Model 1 zonder seasonal dummies of linear trends

*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: Sat, 05 Dec 2009 07:33:00 -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/05/t1260023666hvzkvyl58fbmgf2.htm/, Retrieved Sat, 05 Dec 2009 15:34:38 +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/05/t1260023666hvzkvyl58fbmgf2.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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6.3 2.7 6.1 2.5 6.1 2.2 6.3 2.9 6.3 3.1 6 3 6.2 2.8 6.4 2.5 6.8 1.9 7.5 1.9 7.5 1.8 7.6 2 7.6 2.6 7.4 2.5 7.3 2.5 7.1 1.6 6.9 1.4 6.8 0.8 7.5 1.1 7.6 1.3 7.8 1.2 8 1.3 8.1 1.1 8.2 1.3 8.3 1.2 8.2 1.6 8 1.7 7.9 1.5 7.6 0.9 7.6 1.5 8.3 1.4 8.4 1.6 8.4 1.7 8.4 1.4 8.4 1.8 8.6 1.7 8.9 1.4 8.8 1.2 8.3 1 7.5 1.7 7.2 2.4 7.4 2 8.8 2.1 9.3 2 9.3 1.8 8.7 2.7 8.2 2.3 8.3 1.9 8.5 2 8.6 2.3 8.5 2.8 8.2 2.4 8.1 2.3 7.9 2.7 8.6 2.7 8.7 2.9 8.7 3 8.5 2.2 8.4 2.3 8.5 2.8 8.7 2.8

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 Werkl[t] = + 8.36169075740133 -0.262638752682671Inflatie[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) 8.36169075740133 0.366531 22.8131 0 0 Inflatie -0.262638752682671 0.175566 -1.496 0.139996 0.069998 Multiple Linear Regression - Regression Statistics Multiple R 0.191164899616969 R-squared 0.036544018845566 Adjusted R-squared 0.0202142564531180 F-TEST (value) 2.23787817405514 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 0.139995986867343 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.843238634534586 Sum Squared Residuals 41.9520322915334 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 6.3 7.65256612515811 -1.35256612515811 2 6.1 7.70509387569465 -1.60509387569465 3 6.1 7.78388550149945 -1.68388550149945 4 6.3 7.60003837462158 -1.30003837462158 5 6.3 7.54751062408505 -1.24751062408505 6 6 7.57377449935332 -1.57377449935332 7 6.2 7.62630224988985 -1.42630224988985 8 6.4 7.70509387569465 -1.30509387569465 9 6.8 7.86267712730425 -1.06267712730425 10 7.5 7.86267712730425 -0.362677127304254 11 7.5 7.88894100257252 -0.388941002572521 12 7.6 7.83641325203599 -0.236413252035987 13 7.6 7.67883000042638 -0.078830000426385 14 7.4 7.70509387569465 -0.305093875694651 15 7.3 7.70509387569465 -0.405093875694652 16 7.1 7.94146875310905 -0.841468753109055 17 6.9 7.99399650364559 -1.09399650364559 18 6.8 8.1515797552552 -1.35157975525519 19 7.5 8.0727881294504 -0.57278812945039 20 7.6 8.02026037891386 -0.420260378913856 21 7.8 8.04652425418212 -0.246524254182123 22 8 8.02026037891386 -0.0202603789138561 23 8.1 8.0727881294504 0.0272118705496095 24 8.2 8.02026037891386 0.179739621086143 25 8.3 8.04652425418212 0.253475745817878 26 8.2 7.94146875310905 0.258531246890944 27 8 7.91520487784079 0.084795122159212 28 7.9 7.96773262837732 -0.0677326283773217 29 7.6 8.12531587998692 -0.525315879986925 30 7.6 7.96773262837732 -0.367732628377322 31 8.3 7.99399650364559 0.306003496354412 32 8.4 7.94146875310905 0.458531246890945 33 8.4 7.91520487784079 0.484795122159212 34 8.4 7.99399650364559 0.406003496354411 35 8.4 7.88894100257252 0.51105899742748 36 8.6 7.91520487784079 0.684795122159212 37 8.9 7.99399650364559 0.906003496354411 38 8.8 8.04652425418212 0.753475745817877 39 8.3 8.09905200471866 0.200947995281343 40 7.5 7.91520487784079 -0.415204877840788 41 7.2 7.73135775096292 -0.531357750962919 42 7.4 7.83641325203599 -0.436413252035987 43 8.8 7.81014937676772 0.98985062323228 44 9.3 7.83641325203599 1.46358674796401 45 9.3 7.88894100257252 1.41105899742748 46 8.7 7.65256612515812 1.04743387484188 47 8.2 7.75762162623119 0.442378373768813 48 8.3 7.86267712730425 0.437322872695747 49 8.5 7.83641325203599 0.663586747964013 50 8.6 7.75762162623119 0.842378373768814 51 8.5 7.62630224988985 0.87369775011015 52 8.2 7.73135775096292 