*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 05:57:57 -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/t12587219870rqz8qephuwt9eo.htm/, Retrieved Fri, 20 Nov 2009 13:59:59 +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/t12587219870rqz8qephuwt9eo.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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1,4816 133,91 1,4562 133,14 1,4268 135,31 1,4088 133,09 1,4016 135,39 1,3650 131,85 1,3190 130,25 1,3050 127,65 1,2785 118,30 1,3239 119,73 1,3449 122,51 1,2732 123,28 1,3322 133,52 1,4369 153,20 1,4975 163,63 1,5770 168,45 1,5553 166,26 1,5557 162,31 1,5750 161,56 1,5527 156,59 1,4748 157,97 1,4718 158,68 1,4570 163,55 1,4684 162,89 1,4227 164,95 1,3896 159,82 1,3622 159,05 1,3716 166,76 1,3419 164,55 1,3511 163,22 1,3516 160,68 1,3242 155,24 1,3074 157,60 1,2999 156,56 1,3213 154,82 1,2881 151,11 1,2611 149,65 1,2727 148,99 1,2811 148,53 1,2684 146,70 1,2650 145,11 1,2770 142,70 1,2271 143,59 1,2020 140,96 1,1938 140,77 1,2103 139,81 1,1856 140,58 1,1786 139,59 1,2015 138,05 1,2256 136,06 1,2292 135,98 1,2037 134,75 1,2165 132,22 1,2694 135,37 1,2938 138,84 1,3201 138,83 1,3014 136,55 1,3119 135,63 1,3408 139,14 1,2991 136,09

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 dollar/euro[t] = + 0.736379297486959 + 0.00396306035761503`japanseyen/euro`[t] + 0.0326966020507571M1[t] + 0.0402548296947051M2[t] + 0.0344662394072104M3[t] + 0.0352598018886685M4[t] + 0.0303498489735416M5[t] + 0.0443341545114475M6[t] + 0.0344142389093547M7[t] + 0.0343186178286897M8[t] + 0.0111029233665956M9[t] + 0.0241011607823836M10[t] + 0.0223844437735641M11[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) 0.736379297486959 0.148558 4.9568 1e-05 5e-06 `japanseyen/euro` 0.00396306035761503 0.000993 3.9925 0.000228 0.000114 M1 0.0326966020507571 0.063824 0.5123 0.610848 0.305424 M2 0.0402548296947051 0.063912 0.6299 0.531842 0.265921 M3 0.0344662394072104 0.064078 0.5379 0.593197 0.296598 M4 0.0352598018886685 0.064225 0.549 0.585604 0.292802 M5 0.0303498489735416 0.064097 0.4735 0.638048 0.319024 M6 0.0443341545114475 0.063965 0.6931 0.491657 0.245828 M7 0.0344142389093547 0.063958 0.5381 0.593062 0.296531 M8 0.0343186178286897 0.063821 0.5377 0.5933 0.29665 M9 0.0111029233665956 0.06381 0.174 0.862613 0.431306 M10 0.0241011607823836 0.063811 0.3777 0.707356 0.353678 M11 0.0223844437735641 0.063827 0.3507 0.727376 0.363688 Multiple Linear Regression - Regression Statistics Multiple R 0.5301310807215 R-squared 0.281038962746945 Adjusted R-squared 0.0974744425972294 F-TEST (value) 1.53100916515745 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0.146801831859654 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.100890615063093 Sum Squared Residuals 0.478409061767028 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.4816 1.29976931202594 0.181830687974062 2 1.4562 1.30427598319453 0.151924016805472 3 1.4268 1.30708723388306 0.119712766116942 4 1.4088 1.29908280237061 0.109717197629389 5 1.4016 1.30328788827800 0.0983121117220017 6 1.365 1.30324296014995 0.061757039850053 7 1.319 1.28698214797567 0.0320178520243298 8 1.305 1.27658256996521 0.0284174300347939 9 1.2785 1.21631226115941 0.0621877388405885 10 1.3239 1.23497767488659 0.0889223251134111 11 1.3449 1.24427826567194 0.100621734328061 12 1.2732 1.22494537837374 0.0482546216262614 13 1.3322 1.29822371848647 0.0339762815135263 14 1.4369 1.38377497396829 0.0531250260317147 15 1.4975 1.41932110321072 0.0781788967892847 16 1.577 1.43921661661588 0.137783383384122 17 1.5553 1.42562756151757 0.129672438482426 18 1.5557 1.4239577786429 0.131742221357099 19 1.575 1.41106556777260 0.163934432227403 20 1.5527 1.39127353671458 0.161426463285415 21 1.4748 1.373526865546 