Multiple Regression Model 3

*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 13:36:21 -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/t1258663037p36l2i1gubvv661.htm/, Retrieved Thu, 19 Nov 2009 21:37:29 +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/t1258663037p36l2i1gubvv661.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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10 24.1 9.2 24.1 9.2 24.1 9.5 21.3 9.6 21.3 9.5 21.3 9.1 19.1 8.9 19.1 9 19.1 10.1 26.2 10.3 26.2 10.2 26.2 9.6 21.7 9.2 21.7 9.3 21.7 9.4 19.4 9.4 19.4 9.2 19.4 9 19.5 9 19.5 9 19.5 9.8 28.7 10 28.7 9.8 28.7 9.3 21.8 9 21.8 9 21.8 9.1 20 9.1 20 9.1 20 9.2 22.6 8.8 22.6 8.3 22.6 8.4 22.4 8.1 22.4 7.7 22.4 7.9 18.6 7.9 18.6 8 18.6 7.9 16.2 7.6 16.2 7.1 16.2 6.8 13.8 6.5 13.8 6.9 13.8 8.2 24.1 8.7 24.1 8.3 24.1 7.9 19.9 7.5 19.9 7.8 19.9 8.3 22.3 8.4 22.3 8.2 22.3 7.7 20.9 7.2 20.9 7.3 20.9 8.1 25.5 8.5 25.5 8.4 25.5

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation TWV[t] = + 5.8018408319185 + 0.169475099037918`WV-25`[t] + 0.391285653650254M1[t] + 0.0452611771363897M2[t] + 0.179236700622525M3[t] + 0.627087860780986M4[t] + 0.641063384267121M5[t] + 0.475038907753255M6[t] + 0.360867996604415M7[t] + 0.114843520090550M8[t] + 0.168819043576685M9[t] -0.027951046972269M10[t] + 0.206024476513865M11[t] -0.0339755234861347t + 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) 5.8018408319185 0.546032 10.6255 0 0 `WV-25` 0.169475099037918 0.019263 8.798 0 0 M1 0.391285653650254 0.22511 1.7382 0.088866 0.044433 M2 0.0452611771363897 0.224496 0.2016 0.841109 0.420554 M3 0.179236700622525 0.223911 0.8005 0.427547 0.213773 M4 0.627087860780986 0.234967 2.6688 0.010479 0.00524 M5 0.641063384267121 0.234371 2.7353 0.008824 0.004412 M6 0.475038907753255 0.233802 2.0318 0.047973 0.023987 M7 0.360867996604415 0.239567 1.5063 0.138818 0.069409 M8 0.114843520090550 0.239023 0.4805 0.63317 0.316585 M9 0.168819043576685 0.238506 0.7078 0.482629 0.241314 M10 -0.027951046972269 0.204914 -0.1364 0.892097 0.446049 M11 0.206024476513865 0.204864 1.0057 0.319838 0.159919 t -0.0339755234861347 0.002601 -13.06 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.94897538063781 R-squared 0.900554273056676 Adjusted R-squared 0.872450045877041 F-TEST (value) 32.0433743756968 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.323892318772898 Sum Squared Residuals 4.8256867713639 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 10 10.2435008488964 -0.243500848896442 2 9.2 9.86350084889643 -0.663500848896434 3 9.2 9.96350084889643 -0.763500848896435 4 9.5 9.9028462082626 -0.402846208262591 5 9.6 9.8828462082626 -0.282846208262592 6 9.5 9.6828462082626 -0.182846208262592 7 9.1 9.1618545557442 -0.0618545557441999 8 8.9 8.8818545557442 0.0181454442558009 9 9 8.9018545557442 0.0981454442558003 10 10.1 9.87438214487832 0.225617855121676 11 10.3 10.0743821448783 0.225617855121676 12 10.2 9.83438214487832 0.365617855121675 13 9.6 9.42905432937182 0.170945670628185 14 9.2 9.04905432937182 0.150945670628183 15 9.3 9.14905432937182 0.150945670628184 16 9.4 9.17313723825693 0.226862761743068 17 9.4 9.15313723825693 0.246862761743068 18 9.2 8.95313723825693 0.246862761743068 19 9 8.82193831352575 0.178061686474251 20 9 8.54193831352575 0.45806168647425 21 9 8.56193831352575 0.438061686474250 22 9.8 9.8903636106395 -0.0903636106395012 23 10 10.0903636106395 -0.0903636106395018 24 9.8 9.8503636106395 -0.0503636106395007 