WS 8 multiple regression

*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: Mon, 29 Nov 2010 11:11:13 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Nov/29/t12910290671dmrmxzd0xw9jel.htm/, Retrieved Mon, 29 Nov 2010 12:11:27 +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/2010/Nov/29/t12910290671dmrmxzd0xw9jel.htm/}, year = {2010}, } @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 = {2010}, 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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37 1 30 1 47 1 35 1 30 1 43 1 82 1 40 1 47 1 19 0 52 1 136 1 80 1 42 1 54 1 66 1 81 1 63 1 137 1 72 1 107 1 58 1 36 1 52 1 79 1 77 1 54 1 84 1 48 1 96 1 83 1 66 1 61 1 53 1 30 1 74 1 69 1 59 1 42 1 65 1 70 1 100 1 63 1 105 1 82 1 81 1 75 1 102 1 121 1 98 1 76 1 77 1 63 1 37 1 35 1 23 0 40 1 29 0 37 1 51 1 20 0 28 0 13 0 22 0 25 0 13 0 16 0 13 0 16 0 17 0 9 0 17 0 25 0 14 0 8 0 7 0 10 0 7 0 10 0 3 0

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 9 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Sol.KIA[t] = + 18.434748427673 + 56.6415094339623dummy[t] -2.92791629230297M1[t] -14.9365828092243M2[t] -22.8023921832884M3[t] -14.0967729859239M4[t] -18.3911537885594M5[t] -13.9712488769092M6[t] -4.55134396525905M7[t] -11.4683662773285M8[t] -12.7120956873315M9[t] -9.98311620245583M10[t] -32.0151430068883M11[t] + 0.151523659778377t + 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) 18.434748427673 15.66021 1.1772 0.243355 0.121677 dummy 56.6415094339623 8.319685 6.8081 0 0 M1 -2.92791629230297 12.541119 -0.2335 0.816123 0.408061 M2 -14.9365828092243 12.522574 -1.1928 0.237229 0.118614 M3 -22.8023921832884 12.506057 -1.8233 0.072786 0.036393 M4 -14.0967729859239 12.491574 -1.1285 0.263193 0.131597 M5 -18.3911537885594 12.479134 -1.4738 0.145302 0.072651 M6 -13.9712488769092 12.468742 -1.1205 0.266561 0.13328 M7 -4.55134396525905 12.460404 -0.3653 0.716082 0.358041 M8 -11.4683662773285 12.582191 -0.9115 0.365362 0.182681 M9 -12.7120956873315 12.907499 -0.9849 0.328289 0.164144 M10 -9.98311620245583 13.246653 -0.7536 0.45375 0.226875 M11 -32.0151430068883 12.899535 -2.4819 0.015622 0.007811 t 0.151523659778377 0.160287 0.9453 0.347941 0.173971 Multiple Linear Regression - Regression Statistics Multiple R 0.768582483195594 R-squared 0.590719033475105 Adjusted R-squared 0.510103085523232 F-TEST (value) 7.32757039373595 F-TEST (DF numerator) 13 F-TEST (DF denominator) 66 p-value 1.27160473262222e-08 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 22.3409242967196 Sum Squared Residuals 32941.7152964959 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 37 72.29986522911 -35.29986522911 2 30 60.4427223719676 -30.4427223719676 3 47 52.728436657682 -5.72843665768199 4 35 61.5855795148248 -26.5855795148248 5 30 57.4427223719677 -27.4427223719677 6 43 62.0141509433963 -19.0141509433963 7 82 71.5855795148248 10.4144204851752 8 40 64.8200808625337 -24.8200808625337 9 47 63.7278751123091 -16.7278751123091 10 19 9.96686882300089 9.03313117699911 11 52 44.7278751123091 7.27212488769092 12 136 76.8945417789757 59.1054582210243 13 80 74.1181491464511 5.88185085354886 14 42 62.2610062893082 -20.2610062893082 15 54 54.5467205750225 -0.546720575022459 16 66 63.4038634321653 2.59613656783467 17 81 59.2610062893082 21.7389937106918 18 63 63.8324348607368 -0.832434860736757 19 137 73.4038634321653 63.5961365678347 20 72 66.6383647798742 5.36163522012578 21 107 65.5461590296496 41.4538409703504 22 58 68.4266621743037 -10.4266621743037 23 36 46.5461590296496 -10.5461590296496 24 52 78.7128256963163 -26.7128256963163 25 79 75.9364330637916 3.06356693620836 26 77 64.0792902066487 12.9207097933513 27 54 56.365004492363 -2.36500449236298 28 84 65.2221473495058 18.7778526504942 29 48 61.0792902066487 -13.0792902066487 30 96 65.6507187780773 30.3492812219227 31 83 75.2221473495058 7.77785265049416 32 66 68.4566486972147 -2.45664869721474 33 61 67.3644429469901 -6.36444294699012 34 53 70.2449460916442 -17.2449460916442 35 30 48.3644429469901 -18.3644429469901 36 74 80.5311096136568 -6.53110961365679 37 69 77.7547169811322 -8.75471698113216 38 59 65.8975741239892 -6.89757412398922 39 42 58.1832884097035 -16.1832884097035 40 65 67.0404312668464 -2.04043126684637 41 70 62.8975741239892 7.10242587601078 42 100 67.4690026954178 32.5309973045822 43 63 77.0404312668464 -14.0404312668464 44 105 70.2749326145553 34.7250673854447 45 82 69.1827268643306 12.8172731356694 46 81 72.0632300089847 8.93676999101528 