*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, 29 Dec 2010 14:00:34 +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/Dec/29/t1293631083oduu2agcf91cmi3.htm/, Retrieved Wed, 29 Dec 2010 14:58:06 +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/Dec/29/t1293631083oduu2agcf91cmi3.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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921365 18919 48873 137852 1 987921 19147 52118 145224 2 1132614 21518 60530 163575 3 1332224 20941 55644 190761 4 1418133 22401 57121 196562 5 1411549 22181 55697 204493 6 1695920 22494 56483 259479 7 1636173 21479 51541 259479 8 1539653 22322 56328 223164 9 1395314 21829 54349 194886 10 1127575 20370 59885 160407 11 1036076 18467 55806 151747 12 989236 18780 54559 152448 1 1008380 18815 55590 148388 2 1207763 20881 63442 168510 3 1368839 21443 61258 188041 4 1469798 22333 55829 192020 5 1498721 22944 58023 205250 6 1761769 22536 58887 261642 7 1653214 21658 51510 251614 8 1599104 23035 60006 222726 9 1421179 21969 60831 179039 10 1163995 20297 61559 151462 11 1037735 18564 61325 143653 12 1015407 18844 55222 143762 1 1039210 18762 56370 134580 2 1258049 21757 66063 165273 3 1469445 20501 60864 181016 4 1552346 23181 57596 189079 5 1549144 23015 57650 199266 6 1785895 22828 55324 248742 7 1662335 21597 54203 244139 8 1629440 23005 61155 219777 9 1467430 22243 63908 180679 10 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 6 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation bewegingen[t] = + 8774.99982184822 + 0.00611052995019669passagiers[t] + 0.0862069834129996cargo[t] -0.00329099758702038auto[t] -79.7458365706018maand[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) 8774.99982184822 724.031512 12.1196 0 0 passagiers 0.00611052995019669 0.000903 6.7658 0 0 cargo 0.0862069834129996 0.008944 9.6386 0 0 auto -0.00329099758702038 0.006473 -0.5084 0.612819 0.30641 maand -79.7458365706018 27.921016 -2.8561 0.005705 0.002853 Multiple Linear Regression - Regression Statistics Multiple R 0.910750546235679 R-squared 0.829466557468587 Adjusted R-squared 0.819285456421936 F-TEST (value) 81.4712037202883 F-TEST (DF numerator) 4 F-TEST (DF denominator) 67 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 790.833856282647 Sum Squared Residuals 41903018.6122731 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 18919 18084.8057138182 834.194286181835 2 19147 18667.2327355765 479.767264423454 3 21518 20136.4178568405 1381.58214315951 4 20941 20765.718522272 175.281477727997 5 22401 21319.1588406915 1081.84115930846 6 22181 21050.3216286861 1130.67837131392 7 22494 22595.0332002256 -101.033200225575 8 21479 21724.1666186935 -245.166618693527 9 22322 21586.8178383006 735.182161699386 10 21829 20547.54342884 1281.45657115999 11 20370 19422.4825799109 947.517420089065 12 18467 18460.4911171893 6.50888281074278 13 18780 18941.6709989742 -161.670998974152 14 18815 19081.1459978722 -266.145997872221 15 20881 20830.4117346745 50.5882653254647 16 21443 21482.3730947157 -39.37309471573 17 22333 21538.4276590391 794.572340960893 18 22944 21781.0149037499 1162.98509625011 19 22536 23197.5286472602 -661.528647260202 20 21658 21851.5074391109 -193.507439110941 21 23035 22268.6056963059 766.394303694115 22 21969 21316.5383912464 652.461608753579 23 20297 19818.7765443464 478.223455653643 24 18564 18973.0421623023 -409.042162302321 25 18844 19187.3305133444 -343.330513344427 26 18762 19382.2171779805 -620.217177980502 27 21757 21374.2873064648 382.712693535217 28 20501 22086.2827774693 -1585.28277746932 29 23181 22204.8462489621 976.153751037859 30 23015 22076.6642801763 938.335719823665 31 22828 23080.2496798107 -252.249679810693 32 21597 22163.9971960809 -566.99719608087 33 23005 22562.7317087007 442.268291299288 34 22243 21859.0181638921 383.981836107945 35 20729 20545.3600617243 183.639938275747 36 18310 19402.7896095436 -1092.78960954356 37 19427 19441.2353214703 -14.2353214703136 38 18849 19615.3084143449 -766.308414344876 39 21817 22157.7635218332 -340.763521833167 40 21101 22208.8018315242 -1107.8018315242 41 23546 22648.6749627336 897.32503726638 42 23456 23233.4295597203 222.570440279742 43 23649 24503.345037471 -854.345037470997 44 22432 23669.207788534 -1237.207788534 45 23745 23892.9825720283 -147.982572028294 46 23874 23319.6147031024 554.385296897642 47 22327 22207.3228055646 119.67719443541 48 20143 21204.5435072592 -1061.54350725916 49 21252 21229.7520272251 22.2479727748731 