25 9.3 9.038295557442 0.261704442558010 26 9 8.658295557442 0.341704442558008 27 9 8.758295557442 0.241704442558008 28 9.1 8.86711601584607 0.232883984153932 29 9.1 8.84711601584607 0.252883984153933 30 9.1 8.64711601584607 0.452883984153933 31 9.2 8.93960483870968 0.260395161290322 32 8.8 8.65960483870968 0.140395161290323 33 8.3 8.67960483870968 -0.379604838709677 34 8.4 8.414964204867 -0.0149642048670055 35 8.1 8.614964204867 -0.514964204867006 36 7.7 8.374964204867 -0.674964204867005 37 7.9 8.08826895868704 -0.188268958687038 38 7.9 7.70826895868704 0.19173104131296 39 8 7.80826895868704 0.191731041312959 40 7.9 7.81540435766837 0.0845956423316355 41 7.6 7.79540435766836 -0.195404357668365 42 7.1 7.59540435766836 -0.495404357668364 43 6.8 7.04051768534239 -0.240517685342388 44 6.5 6.76051768534239 -0.260517685342388 45 6.9 6.78051768534239 0.119482314657612 46 8.2 8.29536559139785 -0.0953655913978506 47 8.7 8.49536559139785 0.204634408602149 48 8.3 8.25536559139785 0.0446344086021515 49 7.9 7.90088030560271 -0.00088030560271447 50 7.5 7.52088030560272 -0.0208803056027165 51 7.8 7.62088030560272 0.179119694397283 52 8.3 8.44149617996604 -0.141496179966045 53 8.4 8.42149617996605 -0.0214961799660445 54 8.2 8.22149617996604 -0.0214961799660452 55 7.7 7.83608460667799 -0.136084606677985 56 7.2 7.55608460667799 -0.356084606677986 57 7.3 7.57608460667799 -0.276084606677986 58 8.1 8.12492444821732 -0.0249244482173187 59 8.5 8.32492444821732 0.175075551782682 60 8.4 8.08492444821732 0.315075551782682 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.172779626822282 0.345559253644564 0.827220373177718 18 0.0859977206166429 0.171995441233286 0.914002279383357 19 0.042107397916807 0.084214795833614 0.957892602083193 20 0.0282338908194061 0.0564677816388122 0.971766109180594 21 0.0151356531328832 0.0302713062657664 0.984864346867117 22 0.0159286522869259 0.0318573045738519 0.984071347713074 23 0.00909840878325215 0.0181968175665043 0.990901591216748 24 0.00586030098722628 0.0117206019744526 0.994139699012774 25 0.00640955784415428 0.0128191156883086 0.993590442155846 26 0.00295772033001304 0.00591544066002609 0.997042279669987 27 0.00117729782321747 0.00235459564643493 0.998822702176783 28 0.000509642846894112 0.00101928569378822 0.999490357153106 29 0.000279626406053876 0.000559252812107753 0.999720373593946 30 0.000445376683116328 0.000890753366232657 0.999554623316884 31 0.00536627952735794 0.0107325590547159 0.994633720472642 32 0.0165592171954506 0.0331184343909013 0.98344078280455 33 0.0478143150248297 0.0956286300496594 0.95218568497517 34 0.58533487017873 0.82933025964254 0.41466512982127 35 0.881357898178846 0.237284203642307 0.118642101821154 36 0.99172631068892 0.0165473786221595 0.00827368931107973 37 0.985608619053027 0.0287827618939453 0.0143913809469726 38 0.978797083966787 0.0424058320664252 0.0212029160332126 39 0.957377014407276 0.0852459711854474 0.0426229855927237 40 0.934717861146 0.130564277708 0.065282138854 41 0.886836378894011 0.226327242211977 0.113163621105989 42 0.958541665035612 0.0829166699287756 0.0414583349643878 43 0.952915523254095 0.0941689534918109 0.0470844767459054 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 5 0.185185185185185 NOK 5% type I error level 15 0.555555555555556 NOK 10% type I error level 21 0.777777777777778 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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