CVM Paper: Multiple Linear Regression (monthly dummies en linear trend)

*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, 17 Dec 2009 06:14:43 -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/17/t1261055841tmv9pcl7f0zx7eb.htm/, Retrieved Thu, 17 Dec 2009 14:17:33 +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/17/t1261055841tmv9pcl7f0zx7eb.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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25.6 7.4 1.8 23.7 7.1 2.7 22 6.8 2.3 21.3 6.9 1.9 20.7 7.2 2 20.4 7.4 2.3 20.3 7.3 2.8 20.4 6.9 2.4 19.8 6.9 2.3 19.5 6.8 2.7 23.1 7.1 2.7 23.5 7.2 2.9 23.5 7.1 3 22.9 7 2.2 21.9 6.9 2.3 21.5 7.1 2.8 20.5 7.3 2.8 20.2 7.5 2.8 19.4 7.5 2.2 19.2 7.5 2.6 18.8 7.3 2.8 18.8 7 2.5 22.6 6.7 2.4 23.3 6.5 2.3 23 6.5 1.9 21.4 6.5 1.7 19.9 6.6 2 18.8 6.8 2.1 18.6 6.9 1.7 18.4 6.9 1.8 18.6 6.8 1.8 19.9 6.8 1.8 19.2 6.5 1.3 18.4 6.1 1.3 21.1 6.1 1.3 20.5 5.9 1.2 19.1 5.7 1.4 18.1 5.9 2.2 17 5.9 2.9 17.1 6.1 3.1 17.4 6.3 3.5 16.8 6.2 3.6 15.3 5.9 4.4 14.3 5.7 4.1 13.4 5.4 5.1 15.3 5.6 5.8 22.1 6.2 5.9 23.7 6.3 5.4 22.2 6 5.5 19.5 5.6 4.8 16.6 5.5 3.2 17.3 5.9 2.7 19.8 6.5 2.1 21.2 6.8 1.9 21.5 6.8 0.6 20.6 6.5 0.7 19.1 6.2 -0.2 19.6 6.2 -1 23.5 6.5 -1.7 24 6.7 -0.7

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 7 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation W<25j[t] = -0.85984552928061 + 3.54483298931266`W>25j`[t] -0.171607853890170Inflatie[t] + 0.0397278955082736M1[t] -1.12623947562415M2[t] -2.54488958002931M3[t] -3.63953232460585M4[t] -4.48059367685237M5[t] -4.92702449418643M6[t] -5.00448146757193M7[t] -4.54462317350121M8[t] -4.62640371693578M9[t] -3.97237108806821M10[t] -0.505813455539065M11[t] + 0.0313473298499503t + 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) -0.85984552928061 3.749455 -0.2293 0.819655 0.409828 `W>25j` 3.54483298931266 0.48463 7.3145 0 0 Inflatie -0.171607853890170 0.113298 -1.5147 0.13685 0.068425 M1 0.0397278955082736 0.723448 0.0549 0.95645 0.478225 M2 -1.12623947562415 0.729722 -1.5434 0.129742 0.064871 M3 -2.54488958002931 0.735711 -3.4591 0.001197 0.000598 M4 -3.63953232460585 0.716151 -5.0821 7e-06 3e-06 M5 -4.48059367685237 0.717846 -6.2417 0 0 M6 -4.92702449418643 0.729603 -6.753 0 0 M7 -5.00448146757193 0.720622 -6.9447 0 0 M8 -4.54462317350121 0.711902 -6.3838 0 0 M9 -4.62640371693578 0.71322 -6.4866 0 0 M10 -3.97237108806821 0.718493 -5.5288 2e-06 1e-06 M11 -0.505813455539065 0.710413 -0.712 0.48014 0.24007 t 0.0313473298499503 0.014382 2.1796 0.034558 0.017279 Multiple Linear Regression - Regression Statistics Multiple R 0.922272475482965 R-squared 0.850586519033475 Adjusted R-squared 0.804102324955001 F-TEST (value) 18.2984030571235 F-TEST (DF numerator) 14 F-TEST (DF denominator) 45 p-value 4.45199432874688e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.12274196540784 Sum Squared Residuals 56.7247284399538 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 25.6 25.1340996799891 0.465900320010903 2 23.7 22.7815826734116 0.918417326588387 3 22 20.3994731436187 1.60052685638134 4 21.3 19.7593041693794 1.54069583062059 5 20.7 19.9958792583876 0.704120741612372 6 20.4 20.238280012599 0.161719987401000 7 20.3 19.7518831431871 0.548116856812906 8 20.4 18.8937987129388 1.50620128706123 9 19.8 18.8605262847432 0.93947371525684 10 19.5 19.1227798029733 0.377220197026652 11 23.1 23.6841346621462 -0.584134662146236 12 23.5 24.5414571756885 -1.04145717568849 13 23.5 24.2408883167264 -0.740888316726426 14 22.9 22.8890712596248 0.0109287403751691 15 21.9 21.1301244007493 0.769875599250668 16 21.5 20.6899916569402 0.810008343059813 17 