Deterministische trend paper

*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: Tue, 30 Nov 2010 19:43:17 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Nov/30/t1291146845m8p2jk5bjwljbfn.htm/, Retrieved Tue, 30 Nov 2010 20:54:05 +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/2010/Nov/30/t1291146845m8p2jk5bjwljbfn.htm/}, year = {2010}, } @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 = {2010}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

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Dataseries X:

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13768040.14 14731798.37 17487530.67 16471559.62 16198106.13 15213975.95 17535166.38 17637387.4 16571771.60 17972385.83 16198892.67 16896235.55 16554237.93 16697955.94 19554176.37 19691579.52 15903762.33 15930700.75 18003781.65 17444615.98 18329610.38 17699369.88 16260733.42 15189796.81 14851949.20 15672722.75 18174068.44 17180794.3 18406552.23 17664893.45 18466459.42 17862884.98 16016524.60 16162288.88 17428458.32 17463628.82 17167191.42 16772112.17 19629987.60 19106861.48 17183629.01 16721314.25 18344657.85 18161267.85 19301440.71 18509941.2 18147463.68 17802737.97 16192909.22 16409869.75 18374420.60 17967742.04 20515191.95 20286602.27 18957217.20 19537280.81 16471529.53 18021889.62 18746813.27 20194317.23 19009453.59 19049596.62 19211178.55 20244720.94 20547653.75 21473302.24 19325754.03 19673603.19 20605542.58 21053177.29 20056915.06 20159479.84 16141449.72 18203628.31 20359793.22 21289464.94 19711553.27 20432335.71 15638580.70 17180395.07 14384486.00 15816786.32 13 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 5 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Invoer[t] = + 1285771.56960543 + 0.919497215404327Uitvoer[t] + 10319.8527765871t + 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) 1285771.56960543 854689.596505 1.5044 0.138007 0.069003 Uitvoer 0.919497215404327 0.047135 19.5076 0 0 t 10319.8527765871 5369.232627 1.922 0.059605 0.029802 Multiple Linear Regression - Regression Statistics Multiple R 0.932786144104306 R-squared 0.87008999063298 Adjusted R-squared 0.865531744690277 F-TEST (value) 190.882633708248 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 711613.042257083 Sum Squared Residuals 28864407948891.7 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 14731798.37 13955765.9926870 776032.377312977 2 16471559.62 17386147.0305214 -914587.41052137 3 15213975.95 16210844.6092940 -996868.65929395 4 17637387.4 17450587.6387733 186799.761226651 5 17972385.83 16575068.6740049 1397317.15599513 6 16896235.55 16242527.3889635 653708.161036486 7 16697955.94 16579586.2188172 118369.721182772 8 19691579.52 19348341.1135582 343238.406441784 9 15930700.75 16002115.4214819 -71414.6714819439 10 17444615.98 17943397.1912938 -498781.211293817 11 17699369.88 18253315.6540041 -553945.774004134 12 15189796.81 16361308.9030466 -1171512.09304655 13 15672722.75 15076255.5884276 596467.161572417 14 17180794.3 18141254.8316253 -960460.531625309 15 17664893.45 18365342.8819335 -700449.431933541 16 17862884.98 18430747.2290978 -567862.249097826 17 16162288.88 16188358.8369623 -26069.9569623104 18 17463628.82 17496947.8136144 -33318.9936143708 19 16772112.17 17267033.4793636 -494921.309363639 20 19106861.48 19541887.5617586 -435026.081758638 21 16721314.25 17302787.5031498 -581473.253149771 22 18161267.85 18380670.1413105 -219402.291310472 23 18509941.2 19270749.1696036 -760807.969603649 24 17802737.97 18219990.3566547 -417252.38665468 25 16409869.75 16433102.8261052 -23233.0761051595 26 17967742.04 18449316.3181646 -481574.278164598 27 20286602.27 20428069.4660835 -141467.196083544 28 19537280.81 19005835.8745649 531444.93543512 29 18021889.62 16730572.8364116 1291316.78358840 30 20194317.23 18833009.7523729 1361307.47762707 31 19049596.62 19084826.6480424 -35230.028042414 32 20244720.94 19280632.0398166 964088.90018345 33 21473302.24 20519837.1174501 953465.122549919 34 19673603.19 19406623.5801833 266979.60981666 35 21053177.29 20593705.4409913 459471.849008733 36 20159479.84 20099563.8168337 59916.0231663276 37 18203628.31 16509624.1924681 1694004.11753189 38 21289464.94 20398699.1471136 890765.792886368 