WS7-2

*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: Fri, 20 Nov 2009 02:43:44 -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/Nov/20/t1258710298l8w6f17ag8ht0i5.htm/, Retrieved Fri, 20 Nov 2009 10:45:10 +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/Nov/20/t1258710298l8w6f17ag8ht0i5.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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10.9 96.8 10 114.1 9.2 110.3 9.2 103.9 9.5 101.6 9.6 94.6 9.5 95.9 9.1 104.7 8.9 102.8 9 98.1 10.1 113.9 10.3 80.9 10.2 95.7 9.6 113.2 9.2 105.9 9.3 108.8 9.4 102.3 9.4 99 9.2 100.7 9 115.5 9 100.7 9 109.9 9.8 114.6 10 85.4 9.8 100.5 9.3 114.8 9 116.5 9 112.9 9.1 102 9.1 106 9.1 105.3 9.2 118.8 8.8 106.1 8.3 109.3 8.4 117.2 8.1 92.5 7.7 104.2 7.9 112.5 7.9 122.4 8 113.3 7.9 100 7.6 110.7 7.1 112.8 6.8 109.8 6.5 117.3 6.9 109.1 8.2 115.9 8.7 96 8.3 99.8 7.9 116.8 7.5 115.7 7.8 99.4 8.3 94.3 8.4 91 8.2 93.2 7.7 103.1 7.2 94.1 7.3 91.8 8.1 102.7 8.5 82.6

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] = + 11.6689654284348 -0.0291376935120574X[t] + 0.607321306663722M1[t] + 0.600890186123138M2[t] + 0.217393662901691M3[t] + 0.127998655073318M4[t] + 0.0859694305114406M5[t] + 0.0723797230840934M6[t] -0.0891585214799912M7[t] -0.0927468185738858M8[t] -0.5528177644784M9[t] -0.549134872845152M10[t] + 0.539514661336017M11[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) 11.6689654284348 1.947937 5.9904 0 0 X -0.0291376935120574 0.021719 -1.3416 0.186175 0.093087 M1 0.607321306663722 0.660507 0.9195 0.362541 0.181271 M2 0.600890186123138 0.841453 0.7141 0.478691 0.239345 M3 0.217393662901691 0.839653 0.2589 0.796836 0.398418 M4 0.127998655073318 0.749226 0.1708 0.865082 0.432541 M5 0.0859694305114406 0.666078 0.1291 0.897855 0.448927 M6 0.0723797230840934 0.668049 0.1083 0.914183 0.457092 M7 -0.0891585214799912 0.68046 -0.131 0.896313 0.448157 M8 -0.0927468185738858 0.785247 -0.1181 0.906483 0.453241 M9 -0.5528177644784 0.707895 -0.7809 0.438759 0.219379 M10 -0.549134872845152 0.701734 -0.7825 0.437822 0.218911 M11 0.539514661336017 0.820422 0.6576 0.513999 0.257 Multiple Linear Regression - Regression Statistics Multiple R 0.432682466581798 R-squared 0.187214116887309 Adjusted R-squared -0.0203056830563568 F-TEST (value) 0.902150623401385 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0.551517837201219 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.960789905971162 Sum Squared Residuals 43.3865104405554 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 10.9 9.45575800313136 1.44424199686864 2 10 8.94524478483217 1.05475521516783 3 9.2 8.67247149695654 0.527528503043458 4 9.2 8.76955772760533 0.430442272394663 5 9.5 8.79454519812119 0.70545480187881 6 9.6 8.98491934527824 0.615080654721755 7 9.5 8.78550209914848 0.714497900851515 8 9.1 8.52550209914849 0.574497900851514 9 8.9 8.12079277091688 0.77920722908312 10 9 8.2614228220568 0.738577177943202 11 10.1 8.88969679874746 1.21030320125254 12 10.3 9.31172602330934 0.988273976690663 13 10.2 9.48780946599461 0.712190534005389 14 9.6 8.97146870899302 0.628531291006978 15 9.2 8.8006773484096 0.399322651590406 16 9.3 8.62678302939625 0.673216970603746 17 9.4 8.77414881266275 0.62585118733725 18 9.4 8.8567134938252 0.543286506174808 19 9.2 8.64564117029061 0.554358829709389 20 9 8.21081500921827 0.789184990781734 21 9 8.1819819272922 0.818018072707799 22 9 7.91759803861452 1.08240196138548 23 9.8 8.86930041328902 0.93069958671098 24 10 9.18060640250508 0.81939359749492 25 9.8 9.34794853713674 0.452051462863265 26 