*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: Wed, 18 Nov 2009 11:46:54 -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/t1258570092fi8qoix2j7io0tm.htm/, Retrieved Wed, 18 Nov 2009 19:48:24 +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/18/t1258570092fi8qoix2j7io0tm.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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3030.29 101.2 2803.47 101.1 2767.63 100.7 2882.6 100.1 2863.36 99.9 2897.06 99.7 3012.61 99.5 3142.95 99.2 3032.93 99 3045.78 99 3110.52 99.3 3013.24 99.5 2987.1 99.7 2995.55 100 2833.18 100.4 2848.96 100.6 2794.83 100.7 2845.26 100.7 2915.02 100.6 2892.63 100.5 2604.42 100.6 2641.65 100.5 2659.81 100.4 2638.53 100.3 2720.25 100.4 2745.88 100.4 2735.7 100.4 2811.7 100.4 2799.43 100.4 2555.28 100.5 2304.98 100.6 2214.95 100.6 2065.81 100.5 1940.49 100.5 2042.00 100.7 1995.37 101.1 1946.81 101.5 1765.9 101.9 1635.25 102.1 1833.42 102.1 1910.43 102.1 1959.67 102.4 1969.6 102.8 2061.41 103.1 2093.48 103.1 2120.88 102.9 2174.56 102.4 2196.72 101.9 2350.44 101.3 2440.25 100.7 2408.64 100.6 2472.81 101 2407.6 101.5 2454.62 101.9 2448.05 102.1 2497.84 102.3 2645.64 102.5 2756.76 102.9 2849.27 103.6 2921.44 104.3

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 Gzhidx[t] = + 100.445809918502 -0.0003448988264416Bel20[t] + 0.0483895407575485t + 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) 100.445809918502 0.900743 111.5143 0 0 Bel20 -0.0003448988264416 0.000298 -1.1569 0.252121 0.12606 t 0.0483895407575485 0.006859 7.055 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.784739150333494 R-squared 0.615815534066134 Adjusted R-squared 0.6023353773667 F-TEST (value) 45.6831139130597 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 1.44340095431517e-12 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.749246965713169 Sum Squared Residuals 31.9981478909323 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 101.2 99.449055994482 1.75094400551790 2 101.1 99.5756754870528 1.52432451294723 3 100.7 99.63642620175 1.06357379825002 4 100.1 99.6451627244315 0.454837275568457 5 99.9 99.7001881186098 0.199811881390184 6 99.7 99.7369545689163 -0.036954568916286 7 99.5 99.7454910502785 -0.245491050278510 8 99.2 99.7489264779977 -0.548926477997658 9 99 99.8352617876403 -0.835261787640314 10 99 99.879219378478 -0.879219378478088 11 99.3 99.9052801692118 -0.60528016921181 12 99.5 99.9872214678056 -0.487221467805595 13 99.7 100.044626663886 -0.344626663886324 14 100 100.090101809560 -0.0901018095604436 15 100.4 100.194492572767 0.205507427232691 16 100.6 100.237439610044 0.362560389956380 17 100.7 100.304498524276 0.395501475723556 18 100.7 100.335494817217 0.364505182783457 19 100.6 100.359824215842 0.240175784158466 20 100.5 100.415936041323 0.084063958676896 21 100.6 100.563728872849 0.0362711271506085 22 100.5 100.599277830299 -0.0992778302985136 23 100.4 100.641404008368 -0.241404008367877 24 100.3 100.697132996152 -0.397132996152111 25 100.4 100.717337404813 -0.317337404812844 26 100.4 100.756887188649 -0.356887188648694 27 100.4 100.808787799459 -0.408787799459418 28 100.4 100.830965029407 -0.430965029407405 29 100.4 100.883586478765 -0.483586478765392 30 100.5 101.016183067999 -0.516183067998663 31 100.6 101.150900785015 -0.550900785014549 32 100.6 101.230341567117 -0.630341567116635 33 100.5 101.330169318850 -0.830169318849678 34 100.5 101.421781580537 -0.921781580536887 35 100.7 101.435160441422 -0.735160441422346 36 101.1 101.499632614457 -0.399632614456875 37 101.5 101.564770442226 -0.0647704422264223 38 101.9 101.675555629676 0.224444370324485 39 102.1 101.769006202108 0.330993797892331 40 102.1 101.749047142429 0.350952857570714 41 102.1 101.770876024563 0.329123975437433 42 102.4 101.802282747106 0.59771725289388 43 102.8 101.847247442517 0.952752557482888 44 103.1 101.863971822019 1.23602817798094 45 103.1 101.901300457413 