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 02:03:34 -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/t1258707988rm7jtmmccfjs6ym.htm/, Retrieved Fri, 20 Nov 2009 10:06:40 +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/t1258707988rm7jtmmccfjs6ym.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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0.6348 1.5291 0.634 1.5358 0.62915 1.5355 0.62168 1.5287 0.61328 1.5334 0.6089 1.5225 0.60857 1.5135 0.62672 1.5144 0.62291 1.4913 0.62393 1.4793 0.61838 1.4663 0.62012 1.4749 0.61659 1.4745 0.6116 1.4775 0.61573 1.4678 0.61407 1.4658 0.62823 1.4572 0.64405 1.4721 0.6387 1.4624 0.63633 1.4636 0.63059 1.4649 0.62994 1.465 0.63709 1.4673 0.64217 1.4679 0.65711 1.4621 0.66977 1.4674 0.68255 1.4695 0.68902 1.4964 0.71322 1.5155 0.70224 1.5411 0.70045 1.5476 0.69919 1.54 0.69693 1.5474 0.69763 1.5485 0.69278 1.559 0.70196 1.5544 0.69215 1.5657 0.6769 1.5734 0.67124 1.567 0.66532 1.5547 0.67157 1.54 0.66428 1.5192 0.66576 1.527 0.66942 1.5387 0.6813 1.5431 0.69144 1.5426 0.69862 1.5216 0.695 1.5364 0.69867 1.5469 0.68968 1.5501 0.69233 1.5494 0.68293 1.5475 0.68399 1.5448 0.66895 1.5391 0.68756 1.5578 0.68527 1.5528 0.6776 1.5496 0.68137 1.549 0.67933 1.5449 0.67922 1.5479

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 Britse_pond[t] = + 0.261141331976786 + 0.241000719518527Zwitserse_frank[t] + 0.00488968621576873M1[t] -0.000975010761816689M2[t] + 0.000415678146259142M3[t] -0.00451061566544505M4[t] + 0.00190711140066335M5[t] -0.00375862229591785M6[t] -0.00306619760422058M7[t] -0.00108835102738477M8[t] -0.00311442237833564M9[t] -0.00068715391636125M10[t] -0.000232003526077248M11[t] + 0.00114231325047974t + 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) 0.261141331976786 0.138032 1.8919 0.064811 0.032406 Zwitserse_frank 0.241000719518527 0.09358 2.5754 0.013294 0.006647 M1 0.00488968621576873 0.013263 0.3687 0.714066 0.357033 M2 -0.000975010761816689 0.013283 -0.0734 0.941804 0.470902 M3 0.000415678146259142 0.013222 0.0314 0.975057 0.487528 M4 -0.00451061566544505 0.013202 -0.3417 0.734168 0.367084 M5 0.00190711140066335 0.013175 0.1448 0.885539 0.442769 M6 -0.00375862229591785 0.013158 -0.2856 0.77643 0.388215 M7 -0.00306619760422058 0.013159 -0.233 0.816792 0.408396 M8 -0.00108835102738477 0.013143 -0.0828 0.934364 0.467182 M9 -0.00311442237833564 0.013114 -0.2375 0.813329 0.406665 M10 -0.00068715391636125 0.013097 -0.0525 0.958384 0.479192 M11 -0.000232003526077248 0.013094 -0.0177 0.985941 0.49297 t 0.00114231325047974 0.000197 5.7852 1e-06 0 Multiple Linear Regression - Regression Statistics Multiple R 0.818420057509127 R-squared 0.669811390533243 Adjusted R-squared 0.576497218292637 F-TEST (value) 7.17802424272887 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 2.32630924967836e-07 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0206969578926357 Sum Squared Residuals 0.0197047470364387 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 0.6348 0.635687531658816 -0.000887531658816348 2 0.634 0.632579852752483 0.00142014724751673 3 0.62915 0.635040554695183 -0.00589055469518334 4 0.62168 0.629617769241233 -0.00793776924123293 5 0.61328 0.638310512939558 -0.0250305129395581 6 0.6089 0.631160184650705 -0.0222601846507047 7 0.60857 0.630825916117215 -0.0222559161172149 8 0.62672 0.634162976592097 -0.00744297659209714 9 0.62291 0.627712101870748 -0.00480210187074816 10 0.62393 0.62838967494898 -0.00445967494897994 11 0.61838 0.626854129236003 -0.00847412923600273 12 0.62012 0.63030105220042 -0.0101810522004191 13 0.61659 0.63623665137886 -0.0196466513788602 14 0.6116 0.63223726981031 -0.0206372698103101 15 0.61573 0.632432564989536 -0.0167025649895359 16 0.61407 0.628166582989274 -0.0140965829892745 17 0.62823 0.633654017118003 -0.00542401711800331 18 0.64405 0.632721507392728 