*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: Thu, 19 Nov 2009 11:27:34 -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/19/t12586553795zf596b5iq439t3.htm/, Retrieved Thu, 19 Nov 2009 19:29:51 +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/19/t12586553795zf596b5iq439t3.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:

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Dataseries X:

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7.2 2.4 7.5 8.3 8.8 8.9 7.4 2 7.2 7.5 8.3 8.8 8.8 2.1 7.4 7.2 7.5 8.3 9.3 2 8.8 7.4 7.2 7.5 9.3 1.8 9.3 8.8 7.4 7.2 8.7 2.7 9.3 9.3 8.8 7.4 8.2 2.3 8.7 9.3 9.3 8.8 8.3 1.9 8.2 8.7 9.3 9.3 8.5 2 8.3 8.2 8.7 9.3 8.6 2.3 8.5 8.3 8.2 8.7 8.5 2.8 8.6 8.5 8.3 8.2 8.2 2.4 8.5 8.6 8.5 8.3 8.1 2.3 8.2 8.5 8.6 8.5 7.9 2.7 8.1 8.2 8.5 8.6 8.6 2.7 7.9 8.1 8.2 8.5 8.7 2.9 8.6 7.9 8.1 8.2 8.7 3 8.7 8.6 7.9 8.1 8.5 2.2 8.7 8.7 8.6 7.9 8.4 2.3 8.5 8.7 8.7 8.6 8.5 2.8 8.4 8.5 8.7 8.7 8.7 2.8 8.5 8.4 8.5 8.7 8.7 2.8 8.7 8.5 8.4 8.5 8.6 2.2 8.7 8.7 8.5 8.4 8.5 2.6 8.6 8.7 8.7 8.5 8.3 2.8 8.5 8.6 8.7 8.7 8 2.5 8.3 8.5 8.6 8.7 8.2 2.4 8 8.3 8.5 8.6 8.1 2.3 8.2 8 8.3 8.5 8.1 1.9 8.1 8.2 8 8.3 8 1.7 8.1 8.1 8.2 8 7.9 2 8 8.1 8.1 8.2 7.9 2.1 7.9 8 8.1 8.1 8 1.7 7.9 7.9 8 8.1 8 1.8 8 7.9 7.9 8 7.9 1.8 8 8 7.9 7.9 8 1.8 7.9 8 8 7.9 7.7 1.3 8 7.9 8 8 7.2 1.3 7.7 8 7.9 8 7.5 1.3 7.2 7.7 8 7.9 7.3 1.2 7.5 7.2 7.7 8 7 1.4 7.3 7.5 7.2 7.7 7 2.2 7 7.3 7.5 7.2 7 2.9 7 7 7.3 7.5 7.2 3.1 7 7 7 7.3 7. 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 'Gwilym Jenkins' @ 72.249.127.135 R Framework error message The field 'Names of X columns' contains a hard return which cannot be interpreted. Please, resubmit your request without hard returns in the 'Names of X columns'. Multiple Linear Regression - Estimated Regression Equation Y(t)[t] = + 0.998292328693155 + 0.0376630226888407`X(t)`[t] + 1.46957457460690`Y(t-1)`[t] -0.801806872557724`Y(t-2)`[t] -0.115732461513392`Y(t-3)`[t] + 0.329690799318938`Y(t-4) `[t] -0.144354348061105M1[t] -0.120706258608075M2[t] + 0.608263839578986M3[t] -0.390327570481754M4[t] + 0.0102407157144446M5[t] + 0.117272582800866M6[t] + 0.0204583804677823M7[t] + 0.172191674221844M8[t] + 0.0134046809351707M9[t] -0.0958526420642873M10[t] -0.0193853350638593M11[t] -0.00677880468999083t + 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) 0.998292328693155 0.668951 1.4923 0.143871 0.071935 `X(t)` 0.0376630226888407 0.024701 1.5248 0.135598 0.067799 `Y(t-1)` 1.46957457460690 0.137941 10.6537 0 0 `Y(t-2)` -0.801806872557724 0.263589 -3.0419 0.004247 0.002123 `Y(t-3)` -0.115732461513392 0.263607 -0.439 0.663123 0.331562 `Y(t-4) ` 0.329690799318938 0.143787 2.2929 0.027478 0.013739 M1 -0.144354348061105 0.103653 -1.3927 0.171813 0.085907 M2 -0.120706258608075 0.107018 -1.1279 0.266432 0.133216 M3 0.608263839578986 0.108544 5.6039 2e-06 1e-06 M4 -0.390327570481754 0.141671 -2.7552 0.008956 0.004478 M5 0.0102407157144446 0.155634 0.0658 0.947882 0.473941 M6 0.117272582800866 0.124325 0.9433 0.351498 0.175749 M7 0.0204583804677823 0.10111 0.2023 0.840732 0.420366 M8 0.172191674221844 0.103872 1.6577 0.105608 0.052804 M9 0.0134046809351707 0.112782 0.1189 0.906016 0.453008 M10 -0.0958526420642873 0.11389 -0.8416 0.405264 0.202632 M11 -0.0193853350638593 0.107819 -0.1798 0.858268 0.429134 t -0.00677880468999083 0.002425 -2.7957 0.008077 0.004038 Multiple Linear Regression - Regression Statistics Multiple R 0.985993431228861 R-squared 0.972183046426463 Adjusted R-squared 0.959738619827776 F-TEST (value) 78.121963974539 F-TEST (DF numerator) 17 F-TEST (DF denominator) 38 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.149249431800875 