*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: Sun, 13 Dec 2009 08:36:08 -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/13/t1260718816w2ft18fudiy4c3y.htm/, Retrieved Sun, 13 Dec 2009 16:40:34 +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/13/t1260718816w2ft18fudiy4c3y.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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101.9 96.4 110.4 100.5 98.8 93.7 106.2 101.9 96.4 110.4 100.5 106.7 81 106.2 101.9 96.4 110.4 86.7 94.7 81 106.2 101.9 96.4 95.3 101 94.7 81 106.2 101.9 99.3 109.4 101 94.7 81 106.2 101.8 102.3 109.4 101 94.7 81 96 90.7 102.3 109.4 101 94.7 91.7 96.2 90.7 102.3 109.4 101 95.3 96.1 96.2 90.7 102.3 109.4 96.6 106 96.1 96.2 90.7 102.3 107.2 103.1 106 96.1 96.2 90.7 108 102 103.1 106 96.1 96.2 98.4 104.7 102 103.1 106 96.1 103.1 86 104.7 102 103.1 106 81.1 92.1 86 104.7 102 103.1 96.6 106.9 92.1 86 104.7 102 103.7 112.6 106.9 92.1 86 104.7 106.6 101.7 112.6 106.9 92.1 86 97.6 92 101.7 112.6 106.9 92.1 87.6 97.4 92 101.7 112.6 106.9 99.4 97 97.4 92 101.7 112.6 98.5 105.4 97 97.4 92 101.7 105.2 102.7 105.4 97 97.4 92 104.6 98.1 102.7 105.4 97 97.4 97.5 104.5 98.1 102.7 105.4 97 108.9 87.4 104.5 98.1 102.7 105.4 86.8 89.9 87.4 104.5 98.1 102.7 88.9 109.8 89.9 87.4 104.5 98.1 110.3 111.7 109.8 89.9 87.4 104.5 114.8 98.6 111.7 109.8 89.9 87.4 94.6 96.9 98.6 111.7 109.8 89.9 92 95.1 96.9 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 Y[t] = + 95.4128864396907 -0.373522045942315`Y(t-1)`[t] -0.0751593337637079`Y(t-2)`[t] + 0.35885170198808`Y(t-3)`[t] -0.0788126508587618`Y(t-4)`[t] + 0.240211501967378X[t] -2.86774271905553M1[t] -1.32952280976387M2[t] -14.2114323871744M3[t] -14.3746263866844M4[t] -4.43378811719101M5[t] + 11.5304356672672M6[t] + 2.77911200658021M7[t] -9.08766106143715M8[t] -13.6044063583265M9[t] -9.75894265732346M10[t] + 0.542826991949121M11[t] + 0.118834153023582t + 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) 95.4128864396907 29.571546 3.2265 0.00258 0.00129 `Y(t-1)` -0.373522045942315 0.152926 -2.4425 0.019347 0.009674 `Y(t-2)` -0.0751593337637079 0.143199 -0.5249 0.60273 0.301365 `Y(t-3)` 0.35885170198808 0.13368 2.6844 0.010706 0.005353 `Y(t-4)` -0.0788126508587618 0.140797 -0.5598 0.578927 0.289464 X 0.240211501967378 0.083725 2.869 0.006688 0.003344 M1 -2.86774271905553 2.432236 -1.1791 0.245706 0.122853 M2 -1.32952280976387 2.729066 -0.4872 0.628936 0.314468 M3 -14.2114323871744 2.997538 -4.741 3e-05 1.5e-05 M4 -14.3746263866844 3.897917 -3.6878 0.000705 0.000352 M5 -4.43378811719101 3.968807 -1.1172 0.27094 0.13547 M6 11.5304356672672 2.95942 3.8962 0.000384 0.000192 M7 2.77911200658021 3.337615 0.8327 0.410237 0.205119 M8 -9.08766106143715 3.568795 -2.5464 0.015058 0.007529 M9 -13.6044063583265 4.064238 -3.3473 0.001849 0.000924 M10 -9.75894265732346 3.861444 -2.5273 0.015776 0.007888 M11 0.542826991949121 2.913759 0.1863 0.853202 0.426601 t 0.118834153023582 0.032118 3.6999 0.00068 0.00034 Multiple Linear Regression - Regression Statistics Multiple R 0.965222670470673 R-squared 0.931654803590538 Adjusted R-squared 0.901079320986305 F-TEST (value) 30.4706491684797 F-TEST (DF numerator) 17 F-TEST (DF denominator) 38 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 2.53947701998751 Sum Squared Residuals 245.059854331697 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 101.9 99.1445860766064 2.75541392339363 2 106.2 106.340899427729 -0.140899427728557 