47 75 50.1827268643306 24.8172731356694 48 102 82.3493935309973 19.6506064690027 49 121 79.5730008984727 41.4269991015273 50 98 67.7158580413297 30.2841419586703 51 76 60.001572327044 15.9984276729560 52 77 68.8587151841869 8.14128481581311 53 63 64.7158580413297 -1.71585804132975 54 37 69.2872866127583 -32.2872866127583 55 35 78.8587151841869 -43.8587151841869 56 23 15.4517070979335 7.5482929020665 57 40 71.0010107816712 -31.0010107816712 58 29 17.2400044923630 11.7599955076370 59 37 52.0010107816712 -15.0010107816712 60 51 84.1676774483378 -33.1676774483378 61 20 24.7497753818509 -4.74977538185092 62 28 12.8926325247080 15.1073674752920 63 13 5.17834681042225 7.82165318957775 64 22 14.0354896675651 7.96451033243488 65 25 9.89263252470797 15.1073674752920 66 13 14.4640610961365 -1.46406109613655 67 16 24.0354896675651 -8.03548966756511 68 13 17.2699910152740 -4.26999101527402 69 16 16.1777852650494 -0.177785265049395 70 17 19.0582884097035 -2.05828840970348 71 9 -2.82221473495059 11.8222147349506 72 17 29.3444519317161 -12.3444519317161 73 25 26.5680592991914 -1.56805929919144 74 14 14.7109164420485 -0.71091644204849 75 8 6.99663072776277 1.00336927223723 76 7 15.8537735849056 -8.85377358490564 77 10 11.7109164420485 -1.71091644204850 78 7 16.2823450134771 -9.28234501347707 79 10 25.8537735849056 -15.8537735849056 80 3 19.0882749326145 -16.0882749326145 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.381936551975911 0.763873103951823 0.618063448024089 18 0.24037307516221 0.48074615032442 0.75962692483779 19 0.33521721122036 0.67043442244072 0.66478278877964 20 0.21669057379443 0.43338114758886 0.78330942620557 21 0.28611652264928 0.57223304529856 0.71388347735072 22 0.196576970604890 0.393153941209779 0.80342302939511 23 0.457385643000375 0.91477128600075 0.542614356999625 24 0.9867700954771 0.0264598090458003 0.0132299045229002 25 0.978152365805861 0.0436952683882773 0.0218476341941386 26 0.965936807593717 0.0681263848125657 0.0340631924062828 27 0.958558886901715 0.0828822261965702 0.0414411130982851 28 0.937584692536943 0.124830614926114 0.0624153074630569 29 0.947603057669615 0.104793884660769 0.0523969423303847 30 0.939631807533893 0.120736384932214 0.0603681924661068 31 0.962135288338956 0.075729423322088 0.037864711661044 32 0.946966647260492 0.106066705479017 0.0530333527395084 33 0.943183195735576 0.113633608528847 0.0568168042644236 34 0.9393191456055 0.121361708788999 0.0606808543944997 35 0.957218188187975 0.0855636236240509 0.0427818118120254 36 0.953989302647478 0.0920213947050431 0.0460106973525216 37 0.956124500516595 0.087750998966809 0.0438754994834045 38 0.96362719981421 0.0727456003715791 0.0363728001857895 39 0.981944030523772 0.0361119389524552 0.0180559694762276 40 0.98078222668758 0.0384355466248388 0.0192177733124194 41 0.976015063707327 0.0479698725853465 0.0239849362926732 42 0.978801699447891 0.0423966011042171 0.0211983005521086 43 0.983857291182053 0.0322854176358931 0.0161427088179465 44 0.98954580643183 0.0209083871363413 0.0104541935681707 45 0.984986996898584 0.0300260062028325 0.0150130031014162 46 0.976387801802037 0.0472243963959264 0.0236121981979632 47 0.970192847973634 0.0596143040527326 0.0298071520263663 48 0.978728271269404 0.0425434574611914 0.0212717287305957 49 0.998887048190912 0.00222590361817529 0.00111295180908764 50 0.999858628220736 0.000282743558528988 0.000141371779264494 51 0.999971607003784 5.67859924319369e-05 2.83929962159684e-05 52 0.99999952122614 9.57547721325763e-07 4.78773860662881e-07 53 0.999999879255944 2.41488111227182e-07 1.20744055613591e-07 54 0.999999716385217 5.67229566311273e-07 2.83614783155637e-07 55 0.999999839097784 3.21804431011546e-07 1.60902215505773e-07 56 0.999999091586265 1.81682746963561e-06 9.08413734817806e-07 57 0.999997173011345 5.65397731053823e-06 2.82698865526911e-06 58 0.999985243661943 2.95126761134503e-05 1.47563380567252e-05 59 0.99993257169073 0.000134856618539556 6.74283092697778e-05 60 0.999690718627975 0.000618562744050809 0.000309281372025405 61 0.999850143739166 0.000299712521668523 0.000149856260834261 62 0.9991457325925 0.00170853481499908 0.00085426740749954 63 0.995657755863415 0.00868448827316934 0.00434224413658467 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 15 0.319148936170213 NOK 5% type I error level 26 0.553191489361702 NOK 10% type I error level 34 0.723404255319149 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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