50 21094 21839.7888920226 -745.788892022597 51 21800 23484.7349864933 -1684.7349864933 52 22480 22615.601050989 -135.601050989045 53 23055 22982.6594269886 72.3405730114144 54 23352 22677.5465889002 674.453411099767 55 23171 23227.2882490188 -56.2882490187916 56 20691 22510.8718793965 -1819.87187939646 57 23183 22292.5455153154 890.454484684635 58 22412 21376.855945137 1035.14405486296 59 18958 19310.8089087932 -352.808908793192 60 17347 18068.2077497294 -721.207749729359 61 17353 17628.1041200714 -275.104120071367 62 17153 17772.2259671427 -619.225967142705 63 20141 19370.0660536409 770.933946359075 64 19699 20309.4679516824 -610.467951682433 65 20780 20275.9705658498 504.029434150223 66 21101 20363.9051103387 737.094889661273 67 20871 21676.4611262528 -805.461126252807 68 19574 20190.9499074208 -616.949907420819 69 21002 20365.9025668592 636.097433140803 70 20105 19892.5965559544 212.403444045568 71 17772 18383.1951922813 -611.195192281254 72 16117 17609.2555621185 -1492.25556211847 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.170409424107153 0.340818848214306 0.829590575892847 9 0.117448530737362 0.234897061474725 0.882551469262638 10 0.0597008261568584 0.119401652313717 0.940299173843142 11 0.0699092926789043 0.139818585357809 0.930090707321096 12 0.085143596566366 0.170287193132732 0.914856403433634 13 0.0724810401504039 0.144962080300808 0.927518959849596 14 0.0977185845945543 0.195437169189109 0.902281415405446 15 0.103929895722309 0.207859791444618 0.896070104277691 16 0.132570480593507 0.265140961187014 0.867429519406493 17 0.121955856807227 0.243911713614455 0.878044143192773 18 0.113206356664415 0.22641271332883 0.886793643335585 19 0.119502361748839 0.239004723497678 0.880497638251161 20 0.0906537445669274 0.181307489133855 0.909346255433073 21 0.0746426441950936 0.149285288390187 0.925357355804906 22 0.0813661476293051 0.16273229525861 0.918633852370695 23 0.0709628797037927 0.141925759407585 0.929037120296207 24 0.079047576689335 0.15809515337867 0.920952423310665 25 0.0908089426231899 0.18161788524638 0.90919105737681 26 0.165778471841361 0.331556943682723 0.834221528158638 27 0.136033852935008 0.272067705870016 0.863966147064992 28 0.560036894687198 0.879926210625604 0.439963105312802 29 0.545724132667496 0.908551734665007 0.454275867332504 30 0.555344063174743 0.889311873650514 0.444655936825257 31 0.522165616103261 0.955668767793477 0.477834383896739 32 0.510089255179419 0.979821489641162 0.489910744820581 33 0.492069259895424 0.984138519790848 0.507930740104576 34 0.435639180073328 0.871278360146656 0.564360819926672 35 0.403029712904504 0.806059425809007 0.596970287095496 36 0.455215235174958 0.910430470349916 0.544784764825042 37 0.439900141387686 0.879800282775372 0.560099858612314 38 0.430001388803638 0.860002777607276 0.569998611196362 39 0.362929756636935 0.72585951327387 0.637070243363065 40 0.439850736852253 0.879701473704505 0.560149263147747 41 0.482218945643221 0.964437891286442 0.517781054356779 42 0.465630257652943 0.931260515305886 0.534369742347057 43 0.423056432917137 0.846112865834274 0.576943567082863 44 0.405581804889639 0.811163609779278 0.594418195110361 45 0.337635149911586 0.675270299823172 0.662364850088414 46 0.312068945097746 0.624137890195492 0.687931054902254 47 0.317100506526178 0.634201013052357 0.682899493473822 48 0.282714293007945 0.565428586015891 0.717285706992055 49 0.288742434781176 0.577484869562352 0.711257565218824 50 0.233356595350517 0.466713190701035 0.766643404649483 51 0.376931541530094 0.753863083060188 0.623068458469906 52 0.345060542540121 0.690121085080242 0.654939457459879 53 0.352805589614377 0.705611179228754 0.647194410385623 54 0.278837741576036 0.557675483152073 0.721162258423964 55 0.238562984303126 0.477125968606253 0.761437015696874 56 0.821991037729373 0.356017924541255 0.178008962270627 57 0.751894554400153 0.496210891199693 0.248105445599847 58 0.683990358920111 0.632019282159777 0.316009641079889 59 0.608860642304125 0.78227871539175 0.391139357695875 60 0.533933462662126 0.932133074675748 0.466066537337874 61 0.426641230948471 0.853282461896943 0.573358769051529 62 0.332113649717764 0.664227299435528 0.667886350282236 63 0.275618055117996 0.551236110235993 0.724381944882004 64 0.449209486213483 0.898418972426966 0.550790513786517 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 = No Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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