3 81 80.9499037594068 0.0500962405931573 4 94.7 95.1379947253594 -0.437994725359363 5 101 104.044869076007 -3.04486907600714 6 109.4 107.963626717616 1.43637328238427 7 102.3 101.229168628792 1.07083137120781 8 90.7 91.6500207836738 -0.950020783673838 9 96.2 95.5011926458335 0.698807354166547 10 96.1 95.385369120065 0.714630879934984 11 106 104.373080790145 1.62691920985529 12 103.1 103.338815942237 -0.238815942236608 13 102 98.1536587363678 3.84634126363217 14 104.7 105.129056291149 -0.429056291148919 15 86 84.3345783873083 1.66542161269171 16 92.1 94.629248694579 -2.52924869457905 17 106.9 106.577011353510 0.322988646489924 18 112.6 110.446763446297 2.15323655370333 19 101.7 100.073728572539 1.62627142746112 20 92 94.3969047553746 -2.39690475537456 21 97.4 98.1549173870111 -0.754917387011089 22 97 96.2543357171215 0.745664282878467 23 105.4 105.406101383728 -0.0061013837281591 24 102.7 104.432742095278 -1.73274209527785 25 98.1 99.786373990274 -1.68637399027409 26 104.5 109.148850144557 -4.64885014455749 27 87.4 87.4013665053923 -0.00136650539228322 28 89.9 92.3297343907367 -2.42973439073671 29 109.8 110.540541534533 -0.740541534533216 30 111.7 113.442799112715 -1.74279911271466 31 98.6 99.997500220777 -1.39750022077703 32 96.9 99.3194647107772 -2.41946471077723 33 95.1 96.0869555025472 -0.986955502547245 34 97 96.0006625739928 0.99933742600716 35 112.7 109.584177849793 3.11582215020686 36 102.9 101.055738685319 1.84426131468113 37 97.4 100.265841223737 -2.8658412237372 38 111.4 110.462288346365 0.937711653635157 39 87.4 86.3756313521583 1.02436864784174 40 96.8 92.5378053989454 4.26219460105456 41 114.1 111.968537153527 2.13146284647340 42 110.3 111.239411449681 -0.939411449681161 43 103.9 106.405362948800 -2.50536294879989 44 101.6 99.1256301423596 2.47436985764043 45 94.6 93.5569344646082 1.04306553539179 46 95.9 98.3596325888206 -2.45963258882062 47 104.7 109.436639976334 -4.73663997633399 48 102.8 102.672703277167 0.127296722833328 49 98.1 100.149539973015 -2.04953997301452 50 113.9 109.618905790200 4.28109420979980 51 80.9 83.6385199957343 -2.73851999573432 52 95.7 94.5652167903794 1.13478320962056 53 113.2 111.869040882423 1.33095911757704 54 105.9 106.807399273692 -0.907399273691782 55 108.8 107.594239629092 1.20576037090798 56 102.3 99.0079796078148 3.2920203921852 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.134811766897739 0.269623533795478 0.865188233102261 22 0.0530113740785981 0.106022748157196 0.946988625921402 23 0.0259960279399008 0.0519920558798016 0.9740039720601 24 0.00869164270676587 0.0173832854135317 0.991308357293234 25 0.0392743555387186 0.0785487110774372 0.960725644461281 26 0.142077264177300 0.284154528354600 0.8579227358227 27 0.147039787979451 0.294079575958902 0.85296021202055 28 0.300637984065727 0.601275968131454 0.699362015934273 29 0.245606992646674 0.491213985293347 0.754393007353326 30 0.236046778514818 0.472093557029636 0.763953221485182 31 0.162639601423131 0.325279202846262 0.83736039857687 32 0.286608429185301 0.573216858370602 0.713391570814699 33 0.695066113110858 0.609867773778284 0.304933886889142 34 0.753568750524416 0.492862498951167 0.246431249475584 35 0.645597182373599 0.708805635252802 0.354402817626401 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 1 0.0666666666666667 NOK 10% type I error level 3 0.2 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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