Paper: 2 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: Fri, 11 Dec 2009 07:53:22 -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/t1260543259msi2nc32epjkxq5.htm/, Retrieved Fri, 11 Dec 2009 15:54:32 +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/t1260543259msi2nc32epjkxq5.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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118.7 0 110.1 0 110.3 0 112.9 0 102.2 0 99.4 0 116.1 0 103.8 0 101.8 0 113.7 0 89.7 0 99.5 0 122.9 0 108.6 0 114.4 0 110.5 0 104.1 0 103.6 0 121.6 0 101.1 0 116.0 0 120.1 0 96.0 0 105.0 0 124.7 0 123.9 0 123.6 0 114.8 0 108.8 0 106.1 0 123.2 0 106.2 0 115.2 0 120.6 0 109.5 0 114.4 0 121.4 0 129.5 0 124.3 0 112.6 0 125.1 1 117.9 1 116.4 1 126.4 1 93.3 1 102.9 1 97.8 1 97.1 1 110.7 1 109.3 1 103.2 1 106.2 1 81.3 1 84.5 1 92.7 1 85.0 1 79.1 1 92.6 1 78.1 1 76.9 1 92.5 1

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 9 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y(t)_Bruto_index_consumptiegoederen[t] = + 102.039154411765 -13.9609375000000Dummyvariabele[t] + 15.9344403594772M1[t] + 15.4981515522876M2[t] + 14.3191176470588M3[t] + 10.5000837418301M4[t] + 6.13323733660131M5[t] + 4.07420343137254M6[t] + 15.7151695261438M7[t] + 6.15613562091502M8[t] + 2.67710171568627M9[t] + 11.5180678104575M10[t] -4.30096609477125M11[t] + 0.0590339052287572t + 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) 102.039154411765 5.940292 17.1775 0 0 Dummyvariabele -13.9609375000000 5.181147 -2.6946 0.009744 0.004872 M1 15.9344403594772 6.496566 2.4527 0.017944 0.008972 M2 15.4981515522876 6.815521 2.2739 0.027578 0.013789 M3 14.3191176470588 6.805847 2.1039 0.040761 0.02038 M4 10.5000837418301 6.799026 1.5444 0.129211 0.064605 M5 6.13323733660131 6.838822 0.8968 0.374382 0.187191 M6 4.07420343137254 6.820268 0.5974 0.55313 0.276565 M7 15.7151695261438 6.804529 2.3095 0.025357 0.012679 M8 6.15613562091502 6.791624 0.9064 0.369332 0.184666 M9 2.67710171568627 6.78157 0.3948 0.694805 0.347402 M10 11.5180678104575 6.77438 1.7002 0.095695 0.047848 M11 -4.30096609477125 6.770062 -0.6353 0.528319 0.26416 t 0.0590339052287572 0.139624 0.4228 0.674364 0.337182 Multiple Linear Regression - Regression Statistics Multiple R 0.69588105959103 R-squared 0.484250449097534 Adjusted R-squared 0.341596317996852 F-TEST (value) 3.39457711712366 F-TEST (DF numerator) 13 F-TEST (DF denominator) 47 p-value 0.00101883741500064 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 10.7021310727748 Sum Squared Residuals 5383.17364644608 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 118.7 118.032628676470 0.667371323529671 2 110.1 117.655373774510 -7.55537377450982 3 110.3 116.535373774510 -6.23537377450984 4 112.9 112.775373774510 0.124626225490167 5 102.2 108.467561274510 -6.2675612745098 6 99.4 106.467561274510 -7.06756127450984 7 116.1 118.167561274510 -2.06756127450983 8 103.8 108.667561274510 -4.86756127450983 9 101.8 105.247561274510 -3.44756127450983 10 113.7 114.147561274510 -0.447561274509814 11 89.7 98.3875612745098 -8.68756127450982 12 99.5 102.747561274510 -3.24756127450982 13 122.9 118.741035539216 4.15896446078424 14 108.6 118.363780637255 -9.76378063725492 15 114.4 117.243780637255 -2.84378063725490 16 110.5 113.483780637255 -2.9837806372549 17 104.1 109.175968137255 -5.07596813725492 18 103.6 107.175968137255 -3.5759681372549 19 121.6 118.875968137255 2.72403186274509 20 101.1 109.375968137255 -8.2759681372549 21 116 105.955968137255 10.0440318627451 22 120.1 114.855968137255 5.24403186274509 23 96 99.0959681372549 -3.09596813725490 24 105 103.455968137255 1.54403186274510 25 124.7 119.449442401961 