Sum Squared Residuals 0.846464929929589 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.2 7.22016515033857 -0.0201651503385712 2 7.4 7.447439502515 -0.0474395025150006 3 8.8 8.6355946445209 0.164405355479107 4 9.3 9.29446825643826 0.00553174356174162 5 9.3 9.17092906703097 0.129070932969028 6 8.7 8.80808812731354 -0.108088127313538 7 8.2 8.2113860547406 -0.0113860547406046 8 8.3 8.2524175706198 0.0475824293802062 9 8.5 8.7079184455596 -0.207918445559603 10 8.6 8.67696720350775 -0.0769672035077447 11 8.5 8.57566465430094 -0.0756646543009385 12 8.2 8.35589041851203 -0.155890418512026 13 8.1 7.8946641920782 0.205335807921803 14 7.9 8.06472561630663 -0.164725616306633 15 8.6 8.57493334066022 0.0250666593397786 16 8.7 8.67882531353929 0.0211746864607128 17 8.7 8.65225115635545 0.0477488436445533 18 8.5 8.49524223042187 0.0047577695781299 19 8.4 8.32071092411822 0.079289075881782 20 8.5 8.53086992150946 -0.0308699215094584 21 8.7 8.61558876055193 0.0844112394480652 22 8.7 8.65892194681564 0.0410780531843564 23 8.6 8.501108934918 0.098891065082001 24 8.5 8.39164580453593 0.108354195464070 25 8.3 8.24720664598147 0.0527933540185279 26 8 8.0506160424236 -0.0506160424235927 27 8.2 8.4671342020007 -0.267134202000699 28 8.1 7.98263207404056 0.117367925959436 29 8.1 8.02281909308923 0.0771809069107674 30 8 8.0736665061053 -0.0736665061053069 31 7.9 7.91192635444332 -0.0119263544433215 32 7.9 7.96090129563947 -0.0609012956394656 33 8 7.87202422199438 0.127975778005623 34 8 7.88531602025395 0.114683979746053 35 7.9 7.84185475537672 0.0581452446232822 36 8 7.69593058213856 0.304069417861442 37 7.7 7.7860731426914 -0.0860731426913972 38 7.2 7.29346261396794 -0.0934626139679346 39 7.5 7.47686635584564 0.0231336441543587 40 7.3 7.37719446587287 -0.0771944658728678 41 7 7.20301856618916 -0.203018566189162 42 7 6.85332591075266 0.146674089247345 43 7 7.13869281347745 -0.138692813477446 44 7.2 7.25996148566951 -0.0599614856695144 45 7.3 7.30446857189408 -0.00446857189408546 46 7.1 7.17879482942266 -0.078794829422665 47 6.8 6.88137165540434 -0.0813716554043448 48 6.4 6.65653319481349 -0.256533194813486 49 6.1 6.25189086891036 -0.151890868910363 50 6.5 6.14375622478684 0.356243775213161 51 7.7 7.64547145697255 0.0545285430274547 52 7.9 7.96687989010902 -0.0668798901090223 53 7.5 7.55098211733519 -0.0509821173351869 54 6.9 6.86967722540663 0.0303227745933704 55 6.6 6.51728385322041 0.0827161467795902 56 6.9 6.79584972656177 0.104150273438232 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.0996428273849502 0.199285654769900 0.90035717261505 22 0.124685722626046 0.249371445252092 0.875314277373954 23 0.0544787771443442 0.108957554288688 0.945521222855656 24 0.0518654621740017 0.103730924348003 0.948134537825998 25 0.025174492223657 0.050348984447314 0.974825507776343 26 0.0141609427848188 0.0283218855696376 0.985839057215181 27 0.313685097781473 0.627370195562946 0.686314902218527 28 0.213639456559031 0.427278913118063 0.786360543440969 29 0.14054950096741 0.28109900193482 0.85945049903259 30 0.165451450806419 0.330902901612838 0.834548549193581 31 0.121622593230636 0.243245186461273 0.878377406769363 32 0.105766462983213 0.211532925966425 0.894233537016787 33 0.0711773523594984 0.142354704718997 0.928822647640502 34 0.0437357082036325 0.087471416407265 0.956264291796368 35 0.0204409323100288 0.0408818646200576 0.97955906768997 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 2 0.133333333333333 NOK 10% type I error level 4 0.266666666666667 NOK

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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