0.0113284926072722 19 0.6387 0.632218538355575 0.0064814616444249 20 0.63633 0.635627899046313 0.000702100953687009 21 0.63059 0.635057441881216 -0.00446744188121595 22 0.62994 0.638651123665622 -0.00871112366562185 23 0.63709 0.640802888961278 -0.00371288896127822 24 0.64217 0.642321806169546 -0.000151806169546329 25 0.65711 0.646956001462587 0.0101539985374126 26 0.66977 0.64351092154893 0.0262590784510701 27 0.68255 0.646550025218474 0.0359999747815256 28 0.68902 0.649248964012298 0.0397710359877017 29 0.71322 0.66141211807169 0.0518078819283096 30 0.70224 0.663058316045263 0.0391816839547368 31 0.70045 0.66645955866431 0.0339904413356894 32 0.69919 0.667748113023285 0.0314418869767146 33 0.69693 0.668647760247251 0.0282822397527487 34 0.69763 0.672482442751176 0.0251475572488241 35 0.69278 0.676610413946884 0.0161695860531158 36 0.70196 0.676876127413656 0.0250838725863441 37 0.69215 0.685631435010464 0.00651856498953627 38 0.6769 0.68276475682365 -0.00586475682365074 39 0.67124 0.683755354377288 -0.0125153543772878 40 0.66532 0.677007064965985 -0.0116870649659854 41 0.67157 0.681024394705651 -0.00945439470565118 42 0.66428 0.671488159293564 -0.0072081592935644 43 0.66576 0.675202702847986 -0.00944270284798583 44 0.66942 0.681142571093668 -0.0117225710936682 45 0.6813 0.681319216159079 -1.92161590785556e-05 46 0.69144 0.684768297511773 0.0066717024882266 47 0.69862 0.681304746042648 0.0173152539573519 48 0.695 0.68624587346808 0.0087541265319206 49 0.69867 0.694808380489272 0.00386161951072767 50 0.68968 0.690857199064626 -0.00117719906462599 51 0.69233 0.693221500719519 -0.000891500719518575 52 0.68293 0.688979618791209 -0.00604961879120892 53 0.68399 0.695888957165097 -0.0118989571650970 54 0.66895 0.68999183261774 -0.0210418326177399 55 0.68756 0.696333284014913 -0.00877328401491351 56 0.68527 0.698248440244636 -0.0129784402446363 57 0.6776 0.696593479841706 -0.018993479841706 58 0.68137 0.700018461122449 -0.0186484611224489 59 0.67933 0.700627821813187 -0.0212978218131867 60 0.67922 0.7027251407483 -0.0235051407482993 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0413784945245326 0.0827569890490652 0.958621505475467 18 0.570104167688176 0.85979166462365 0.429895832311824 19 0.55087019966042 0.898259600679161 0.449129800339580 20 0.427835444792791 0.855670889585582 0.572164555207209 21 0.359340170769428 0.718680341538855 0.640659829230572 22 0.339169708176823 0.678339416353647 0.660830291823177 23 0.344711387733875 0.68942277546775 0.655288612266125 24 0.428564598696034 0.857129197392069 0.571435401303966 25 0.507941723209632 0.984116553580735 0.492058276790368 26 0.590663239319394 0.818673521361212 0.409336760680606 27 0.675163009528125 0.64967398094375 0.324836990471875 28 0.626448849129152 0.747102301741695 0.373551150870848 29 0.672891666023559 0.654216667952881 0.327108333976441 30 0.764733982479376 0.470532035041248 0.235266017520624 31 0.787890571098882 0.424218857802235 0.212109428901118 32 0.80634727817509 0.387305443649819 0.193652721824910 33 0.822023239561195 0.355953520877609 0.177976760438804 34 0.807097678024853 0.385804643950293 0.192902321975147 35 0.793835902793481 0.412328194413038 0.206164097206519 36 0.861637815466357 0.276724369067286 0.138362184533643 37 0.87057656341642 0.25884687316716 0.12942343658358 38 0.889430693205862 0.221138613588277 0.110569306794138 39 0.902192931310464 0.195614137379071 0.0978070686895356 40 0.915596331954314 0.168807336091372 0.084403668045686 41 0.916236848510365 0.16752630297927 0.083763151489635 42 0.849247929962706 0.301504140074588 0.150752070037294 43 0.846376693944704 0.307246612110593 0.153623306055296 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 1 0.0370370370370370 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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