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: Sun, 19 Dec 2010 22:14:18 +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/Dec/19/t1292796739d5wiexwar949yrc.htm/, Retrieved Sun, 19 Dec 2010 23:12:22 +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/Dec/19/t1292796739d5wiexwar949yrc.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:

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No (this computation is public)

User-defined keywords:

Dataseries X:

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0,762253 1,3119 35.16 0,768403 1,3014 41.54 0,757518 1,3201 45.07 0,772917 1,2938 46.84 0,787774 1,2694 45.20 0,82203 1,2165 46.65 0,830772 1,2037 52.55 0,813537 1,2292 55.05 0,815927 1,2256 60.75 0,832293 1,2015 55.99 0,848464 1,1786 53.39 0,843455 1,1856 49.42 0,826241 1,2103 55.12 0,837661 1,1938 59.84 0,831947 1,202 55.98 0,81493 1,2271 61.27 0,783085 1,277 66.94 0,790514 1,265 64.67 0,788395 1,2684 67.74 0,780579 1,2811 69.79 0,785731 1,2727 64.49 0,792959 1,2611 54.92 0,776337 1,2881 53.32 0,75683 1,3213 56.13 0,76929 1,2999 54.63 0,764877 1,3074 52.11 0,755173 1,3242 57.83 0,739864 1,3516 64.93 0,740138 1,3511 63.40 0,745212 1,3419 65.37 0,729076 1,3716 69.91 0,734107 1,3622 73.81 0,719632 1,3896 71.42 0,702889 1,4227 75.57 0,681013 1,4684 86.02 0,686342 1,457 85.91 0,67944 1,4718 92.93 0,678058 1,4748 88.71 0,644039 1,5527 98.01 0,63488 1,5751 98.39 0,642797 1,5557 110.21 0,642963 1,5553 121.36 0,634115 1,577 137.11 0,66778 1 etc...

