Meervoudig regressiemodel Paper

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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: Tue, 21 Dec 2010 16:24:26 +0000

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/21/t1292948621akplqf3ps3shu7u.htm/, Retrieved Tue, 21 Dec 2010 17:23:41 +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/21/t1292948621akplqf3ps3shu7u.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}, }

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Dataseries X:

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8,8 8,1 0 8,5 9,9 0 8,6 11,5 0 8,7 23,4 0 9,1 25,4 0 8,8 27,9 0 6,3 26,1 0 2,5 18,8 0 -2,7 14,1 0 -4,5 11,5 0 -7 15,8 0 -9,3 12,4 0 -12,2 4,5 0 -13,2 -2,2 1 -13,7 -4,2 1 -15 -9,4 1 -16,9 -14,5 1 -16,3 -17,9 1 -16,7 -15,1 1 -16 -15,2 1 -14,5 -15,7 1 -12,2 -18 1 -7,5 -18,1 1 -4,4 -13,5 1 -1,1 -9,9 1 1,3 -4,8 1 -0,1 -1,7 0 0,4 -0,1 0 2,4 2,2 0 1 10,2 0 3,3 7,6 0 1,8 10,8 0 3,2 3,8 0 1,3 11 0 1,5 10,8 0 1,3 20,1 0 2 14,9 0 3 13 0 4,4 10,9 0 3,1 9,6 0 2,6 4 0 2,7 -1,1 0 4 -7,7 0 4,1 -8,9 0 3 -8 0 2,7 -7,1 0 4 -5,3 0 4,8 -2,5 0 6 -2,4 0 4,6 -2,9 0 4,4 -4,8 0 6,6 -7,2 0 4,7 1,7 0 7,6 2,2 0 5,3 13,4 0 6,6 12,3 0 4 13,7 0 3,8 4,4 0 1,2 -2,5 0