0.468642249037081 53 8.1 7.75762162623119 0.342378373768814 54 7.9 7.65256612515812 0.247433874841883 55 8.6 7.65256612515812 0.947433874841882 56 8.7 7.60003837462158 1.09996162537842 57 8.7 7.57377449935332 1.12622550064668 58 8.5 7.78388550149945 0.716114498500547 59 8.4 7.75762162623119 0.642378373768814 60 8.5 7.62630224988985 0.87369775011015 61 8.7 7.62630224988985 1.07369775011015 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.000915921582838224 0.00183184316567645 0.999084078417162 6 0.00341541013459013 0.00683082026918027 0.99658458986541 7 0.000880612172794921 0.00176122434558984 0.999119387827205 8 0.000964943562243102 0.00192988712448620 0.999035056437757 9 0.00493473627427877 0.00986947254855754 0.995065263725721 10 0.0655840983506843 0.131168196701369 0.934415901649316 11 0.0725569919738215 0.145113983947643 0.927443008026179 12 0.105867682088114 0.211735364176227 0.894132317911886 13 0.362961072586656 0.725922145173313 0.637038927413344 14 0.465743354914726 0.931486709829451 0.534256645085274 15 0.542294095693169 0.915411808613661 0.457705904306831 16 0.564205040081485 0.87158991983703 0.435794959918515 17 0.670460870493706 0.659078259012587 0.329539129506294 18 0.813932215527736 0.372135568944529 0.186067784472264 19 0.792401764985612 0.415196470028776 0.207598235014388 20 0.7815420904688 0.436915819062399 0.218457909531199 21 0.765080682719726 0.469838634560548 0.234919317280274 22 0.767162787509494 0.465674424981013 0.232837212490507 23 0.748226493551525 0.503547012896949 0.251773506448475 24 0.753285996570436 0.493428006859127 0.246714003429564 25 0.749600785608527 0.500798428782945 0.250399214391473 26 0.76604043839473 0.46791912321054 0.23395956160527 27 0.761772279560462 0.476455440879077 0.238227720439538 28 0.732931133018293 0.534137733963414 0.267068866981707 29 0.74266290383136 0.514674192337281 0.257337096168640 30 0.763179534561672 0.473640930876657 0.236820465438328 31 0.753420575072914 0.493158849854173 0.246579424927086 32 0.76979413240304 0.46041173519392 0.23020586759696 33 0.785333602657663 0.429332794684674 0.214666397342337 34 0.76276703060117 0.47446593879766 0.23723296939883 35 0.77171856782162 0.456562864356761 0.228281432178380 36 0.78727709373148 0.425445812537039 0.212722906268519 37 0.820071978001223 0.359856043997555 0.179928021998777 38 0.821117403638192 0.357765192723616 0.178882596361808 39 0.765590838281073 0.468818323437853 0.234409161718927 40 0.800926897186427 0.398146205627147 0.199073102813573 41 0.934062898516489 0.131874202967022 0.0659371014835111 42 0.990377263666324 0.0192454726673530 0.00962273633367652 43 0.99248955323257 0.0150208935348605 0.00751044676743024 44 0.998934036337923 0.00213192732415369 0.00106596366207684 45 0.99998504390682 2.99121863621365e-05 1.49560931810683e-05 46 0.999983120367513 3.37592649739792e-05 1.68796324869896e-05 47 0.999961910410402 7.61791791951082e-05 3.80895895975541e-05 48 0.999879584048143 0.000240831903713949 0.000120415951856974 49 0.999751949679133 0.000496100641734812 0.000248050320867406 50 0.999628118544116 0.00074376291176752 0.00037188145588376 51 0.998992874714743 0.00201425057051404 0.00100712528525702 52 0.997240526883763 0.00551894623247407 0.00275947311623704 53 0.9940266024018 0.0119467951963983 0.00597339759819916 54 0.999969288085103 6.14238297947148e-05 3.07119148973574e-05 55 0.999703071344837 0.000593857310325171 0.000296928655162585 56 0.997619192752246 0.00476161449550861 0.00238080724775431 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 17 0.326923076923077 NOK 5% type I error level 20 0.384615384615385 NOK 10% type I error level 20 0.384615384615385 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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