Paper statistiek: Multiple Regression Analysis

*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, 11 Dec 2009 09:03:36 -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/Dec/11/t1260547571ri7eqkh41tshrd1.htm/, Retrieved Fri, 11 Dec 2009 17:06:23 +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/Dec/11/t1260547571ri7eqkh41tshrd1.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:

ETP(35)

Dataseries X:

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1.43 0 0 0 0 0.51 1.43 0 0 0 0 0.51 1.43 0 0 0 0 0.51 1.43 0 0 0 0 0.51 1.43 0 0 0 0 0.52 1.43 0 0 0 0 0.52 1.44 0 0 0 0 0.52 1.48 1 0 0 0 0.53 1.48 1 0 0 0 0.53 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.52 1.48 1 0 0 0 0.53 1.48 1 0 0 0 0.53 1.48 1 0 0 0 0.53 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.54 1.48 1 0 0 0 0.53 1.48 1 0 0 0 0.53 1.48 1 0 0 0 0.53 1.48 1 0 0 0 0.53 1.48 1 0 0 0 0.53 1.57 1 1 0 0 0.54 1.58 1 1 0 0 0.55 1.58 1 1 0 0 0.55 1.58 1 1 0 0 0.55 1.58 1 1 0 0 0.55 1.59 1 1 1 1 0.55 1.6 1 1 1 2 0.55 1.6 1 1 1 3 0.55 1.61 1 1 1 4 0.55 1.61 1 1 1 5 0.56 1.61 1 1 1 6 0.56 1.62 1 1 1 7 0.56 1.63 1 1 1 8 0.56 1.63 1 1 1 9 0.56 1.64 1 1 1 10 0.55 1.64 1 1 1 11 0.56 1.64 1 1 1 12 0.5 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Broodprijzen[t] = + 1.37674903761831 + 0.0476820908725279Dummy1[t] + 0.0968243660889128Dummy2[t] + 0.0149811800172971Dummy3[t] + 0.00364815973913286Dummy4[t] + 0.106652028005743Bakmeel[t] -4.25201481723485e-05t + 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) 1.37674903761831 0.040399 34.0785 0 0 Dummy1 0.0476820908725279 0.001963 24.2861 0 0 Dummy2 0.0968243660889128 0.002132 45.4254 0 0 Dummy3 0.0149811800172971 0.0023 6.5124 0 0 Dummy4 0.00364815973913286 0.000179 20.3979 0 0 Bakmeel 0.106652028005743 0.078884 1.352 0.182111 0.091056 t -4.25201481723485e-05 8.5e-05 -0.4987 0.620083 0.310042 Multiple Linear Regression - Regression Statistics Multiple R 0.999089522014467 R-squared 0.998179872999095 Adjusted R-squared 0.997973820885786 F-TEST (value) 4844.30786447532 F-TEST (DF numerator) 6 F-TEST (DF denominator) 53 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0033811833505616 Sum Squared Residuals 0.000605917245056093 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.43 1.43109905175307 -0.00109905175307049 2 1.43 1.43105653160489 -0.00105653160489032 3 1.43 1.43101401145672 -0.00101401145671809 4 1.43 1.43097149130855 -0.000971491308545764 5 1.43 1.43199549144043 -0.00199549144043077 6 1.43 1.43195297129226 -0.00195297129225846 7 1.44 1.43191045114409 0.00808954885591388 8 1.48 1.4806165421485 -0.000616542148499051 9 1.48 1.48057402200033 -0.000574022000326703 10 1.48 1.47946498157210 0.000535018427903076 11 1.48 1.47942246142392 0.000577538576075424 12 1.48 1.47937994127575 0.000620058724247772 13 1.48 1.47933742112758 0.000662578872420121 14 1.48 1.47929490097941 0.000705099020592469 15 1.48 1.47925238083124 0.000747619168764818 16 1.48 1.47920986068306 0.000790139316937166 17 1.48 1.47916734053489 0.000832659465109514 18 1.48 1.47912482038672 0.000875179613281863 19 1.48 1.47908230023855 0.000917699761454211 20 1.48 1.48010630037043 -0.000106300370430871 21 1.48 1.48006378022226 -6.37802222585223e-05 22 1.48 1.48002126007409 -2.12600740861740e-05 23 1.48 1.48104526020597 -0.00104526020597126 24 1.48 1.4810027400578 -0.00100274005779891 25 1.48 1.48096021990963 -0.00096021990962656 26 1.48 1.48091769976145 -0.00091769976145421 27 1.48 1.48087517961328 -0.000875179613281862 28 1.48 1.48083265946511 -0.000832659465109514 29 1.48 1.48079013931694 -0.000790139316937166 30 1.48 1.48074761916876 -0.000747619168764817 31 1.48 1.48070509902059 -0.000705099020592469 32 1.48 1.48066257887242 -0.00066257887242012 33 1.48 1.47955353844419 0.000446461555809658 34 1.48 1.47951101829602 0.000488981703982006 35 1.48 1.47946849814785 0.000531501852154354 36 1.48 1.47942597799967 