Paper Multiple Regression monthly dummies, lineaire trend en autoregressie van 4 lags

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Title produced by software: Multiple Regression

Date of computation: Thu, 17 Dec 2009 10:01:01 -0700

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/17/t1261069329sg7poeombh667q8.htm/, Retrieved Thu, 17 Dec 2009 18:02:21 +0100

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@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/t1261069329sg7poeombh667q8.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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8.4 1.58 8.4 8.4 8.3 7.6 8.4 1.86 8.4 8.4 8.4 8.3 8.6 1.74 8.4 8.4 8.4 8.4 8.9 1.59 8.6 8.4 8.4 8.4 8.8 1.26 8.9 8.6 8.4 8.4 8.3 1.13 8.8 8.9 8.6 8.4 7.5 1.92 8.3 8.8 8.9 8.6 7.2 2.61 7.5 8.3 8.8 8.9 7.4 2.26 7.2 7.5 8.3 8.8 8.8 2.41 7.4 7.2 7.5 8.3 9.3 2.26 8.8 7.4 7.2 7.5 9.3 2.03 9.3 8.8 7.4 7.2 8.7 2.86 9.3 9.3 8.8 7.4 8.2 2.55 8.7 9.3 9.3 8.8 8.3 2.27 8.2 8.7 9.3 9.3 8.5 2.26 8.3 8.2 8.7 9.3 8.6 2.57 8.5 8.3 8.2 8.7 8.5 3.07 8.6 8.5 8.3 8.2 8.2 2.76 8.5 8.6 8.5 8.3 8.1 2.51 8.2 8.5 8.6 8.5 7.9 2.87 8.1 8.2 8.5 8.6 8.6 3.14 7.9 8.1 8.2 8.5 8.7 3.11 8.6 7.9 8.1 8.2 8.7 3.16 8.7 8.6 7.9 8.1 8.5 2.47 8.7 8.7 8.6 7.9 8.4 2.57 8.5 8.7 8.7 8.6 8.5 2.89 8.4 8.5 8.7 8.7 8.7 2.63 8.5 8.4 8.5 8.7 8.7 2.38 8.7 8.5 8.4 8.5 8.6 1.69 8.7 8.7 8.5 8.4 8.5 1.96 8.6 8.7 8.7 8.5 8.3 2.19 8.5 8.6 8.7 8.7 8 1.87 8.3 8.5 8.6 8.7 8.2 1.6 8 8.3 8.5 8.6 8.1 1.63 8.2 8 8.3 8.5 8.1 1.22 8.1 8.2 8 8.3 8 1.21 8.1 8.1 8.2 8 7.9 1.49 8 8.1 8.1 8.2 7.9 1.64 7.9 8 8.1 8.1 8 1.66 7.9 7.9 8 8.1 8 1.7 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.881777647983385 -0.0113231086563806X[t] + 1.63806464568807Y1[t] -1.00286926978010Y2[t] + 0.0145595834654874Y3[t] + 0.251295000293058Y4[t] + 0.0625666023511709M1[t] -0.00068203250343602M2[t] + 0.149463960484866M3[t] + 0.0540894234403781M4[t] -0.176549045058484M5[t] -0.06307426185533M6[t] -0.0851874528307616M7[t] -0.113415631987668M8[t] -0.0279778267138532M9[t] + 0.625142574401157M10[t] -0.521386048081949M11[t] -0.00183070169537278t + 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) 0.881777647983385 0.698477 1.2624 0.212537 0.106268 X -0.0113231086563806 0.019941 -0.5678 0.572649 0.286325 Y1 1.63806464568807 0.142074 11.5297 0 0 Y2 -1.00286926978010 0.277449 -3.6146 0.000688 0.000344 Y3 0.0145595834654874 0.27799 0.0524 0.958435 0.479217 Y4 0.251295000293058 0.144383 1.7405 0.087807 0.043903 M1 0.0625666023511709 0.214197 0.2921 0.771396 0.385698 M2 -0.00068203250343602 0.165888 -0.0041 0.996736 0.498368 M3 0.149463960484866 0.170434 0.877 0.384619 0.19231 M4 0.0540894234403781 0.173733 0.3113 0.756814 0.378407 M5 -0.176549045058484 0.152101 -1.1607 0.251153 0.125576 M6 -0.06307426185533 0.141031 -0.4472 0.656598 0.328299 M7 -0.0851874528307616 0.170104 -0.5008 0.618669 0.309335 M8 -0.113415631987668 0.164323 -0.6902 0.493199 0.2466 M9 -0.0279778267138532 0.165785 -0.1688 0.866654 0.433327 M10 0.625142574401157 0.165394 3.7797 0.000413 0.000206 M11 -0.521386048081949 0.224014 -2.3275 0.023946 0.011973 t -0.00183070169537278 0.002 -0.9151 0.364429 0.182215 Multiple Linear Regression - Regression Statistics Multiple R 0.977655491510764 R-squared 0.955810260081153 Adjusted R-squared 0.94108034677487 F-TEST (value) 64.8890621558174 F-TEST (DF numerator) 17 F-TEST (DF denominator) 51 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.172370906703546 Sum Squared Residuals 1.51529820336793 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.4 8.29095073957985 0.109049260420154 2 8.4 8.40006339115775 -6.33911577450543e-05 3 8.6 8.57486695551875 0.0251330444812547 4 8.9 8.80697311221495 0.0930268877850465 5 8.8 8.86908610762773 -0.0690861076277269 6 8.3 8.5204468644511 -0.220446864451098 7 7.5 7.82343919517399 -0.323439195173987 8 7.2 7.05048282942977 0.149517170570232 9 7.4 7.41651975139356 -0.0165197513935551 10 8.8 8.55728952766746 0.242710472332539 11 9.3 9.29794144452062 0.00205855547937821 12 9.3 9.16263986765524 0.137360132344761 13 8.7 8.78318537014649 -0.083185370146486 14 8.2 8.09787020201016 0.102129797989835 15 8.3 8.15769270289744 0.142307297102564 16 8.5 