39 20432335.71 19812964.1709514 619371.53904862 40 17180395.07 16078197.0871948 1102197.98280524 41 15816786.32 14935380.3554680 881405.964531973 42 15071819.75 14459405.8262734 612413.923726607 43 14521120.61 14886000.7710369 -364880.161036882 44 15668789.39 15799528.9276570 -130739.537656962 45 14346884.11 15012767.9900596 -665883.880059612 46 13881008.13 14430863.5565420 -549855.426541984 47 15465943.69 16194358.8202956 -728415.13029564 48 14238232.92 15328838.4774424 -1090605.55744244 49 13557713.21 13305846.3413946 251866.868605382 50 16127590.29 16687857.3201623 -560267.030162348 51 16793894.2 16578403.1714213 215491.028578742 52 16014007.43 16540909.1781918 -526901.748191792 53 16867867.15 16367071.2735348 500795.876465208 54 16014583.21 15782576.4692418 232006.740758186 55 15878594.85 16282246.3403560 -403651.490355965 56 18664899.14 19250335.8181842 -585436.6781842 57 17962530.06 17477761.8893025 484768.170697477 58 17332692.2 17513456.5182544 -180764.318254405 59 19542066.35 20154823.7446599 -612757.394659859 60 17203555.19 18162749.4358258 -959194.245825815 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.795690332014852 0.408619335970296 0.204309667985148 7 0.797300724296087 0.405398551407827 0.202699275703913 8 0.712942302455311 0.574115395089377 0.287057697544689 9 0.774853243317343 0.450293513365314 0.225146756682657 10 0.75197173661393 0.49605652677214 0.24802826338607 11 0.69946537172471 0.60106925655058 0.30053462827529 12 0.782991475769946 0.434017048460108 0.217008524230054 13 0.764367002339448 0.471265995321104 0.235632997660552 14 0.744629969532575 0.510740060934851 0.255370030467425 15 0.689816220493306 0.620367559013388 0.310183779506694 16 0.629823934622245 0.74035213075551 0.370176065377755 17 0.5517223681847 0.8965552636306 0.4482776318153 18 0.488600238266448 0.977200476532896 0.511399761733552 19 0.430918807917596 0.861837615835192 0.569081192082404 20 0.393598891581860 0.787197783163719 0.60640110841814 21 0.366728546760611 0.733457093521222 0.633271453239389 22 0.331926279903736 0.663852559807471 0.668073720096265 23 0.357472090915187 0.714944181830374 0.642527909084813 24 0.360992377495264 0.721984754990528 0.639007622504736 25 0.334710060665959 0.669420121331918 0.665289939334041 26 0.395269670193273 0.790539340386545 0.604730329806727 27 0.482334180916793 0.964668361833587 0.517665819083207 28 0.548251376600301 0.903497246799399 0.451748623399699 29 0.643283881364425 0.71343223727115 0.356716118635575 30 0.765894892599887 0.468210214800225 0.234105107400113 31 0.765129481263787 0.469741037472427 0.234870518736213 32 0.760616362122812 0.478767275754377 0.239383637877188 33 0.75930924280845 0.4813815143831 0.24069075719155 34 0.711380140773148 0.577239718453703 0.288619859226851 35 0.65559170521127 0.688816589577461 0.344408294788731 36 0.651905116103081 0.696189767793838 0.348094883896919 37 0.747403502112904 0.505192995774192 0.252596497887096 38 0.704825299797407 0.590349400405185 0.295174700202592 39 0.640420677742922 0.719158644514157 0.359579322257078 40 0.732487463982028 0.535025072035945 0.267512536017972 41 0.840459384440215 0.319081231119570 0.159540615559785 42 0.912025139206202 0.175949721587595 0.0879748607937976 43 0.912565386555537 0.174869226888927 0.0874346134444633 44 0.921032359047514 0.157935281904971 0.0789676409524857 45 0.90936449656753 0.181271006864940 0.0906355034324701 46 0.88682694754074 0.226346104918521 0.113173052459261 47 0.854412748165017 0.291174503669967 0.145587251834983 48 0.938386378884351 0.123227242231298 0.061613621115649 49 0.897828828486568 0.204342343026864 0.102171171513432 50 0.890969326597054 0.218061346805892 0.109030673402946 51 0.817583851016629 0.364832297966742 0.182416148983371 52 0.85334700860718 0.293305982785639 0.146652991392819 53 0.76059495363166 0.47881009273668 0.23940504636834 54 0.60097122550149 0.79805754899702 0.39902877449851 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 0 0 OK 10% type I error level 0 0 OK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; par2 = Do not include Seasonal 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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