Multiple Regression Analysis 3 Paper

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Title produced by software: Multiple Regression

Date of computation: Sun, 20 Dec 2009 05:23:34 -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/20/t12613138645a0dapt181sssrp.htm/, Retrieved Sun, 20 Dec 2009 13:57:57 +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/20/t12613138645a0dapt181sssrp.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}, }

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Dataseries X:

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-1,2 23,6 -2,4 25,7 0,8 32,5 -0,1 33,5 -1,5 34,5 -4,4 27,9 -4,2 45,3 3,5 40,8 10 58,5 8,6 32,5 9,5 35,5 9,9 46,7 10,4 53,2 16 36,1 12,7 54 10,2 58,1 8,9 41,8 12,6 43,1 13,6 76 14,8 42,8 9,5 41 13,7 61,4 17 34,2 14,7 53,8 17,4 80,7 9 79,5 9,1 96,5 12,2 108,3 15,9 100,1 12,9 108,5 10,9 127,4 10,6 86,5 13,2 71,4 9,6 88,2 6,4 135,6 5,8 70,5 -1 87,5 -0,2 73,3 2,7 92,2 3,6 61,1 -0,9 45,7 0,3 30,5 -1,1 34,8 -2,5 29,2 -3,4 56,7 -3,5 67,1 -3,9 41,8 -4,6 46,8 -0,1 50,1 4,3 81,9 10,2 115,8 8,7 102,5 13,3 106,6 15 101,4 20,7 136,1 20,7 143,4 26,4 127,5 31,2 113,8 31,4 75,3 26,6 98,5 26,6 113,7 19,2 103,7 6,5 73,9 3,1 52,5 -0,2 63,9 -4 44,9 -12,6 31,3 -13 24,9 -17,6 22,8 -21,7 24,8 -23,2 22,8 -16,8 20,9 -19,8 21,5

