Paper statistiek: Multiple Regression Analysis

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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 03:09:13 -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/t1260526270q492sqykw0zjmcx.htm/, Retrieved Fri, 11 Dec 2009 11:11:22 +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/2009/Dec/11/t1260526270q492sqykw0zjmcx.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:

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User-defined keywords:

ETP(31)

Dataseries X:

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1.43 0 0 0 0 1.43 0 0 0 0 1.43 0 0 0 0 1.43 0 0 0 0 1.43 0 0 0 0 1.43 0 0 0 0 1.44 0 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.48 1 0 0 0 1.57 1 1 0 0 1.58 1 1 0 0 1.58 1 1 0 0 1.58 1 1 0 0 1.58 1 1 0 0 1.59 1 1 1 1 1.6 1 1 1 2 1.6 1 1 1 3 1.61 1 1 1 4 1.61 1 1 1 5 1.61 1 1 1 6 1.62 1 1 1 7 1.63 1 1 1 8 1.63 1 1 1 9 1.64 1 1 1 10 1.64 1 1 1 11 1.64 1 1 1 12 1.64 1 1 1 13 1.64 1 1 1 14 1.65 1 1 1 15 1.65 1 1 1 16 1.65 1 1 1 17 1.65 1 1 1 18

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 Broodprijzen[t] = + 1.42739788145102 + 0.0493205345320607Dummy2[t] + 0.0961175746131058Dummy3[t] + 0.0170539138171548Dummy1[t] + 0.0035911662386438Dummy4[t] + 0.00294529473975772M1[t] + 0.00298200793887241M2[t] + 0.00424223606060671M3[t] + 0.00550246418234103M4[t] + 0.00476269230407537M5[t] + 0.00402292042580968M6[t] + 0.00315413253638430M7[t] + 0.00183202050397774M8[t] + 0.00237401537798331M9[t] + 0.00291601025198887M10[t] + 0.00145800512599444M11[t] + 2.15386305369157e-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) 1.42739788145102 0.002198 649.3756 0 0 Dummy2 0.0493205345320607 0.002075 23.7735 0 0 Dummy3 0.0961175746131058 0.002234 43.0306 0 0 Dummy1 0.0170539138171548 0.002458 6.9377 0 0 Dummy4 0.0035911662386438 0.000177 20.3441 0 0 M1 0.00294529473975772 0.002236 1.317 0.194814 0.097407 M2 0.00298200793887241 0.002277 1.3098 0.197205 0.098602 M3 0.00424223606060671 0.002269 1.8693 0.068398 0.034199 M4 0.00550246418234103 0.002264 2.4302 0.019336 0.009668 M5 0.00476269230407537 0.002261 2.1065 0.041033 0.020516 M6 0.00402292042580968 0.00226 1.7803 0.082097 0.041048 M7 0.00315413253638430 0.002244 1.4054 0.167098 0.083549 M8 0.00183202050397774 0.00222 0.8251 0.413874 0.206937 M9 0.00237401537798331 0.002211 1.0739 0.288863 0.144431 M10 0.00291601025198887 0.002204 1.3232 0.192754 0.096377 M11 0.00145800512599444 0.0022 0.6629 0.51095 0.255475 t 2.15386305369157e-05 7.4e-05 0.2916 0.77196 0.38598 Multiple Linear Regression - Regression Statistics Multiple R 0.999219553376748 R-squared 0.998439715850428 Adjusted R-squared 0.997859145004076 F-TEST (value) 1719.75517221279 F-TEST (DF numerator) 16 F-TEST (DF denominator) 43 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.00347554912147548 Sum Squared Residuals 0.000519415992918925 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.43 1.43036471482132 -0.000364714821318268 2 1.43 1.43042296665096 -0.000422966650962066 3 1.43 1.43170473340323 -0.0017047334032334 4 1.43 1.43298650015550 -0.00298650015550467 5 1.43 1.43226826690778 -0.00226826690777584 6 1.43 1.43155003366005 -0.00155003366004712 7 1.44 1.43070278440116 0.00929721559884135 8 1.48 1.47872274553135 0.00127725446865037 9 1.48 1.47928627903589 0.000713720964107889 10 1.48 1.47984981254043 0.000150187459565413 11 1.48 1.47841334604498 0.00158665395502293 12 1.48 1.47697687954952 0.00302312045048046 13 1.48 1.47994371291981 5.62870801858207e-05 14 1.48 1.48000196474947 -1.96474946578098e-06 15 1.48 1.48128373150174 -0.00128373150173700 