1.19869954258737 46 102.9 101.940239770326 0.959760229674336 47 102.4 101.970115142080 0.429884857920173 48 101.9 102.010861724843 -0.11086172484343 49 101.3 102.006233418000 -0.706233418000384 50 100.7 102.023647595155 -1.32364759515521 51 100.6 102.082939387817 -1.48293938781658 52 101 102.109196770881 -1.10919677088137 53 101.5 102.180077164111 -0.680077164111174 54 101.9 102.212249562049 -0.312249562049432 55 102.1 102.262905088097 -0.162905088096713 56 102.3 102.294122116286 0.00587788371426798 57 102.5 102.291535610495 0.208464389504791 58 102.9 102.301599993659 0.598400006341438 59 103.6 102.318082943982 1.28191705601799 60 104.3 102.341581136435 1.95841886356474 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.0107768402339661 0.0215536804679322 0.989223159766034 7 0.00608838554383647 0.0121767710876729 0.993911614456163 8 0.00155783697921221 0.00311567395842443 0.998442163020788 9 0.00047728704873582 0.00095457409747164 0.999522712951264 10 0.000645273789006602 0.00129054757801320 0.999354726210993 11 0.0077001871799187 0.0154003743598374 0.992299812820081 12 0.0318209203040236 0.0636418406080473 0.968179079695976 13 0.06308804850926 0.12617609701852 0.93691195149074 14 0.113560588840654 0.227121177681307 0.886439411159346 15 0.114409504250112 0.228819008500224 0.885590495749888 16 0.114853774630169 0.229707549260338 0.885146225369831 17 0.0915666895254915 0.183133379050983 0.908433310474508 18 0.0799874164136122 0.159974832827224 0.920012583586388 19 0.0763138749979186 0.152627749995837 0.923686125002081 20 0.0570226933878343 0.114045386775669 0.942977306612166 21 0.0643204448347024 0.128640889669405 0.935679555165298 22 0.0522827633696943 0.104565526739389 0.947717236630306 23 0.0388294860723995 0.077658972144799 0.9611705139276 24 0.0292549182136868 0.0585098364273736 0.970745081786313 25 0.0184310272496815 0.0368620544993630 0.981568972750319 26 0.0113781618247325 0.022756323649465 0.988621838175267 27 0.00678587536427857 0.0135717507285571 0.993214124635721 28 0.00444258486400467 0.00888516972800935 0.995557415135995 29 0.00284826718308700 0.00569653436617399 0.997151732816913 30 0.00204917532356174 0.00409835064712349 0.997950824676438 31 0.00250665816557411 0.00501331633114822 0.997493341834426 32 0.00245574774370203 0.00491149548740407 0.997544252256298 33 0.00254628733287257 0.00509257466574513 0.997453712667127 34 0.00248579190314352 0.00497158380628704 0.997514208096856 35 0.00158596872775184 0.00317193745550369 0.998414031272248 36 0.00097337692017536 0.00194675384035072 0.999026623079825 37 0.000692626398447790 0.00138525279689558 0.999307373601552 38 0.000474225984717692 0.000948451969435385 0.999525774015282 39 0.000299426271252842 0.000598852542505684 0.999700573728747 40 0.00022566125658265 0.0004513225131653 0.999774338743417 41 0.000174450948700117 0.000348901897400234 0.9998255490513 42 0.000226030988709789 0.000452061977419578 0.99977396901129 43 0.000693844913984343 0.00138768982796869 0.999306155086016 44 0.00476168305440548 0.00952336610881097 0.995238316945595 45 0.0312403785594689 0.0624807571189377 0.968759621440531 46 0.174617838117056 0.349235676234112 0.825382161882944 47 0.536226059484385 0.92754788103123 0.463773940515615 48 0.949417733094874 0.101164533810251 0.0505822669051257 49 0.997265510462471 0.00546897907505795 0.00273448953752898 50 0.995290453802295 0.00941909239541017 0.00470954619770509 51 0.98844202976142 0.0231159404771609 0.0115579702385804 52 0.976526586670427 0.0469468266591469 0.0234734133295734 53 0.936412264539578 0.127175470920843 0.0635877354604216 54 0.908701645919244 0.182596708161513 0.0912983540807564 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 22 0.448979591836735 NOK 5% type I error level 30 0.612244897959184 NOK 10% type I error level 34 0.693877551020408 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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