20.5 20.5892442324062 -0.0892442324061564 18 20.2 20.8831273427846 -0.68312734278458 19 19.4 20.9399824115831 -1.53998241158313 20 19.2 21.3625448939477 -2.16254489394773 21 18.8 20.5688235117225 -1.76882351172254 22 18.8 20.2422359298133 -1.44223592981331 23 22.6 22.6938517807876 -0.093851780787630 24 23.3 22.5392067537031 0.76079324629687 25 23 22.6789251206174 0.321074879382581 26 21.4 21.578626650113 -0.178626650112986 27 19.9 20.494324818322 -0.594324818321984 28 18.8 20.1228352160689 -1.32283521606891 29 18.6 19.7362476341597 -1.13624763415968 30 18.4 19.3040033612866 -0.904003361286556 31 18.6 18.9034104188197 -0.303410418819732 32 19.9 19.3946160427404 0.505383957259591 33 19.2 18.3665368593071 0.83346314069293 34 18.4 17.6339836222995 0.766016377700477 35 21.1 21.1318885846786 -0.031888584678619 36 20.5 20.9772435575941 -0.477243557594123 37 19.1 20.3050306143118 -1.20503061431178 38 18.1 19.7420908877797 -1.64209088777970 39 17 18.2346626155014 -1.23466261550137 40 17.1 17.8460122278593 -0.746012227859276 41 17.4 17.6766216617692 -0.276621661769181 42 16.8 16.8898940899648 -0.0898940899647866 43 15.3 15.6430482665233 -0.343048266523299 44 14.3 15.4767696487485 -1.17676964874849 45 13.4 14.1912786844799 -0.791278684479897 46 15.3 15.4654997433368 -0.165499743336828 47 22.1 21.0731437139145 1.02685628608549 48 23.7 22.0505917251799 1.64940827482012 49 22.2 21.0410562683553 1.15894373164472 50 19.5 18.6086285290709 0.891371470929134 51 16.6 17.1414150218087 -0.541415021808654 52 17.3 17.5818567297522 -0.281856729752217 53 19.8 19.0020072132774 0.797992786722647 54 21.2 19.6846951933651 1.51530480663492 55 21.5 19.8616757598867 1.63832424011326 56 20.6 19.2722707016246 1.3277292983754 57 19.1 18.3128346597473 0.787165340252669 58 19.6 19.135500901577 0.464499098423013 59 23.5 23.816981258473 -0.316981258473003 60 24 24.8915007878344 -0.891500787834384 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 3.81648950483327e-05 7.63297900966653e-05 0.999961835104952 19 0.032096708147092 0.064193416294184 0.967903291852908 20 0.153824221159038 0.307648442318077 0.846175778840962 21 0.131458381615298 0.262916763230597 0.868541618384701 22 0.115192954468204 0.230385908936409 0.884807045531796 23 0.0672808936404348 0.134561787280870 0.932719106359565 24 0.0531745335877493 0.106349067175499 0.94682546641225 25 0.0386331575387983 0.0772663150775966 0.961366842461202 26 0.0320427630591567 0.0640855261183134 0.967957236940843 27 0.0273410701098955 0.054682140219791 0.972658929890104 28 0.0469535691529899 0.0939071383059797 0.95304643084701 29 0.0364119849486726 0.0728239698973452 0.963588015051327 30 0.0310862938443072 0.0621725876886144 0.968913706155693 31 0.0262048094081598 0.0524096188163196 0.97379519059184 32 0.0640144152826126 0.128028830565225 0.935985584717387 33 0.0684932271440438 0.136986454288088 0.931506772855956 34 0.0633656527420648 0.126731305484130 0.936634347257935 35 0.06836541614117 0.13673083228234 0.93163458385883 36 0.326329439368213 0.652658878736426 0.673670560631787 37 0.670628876188109 0.658742247623782 0.329371123811891 38 0.644606756384774 0.710786487230452 0.355393243615226 39 0.566441097565652 0.867117804868697 0.433558902434348 40 0.484971184241663 0.969942368483326 0.515028815758337 41 0.538029654744658 0.923940690510685 0.461970345255342 42 0.484545793828947 0.969091587657894 0.515454206171053 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.04 NOK 5% type I error level 1 0.04 OK 10% type I error level 9 0.36 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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