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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Wed, 18 Nov 2009 10:25:36 -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/18/t1258565553xz6u50qdxmlbmwb.htm/, Retrieved Wed, 18 Nov 2009 18:32:45 +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/18/t1258565553xz6u50qdxmlbmwb.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}, }

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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 3 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] = + 2.20711870010436 + 0.0300708114127846`X(t)`[t] + 1.14109921130441`Y(t-1)`[t] -0.46973056134648`Y(t-2)`[t] -0.244934881684172`Y(t-3)`[t] + 0.321061204660699`Y(t-4) `[t] -0.0107759424689206t + 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) 2.20711870010436 1.058326 2.0855 0.042256 0.021128 `X(t)` 0.0300708114127846 0.041108 0.7315 0.467955 0.233977 `Y(t-1)` 1.14109921130441 0.128234 8.8986 0 0 `Y(t-2)` -0.46973056134648 0.201695 -2.3289 0.024031 0.012016 `Y(t-3)` -0.244934881684172 0.201658 -1.2146 0.230337 0.115169 `Y(t-4) ` 0.321061204660699 0.134889 2.3802 0.021238 0.010619 t -0.0107759424689206 0.003909 -2.7566 0.008182 0.004091 Multiple Linear Regression - Regression Statistics Multiple R 0.942705450612171 R-squared 0.888693566613897 Adjusted R-squared 0.875064207423762 F-TEST (value) 65.2043543805884 F-TEST (DF numerator) 6 F-TEST (DF denominator) 49 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.262912839277478 Sum Squared Residuals 3.3870348917903 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.2 7.63001089329295 -0.430010893292947 2 7.4 7.7310226323208 -0.331022632320797 3 8.8 8.12781008467497 0.672189915325033 4 9.3 9.43425134539835 -0.134251345398347 5 9.3 9.18508272267896 0.114917277321039 6 8.7 8.6878086363826 0.0121913636173924 7 8.2 8.30736308824881 -0.107363088248815 8 8.3 8.15637815470081 0.143621845299190 9 8.5 8.64454542418735 -0.144545424187355 10 8.6 8.75386822931417 -0.15386822931417 11 8.5 8.59326741091402 -0.0932674109140198 12 8.2 8.39249931074413 -0.192499310744133 13 8.1 8.12307833264098 -0.0230783326409790 14 7.9 8.20773957064516 -0.307739570645162 15 8.6 8.05709118608919 0.542908813910811 16 8.7 8.87322009285542 -0.173220092855418 17 8.7 8.66763061558644 0.0323693844135539 18 8.5 8.35015830974159 0.149841690258411 19 8.4 8.31441896124714 0.0855810387528632 20 8.5 8.33062073608953 0.169379263910467 21 8.7 8.52991474722254 0.170085252777464 22 8.7 8.66066683811613 0.0393331618838724 23 8.6 8.48130268789575 0.118697312104247 24 8.5 8.35156429299074 0.148435707009259 25 8.3 8.34387788874072 -0.0438778887407227 26 8 8.16732740489015 -0.167327404890150 27 8.2 7.89754809786027 0.302451902139730 28 8.1 8.26978494078566 -0.169784940785661 29 8.1 8.048192863925 0.0518071360749982 30 8 7.93307047757313 0.0669295224268723 31 7.9 7.90591158649816 -0.00591158649815773 32 7.9 7.79889973970865 0.101100260291347 33 8 7.84756201697768 0.152437983022317 34 8 7.94629044448283 0.0537095555171705 35 7.9 7.85643532541319 0.043564674586809 36 8 7.70705597364541 0.292944026354587 37 7.7 7.87443372320126 -0.174433723201258 38 7.2 7.49884844937478 -0.298848449374782 39 7.5 7.00184246102311 0.498157538976888 40 7.3 7.6708410664488 -0.370841066448798 41 7 7.32308935504148 -0.323089355041484 42 7 6.85397534374516 0.146024656254838 43 7 7.15047347540418 -0.150473475404178 44 7.2 7.15497991879093 0.0450200812090737 45 7.3 7.2881337817498 0.0118662182502067 46 7.1 7.3005287292833 -0.200528729283296 47 6.8 6.98962956121224 -0.189629561212238 48 6.4 6.76116747696118 -0.361167476961176 49 6.1 6.54603492659012 -0.446034926590124 50 6.5 6.41113923683053 0.0888607631694698 51 7.7 7.00238481970406 0.697615180295943 52 7.9 8.10305628319642 -0.203056283196422 53 7.5 7.56553827644201 -0.0655382764420088 54 6.9 6.81782959303635 0.0821704069636503 55 6.6 6.59845951931892 0.00154048068107764 56 6.9 6.67434293816598 0.22565706183402 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 10 0.920939338002278 0.158121323995445 0.0790606619977223 11 0.895497270303237 0.209005459393526 0.104502729696763 12 0.925639570102256 0.148720859795488 0.0743604298977441 13 0.881575246473373 0.236849507053254 0.118424753526627 14 0.883438798630877 0.233122402738245 0.116561201369123 15 0.944998252736588 0.110003494526825 0.0550017472634124 16 0.92690530134784 0.146189397304321 0.0730946986521606 17 0.885946188266166 0.228107623467667 0.114053811733834 18 0.835603992180046 0.328792015639909 0.164396007819954 19 0.769506837562787 0.460986324874426 0.230493162437213 20 0.709896069332657 0.580207861334687 0.290103930667343 21 0.639070600228055 0.72185879954389 0.360929399771945 22 0.551122852309985 0.89775429538003 0.448877147690015 23 0.479929643595855 0.95985928719171 0.520070356404145 24 0.394427561258992 0.788855122517985 0.605572438741008 25 0.329429711078184 0.658859422156368 0.670570288921816 26 0.357107911185936 0.714215822371872 0.642892088814064 27 0.304281135686185 0.60856227137237 0.695718864313815 28 0.329224652876895 0.65844930575379 0.670775347123105 29 0.286336116663401 0.572672233326803 0.713663883336599 30 0.219847857849408 0.439695715698816 0.780152142150592 31 0.170949084044829 0.341898168089659 0.829050915955171 32 0.121396547622880 0.242793095245760 0.87860345237712 33 0.0878571850952997 0.175714370190599 0.9121428149047 34 0.0612885371275736 0.122577074255147 0.938711462872426 35 0.0439550725470255 0.087910145094051 0.956044927452975 36 0.073684878507298 0.147369757014596 0.926315121492702 37 0.0660376063726257 0.132075212745251 0.933962393627374 38 0.0742289544768996 0.148457908953799 0.9257710455231 39 0.571339284547532 0.857321430904936 0.428660715452468 40 0.584459996050552 0.831080007898896 0.415540003949448 41 0.55935292623742 0.88129414752516 0.44064707376258 42 0.497874492666588 0.995748985333175 0.502125507333412 43 0.460940849138019 0.921881698276037 0.539059150861981 44 0.389683438664393 0.779366877328787 0.610316561335607 45 0.265845876695898 0.531691753391796 0.734154123304102 46 0.168099152823999 0.336198305647998 0.831900847176001 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 0.0270270270270270 OK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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