0.000574022000326703 37 1.48 1.4793834578515 0.000616542148499051 38 1.57 1.5772318240723 -0.00723182407229876 39 1.58 1.57825582420418 0.00174417579581617 40 1.58 1.57821330405601 0.00178669594398852 41 1.58 1.57817078390784 0.00182921609216086 42 1.58 1.57812826375967 0.00187173624033321 43 1.59 1.59671508336792 -0.00671508336792435 44 1.6 1.60032072295888 -0.000320722958884853 45 1.6 1.60392636254985 -0.00392636254984536 46 1.61 1.60753200214081 0.00246799785919413 47 1.61 1.61220416201182 -0.00220416201182380 48 1.61 1.61580980160278 -0.00580980160278431 49 1.62 1.61941544119374 0.000584558806255193 50 1.63 1.62302108078471 0.00697891921529447 51 1.63 1.62662672037567 0.00337327962433396 52 1.64 1.62916583968657 0.0108341603134309 53 1.64 1.63383799955759 0.00616200044241296 54 1.64 1.63637711886849 0.00362288113150988 55 1.64 1.63998275845945 1.72415405493680e-05 56 1.64 1.64465491833047 -0.00465491833046857 57 1.65 1.64719403764137 0.00280596235862836 58 1.65 1.65079967723233 -0.000799677232332149 59 1.65 1.65440531682329 -0.00440531682329266 60 1.65 1.65801095641425 -0.00801095641425316 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 10 0.773918069688699 0.452163860622603 0.226081930311301 11 0.643232321851969 0.713535356296061 0.356767678148031 12 0.510520378975068 0.978959242049865 0.489479621024932 13 0.38916220748999 0.77832441497998 0.61083779251001 14 0.285798194903708 0.571596389807416 0.714201805096292 15 0.202385693934280 0.404771387868561 0.79761430606572 16 0.138234057172912 0.276468114345823 0.861765942827088 17 0.0912240757023639 0.182448151404728 0.908775924297636 18 0.058544873601678 0.117089747203356 0.941455126398322 19 0.03737224173965 0.0747444834793 0.96262775826035 20 0.0295400734854921 0.0590801469709842 0.970459926514508 21 0.0200701961748158 0.0401403923496316 0.979929803825184 22 0.0132648190576765 0.0265296381153531 0.986735180942323 23 0.0084117326972545 0.016823465394509 0.991588267302746 24 0.00473329946547061 0.00946659893094122 0.99526670053453 25 0.00249582234972956 0.00499164469945912 0.99750417765027 26 0.00125050547052484 0.00250101094104967 0.998749494529475 27 0.00059782480330381 0.00119564960660762 0.999402175196696 28 0.000272714452270478 0.000545428904540955 0.99972728554773 29 0.000118438365015438 0.000236876730030876 0.999881561634985 30 4.88151434487691e-05 9.76302868975382e-05 0.999951184856551 31 1.90910209652536e-05 3.81820419305071e-05 0.999980908979035 32 7.21299868379037e-06 1.44259973675807e-05 0.999992787001316 33 2.75396185755998e-06 5.50792371511996e-06 0.999997246038142 34 1.01473542438012e-06 2.02947084876023e-06 0.999998985264576 35 3.52732125553602e-07 7.05464251107204e-07 0.999999647267874 36 1.13701449030775e-07 2.2740289806155e-07 0.99999988629855 37 3.41751354709079e-08 6.83502709418157e-08 0.999999965824865 38 1.79021519541505e-08 3.58043039083009e-08 0.999999982097848 39 8.58522447591033e-07 1.71704489518207e-06 0.999999141477552 40 9.57314447626582e-07 1.91462889525316e-06 0.999999042685552 41 5.56025034460861e-07 1.11205006892172e-06 0.999999443974966 42 2.47800894435532e-07 4.95601788871065e-07 0.999999752199106 43 3.35096191529875e-07 6.7019238305975e-07 0.999999664903809 44 1.17036784044181e-07 2.34073568088362e-07 0.999999882963216 45 8.8004149935026e-07 1.76008299870052e-06 0.9999991199585 46 6.79646240553768e-07 1.35929248110754e-06 0.99999932035376 47 2.54632883548039e-06 5.09265767096078e-06 0.999997453671164 48 0.00278404280733589 0.00556808561467178 0.997215957192664 49 0.0379476983417106 0.0758953966834212 0.96205230165829 50 0.0289388238613817 0.0578776477227634 0.971061176138618 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 25 0.609756097560976 NOK 5% type I error level 28 0.682926829268293 NOK 10% type I error level 32 0.780487804878049 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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