Paper - Multiple Regression VOORUITZICHTEN WERKLOOSHEID 3

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R Software Module: rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Sun, 07 Dec 2008 10:15:24 -0700

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/07/t1228670216enkjuen1drm93bz.htm/, Retrieved Sun, 07 Dec 2008 17:16:56 +0000

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/2008/Dec/07/t1228670216enkjuen1drm93bz.htm/}, year = {2008}, } @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 = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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Dataseries X:

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41 0 35 0 34 0 36 0 39 0 40 0 30 0 33 0 30 0 32 0 41 0 40 0 41 0 40 0 39 0 34 0 34 0 46 0 45 0 44 0 40 0 39 0 37 0 39 0 35 0 26 0 26 0 33 0 27 0 30 0 26 0 27 0 18 0 19 0 13 0 14 0 41 0 21 0 16 0 17 0 9 0 14 0 14 0 16 0 11 0 10 0 6 0 9 0 5 0 7 0 2 0 0 0 8 0 13 0 11 0 19 1 23 1 23 1 43 1 59 1

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 6 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Wer[t] = + 51.8842105263158 + 28.6842105263158Val[t] -1.63070175438597M1[t] -7.72456140350877M2[t] -9.41842105263158M3[t] -8.11228070175438M4[t] -8.00614035087719M5[t] -2.1M6[t] -4.79385964912281M7[t] -7.22456140350877M8[t] -9.91842105263158M9[t] -9.01228070175438M10[t] -4.90614035087719M11[t] -0.706140350877193t + 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) 51.8842105263158 4.421039 11.7358 0 0 Val 28.6842105263158 4.730797 6.0633 0 0 M1 -1.63070175438597 5.40248 -0.3018 0.764132 0.382066 M2 -7.72456140350877 5.398163 -1.431 0.1592 0.0796 M3 -9.41842105263158 5.394803 -1.7458 0.087516 0.043758 M4 -8.11228070175438 5.392402 -1.5044 0.139316 0.069658 M5 -8.00614035087719 5.39096 -1.4851 0.144337 0.072169 M6 -2.1 5.39048 -0.3896 0.698648 0.349324 M7 -4.79385964912281 5.39096 -0.8892 0.378501 0.18925 M8 -7.22456140350877 5.332115 -1.3549 0.182062 0.091031 M9 -9.91842105263158 5.328713 -1.8613 0.069097 0.034548 M10 -9.01228070175438 5.326282 -1.692 0.097403 0.048701 M11 -4.90614035087719 5.324823 -0.9214 0.361664 0.180832 t -0.706140350877193 0.071977 -9.8107 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.83676531450093 R-squared 0.700176191551841 Adjusted R-squared 0.615443376120839 F-TEST (value) 8.26334151639277 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 3.17781719827437e-08 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 8.4185150638938 Sum Squared Residuals 3260.08421052632 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 41 49.5473684210527 -8.54736842105267 2 35 42.7473684210526 -7.74736842105264 3 34 40.3473684210526 -6.34736842105263 4 36 40.9473684210526 -4.94736842105263 5 39 40.3473684210526 -1.34736842105264 6 40 45.5473684210526 -5.54736842105263 7 30 42.1473684210526 -12.1473684210526 8 33 39.0105263157895 -6.01052631578947 9 30 35.6105263157895 -5.61052631578947 10 32 35.8105263157895 -3.81052631578947 11 41 39.2105263157895 1.78947368421053 12 40 43.4105263157895 -3.41052631578947 13 41 41.0736842105263 -0.0736842105263049 14 40 34.2736842105263 5.72631578947369 15 39 31.8736842105263 7.12631578947368 16 34 32.4736842105263 1.52631578947369 17 34 31.8736842105263 2.12631578947368 18 46 37.0736842105263 8.92631578947369 19 45 33.6736842105263 11.3263157894737 20 44 30.5368421052632 13.4631578947368 21 40 27.1368421052632 12.8631578947368 22 39 27.3368421052632 11.6631578947368 23 37 30.7368421052632 6.26315789473685 24 39 34.9368421052632 4.06315789473684 25 35 32.6 2.40000000000001 26 26 25.8 0.200000000000001 27 26 23.4 2.6 28 33 24 9 29 27 23.4 3.6 30 30 28.6 1.40000000000000 31 26 25.2 0.799999999999998 32 27 22.0631578947368 4.93684210526316 33 18 18.6631578947368 -0.663157894736844 34 19 18.8631578947368 0.136842105263159 35 13 22.2631578947368 -9.26315789473684 36 14 26.4631578947368 -12.4631578947368 37 41 24.1263157894737 16.8736842105263 38 21 17.3263157894737 3.67368421052631 39 16 14.9263157894737 1.07368421052631 40 17 15.5263157894737 1.47368421052631 41 9 14.9263157894737 -5.92631578947368 42 14 20.1263157894737 -6.12631578947369 43 14 16.7263157894737 -2.72631578947368 44 16 13.5894736842105 2.41052631578947 45 11 10.1894736842105 0.810526315789464 46 10 10.3894736842105 -0.389473684210532 47 6 13.7894736842105 -7.78947368421053 48 9 17.9894736842105 -8.98947368421053 49 5 15.6526315789474 -10.6526315789474 50 7 8.85263157894737 -1.85263157894737 51 2 6.45263157894737 -4.45263157894737 52 0 7.05263157894737 -7.05263157894737 53 8 6.45263157894737 1.54736842105263 54 13 11.6526315789474 1.34736842105263 55 11 8.25263157894737 2.74736842105263 56 19 33.8 -14.8 57 23 30.4 -7.4 58 23 30.6 -7.6 59 43 34 9 60 59 38.2 20.8 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0948721726336936 0.189744345267387 0.905127827366306 18 0.0516726527962453 0.103345305592491 0.948327347203755 19 0.120813056720715 0.241626113441429 0.879186943279285 20 0.0950457612117444 0.190091522423489 0.904954238788256 21 0.0655916495308173 0.131183299061635 0.934408350469183 22 0.0376738131051856 0.0753476262103713 0.962326186894814 23 0.0358949101148537 0.0717898202297075 0.964105089885146 24 0.0216242892804407 0.0432485785608815 0.97837571071956 25 0.0312863156602796 0.0625726313205592 0.96871368433972 26 0.0600876200698153 0.120175240139631 0.939912379930185 27 0.0614316377551785 0.122863275510357 0.938568362244822 28 0.0389695959219332 0.0779391918438664 0.961030404078067 29 0.0308913738926161 0.0617827477852321 0.969108626107384 30 0.0294102691351343 0.0588205382702685 0.970589730864866 31 0.0218858594939977 0.0437717189879954 0.978114140506002 32 0.0197665077831060 0.0395330155662121 0.980233492216894 33 0.0203132505027394 0.0406265010054788 0.97968674949726 34 0.0184109610413409 0.0368219220826819 0.981589038958659 35 0.0345002131534853 0.0690004263069706 0.965499786846515 36 0.0767585477770433 0.153517095554087 0.923241452222957 37 0.171582169706688 0.343164339413377 0.828417830293312 38 0.118205198665175 0.236410397330350 0.881794801334825 39 0.0845388337927263 0.169077667585453 0.915461166207274 40 0.0734804648232308 0.146960929646462 0.92651953517677 41 0.0504872462512742 0.100974492502548 0.949512753748726 42 0.0305616628786126 0.0611233257572253 0.969438337121387 43 0.0133884288567300 0.0267768577134599 0.98661157114327 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 6 0.222222222222222 NOK 10% type I error level 14 0.518518518518518 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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