8.7171060446237 -0.217106044623708 17 8.6 8.55039592099702 0.0496040790029831 18 8.5 8.49541951698941 0.00458048301058582 19 8.2 8.23893381317768 -0.038933813177676 20 8.1 7.87228820116624 0.227711798833760 21 7.9 8.11254684367637 -0.212546843676366 22 8.6 8.50395592653023 0.0960440734697732 23 8.7 8.62631094311464 0.0736890568853605 24 8.7 8.57905669306873 0.120943306931270 25 8.5 8.50725132008665 -0.00725132008665006 26 8.4 8.2907892020851 0.109210797914892 27 8.5 8.49737798802452 0.00262201197548507 28 8.7 8.66429823238903 0.0357017676109676 29 8.7 8.61027088311334 0.0897291168866645 30 8.6 8.50548051395524 0.0945194860447574 31 8.5 8.34271433410081 0.157285665899188 32 8.3 8.29679060072538 0.00320939927462011 33 8 8.15523913856771 -0.155239138567712 34 8.2 8.49215507919832 -0.292155079198317 35 8.1 7.94388835510939 0.156111644890614 36 8.1 8.049078982422 0.0509210175780046 37 8 8.13773845774755 -0.137738457747547 38 7.9 7.95448522791704 -0.0544852279170366 39 7.9 8.0124530152914 -0.112453015291406 40 8 8.01385228300988 -0.0138522830098793 41 8 7.9173585770564 0.0826414229436056 42 7.9 7.90302007612404 -0.00302007612404089 43 8 7.71717860157723 0.282821398422765 44 7.7 7.98200416662927 -0.282004166629268 45 7.2 7.47233576009017 -0.272335760090167 46 7.5 7.58087452722453 -0.0808745272245331 47 7.3 7.44896163379628 -0.148961633796279 48 7 7.25295896591456 -0.252958965914562 49 7 6.89330383239392 0.106696167606080 50 7 7.19363568411332 -0.193635684113323 51 7.2 7.28562563400954 -0.0856256340095376 52 7.3 7.43645527411651 -0.136455274116513 53 7.1 7.16518055497692 -0.0651805549769159 54 6.8 6.84334436556988 -0.0433443655698851 55 6.4 6.58298743763137 -0.182987437631372 56 6.1 6.20877856759834 -0.108778567598342 57 6.5 6.14080647617688 0.359193523823121 58 7.7 7.66572493937946 0.0342750606205383 59 7.9 7.98289762345907 -0.082897623459073 60 7.5 7.55626549093947 -0.0562654909394732 61 6.9 6.88757028004555 0.0124297199544479 62 6.6 6.56315629271662 0.0368437072833785 63 6.9 6.87198370425836 0.0280162957416406 64 7.7 7.46131505364591 0.238684946354086 65 8 8.08770795622861 -0.0877079562286104 66 8 7.83228866291032 0.167711337089681 67 7.7 7.59474661833892 0.105253381661081 68 7.3 7.289655634451 0.0103443655489979 69 7.4 7.10255203009532 0.297447969904679 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.04566635979571 0.09133271959142 0.95433364020429 22 0.105742228334724 0.211484456669448 0.894257771665276 23 0.0786259251576242 0.157251850315248 0.921374074842376 24 0.0433719751061164 0.0867439502122328 0.956628024893884 25 0.0864706209825645 0.172941241965129 0.913529379017435 26 0.0808870886838879 0.161774177367776 0.919112911316112 27 0.122869210648916 0.245738421297832 0.877130789351084 28 0.0750158587633661 0.150031717526732 0.924984141236634 29 0.0463512142068235 0.0927024284136469 0.953648785793177 30 0.0328825192601484 0.0657650385202967 0.967117480739852 31 0.0514249735843737 0.102849947168747 0.948575026415626 32 0.0462500244185428 0.0925000488370856 0.953749975581457 33 0.0314691877507664 0.0629383755015328 0.968530812249234 34 0.204268378089120 0.408536756178241 0.79573162191088 35 0.190265568576239 0.380531137152478 0.809734431423761 36 0.185644792195733 0.371289584391466 0.814355207804267 37 0.196323382145476 0.392646764290953 0.803676617854524 38 0.194226864948853 0.388453729897706 0.805773135051147 39 0.166009717660905 0.332019435321811 0.833990282339095 40 0.110854125432947 0.221708250865893 0.889145874567053 41 0.110377251582977 0.220754503165954 0.889622748417023 42 0.0693302693153852 0.138660538630770 0.930669730684615 43 0.659478798579901 0.681042402840198 0.340521201420099 44 0.880728325877438 0.238543348245124 0.119271674122562 45 0.811740295760554 0.376519408478892 0.188259704239446 46 0.733453582026012 0.533092835947975 0.266546417973988 47 0.618732210915877 0.762535578168246 0.381267789084123 48 0.540126973931229 0.919746052137542 0.459873026068771 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 6 0.214285714285714 NOK

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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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