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 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation Industriële_productie[t] = + 2.63558186333943 + 0.0483731756837123registratie_personenwagens[t] -13.2919550675034crisis[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) 2.63558186333943 0.884213 2.9807 0.004249 0.002125 registratie_personenwagens 0.0483731756837123 0.072115 0.6708 0.505119 0.252559 crisis -13.2919550675034 2.074186 -6.4083 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.777293055357366 R-squared 0.60418449390679 Adjusted R-squared 0.590048225832032 F-TEST (value) 42.7400280407567 F-TEST (DF numerator) 2 F-TEST (DF denominator) 56 p-value 5.36781730176017e-12 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 4.9102797863297 Sum Squared Residuals 1350.20746448213 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.8 3.02740458637750 5.7725954136225 2 8.5 3.11447630260818 5.38552369739182 3 8.6 3.19187338370213 5.40812661629787 4 8.7 3.7675141743383 4.9324858256617 5 9.1 3.86426052570572 5.23573947429427 6 8.8 3.98519346491501 4.81480653508499 7 6.3 3.89812174868432 2.40187825131567 8 2.5 3.54499756619322 -1.04499756619322 9 -2.7 3.31764364047978 -6.01764364047978 10 -4.5 3.19187338370212 -7.69187338370212 11 -7 3.39987803914209 -10.3998780391421 12 -9.3 3.23540924181747 -12.5354092418175 13 -12.2 2.85326115391614 -15.0532611539161 14 -13.2 -10.7627941906681 -2.43720580933186 15 -13.7 -10.8595405420356 -2.84045945796444 16 -15 -11.1110810555909 -3.88891894440913 17 -16.9 -11.3577842515778 -5.5422157484222 18 -16.3 -11.5222530489024 -4.77774695109758 19 -16.7 -11.3868081569880 -5.31319184301197 20 -16 -11.3916454745564 -4.6083545254436 21 -14.5 -11.4158320623983 -3.08416793760175 22 -12.2 -11.5270903664708 -0.672909633529208 23 -7.5 -11.5319276840392 4.03192768403916 24 -4.4 -11.3094110758941 6.90941107589409 25 -1.1 -11.1352676434327 10.0352676434327 26 1.3 -10.8885644474458 12.1885644474458 27 -0.1 2.55334746467712 -2.65334746467712 28 0.4 2.63074454577106 -2.23074454577106 29 2.4 2.7420028498436 -0.342002849843601 30 1 3.1289882553133 -2.1289882553133 31 3.3 3.00321799853565 0.296782001464352 32 1.8 3.15801216072353 -1.35801216072353 33 3.2 2.81939993093754 0.380600069062459 34 1.3 3.16768679586027 -1.86768679586027 35 1.5 3.15801216072353 -1.65801216072353 36 1.3 3.60788269458205 -2.30788269458205 37 2 3.35634218102675 -1.35634218102675 38 3 3.26443314722769 -0.264433147227694 39 4.4 3.1628494782919 1.23715052170810 40 3.1 3.09996434990307 3.56500969282048e-05 41 2.6 2.82907456607428 -0.229074566074283 42 2.7 2.58237137008735 0.117628629912650 43 4 2.26310841057485 1.73689158942515 44 4.1 2.20506059975439 1.89493940024561 45 3 2.24859645786974 0.751403542130264 46 2.7 2.29213231598508 0.407867684014924 47 4 2.37920403221576 1.62079596778424 48 4.8 2.51464892413015 2.28535107586985 49 6 2.51948624169852 3.48051375830148 50 4.6 2.49529965385667 2.10470034614333 51 4.4 2.40339062005762 1.99660937994238 52 6.6 2.28729499841670 4.31270500158330 53 4.7 2.71781626200174 1.98218373799826 54 7.6 2.7420028498436 4.8579971501564 55 5.3 3.28378241750118 2.01621758249882 56 6.6 3.23057192424909 3.36942807575090 57 4 3.29829437020629 0.701705629793708 58 3.8 2.84842383634777 0.951576163652232 59 1.2 2.51464892413015 -1.31464892413015 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.000136104003007405 0.000272208006014811 0.999863895996993 7 0.00617729812059326 0.0123545962411865 0.993822701879407 8 0.098749562240049 0.197499124480098 0.901250437759951 9 0.52976615760369 0.94046768479262 0.47023384239631 10 0.775370746346423 0.449258507307154 0.224629253653577 11 0.941714688491485 0.116570623017029 0.0582853115085147 12 0.991746136728714 0.016507726542572 0.008253863271286 13 0.999644481365186 0.000711037269628169 0.000355518634814084 14 0.999271230066078 0.00145753986784454 0.000728769933922272 15 0.99866564159954 0.00266871680092127 0.00133435840046064 16 0.998069574942457 0.0038608501150853 0.00193042505754265 17 0.998206268792353 0.00358746241529338 0.00179373120764669 18 0.998628357684503 0.00274328463099478 0.00137164231549739 19 0.999348107390113 0.00130378521977370 0.000651892609886848 20 0.999842306626754 0.000315386746491409 0.000157693373245704 21 0.999985369036093 2.92619278142926e-05 1.46309639071463e-05 22 0.999999695335824 6.09328351582101e-07 3.04664175791050e-07 23 0.999999987151329 2.56973428721860e-08 1.28486714360930e-08 24 0.99999999828364 3.43271906227741e-09 1.71635953113871e-09 25 0.999999999257347 1.48530585200646e-09 7.42652926003229e-10 26 0.999999999236245 1.52751064500025e-09 7.63755322500123e-10 27 0.999999999681584 6.36831651119084e-10 3.18415825559542e-10 28 0.999999999798995 4.0201075523524e-10 2.0100537761762e-10 29 0.99999999950365 9.92698320863678e-10 4.96349160431839e-10 30 0.999999999126374 1.74725114472037e-09 8.73625572360183e-10 31 0.999999996959568 6.08086356527417e-09 3.04043178263709e-09 32 0.999999992096858 1.58062850412801e-08 7.90314252064004e-09 33 0.999999975932402 4.81351962223277e-08 2.40675981111638e-08 34 0.999999957449194 8.51016124658877e-08 4.25508062329438e-08 35 0.999999925254821 1.49490357945552e-07 7.47451789727758e-08 36 0.999999902827653 1.94344693201177e-07 9.71723466005886e-08 37 0.999999852369131 2.95261738002870e-07 1.47630869001435e-07 38 0.999999650495939 6.9900812230989e-07 3.49504061154945e-07 39 0.999998820960459 2.35807908242113e-06 1.17903954121057e-06 40 0.999997422204179 5.15559164205539e-06 2.57779582102769e-06 41 0.999995786510766 8.42697846870645e-06 4.21348923435322e-06 42 0.99999204175397 1.59164920581931e-05 7.95824602909656e-06 43 0.999976637752 4.67244959990693e-05 2.33622479995347e-05 44 0.999929901281111 0.000140197437778007 7.00987188890036e-05 45 0.999820683579502 0.000358632840994998 0.000179316420497499 46 0.999641553633626 0.000716892732747996 0.000358446366373998 47 0.99898833921049 0.00202332157902166 0.00101166078951083 48 0.997060037167364 0.00587992566527228 0.00293996283263614 49 0.99311653989372 0.0137669202125604 0.00688346010628021 50 0.981527714343873 0.0369445713122534 0.0184722856561267 51 0.953979048301697 0.0920419033966058 0.0460209516983029 52 0.932075880551786 0.135848238896427 0.0679241194482137 53 0.840268670937205 0.319462658125590 0.159731329062795 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 37 0.770833333333333 NOK 5% type I error level 41 0.854166666666667 NOK 10% type I error level 42 0.875 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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