Paper/6

*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: Wed, 30 Dec 2009 06:24:30 -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/30/t126217952467ukxb2f0r43v74.htm/, Retrieved Wed, 30 Dec 2009 14:25:36 +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/30/t126217952467ukxb2f0r43v74.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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10519,20 1154,80 10414,90 1206,70 12476,80 1199,00 12384,60 1265,00 12266,70 1247,10 12919,90 1116,50 11497,30 1153,90 12142,00 1077,40 13919,40 1132,50 12656,80 1058,80 12034,10 1195,10 13199,70 1263,40 10881,30 1023,10 11301,20 1141,00 13643,90 1116,30 12517,00 1135,60 13981,10 1210,50 14275,70 1230,00 13425,00 1136,50 13565,70 1068,70 16216,30 1372,50 12970,00 1049,90 14079,90 1302,20 14235,00 1305,90 12213,40 1173,50 12581,00 1277,40 14130,40 1238,60 14210,80 1508,60 14378,50 1423,40 13142,80 1375,10 13714,70 1344,10 13621,90 1287,50 15379,80 1446,90 13306,30 1451,00 14391,20 1604,40 14909,90 1501,50 14025,40 1522,80 12951,20 1328,00 14344,30 1420,50 16093,40 1648,00 15413,60 1631,10 14705,70 1396,60 15972,80 1663,40 16241,40 1283,00 16626,40 1582,40 17136,20 1785,20 15622,90 1853,60 18003,90 1994,10 16136,10 2042,80 14423,70 1586,10 16789,40 1942,40 16782,20 1763,60 14133,80 1819,90 12607,00 1836,00 12004,50 1447,50 12175,40 1509,50 13268,00 1661,20 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation InvoerEU[t] = + 7568.58743833104 + 4.42326955870858InvoerAM[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) 7568.58743833104 926.758979 8.1667 0 0 InvoerAM 4.42326955870858 0.651658 6.7877 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.662182690045599 R-squared 0.438485914996026 Adjusted R-squared 0.428968727114602 F-TEST (value) 46.0730544000199 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 6.18214590630828e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1291.65973556373 Sum Squared Residuals 98434707.476118 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 10519.2 12676.5791247278 -2157.37912472776 2 10414.9 12906.1468148247 -2491.2468148247 3 12476.8 12872.0876392226 -395.287639222641 4 12384.6 13164.0234300974 -779.423430097407 5 12266.7 13084.8469049965 -818.146904996522 6 12919.9 12507.1679006292 412.732099370817 7 11497.3 12672.5981821249 -1175.29818212488 8 12142 12334.2180608837 -192.218060883677 9 13919.4 12577.9402135685 1341.45978643148 10 12656.8 12251.9452470917 404.854752908302 11 12034.1 12854.8368879437 -820.736887943676 12 13199.7 13156.9461988035 42.7538011965271 13 10881.3 12094.0345238458 -1212.73452384580 14 11301.2 12615.5380048175 -1314.33800481754 15 13643.9 12506.2832467174 1137.61675328256 16 12517 12591.6523492005 -74.652349200516 17 13981.1 12922.9552391478 1058.14476085221 18 14275.7 13009.2089955426 1266.49100445739 19 13425 12595.6332918034 829.366708196646 20 13565.7 12295.7356157229 1269.96438427709 21 16216.3 13639.5249076586 2576.77509234142 22 12970 12212.5781480192 757.421851980809 23 14079.9 13328.5690576814 751.330942318633 24 14235 13344.9351550486 890.064844951411 25 12213.4 12759.2942654756 -545.894265475572 26 12581 13218.8719726254 -637.871972625394 27 14130.4 13047.2491137475 1083.1508862525 28 14210.8 14241.5318945988 -30.7318945988184 29 14378.5 13864.6693281968 513.830671803153 30 13142.8 13651.0254085112 -508.225408511222 31 13714.7 13513.9040521913 200.795947808745 32 13621.9 13263.5469951683 358.35300483165 33 15379.8 13968.6161628265 1411.1838371735 34 13306.3 13986.7515680172 -680.451568017204 35 14391.2 14665.2811183231 -274.0811183231 36 14909.9 14210.126680732 699.773319268013 37 14025.4 14304.3423223325 -278.94232233248 38 12951.2 13442.6894122960 -491.489412296047 39 14344.3 13851.8418464766 492.458153523408 40 16093.4 14858.1356710828 1235.26432891721 41 15413.6 14783.3824155406 630.217584459381 42 14705.7 13746.1257040235 959.574295976545 43 15972.8 14926.2540222869 1046.54597771309 44 16241.4 13243.6422821542 2997.75771784584 45 16626.4 14567.9691880315 