0.101273134454001 22 1.4718 1.38933887581569 0.0824611241843057 23 1.457 1.40692226274846 0.0500777372515401 24 1.4684 1.38192219913887 0.08647780086113 25 1.4227 1.42278270552631 -8.2705526313915e-05 26 1.3896 1.41001043353570 -0.0204104335356969 27 1.3622 1.40117028677284 -0.0389702867728385 28 1.3716 1.43251904461151 -0.0609190446115086 29 1.3419 1.41885072830605 -0.0769507283060524 30 1.3511 1.42756416356833 -0.0764641635683304 31 1.3516 1.40757807465790 -0.0559780746578955 32 1.3242 1.38592340523180 -0.0617234052318047 33 1.3074 1.37206053321368 -0.0646605332136821 34 1.2999 1.38093718785755 -0.0810371878575503 35 1.3213 1.37232474582648 -0.0510247458264808 36 1.2881 1.33523734812617 -0.0471373481261649 37 1.2611 1.36214788205480 -0.101047882054804 38 1.2727 1.36709048986273 -0.0943904898627262 39 1.2811 1.35947889181073 -0.0783788918107285 40 1.2684 1.35302005383775 -0.0846200538377511 41 1.265 1.34180883495402 -0.0768088349540164 42 1.277 1.34624216503007 -0.06924216503007 43 1.2271 1.33984937314625 -0.112749373146255 44 1.202 1.32933090332506 -0.127330903325062 45 1.1938 1.30536222739502 -0.111562227395021 46 1.2103 1.3145559268675 -0.104255926867499 47 1.1856 1.31589076633404 -0.130290766334043 48 1.1786 1.28958289280644 -0.110982892806440 49 1.2015 1.31617638190647 -0.114676381906470 50 1.2256 1.31584811943876 -0.0902481194387638 51 1.2292 1.30974248432266 -0.0805424843226598 52 1.2037 1.30566148256425 -0.101961482564252 53 1.2165 1.29072498694436 -0.0742249869443587 54 1.2694 1.31719293260875 -0.0477929326087519 55 1.2938 1.32102483644758 -0.0272248364475831 56 1.3201 1.32088958476334 -0.000789584763342042 57 1.3014 1.28863811268589 0.0127618873141142 58 1.3119 1.29799033457267 0.0139096654273322 59 1.3408 1.31018395941908 0.0306160405809229 60 1.2991 1.27571218155479 0.0233878184452127 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.525752337811907 0.948495324376186 0.474247662188093 17 0.419038091692187 0.838076183384375 0.580961908307813 18 0.381896069837683 0.763792139675366 0.618103930162317 19 0.505730232913822 0.988539534172356 0.494269767086178 20 0.647564074075112 0.704871851849776 0.352435925924888 21 0.629716034794542 0.740567930410917 0.370283965205458 22 0.640721451047226 0.718557097905547 0.359278548952774 23 0.66139690694252 0.677206186114961 0.338603093057480 24 0.672116864137040 0.655766271725921 0.327883135862960 25 0.823046767778551 0.353906464442898 0.176953232221449 26 0.890677693353915 0.218644613292170 0.109322306646085 27 0.925559417377834 0.148881165244333 0.0744405826221665 28 0.9647854376943 0.0704291246114 0.0352145623057 29 0.977648539918905 0.0447029201621896 0.0223514600810948 30 0.978953254517737 0.0420934909645269 0.0210467454822635 31 0.976951917305994 0.0460961653880115 0.0230480826940057 32 0.972444117592336 0.0551117648153274 0.0275558824076637 33 0.962025003012479 0.0759499939750426 0.0379749969875213 34 0.95015721806025 0.0996855638794993 0.0498427819397497 35 0.928509928582564 0.142980142834873 0.0714900714174363 36 0.89709476076422 0.205810478471561 0.102905239235780 37 0.880857877735305 0.23828424452939 0.119142122264695 38 0.851384567837919 0.297230864324162 0.148615432162081 39 0.813520912030097 0.372958175939806 0.186479087969903 40 0.805987719078093 0.388024561843814 0.194012280921907 41 0.884112294249349 0.231775411501303 0.115887705750651 42 0.954764150806363 0.0904716983872748 0.0452358491936374 43 0.941718914044234 0.116562171911532 0.0582810859557661 44 0.89153923538632 0.216921529227359 0.108460764613680 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 3 0.103448275862069 NOK 10% type I error level 8 0.275862068965517 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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