workshop 7,5

*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 06:30:30 -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/t1258723931jgv5nrkxfvkbz9i.htm/, Retrieved Fri, 20 Nov 2009 14:32:24 +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/t1258723931jgv5nrkxfvkbz9i.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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No (this computation is public)

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Dataseries X:

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8,6 10 8,9 8,9 8,3 9,2 8,6 8,9 8,3 9,2 8,3 8,6 8,3 9,5 8,3 8,3 8,4 9,6 8,3 8,3 8,5 9,5 8,4 8,3 8,4 9,1 8,5 8,4 8,6 8,9 8,4 8,5 8,5 9 8,6 8,4 8,5 10,1 8,5 8,6 8,4 10,3 8,5 8,5 8,5 10,2 8,4 8,5 8,5 9,6 8,5 8,4 8,5 9,2 8,5 8,5 8,5 9,3 8,5 8,5 8,5 9,4 8,5 8,5 8,5 9,4 8,5 8,5 8,5 9,2 8,5 8,5 8,5 9 8,5 8,5 8,6 9 8,5 8,5 8,4 9 8,6 8,5 8,1 9,8 8,4 8,6 8,0 10 8,1 8,4 8,0 9,8 8,0 8,1 8,0 9,3 8,0 8,0 8,0 9 8,0 8,0 7,9 9 8,0 8,0 7,8 9,1 7,9 8,0 7,8 9,1 7,8 7,9 7,9 9,1 7,8 7,8 8,1 9,2 7,9 7,8 8,0 8,8 8,1 7,9 7,6 8,3 8,0 8,1 7,3 8,4 7,6 8,0 7,0 8,1 7,3 7,6 6,8 7,7 7,0 7,3 7,0 7,9 6,8 7,0 7,1 7,9 7,0 6,8 7,2 8 7,1 7,0 7,1 7,9 7,2 7,1 6,9 7,6 7,1 7,2 6,7 7,1 6,9 7,1 6,7 6,8 6,7 6,9 6,6 6,5 6,7 6,7 6,9 6,9 6,6 6,7 7,3 8,2 6,9 6,6 7,5 8,7 7,3 6,9 7,3 8,3 7,5 7,3 7,1 7,9 7,3 7,5 6,9 7,5 7,1 7,3 7,1 7,8 6,9 7,1 7,5 8,3 7,1 6,9 7,7 8,4 7,5 7,1 7,8 8,2 7,7 7,5 7,8 7,7 7,8 7,7 7,7 7,2 7,8 7,8 7,8 7,3 7,7 7,8 7,8 8,1 7,8 7,7 7,9 8,5 7,8 7,8