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 6 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation Olieprijzen[t] = -2393.44635741823 + 1530.92424367196PeriodegemiddeldeEUR[t] + 977.515418385255PeriodegemiddeldeUS[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) -2393.44635741823 493.701744 -4.848 1e-05 5e-06 PeriodegemiddeldeEUR 1530.92424367196 335.211966 4.567 2.7e-05 1.3e-05 PeriodegemiddeldeUS 977.515418385255 180.916131 5.4031 1e-06 1e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.833503259517036 R-squared 0.694727683625524 Adjusted R-squared 0.684016374279051 F-TEST (value) 64.8592680085644 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 2.1094237467878e-15 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 12.3394914909879 Sum Squared Residuals 8678.99386460131 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 35.16 55.9077174730628 -20.7477174730628 2 41.54 55.0589896786029 -13.5189896786029 3 45.07 56.674417610038 -11.604417610038 4 46.84 54.5404645348103 -7.70046453481026 5 45.2 53.4340298144444 -8.23402981444441 6 46.65 54.1668050730911 -7.51680507309109 7 52.55 55.03794745594 -2.48794745594007 8 55.05 53.5791112850779 1.47088871492212 9 60.75 53.7189647212669 7.0310352787331 10 55.99 55.2159493101176 0.774050689882439 11 53.39 57.5874221735147 -4.19742217351468 12 49.42 56.7616305656584 -7.34163056565844 13 55.12 54.5529314692051 0.56706853079489 14 59.84 55.9070819285822 3.93291807141779 15 55.98 55.1750072309997 0.804992769000288 16 61.27 53.658906377904 7.61109362209598 17 66.94 53.6846432155944 13.2553567844056 18 64.67 53.3276944012104 11.3423055987896 19 67.74 53.4072183513793 14.3327816486207 20 69.79 53.855960276332 15.934039723668 21 64.49 53.5321524652938 10.9578475347062 22 54.92 53.2584940452859 1.66150595471406 23 53.32 54.2043875633725 -0.884387563372523 24 56.13 56.7941602324538 -0.664160232453843 25 54.63 54.9506463551622 -0.320646355162208 26 52.11 55.5260433057271 -3.4160433057271 27 57.83 57.0922134740067 0.73778652599326 28 64.93 60.4392166913885 4.4907833086115 29 63.4 60.369932224962 3.03006777503796 30 65.37 59.1446999882094 6.22530001179062 31 69.91 63.4739143183605 6.43608568163953 32 73.81 61.9873492554529 11.8226507445471 33 71.42 66.6111432920572 4.80885670794278 34 75.57 73.3346390288095 2.23536097119046 35 86.02 84.5165948944477 1.5034051055523 36 85.91 81.5312144193839 4.37878558061614 37 92.93 85.4320034816617 7.49799651833829 38 88.71 86.248812432063 2.46118756793702 39 98.01 110.316751678798 -12.3067516787977 40 98.39 118.191361902836 -19.8013619028358 41 110.21 111.347890023313 -1.13789002331287 42 121.36 111.211017280408 10.1489827195919 43 137.11 118.877484151359 18.2325158486412 44 121.29 92.7035730529478 28.5864269470522 45 106.41 76.6042944272333 29.8057055727667 46 93.38 57.9695953760365 35.4104046239635 47 58.66 53.5493855074356 5.11061449256436 48 43.12 59.5328501503823 -16.4128501503823 49 34.57 57.0607468941592 -22.4907468941591 50 41.77 53.7455278874814 -11.9755278874814 51 42.85 55.3340167124489 -12.4840167124489 52 48.09 56.5666947718148 -8.47669477181476 53 48.91 62.4188205159615 -13.5088205159615 54 65.62 68.9077731231737 -3.28777312317366 55 68.47 70.3641343431196 -1.89413434311956 56 71.52 74.2499852719717 -2.72998527197168 57 68.07 81.3263605245398 -13.2563605245398 58 65 88.1316810307755 -23.1316810307755 59 76.34 90.9216831102657 -14.5816831102657 60 76.18 82.6678618486089 -6.48786184860892 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.0807699925858095 0.161539985171619 0.91923000741419 7 0.0335523684603659 0.0671047369207318 0.966447631539634 8 0.0218843380336745 0.0437686760673491 0.978115661966325 9 0.0287350534325526 0.0574701068651051 0.971264946567447 10 0.0119986919086722 0.0239973838173443 0.988001308091328 11 0.00463078994849895 0.0092615798969979 0.9953692100515 12 0.00229450653383979 0.00458901306767959 0.99770549346616 13 0.000847451382229283 0.00169490276445857 0.99915254861777 14 0.000583259719951104 0.00116651943990221 0.999416740280049 15 0.000213619650345421 0.000427239300690842 0.999786380349655 16 0.000184994579390121 0.000369989158780243 0.99981500542061 17 0.00248883056365491 0.00497766112730983 0.997511169436345 18 0.00280506592287483 0.00561013184574967 0.997194934077125 19 0.00438905097209481 0.00877810194418962 0.995610949027905 20 0.0107065395260413 0.0214130790520825 0.989293460473959 21 0.00845210836196781 0.0169042167239356 0.991547891638032 22 0.00527463515225548 0.010549270304511 0.994725364847745 23 0.0027593684275241 0.00551873685504821 0.997240631572476 24 0.00363496967677163 0.00726993935354326 0.996365030323228 25 0.00196630092587536 0.00393260185175071 0.998033699074125 26 0.00100971666678705 0.00201943333357411 0.998990283333213 27 0.0010316447867273 0.00206328957345459 0.998968355213273 28 0.00279358928564804 0.00558717857129608 0.997206410714352 29 0.00225548499931896 0.00451096999863791 0.99774451500068 30 0.00176583471244225 0.00353166942488449 0.998234165287558 31 0.00139516164588892 0.00279032329177784 0.99860483835411 32 0.00143537913892482 0.00287075827784964 0.998564620861075 33 0.000789913434571762 0.00157982686914352 0.999210086565428 34 0.000399006271863093 0.000798012543726185 0.999600993728137 35 0.000193348869936288 0.000386697739872576 0.999806651130064 36 9.41350849712153e-05 0.000188270169942431 0.999905864915029 37 5.03020977425496e-05 0.000100604195485099 0.999949697902257 38 2.26256656377323e-05 4.52513312754646e-05 0.999977374334362 39 2.5556591179626e-05 5.11131823592519e-05 0.99997444340882 40 6.60312590718342e-05 0.000132062518143668 0.999933968740928 41 4.32933070512838e-05 8.65866141025676e-05 0.99995670669295 42 4.75846398765785e-05 9.5169279753157e-05 0.999952415360123 43 0.000100127865396253 0.000200255730792507 0.999899872134604 44 0.00291662625863259 0.00583325251726518 0.997083373741367 45 0.0581644190558304 0.116328838111661 0.94183558094417 46 0.849295088530697 0.301409822938606 0.150704911469303 47 0.954382948617727 0.0912341027645455 0.0456170513822727 48 0.956495719715067 0.0870085605698655 0.0435042802849327 49 0.989715263232747 0.0205694735345058 0.0102847367672529 50 0.984260694743792 0.0314786105124154 0.0157393052562077 51 0.965800809466144 0.0683983810677113 0.0341991905338556 52 0.974601340968821 0.0507973180623574 0.0253986590311787 53 0.949204525195798 0.101590949608404 0.050795474804202 54 0.876037785470484 0.247924429059031 0.123962214529516 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 31 0.63265306122449 NOK 5% type I error level 38 0.775510204081633 NOK 10% type I error level 44 0.897959183673469 NOK

Charts produced by software:

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Parameters (Session):

par1 = 3 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 3 ; 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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