workshop 7

*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 09:13:14 -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/t12587337258xqe0e4mak0srra.htm/, Retrieved Fri, 20 Nov 2009 17:15:37 +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/t12587337258xqe0e4mak0srra.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:

workshop 7

Dataseries X:

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0,634 1,5358 0,6348 0,62915 1,5355 0,634 0,62168 1,5287 0,62915 0,61328 1,5334 0,62168 0,6089 1,5225 0,61328 0,60857 1,5135 0,6089 0,62672 1,5144 0,60857 0,62291 1,4913 0,62672 0,62393 1,4793 0,62291 0,61838 1,4663 0,62393 0,62012 1,4749 0,61838 0,61659 1,4745 0,62012 0,6116 1,4775 0,61659 0,61573 1,4678 0,6116 0,61407 1,4658 0,61573 0,62823 1,4572 0,61407 0,64405 1,4721 0,62823 0,6387 1,4624 0,64405 0,63633 1,4636 0,6387 0,63059 1,4649 0,63633 0,62994 1,465 0,63059 0,63709 1,4673 0,62994 0,64217 1,4679 0,63709 0,65711 1,4621 0,64217 0,66977 1,4674 0,65711 0,68255 1,4695 0,66977 0,68902 1,4964 0,68255 0,71322 1,5155 0,68902 0,70224 1,5411 0,71322 0,70045 1,5476 0,70224 0,69919 1,54 0,70045 0,69693 1,5474 0,69919 0,69763 1,5485 0,69693 0,69278 1,559 0,69763 0,70196 1,5544 0,69278 0,69215 1,5657 0,70196 0,6769 1,5734 0,69215 0,67124 1,567 0,6769 0,66532 1,5547 0,67124 0,67157 1,54 0,66532 0,66428 1,5192 0,67157 0,66576 1,527 0,66428 0,66942 1,5387 0,66576 0,6813 1,5431 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 11 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation britse_pond[t] = + 0.145412170967563 -0.0985427050412497Zwitserse_frank[t] + 1.00268878936414`Britse_pond_-1`[t] -0.00341265297691773M1[t] + 0.00148026239263723M2[t] -0.00395853837575197M3[t] + 0.006952974351011M4[t] -0.00493876877735603M5[t] + 0.00214802675416841M6[t] + 0.00271409312945089M7[t] -0.00235740195402955M8[t] + 0.0018233559982335M9[t] -0.00140612307178253M10[t] + 0.00100553451485034M11[t] + 0.000104797369572300t + 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) 0.145412170967563 0.05834 2.4925 0.016524 0.008262 Zwitserse_frank -0.0985427050412497 0.045061 -2.1869 0.034112 0.017056 `Britse_pond_-1` 1.00268878936414 0.065673 15.2679 0 0 M1 -0.00341265297691773 0.0057 -0.5987 0.552463 0.276232 M2 0.00148026239263723 0.005693 0.26 0.796077 0.398039 M3 -0.00395853837575197 0.005687 -0.6961 0.490028 0.245014 M4 0.006952974351011 0.005707 1.2183 0.2296 0.1148 M5 -0.00493876877735603 0.005671 -0.8709 0.388561 0.194281 M6 0.00214802675416841 0.005709 0.3763 0.708524 0.354262 M7 0.00271409312945089 0.005696 0.4765 0.636095 0.318048 M8 -0.00235740195402955 0.005674 -0.4155 0.679792 0.339896 M9 0.0018233559982335 0.005682 0.3209 0.749804 0.374902 M10 -0.00140612307178253 0.005673 -0.2479 0.8054 0.4027 M11 0.00100553451485034 0.005681 0.177 0.860326 0.430163 t 0.000104797369572300 0.000108 0.9693 0.33771 0.168855 Multiple Linear Regression - Regression Statistics Multiple R 0.9731156931693 R-squared 0.946954152292366 Adjusted R-squared 0.930075928021755 F-TEST (value) 56.1050817378485 F-TEST (DF numerator) 14 F-TEST (DF denominator) 44 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.00843233357505844 Sum Squared Residuals 0.00312858697892655 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 0.634 0.627269272446225 0.00673072755377538 2 0.62915 0.631494396965373 -0.00234439696537318 3 0.62168 0.621967443332421 -0.000287443332420607 4 0.61328 0.625030517458512 -0.0117505174585119 5 0.6089 0.605895101354008 0.00300489864599194 6 0.60857 0.609581801703061 -0.00101180170306099 7 0.62672 0.609833089712889 0.0168869102871114 8 0.62291 0.625341530012393 -0.00243153001239255 9 0.62393 0.626989353507245 -0.0030593535072454 10 0.61838 0.62616846953749 -0.00778846953748932 11 0.62012 0.622272534449369 -0.00215253444936882 12 0.61659 0.623155892879601 -0.00656589287960091 13 0.6116 0.616012917730676 -0.00441291773067619 14 0.61573 0.616963077649777 -0.00123307764977660 15 0.61407 0.615967264361116 -0.00189726436111608 16 0.62823 0.626166578330462 0.00206342166953835 17 0.64405 0.627109419523948 0.0169405804760515 18 0.6387 0.651119413311686 -0.0124194133116861 19 0.63633 0.646307640787393 -0.00997764078739335 20 0.63059 0.638836465126138 -0.00824646512613843 21 0.62994 0.63735673252652 -0.00741673252651944 22 0.63709 0.633353654891394 0.00373634510860577 23 0.64217 0.642980209068528 -0.000810209068528287 24 0.65711 0.64774467866246 0.00936532133754063 25 0.66977 0.658894717231496 0.0108752827685044 26 0.68255 0.676379530363386 0.00617046963661376 27 0.68902 0.681209090927034 0.00781090907296648 28 0.71322 0.696830631824267 0.0163893681757331 29 0.70224 0.706786061519028 -0.00454606151902844 30 0.70045 0.702327603930139 -0.00187760393013871 31 0.69919 0.701952579300345 -0.00276257930034526 32 0.69693 0.694993277694533 0.00193672230546707 33 0.69763 0.69690435937686 0.000725640623139897 34 0.69278 0.693446861426038 -0.000666861426038097 35 0.70196 0.691553572197017 0.0104064278029832 36 0.69215 0.698743985571136 -0.00659398557113554 37 0.6769 0.68484097411131 -0.00794097411131038 38 0.67124 0.675178356124898 -0.00393835612489835 39 0.66532 0.665381209450288 -6.12094502876953e-05 40 0.67157 0.671910179677694 -0.000340179677693688 41 0.66428 0.668439727117283 -0.00415972711728284 42 0.66576 0.667553085644593 -0.0017930856445932 43 0.66942 0.668554979148724 0.000865020851275675 44 0.6813 0.666824534501707 0.0144754654982926 45 0.69144 0.683071303993709 0.0083686960062906 46 0.69862 0.692183283423284 0.00643671657671561 47 0.695 0.700440611852514 -0.00544061185251366 48 0.69867 0.694875442886804 0.00379455711319583 49 0.68968 0.694932118480293 -0.00525211848029325 50 0.69233 0.690984638896566 0.00134536110343437 51 0.68293 0.688494991929142 -0.0055649919291421 52 0.68399 0.690352092709066 -0.00636209270906589 53 0.66895 0.680189690485732 -0.0112396904857322 54 0.68756 0.670458095410521 0.0171019045894790 55 0.68527 0.690281711050649 -0.00501171105064852 56 0.6776 0.683334192665229 -0.00573419266522864 57 0.68137 0.679988250595666 0.00138174940433434 58 0.67933 0.681047730721794 -0.00171773072179397 59 0.67922 0.681223072432573 -0.00200307243257242 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.721427980163072 0.557144039673856 0.278572019836928 19 0.85371131748415 0.2925773650317 0.14628868251585 20 0.84951658871089 0.300966822578221 0.150483411289110 21 0.851319014682401 0.297361970635197 0.148680985317599 22 0.853890768858869 0.292218462282263 0.146109231141131 23 0.851022658957065 0.29795468208587 0.148977341042935 24 0.88131245196 0.237375096079998 0.118687548039999 25 0.853192463733842 0.293615072532317 0.146807536266158 26 0.818154601794346 0.363690796411308 0.181845398205654 27 0.738685229677218 0.522629540645563 0.261314770322782 28 0.818907348488565 0.362185303022869 0.181092651511435 29 0.926469088136997 0.147061823726006 0.0735309118630028 30 0.888681490891468 0.222637018217063 0.111318509108532 31 0.856192465365592 0.287615069268816 0.143807534634408 32 0.783921103250234 0.432157793499532 0.216078896749766 33 0.69918745880513 0.601625082389739 0.300812541194870 34 0.61489790547897 0.77020418904206 0.38510209452103 35 0.653064213201914 0.693871573596173 0.346935786798087 36 0.633315759348669 0.733368481302662 0.366684240651331 37 0.613104101521205 0.773791796957591 0.386895898478795 38 0.683805779535638 0.632388440928724 0.316194220464362 39 0.67771627543586 0.64456744912828 0.32228372456414 40 0.608879685141083 0.782240629717834 0.391120314858917 41 0.475684106538862 0.951368213077723 0.524315893461138 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 = 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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