Model 1

*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 05:32:50 -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/t1258721490344k64nr1y1qk6a.htm/, Retrieved Fri, 20 Nov 2009 13:51:42 +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/t1258721490344k64nr1y1qk6a.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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108.5 98.71 112.3 98.54 116.6 98.2 115.5 96.92 120.1 99.06 132.9 99.65 128.1 99.82 129.3 99.99 132.5 100.33 131 99.31 124.9 101.1 120.8 101.1 122 100.93 122.1 100.85 127.4 100.93 135.2 99.6 137.3 101.88 135 101.81 136 102.38 138.4 102.74 134.7 102.82 138.4 101.72 133.9 103.47 133.6 102.98 141.2 102.68 151.8 102.9 155.4 103.03 156.6 101.29 161.6 103.69 160.7 103.68 156 104.2 159.5 104.08 168.7 104.16 169.9 103.05 169.9 104.66 185.9 104.46 190.8 104.95 195.8 105.85 211.9 106.23 227.1 104.86 251.3 107.44 256.7 108.23 251.9 108.45 251.2 109.39 270.3 110.15 267.2 109.13 243 110.28 229.9 110.17 187.2 109.99 178.2 109.26 175.2 109.11 192.4 107.06 187 109.53 184 108.92 194.1 109.24 212.7 109.12 217.5 109 200.5 107.23 205.9 109.49 196.5 109.04 206.3 109.02

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] = -873.66330396259 + 10.0004964841749X[t] + 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) -873.66330396259 77.616918 -11.2561 0 0 X 10.0004964841749 0.743018 13.4593 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.868517469066465 R-squared 0.754322594073617 Adjusted R-squared 0.750158570244357 F-TEST (value) 181.152324050830 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 22.2647862679449 Sum Squared Residuals 29247.5217458788 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 108.5 113.485703990315 -4.98570399031535 2 112.3 111.785619588005 0.51438041199518 3 116.6 108.385450783385 8.21454921661466 4 115.5 95.5848152836414 19.9151847163585 5 120.1 116.985877759776 3.11412224022424 6 132.9 122.886170685439 10.0138293145610 7 128.1 124.586255087749 3.51374491225140 8 129.3 126.286339490058 3.01366050994167 9 132.5 129.686508294678 2.81349170532216 10 131 119.486001880819 11.5139981191805 11 124.9 137.386890587492 -12.4868905874925 12 120.8 137.386890587492 -16.5868905874925 13 122 135.686806185183 -13.6868061851829 14 122.1 134.886766466449 -12.7867664664488 15 127.4 135.686806185183 -8.28680618518287 16 135.2 122.38614586123 12.8138541387699 17 137.3 145.187277845149 -7.88727784514891 18 135 144.487243091257 -9.48724309125675 19 136 150.187526087236 -14.1875260872364 20 138.4 153.787704821539 -15.3877048215393 21 134.7 154.587744540273 -19.8877445402733 22 138.4 143.587198407681 -5.18719840768097 23 133.9 161.088067254987 -27.1880672549871 24 133.6 156.187823977741 -22.5878239777414 25 141.2 153.187675032489 -11.9876750324890 26 151.8 155.387784259007 -3.58778425900742 27 155.4 156.687848801950 -1.28784880195012 28 156.6 139.286984919486 17.3130150805142 29 161.6 163.288176481506 -1.68817648150553 30 160.7 163.188171516664 -2.48817151666388 31 156 168.388429688435 -12.3884296884348 32 159.5 167.188370110334 -7.68837011033374 33 168.7 167.988409829068 0.711590170932269 34 169.9 156.887858731634 13.0121412683664 35 169.9 172.988658071155 -3.08865807115517 36 185.9 170.98855877432 14.9114412256798 37 190.8 175.888802051566 14.9111979484341 38 195.8 184.889248887323 10.9107511126767 39 211.9 188.689437551310 23.2105624486902 40 227.1 174.988757367990 52.1112426320098 41 251.3 200.790038297161 50.5099617028386 42 256.7 208.690430519660 48.0095694803403 43 251.9 210.890539746178 41.0094602538219 44 251.2 220.291006441303 30.9089935586975 45 270.3 227.891383769275 42.4086162307245 46 267.2 217.690877355417 49.509122644583 47 243 229.191448312218 13.8085516877818 48 229.9 228.091393698959 1.80860630104105 49 187.2 226.291304331807 -39.0913043318074 50 178.2 218.99094189836 -40.7909418983598 51 175.2 217.490867425734 -42.2908674257335 52 192.4 196.989849633175 -4.589849633175 53 