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] = -0.395712991636488 + 0.269567934887980afzetp[t] -3.79013568721123M1[t] -3.21395655816536M2[t] -6.56078146518992M3[t] -4.87045347166873M4[t] -3.97578920142145M5[t] -2.65155387116502M6[t] -7.54172565371467M7[t] -2.45587483350240M8[t] -2.04195046420927M9[t] -2.31005489925694M10[t] -0.302773237368129M11[t] -0.210015990850824t + 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) -0.395712991636488 4.145033 -0.0955 0.924268 0.462134 afzetp 0.269567934887980 0.031078 8.6738 0 0 M1 -3.79013568721123 4.676195 -0.8105 0.4209 0.21045 M2 -3.21395655816536 4.884167 -0.658 0.513075 0.256537 M3 -6.56078146518992 4.919882 -1.3335 0.187486 0.093743 M4 -4.87045347166873 4.881645 -0.9977 0.322496 0.161248 M5 -3.97578920142145 4.866318 -0.817 0.417214 0.208607 M6 -2.65155387116502 4.852244 -0.5465 0.586811 0.293405 M7 -7.54172565371467 4.889913 -1.5423 0.128347 0.064174 M8 -2.45587483350240 4.849025 -0.5065 0.614416 0.307208 M9 -2.04195046420927 4.849357 -0.4211 0.675229 0.337615 M10 -2.31005489925694 4.850378 -0.4763 0.635646 0.317823 M11 -0.302773237368129 4.841556 -0.0625 0.950347 0.475174 t -0.210015990850824 0.04853 -4.3275 5.9e-05 3e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.76096849416809 R-squared 0.57907304911645 Adjusted R-squared 0.486326432820075 F-TEST (value) 6.24360297162756 F-TEST (DF numerator) 13 F-TEST (DF denominator) 59 p-value 3.46713284971045e-07 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 8.38491643678452 Sum Squared Residuals 4148.10259545969 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 -1.2 1.96593859365773 -3.16593859365773 2 -2.4 2.89819439511755 -5.29819439511755 3 0.8 1.17441545448043 -0.374415454480434 4 -0.1 2.92429539203879 -3.02429539203879 5 -1.5 3.87851160632323 -5.37851160632323 6 -4.4 3.21358257546817 -7.61358257546817 7 -4.2 2.80387686911853 -7.00387686911853 8 3.5 6.46665599148408 -2.96665599148408 9 10 11.4419168174436 -1.44191681744361 10 8.6 3.95503008445764 4.64496991554236 11 9.5 6.56099956015958 2.93900043984042 12 9.9 9.67291767742227 0.227082322577732 13 10.4 7.42495757613207 2.97504242386793 14 16 3.18150902774265 12.8184909722573 15 12.7 4.44993416436212 8.25006583563788 16 10.2 7.0354747000732 3.1645252999268 17 8.9 3.32616564079559 5.57383435920441 18 12.6 4.79082329555557 7.80917670444443 19 13.6 8.55942057996962 5.04057942003038 20 14.8 4.48559997105015 10.3144000289499 21 9.5 4.20428606669409 5.29571393330591 22 13.7 9.22535151251037 4.47464848748963 23 17 3.69036935459531 13.3096306454047 24 14.7 9.06665812491702 5.63334187508298 25 17.4 12.3178838953416 5.08211610465838 26 9 12.3605655116711 -3.36056551167109 27 9.1 13.3863795068914 -4.28637950689136 28 12.2 18.0475931412399 -5.84759314123989 29 15.9 16.5217843545549 -0.621784354554908 30 12.9 19.9003743470195 -7.00037434701954 31 10.9 19.8950205430019 -8.99502054300188 32 10.6 13.7455268354450 -3.14552683544497 33 13.2 9.87895939707878 3.32104060292122 34 9.6 13.9295802772983 -4.32958027729833 35 6.4 28.5043660620266 -22.1043660620266 36 5.8 11.0482507473364 -5.24825074733639 37 -1 11.63075396237 -12.63075396237 38 -0.2 8.16905242515572 -8.36905242515572 39 2.7 9.70704549666316 -7.00704549666316 40 3.6 2.80379472431736 0.796205275682643 41 -0.9 -0.662903193561076 -0.237096806438924 42 0.3 -3.64611646445276 3.94611646445276 43 -1.1 -7.58716211783493 6.48716211783492 44 -2.5 -4.22090772384616 1.72090772384616 45 -3.4 3.39611886401558 -6.79611886401558 46 -3.5 5.72150496095207 -9.22150496095207 47 -3.9 0.698701879324174 -4.59870187932417 48 -4.6 2.13929880028138 -6.73929880028138 49 -0.1 -0.97127869265034 0.871278692650341 50 4.3 7.96714477498246 -3.66714477498246 51 10.2 13.5486568698096 -3.34865686980959 52 8.7 11.4437153384698 -2.74371533846983 53 13.3 13.2335921509070 0.0664078490930123 54 15 12.9460582288951 2.05394177110489 55 20.7 17.1998777961075 3.50012220389248 56 20.7 24.0435585501512 -3.34355855015123 57 26.4 19.9613367638746 6.43866323612535 58 31.2 15.7901356300108 15.4098643699892 59 31.4 7.2090358078616 24.1909641921384 60 26.6 13.5557691437800 13.0442308562200 61 26.6 13.6530500760153 12.9469499239847 62 19.2 11.3235338653305 7.87646613466948 63 6.5 -0.266431492206655 6.76643149220665 64 3.1 -4.55487329613905 7.65487329613905 65 -0.2 -0.797150559019634 0.597150559019634 66 -4 -4.80472198248564 0.80472198248564 67 -12.6 -13.5710336703626 0.971033670362639 68 -13 -10.4204336242843 -2.57956637571574 69 -17.6 -10.7826179091067 -6.81738209089328 70 -21.7 -10.7216024652293 -10.9783975347707 71 -23.2 -9.46347266396723 -13.7365273360328 72 -16.8 -9.88289449373708 -6.91710550626292 73 -19.8 -13.7213054108663 -6.07869458913365 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0269554892614231 0.0539109785228463 0.973044510738577 18 0.0104800889535189 0.0209601779070379 0.98951991104648 19 0.00665306702811855 0.0133061340562371 0.993346932971881 20 0.00229457648338451 0.00458915296676901 0.997705423516616 21 0.00346627964060500 0.00693255928120999 0.996533720359395 22 0.00632975750363037 0.0126595150072607 0.99367024249637 23 0.0037816846383959 0.0075633692767918 0.996218315361604 24 0.00240600934000037 0.00481201868000074 0.99759399066 25 0.00225257153335625 0.00450514306671250 0.997747428466644 26 0.0125587207318685 0.025117441463737 0.987441279268132 27 0.0144658177121489 0.0289316354242978 0.985534182287851 28 0.00776479685666114 0.0155295937133223 0.992235203143339 29 0.00414390456757448 0.00828780913514897 0.995856095432426 30 0.00207434706179193 0.00414869412358386 0.997925652938208 31 0.00129857855631771 0.00259715711263542 0.998701421443682 32 0.00133881227783150 0.00267762455566299 0.998661187722168 33 0.00133409674903054 0.00266819349806108 0.99866590325097 34 0.00173713520495706 0.00347427040991412 0.998262864795043 35 0.00938724341986912 0.0187744868397382 0.99061275658013 36 0.0229985690588749 0.0459971381177499 0.977001430941125 37 0.0990025231509527 0.198005046301905 0.900997476849047 38 0.128507798099518 0.257015596199035 0.871492201900482 39 0.122944689325034 0.245889378650069 0.877055310674966 40 0.0915940317226047 0.183188063445209 0.908405968277395 41 0.074262789671464 0.148525579342928 0.925737210328536 42 0.0566144801933333 0.113228960386667 0.943385519806667 43 0.0506457343918204 0.101291468783641 0.94935426560818 44 0.0733580370577546 0.146716074115509 0.926641962942245 45 0.0658328457246812 0.131665691449362 0.934167154275319 46 0.0594801834462871 0.118960366892574 0.940519816553713 47 0.0426631868431816 0.0853263736863631 0.957336813156818 48 0.0313845767372419 0.0627691534744837 0.968615423262758 49 0.0188846700397340 0.0377693400794679 0.981115329960266 50 0.0112333889602400 0.0224667779204800 0.98876661103976 51 0.0135368868317787 0.0270737736635574 0.986463113168221 52 0.0357932608962189 0.0715865217924378 0.964206739103781 53 0.105008938770074 0.210017877540148 0.894991061229926 54 0.85766776689343 0.284664466213141 0.142332233106570 55 0.87384826060029 0.252303478799418 0.126151739399709 56 0.854302040008583 0.291395919982834 0.145697959991417 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 11 0.275 NOK 5% type I error level 22 0.55 NOK 10% type I error level 26 0.65 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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