 43 0.999630274111636 0.000739451776727719 0.000369725888363860 44 0.999679989580922 0.0006400208381569 0.00032001041907845 45 0.999196640635255 0.00160671872949027 0.000803359364745134 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 15 0.5 NOK 5% type I error level 16 0.533333333333333 NOK 10% type I error level 18 0.6 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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