187 221.691075949087 -34.691075949087 54 184 215.590773093740 -31.5907730937403 55 194.1 218.790931968676 -24.6909319686762 56 212.7 217.590872390575 -4.89087239057535 57 217.5 216.390812812474 1.10918718752570 58 200.5 198.689934035485 1.81006596451524 59 205.9 221.29105608972 -15.3910560897199 60 196.5 216.790832671841 -20.2908326718414 61 206.3 216.590822742158 -10.2908227421577 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0143647311786359 0.0287294623572718 0.985635268821364 6 0.0252211246729953 0.0504422493459907 0.974778875327005 7 0.00800843981809698 0.0160168796361940 0.991991560181903 8 0.00229005668505935 0.00458011337011871 0.997709943314941 9 0.000626141236078194 0.00125228247215639 0.999373858763922 10 0.000278772512765352 0.000557545025530705 0.999721227487235 11 0.000178858635371912 0.000357717270743824 0.999821141364628 12 0.000124132088793737 0.000248264177587473 0.999875867911206 13 4.7801849389359e-05 9.5603698778718e-05 0.99995219815061 14 1.59531565081441e-05 3.19063130162881e-05 0.999984046843492 15 4.15915919858439e-06 8.31831839716879e-06 0.999995840840801 16 4.71777624499913e-06 9.43555248999826e-06 0.999995282223755 17 1.78266967915192e-06 3.56533935830383e-06 0.99999821733032 18 5.15082077065081e-07 1.03016415413016e-06 0.999999484917923 19 1.39216197907072e-07 2.78432395814145e-07 0.999999860783802 20 3.80902030835625e-08 7.6180406167125e-08 0.999999961909797 21 1.08902535520152e-08 2.17805071040303e-08 0.999999989109746 22 3.84208394818238e-09 7.68416789636475e-09 0.999999996157916 23 1.73508238769923e-09 3.47016477539846e-09 0.999999998264918 24 6.11915514520135e-10 1.22383102904027e-09 0.999999999388084 25 2.38911529340144e-10 4.77823058680288e-10 0.999999999761088 26 5.60130907505944e-10 1.12026181501189e-09 0.99999999943987 27 1.44930997885129e-09 2.89861995770258e-09 0.99999999855069 28 2.11692018178332e-08 4.23384036356664e-08 0.999999978830798 29 2.73202714068092e-08 5.46405428136183e-08 0.999999972679729 30 2.29734246525285e-08 4.59468493050569e-08 0.999999977026575 31 1.10707523352930e-08 2.21415046705860e-08 0.999999988929248 32 6.53527651273406e-09 1.30705530254681e-08 0.999999993464723 33 8.05462982439929e-09 1.61092596487986e-08 0.99999999194537 34 2.65085920087019e-08 5.30171840174039e-08 0.999999973491408 35 2.15693688316131e-08 4.31387376632262e-08 0.999999978430631 36 9.50203553411203e-08 1.90040710682241e-07 0.999999904979645 37 2.51350481295656e-07 5.02700962591312e-07 0.999999748649519 38 3.20849929433431e-07 6.41699858866861e-07 0.99999967915007 39 8.9435072159145e-07 1.7887014431829e-06 0.999999105649278 40 4.20379789486717e-05 8.40759578973433e-05 0.999957962021051 41 0.000444152211125601 0.000888304422251203 0.999555847788874 42 0.00223357510906924 0.00446715021813849 0.99776642489093 43 0.00600666616410136 0.0120133323282027 0.993993333835899 44 0.00850882986701048 0.0170176597340210 0.99149117013299 45 0.0351453588866007 0.0702907177732014 0.9648546411134 46 0.353049977670235 0.70609995534047 0.646950022329765 47 0.606409481016483 0.787181037967034 0.393590518983517 48 0.832660436519391 0.334679126961217 0.167339563480608 49 0.877576507542923 0.244846984914153 0.122423492457077 50 0.927094708591794 0.145810582816412 0.0729052914082059 51 0.973756947942697 0.0524861041146052 0.0262430520573026 52 0.952395249986489 0.0952095000270227 0.0476047500135113 53 0.953308498489513 0.0933830030209749 0.0466915015104874 54 0.976746558553191 0.0465068828936173 0.0232534414468087 55 0.974234109445724 0.0515317811085522 0.0257658905542761 56 0.93306994877034 0.133860102459318 0.0669300512296592 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 35 0.673076923076923 NOK 5% type I error level 40 0.769230769230769 NOK 10% type I error level 46 0.884615384615385 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No 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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