5.25055759803916 26 123.9 119.0721875 4.82781250000001 27 123.6 117.9521875 5.64781250000001 28 114.8 114.1921875 0.607812500000011 29 108.8 109.884375 -1.08437500000000 30 106.1 107.884375 -1.78437499999999 31 123.2 119.584375 3.61562500000001 32 106.2 110.084375 -3.88437499999998 33 115.2 106.664375 8.53562500000002 34 120.6 115.564375 5.035625 35 109.5 99.804375 9.69562500000002 36 114.4 104.164375 10.2356250000000 37 121.4 120.157849264706 1.24215073529407 38 129.5 119.780594362745 9.71940563725492 39 124.3 118.660594362745 5.63940563725492 40 112.6 114.900594362745 -2.30059436274508 41 125.1 96.631844362745 28.4681556372549 42 117.9 94.631844362745 23.2681556372549 43 116.4 106.331844362745 10.0681556372549 44 126.4 96.8318443627451 29.5681556372549 45 93.3 93.4118443627451 -0.111844362745098 46 102.9 102.311844362745 0.588155637254907 47 97.8 86.5518443627451 11.2481556372549 48 97.1 90.9118443627451 6.18815563725489 49 110.7 106.905318627451 3.79468137254896 50 109.3 106.528063725490 2.77193627450981 51 103.2 105.408063725490 -2.20806372549018 52 106.2 101.648063725490 4.55193627450982 53 81.3 97.3402512254902 -16.0402512254902 54 84.5 95.3402512254902 -10.8402512254902 55 92.7 107.040251225490 -14.3402512254902 56 85 97.5402512254902 -12.5402512254902 57 79.1 94.1202512254902 -15.0202512254902 58 92.6 103.020251225490 -10.4202512254902 59 78.1 87.2602512254902 -9.1602512254902 60 76.9 91.6202512254902 -14.7202512254902 61 92.5 107.613725490196 -15.1137254901961 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0169415751900328 0.0338831503800656 0.983058424809967 18 0.00493523548350678 0.00987047096701357 0.995064764516493 19 0.00166376909301713 0.00332753818603426 0.998336230906983 20 0.00123904610423935 0.00247809220847870 0.99876095389576 21 0.0069966989375544 0.0139933978751088 0.993003301062446 22 0.00286225660178409 0.00572451320356817 0.997137743398216 23 0.00190199977292229 0.00380399954584458 0.998098000227078 24 0.00101604003860029 0.00203208007720058 0.9989839599614 25 0.00044621292669335 0.0008924258533867 0.999553787073307 26 0.00128863688003200 0.00257727376006400 0.998711363119968 27 0.000986498656547025 0.00197299731309405 0.999013501343453 28 0.00140606965654674 0.00281213931309348 0.998593930343453 29 0.00099537325960158 0.00199074651920316 0.999004626740398 30 0.00103381983139517 0.00206763966279034 0.998966180168605 31 0.000472271449766396 0.000944542899532792 0.999527728550234 32 0.00278419888377843 0.00556839776755685 0.997215801116222 33 0.00134110718083284 0.00268221436166568 0.998658892819167 34 0.000649506432356588 0.00129901286471318 0.999350493567643 35 0.00199208842722839 0.00398417685445677 0.998007911572772 36 0.00132313752220600 0.00264627504441199 0.998676862477794 37 0.00291019594400145 0.0058203918880029 0.997089804055999 38 0.00252356632667707 0.00504713265335413 0.997476433673323 39 0.00192827325102928 0.00385654650205855 0.99807172674897 40 0.00183333004943389 0.00366666009886778 0.998166669950566 41 0.00929491163132854 0.0185898232626571 0.990705088368671 42 0.0123217299018691 0.0246434598037382 0.987678270098131 43 0.0315765929524742 0.0631531859049483 0.968423407047526 44 0.716489100997689 0.567021798004622 0.283510899002311 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 22 0.785714285714286 NOK 5% type I error level 26 0.928571428571429 NOK 10% type I error level 27 0.964285714285714 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = 1 ; par3 = 0 ; par4 = 12 ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ; par4 = 12 ;

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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