9.3 8.92484839937373 0.37515160062627 27 9 8.49181779718178 0.508182202818215 28 9 8.50731848599682 0.492681514003181 29 9.1 8.78289012071637 0.317109879283632 30 9.1 8.65274963924079 0.447250360759209 31 9.1 8.51160778013515 0.588392219864854 32 9.2 8.11466062062848 1.08533937937152 33 8.8 8.02463838232709 0.775361617672909 34 8.3 7.93508065472176 0.364919345278245 35 8.4 8.79354241015767 -0.39354241015767 36 8.1 8.97372877856947 -0.873728778569473 37 7.7 9.24013907114212 -1.54013907114212 38 7.9 8.99186509445146 -1.09186509445146 39 7.9 8.31990540546065 -0.419905405460646 40 8 8.495663408592 -0.495663408591996 41 7.9 8.84116550774048 -0.941165507740482 42 7.6 8.51580247973412 -0.915802479734121 43 7.1 8.29307507879472 -1.19307507879472 44 6.8 8.376899862237 -1.57689986223699 45 6.5 7.69829621499205 -1.19829621499205 46 6.9 7.94090819342417 -1.04090819342417 47 8.2 8.83142141172335 -0.631421411723346 48 8.7 8.87174685127727 -0.171746851277272 49 8.3 9.36834492259518 -1.06834492259518 50 7.9 8.86657301234962 -0.966573012349615 51 7.5 8.51512795199143 -1.01512795199143 52 7.8 8.9006773484096 -1.10067734840959 53 8.3 9.00725036075921 -0.707250360759209 54 8.4 9.08981504192165 -0.689815041921651 55 8.2 8.86417387163104 -0.664173871631041 56 7.7 8.57212240876778 -0.872122408767777 57 7.2 8.37429070447178 -1.17429070447178 58 7.3 8.44499029118276 -1.14499029118276 59 8.1 9.2160389660825 -1.11603896608250 60 8.5 9.26219194433884 -0.762191944338841 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0498302496425607 0.0996604992851213 0.95016975035744 17 0.0146418057354139 0.0292836114708277 0.985358194264586 18 0.00522302863954832 0.0104460572790966 0.994776971360452 19 0.00193967031361146 0.00387934062722291 0.998060329686389 20 0.000519175917824384 0.00103835183564877 0.999480824082176 21 0.000158765815371608 0.000317531630743217 0.999841234184628 22 4.98165637385357e-05 9.96331274770715e-05 0.999950183436261 23 2.65793975185048e-05 5.31587950370097e-05 0.999973420602482 24 1.45243702952421e-05 2.90487405904843e-05 0.999985475629705 25 0.000105811826105529 0.000211623652211059 0.999894188173895 26 0.000138051186280616 0.000276102372561232 0.99986194881372 27 7.36906955640018e-05 0.000147381391128004 0.999926309304436 28 3.59986462488199e-05 7.19972924976397e-05 0.999964001353751 29 2.81432395056632e-05 5.62864790113264e-05 0.999971856760494 30 1.71129168146047e-05 3.42258336292093e-05 0.999982887083185 31 1.68471852572519e-05 3.36943705145037e-05 0.999983152814743 32 0.000231862239657164 0.000463724479314328 0.999768137760343 33 0.00598736939097136 0.0119747387819427 0.994012630609029 34 0.125368425707098 0.250736851414196 0.874631574292902 35 0.583463184585617 0.833073630828765 0.416536815414383 36 0.843760862661957 0.312478274676086 0.156239137338043 37 0.966202592577284 0.0675948148454324 0.0337974074227162 38 0.973122443685408 0.0537551126291846 0.0268775563145923 39 0.967523704320857 0.0649525913582868 0.0324762956791434 40 0.967273199392877 0.0654536012142463 0.0327268006071232 41 0.954540549066303 0.0909189018673943 0.0454594509336971 42 0.914284036542032 0.171431926915935 0.0857159634579677 43 0.902273208460038 0.195453583079923 0.0977267915399617 44 0.957417211526802 0.0851655769463962 0.0425827884731981 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 14 0.482758620689655 NOK 5% type I error level 17 0.586206896551724 NOK 10% type I error level 24 0.827586206896552 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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