16 1.48 1.48256549825401 -0.00256549825400823 17 1.48 1.48184726500628 -0.00184726500627949 18 1.48 1.48112903175855 -0.00112903175855072 19 1.48 1.48028178249966 -0.000281782499662258 20 1.48 1.47898120909779 0.00101879090220739 21 1.48 1.47954474260234 0.000455257397664907 22 1.48 1.48010827610688 -0.000108276106877574 23 1.48 1.47867180961142 0.00132819038857995 24 1.48 1.47723534311596 0.00276465688403747 25 1.48 1.48020217648626 -0.000202176486257168 26 1.48 1.48026042831591 -0.000260428315908769 27 1.48 1.48154219506818 -0.00154219506817999 28 1.48 1.48282396182045 -0.00282396182045122 29 1.48 1.48210572857272 -0.00210572857272248 30 1.48 1.48138749532499 -0.00138749532499371 31 1.48 1.48054024606611 -0.000540246066105246 32 1.48 1.47923967266424 0.0007603273357644 33 1.48 1.47980320616878 0.000196793831221919 34 1.48 1.48036673967332 -0.000366739673320562 35 1.48 1.47893027317786 0.00106972682213696 36 1.48 1.47749380668241 0.00250619331759448 37 1.48 1.4804606400527 -0.000460640052700156 38 1.57 1.57663646649546 -0.00663646649545754 39 1.58 1.57791823324773 0.00208176675227125 40 1.58 1.5792 0.00080000000000001 41 1.58 1.57848176675227 0.00151823324772876 42 1.58 1.57776353350454 0.00223646649545753 43 1.59 1.59756136430145 -0.00756136430145254 44 1.6 1.59985195713823 0.000148042861773317 45 1.6 1.60400665688141 -0.00400665688141297 46 1.61 1.6081613566246 0.00183864337540076 47 1.61 1.61031605636779 -0.000316056367785525 48 1.61 1.61247075611097 -0.0024707561109718 49 1.62 1.61902875571991 0.000971244280089771 50 1.63 1.62267817378821 0.00732182621179416 51 1.63 1.62755110677912 0.00244889322087914 52 1.64 1.63242403977004 0.00757596022996411 53 1.64 1.63529697276095 0.00470302723904905 54 1.64 1.63816990575187 0.00183009424813402 55 1.64 1.64091382273162 -0.00091382273162132 56 1.64 1.64320441556840 -0.00320441556839548 57 1.65 1.64735911531158 0.00264088468841825 58 1.65 1.65151381505477 -0.00151381505476803 59 1.65 1.65366851479795 -0.00366851479795431 60 1.65 1.65582321454114 -0.00582321454114059 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 20 0.601040966536062 0.797918066927877 0.398959033463938 21 0.427733939860089 0.855467879720177 0.572266060139911 22 0.278565378002836 0.557130756005671 0.721434621997164 23 0.182266517029058 0.364533034058115 0.817733482970942 24 0.170728889335890 0.341457778671781 0.82927111066411 25 0.160741096353716 0.321482192707431 0.839258903646284 26 0.111576047155336 0.223152094310671 0.888423952844664 27 0.0638620959113786 0.127724191822757 0.936137904088621 28 0.0385933907725709 0.0771867815451418 0.96140660922743 29 0.0227933187633900 0.0455866375267801 0.97720668123661 30 0.0142701039711522 0.0285402079423045 0.985729896028848 31 0.0273675264163018 0.0547350528326035 0.972632473583698 32 0.0173053028471916 0.0346106056943833 0.982694697152808 33 0.00915572519197798 0.0183114503839560 0.990844274808022 34 0.0040292086455258 0.0080584172910516 0.995970791354474 35 0.00180765391645856 0.00361530783291713 0.998192346083541 36 0.00289307808436593 0.00578615616873187 0.997106921915634 37 0.00110578273309672 0.00221156546619345 0.998894217266903 38 0.00136021723181231 0.00272043446362462 0.998639782768188 39 0.0158295329542694 0.0316590659085387 0.98417046704573 40 0.0100097935584574 0.0200195871169147 0.989990206441543 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 5 0.238095238095238 NOK 5% type I error level 11 0.523809523809524 NOK 10% type I error level 13 0.619047619047619 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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