2058.43081196849 46 17136.2 15465.0082545376 1671.19174546239 47 15622.9 15767.5598923533 -144.659892353279 48 18003.9 16389.0292653518 1614.87073464817 49 16136.1 16604.4424928609 -468.342492860942 50 14423.7 14584.3352853987 -160.635285398732 51 16789.4 16160.3462291666 629.053770833399 52 16782.2 15369.4656320695 1412.73436793049 53 14133.8 15618.4957082248 -1484.6957082248 54 12607 15689.71034812 -3082.71034812001 55 12004.5 13971.2701245617 -1966.77012456172 56 12175.4 14245.5128372017 -2070.11283720166 57 13268 14916.5228292577 -1648.52282925775 58 12299.3 14009.7525697225 -1710.45256972249 59 11800.6 13367.0515028421 -1566.45150284213 60 13873.3 14389.7114248156 -516.411424815556 61 12315 14370.2490387572 -2055.24903875724 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.326175140676064 0.652350281352128 0.673824859323936 6 0.534070915520423 0.931858168959155 0.465929084479577 7 0.401816708303396 0.803633416606791 0.598183291696604 8 0.287607484593559 0.575214969187119 0.71239251540644 9 0.445179614459619 0.890359228919238 0.554820385540381 10 0.33465472283468 0.66930944566936 0.66534527716532 11 0.245869328060575 0.49173865612115 0.754130671939425 12 0.245618356912773 0.491236713825545 0.754381643087227 13 0.256074964728643 0.512149929457285 0.743925035271357 14 0.228795681703035 0.457591363406071 0.771204318296965 15 0.270784672129885 0.541569344259769 0.729215327870115 16 0.207299707821712 0.414599415643424 0.792700292178288 17 0.247459772544129 0.494919545088257 0.752540227455871 18 0.294089945331733 0.588179890663466 0.705910054668267 19 0.265089855301131 0.530179710602261 0.73491014469887 20 0.260600445212913 0.521200890425826 0.739399554787087 21 0.489149109079979 0.978298218159959 0.510850890920021 22 0.446869924785981 0.893739849571963 0.553130075214019 23 0.38423093948662 0.76846187897324 0.61576906051338 24 0.330906097243457 0.661812194486915 0.669093902756543 25 0.275577607015875 0.55115521403175 0.724422392984125 26 0.237547571427433 0.475095142854866 0.762452428572567 27 0.212498646061855 0.42499729212371 0.787501353938145 28 0.168779141865256 0.337558283730512 0.831220858134744 29 0.127862041890102 0.255724083780204 0.872137958109898 30 0.101163133663615 0.202326267327229 0.898836866336385 31 0.0716733943861838 0.143346788772368 0.928326605613816 32 0.0502485726768511 0.100497145353702 0.949751427323149 33 0.0485551691804073 0.0971103383608145 0.951444830819593 34 0.0396619391926733 0.0793238783853466 0.960338060807327 35 0.0280287580134684 0.0560575160269368 0.971971241986532 36 0.0197742701251151 0.0395485402502303 0.980225729874885 37 0.0129644253084185 0.025928850616837 0.987035574691582 38 0.00838595762778336 0.0167719152555667 0.991614042372217 39 0.00528327720040407 0.0105665544008081 0.994716722799596 40 0.00441830106147293 0.00883660212294586 0.995581698938527 41 0.00273057139608923 0.00546114279217847 0.99726942860391 42 0.00219287539688033 0.00438575079376065 0.99780712460312 43 0.00163626003662666 0.00327252007325332 0.998363739963373 44 0.0550698370041066 0.110139674008213 0.944930162995893 45 0.181691544100356 0.363383088200711 0.818308455899645 46 0.273292620113857 0.546585240227715 0.726707379886143 47 0.226507032428549 0.453014064857098 0.773492967571451 48 0.282181732255185 0.564363464510369 0.717818267744815 49 0.240550768872006 0.481101537744012 0.759449231127994 50 0.224394866253691 0.448789732507382 0.775605133746309 51 0.236525922576802 0.473051845153603 0.763474077423198 52 0.89519058690416 0.20961882619168 0.10480941309584 53 0.901222748413654 0.197554503172692 0.098777251586346 54 0.943293869526708 0.113412260946585 0.0567061304732924 55 0.896734926623768 0.206530146752464 0.103265073376232 56 0.835869837506683 0.328260324986633 0.164130162493317 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.0769230769230769 NOK 5% type I error level 8 0.153846153846154 NOK 10% type I error level 11 0.211538461538462 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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