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 Y[t] = + 0.201644028496708 + 0.211282850119338X[t] + 1.07442780328295Y1[t] -0.360757733933598Y2[t] + 0.0825009869871107M1[t] + 0.123583145368402M2[t] + 0.204738091758070M3[t] + 0.132843919065565M4[t] + 0.12649724446896M5[t] + 0.179670038959494M6[t] + 0.238304649778517M7[t] + 0.281164917599927M8[t] + 0.222128886451194M9[t] + 0.0713170887252957M10[t] + 0.00836663797171274M11[t] + 0.00202552882501797t + 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) 0.201644028496708 0.515655 0.391 0.697695 0.348848 X 0.211282850119338 0.06856 3.0817 0.003584 0.001792 Y1 1.07442780328295 0.159756 6.7254 0 0 Y2 -0.360757733933598 0.13376 -2.697 0.00995 0.004975 M1 0.0825009869871107 0.10194 0.8093 0.422794 0.211397 M2 0.123583145368402 0.108378 1.1403 0.260474 0.130237 M3 0.204738091758070 0.103707 1.9742 0.054805 0.027402 M4 0.132843919065565 0.103238 1.2868 0.205056 0.102528 M5 0.12649724446896 0.103685 1.22 0.229107 0.114554 M6 0.179670038959494 0.107244 1.6753 0.101125 0.050562 M7 0.238304649778517 0.114831 2.0753 0.043979 0.021989 M8 0.281164917599927 0.125354 2.243 0.030107 0.015053 M9 0.222128886451194 0.123218 1.8027 0.078442 0.039221 M10 0.0713170887252957 0.100561 0.7092 0.482032 0.241016 M11 0.00836663797171274 0.100645 0.0831 0.934134 0.467067 t 0.00202552882501797 0.002383 0.85 0.400037 0.200018 Multiple Linear Regression - Regression Statistics Multiple R 0.978038480327254 R-squared 0.956559269000845 Adjusted R-squared 0.941405525629047 F-TEST (value) 63.1236286329784 F-TEST (DF numerator) 15 F-TEST (DF denominator) 43 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.148271373550655 Sum Squared Residuals 0.945329209227708 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.6 8.75066266271143 -0.150662662711430 2 8.3 8.30241572883739 -0.00241572883738628 3 8.3 8.17149518324727 0.128504816752731 4 8.3 8.27323871459566 0.0267612854043373 5 8.4 8.29004585383601 0.10995414616399 6 8.5 8.43155867246792 0.0684413275320767 7 8.4 8.47907267899916 -0.0790726789991636 8 8.6 8.33818335190007 0.26181664809993 9 8.5 8.55326246863824 -0.0532624686382377 10 8.5 8.45729300775361 0.0427069922463853 11 8.4 8.47470042924228 -0.0747004292422772 12 8.5 8.33978825475535 0.160211745244646 13 8.5 8.44106361421753 0.058936385782466 14 8.5 8.36358238798275 0.136417612017252 15 8.5 8.46789114820937 0.032108851790632 16 8.5 8.41915078935381 0.0808492106461849 17 8.5 8.41482964358223 0.085170356417772 18 8.5 8.42777139687391 0.0722286031260873 19 8.5 8.44617496649409 0.053825033505914 20 8.6 8.49106076314051 0.108939236859486 21 8.4 8.5414930411451 -0.141493041145093 22 8.1 8.31077171828973 -0.210771718289735 23 8 8.04192657218687 -0.0419265721868712 24 8 7.9941134328681 0.00588656713190568 25 8 8.00907429701391 -0.00907429701391372 26 8 7.98879712918442 0.0112028708155788 27 7.9 8.07197760439911 -0.171977604399106 28 7.8 7.91579446521526 -0.115794465215259 29 7.8 7.84010631250874 -0.0401063125087369 30 7.9 7.93138040921765 -0.0313804092176488 31 8.1 8.12061161420192 -0.0206116142019195 32 8 8.25979405806384 -0.259794058063841 33 7.6 7.91754780356544 -0.317547803565444 34 7.3 7.39619447175668 -0.0961944717566763 35 7 7.09385944738086 -0.0938594473808646 36 6.8 6.78890417738163 0.0110958226183700 37 7 6.80902902274112 0.190970977258884 38 7.1 7.13917381739073 -0.0391738173907347 39 7.2 7.27877381115893 -0.0787738111589281 40 7.1 7.25914388921444 -0.159143889214444 41 6.9 7.0479193346854 -0.147919334685400 42 6.7 6.81866644567805 -0.118666445678054 43 6.7 6.67320771641642 0.026792283583577 44 6.6 6.72686020481377 -0.126860204813769 45 6.9 6.6469200622095 0.253079937790506 46 7.3 7.131205612842 0.168794387158002 47 7.5 7.4974659171062 0.00253408289379855 48 7.3 7.47719413499492 -0.177194134994923 49 7.1 7.190170403316 -0.0901704033160069 50 6.9 7.00603093660471 -0.106030936604710 51 7.1 7.00986225298533 0.090137747014672 52 7.5 7.33267214162082 0.167327858379181 53 7.7 7.70709885538763 -0.00709885538762563 54 7.8 7.79062307576246 0.0093769242375388 55 7.8 7.78093302388841 0.0190669761115921 56 7.7 7.6841016220818 0.0158983779181940 57 7.8 7.54077662444173 0.259223375558269 58 7.8 7.70453518935798 0.0954648106420248 59 7.9 7.69204763408379 0.207952365916215 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.0263931641773778 0.0527863283547556 0.973606835822622 20 0.148271753401231 0.296543506802463 0.851728246598769 21 0.101417661472877 0.202835322945754 0.898582338527123 22 0.399673673115649 0.799347346231298 0.600326326884351 23 0.277025970375291 0.554051940750581 0.72297402962471 24 0.340870320513647 0.681740641027294 0.659129679486353 25 0.256686319972698 0.513372639945397 0.743313680027302 26 0.295508752058801 0.591017504117603 0.704491247941199 27 0.417087829559607 0.834175659119214 0.582912170440393 28 0.365006847268078 0.730013694536155 0.634993152731922 29 0.326289983459356 0.652579966918711 0.673710016540644 30 0.279557744586632 0.559115489173264 0.720442255413368 31 0.240559954899215 0.481119909798429 0.759440045100785 32 0.416296812405157 0.832593624810315 0.583703187594843 33 0.578501184076847 0.842997631846305 0.421498815923152 34 0.722430620438666 0.555138759122669 0.277569379561334 35 0.875117404829653 0.249765190340694 0.124882595170347 36 0.7978204810275 0.404359037945002 0.202179518972501 37 0.904564074857248 0.190871850285504 0.0954359251427522 38 0.963526129967549 0.0729477400649024 0.0364738700324512 39 0.9928106108946 0.0143787782108004 0.00718938910540022 40 0.980736449712495 0.038527100575011 0.0192635502875055 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 0.090909090909091 NOK 10% type I error level 4 0.181818181818182 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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