*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: Mon, 14 Dec 2009 03:00:52 -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/14/t1260784949eym10q5nx4d0fnu.htm/, Retrieved Mon, 14 Dec 2009 11:02:41 +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/2009/Dec/14/t1260784949eym10q5nx4d0fnu.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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112,7 129,9 97,0 95,1 102,9 128,0 112,7 97,0 97,4 123,5 102,9 112,7 111,4 124,0 97,4 102,9 87,4 127,4 111,4 97,4 96,8 127,6 87,4 111,4 114,1 128,4 96,8 87,4 110,3 131,4 114,1 96,8 103,9 135,1 110,3 114,1 101,6 134,0 103,9 110,3 94,6 144,5 101,6 103,9 95,9 147,3 94,6 101,6 104,7 150,9 95,9 94,6 102,8 148,7 104,7 95,9 98,1 141,4 102,8 104,7 113,9 138,9 98,1 102,8 80,9 139,8 113,9 98,1 95,7 145,6 80,9 113,9 113,2 147,9 95,7 80,9 105,9 148,5 113,2 95,7 108,8 151,1 105,9 113,2 102,3 157,5 108,8 105,9 99,0 167,5 102,3 108,8 100,7 172,3 99,0 102,3 115,5 173,5 100,7 99,0 100,7 187,5 115,5 100,7 109,9 205,5 100,7 115,5 114,6 195,1 109,9 100,7 85,4 204,5 114,6 109,9 100,5 204,5 85,4 114,6 114,8 201,7 100,5 85,4 116,5 207,0 114,8 100,5 112,9 206,6 116,5 114,8 102,0 210,6 112,9 116,5 106,0 211,1 102,0 112,9 105,3 215,0 106,0 102,0 118,8 223,9 105,3 106,0 106,1 238,2 118,8 105,3 109,3 238,9 106,1 118,8 117,2 229,6 109,3 106,1 92,5 232,2 117,2 109,3 104,2 222,1 92,5 117,2 112,5 221 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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation TIP[t] = + 144.143675985304 + 0.182263188182482Grondstofprijzen[t] -0.415679192052341Y1[t] -0.280948161450398Y2[t] + 9.07314455214042M1[t] + 6.41427892739759M2[t] + 5.2894989657864M3[t] + 13.8736604675959M4[t] -9.00720960734173M5[t] -5.19098370710964M6[t] + 7.79185997838169M7[t] + 18.0079345104121M8[t] + 16.5789084590517M9[t] + 6.74442140782157M10[t] + 1.58362475706984M11[t] -0.174841418937455t + 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) 144.143675985304 21.336076 6.7559 0 0 Grondstofprijzen 0.182263188182482 0.029271 6.2267 0 0 Y1 -0.415679192052341 0.14164 -2.9348 0.005391 0.002696 Y2 -0.280948161450398 0.151389 -1.8558 0.070508 0.035254 M1 9.07314455214042 2.319518 3.9117 0.000329 0.000165 M2 6.41427892739759 2.742595 2.3388 0.024181 0.012091 M3 5.2894989657864 2.995405 1.7659 0.084686 0.042343 M4 13.8736604675959 2.500292 5.5488 2e-06 1e-06 M5 -9.00720960734173 3.043799 -2.9592 0.005051 0.002525 M6 -5.19098370710964 3.233861 -1.6052 0.115944 0.057972 M7 7.79185997838169 3.079881 2.5299 0.015247 0.007624 M8 18.0079345104121 2.982783 6.0373 0 0 M9 16.5789084590517 4.083701 4.0598 0.000209 0.000105 M10 6.74442140782157 3.581901 1.8829 0.06665 0.033325 M11 1.58362475706984 2.752112 0.5754 0.568076 0.284038 t -0.174841418937455 0.063692 -2.7451 0.008864 0.004432 Multiple Linear Regression - Regression Statistics Multiple R 0.944133064721672 R-squared 0.891387243900737 Adjusted R-squared 0.852596973865287 F-TEST (value) 22.9796607006367 F-TEST (DF numerator) 15 F-TEST (DF denominator) 42 p-value 1.66533453693773e-15 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.44724217155702 Sum Squared Residuals 499.106100753168 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 112.7 109.678915480402 3.02108451959831 2 102.9 99.4389435571974 3.46105644280258 3 97.4 96.9819077771693 0.418092222830749 4 111.4 110.521886992634 0.878113007365686 5 87.4 83.8115765378241 3.58842346217589 6 96.8 93.5324400057059 3.26755999429414 7 114.1 109.321624292323 4.77837570767673 8 110.3 110.077484229824 0.222515770175515 9 103.9 105.867168292509 -1.96716829250874 10 101.6 99.385300157987 2.21469984201304 11 94.6 98.7175559392168 -4.11755593921677 12 95.9 101.025361805823 -5.12536180582271 13 104.7 112.006066596967 -7.30606659696734 14 102.8 104.748171039339 -1.94817103933948 15 98.1 100.435475029195 -2.33547502919468 16 113.9 110.876630851012 3.02336914898774 17 80.9 82.7376813508913 -1.83768135089129 18 95.7 96.7146247104553 -1.01462471045529 19 113.2 113.061069595317 0.138930404682638 20 105.9 111.779241970938 -5.879241970938 21 108.8 108.767124066515 0.0328759334853511 22 102.3 100.769731922351 1.53026807764890 23 99 99.1438908146208 -0.143890814620793 24 100.7 101.458192325090 -0.758192325089718 25 115.5 110.795685590409 4.70431440959101 26 100.7 103.883999264443 -3.18399926444313 27 109.9 107.859134524088 2.04086547591209 28 114.6 114.706701672446 -0.106701672446493 29 85.4 88.8258488594971 -3.42584885949707 30 100.5 103.284609389903 -2.78460938990320 31 114.8 117.509205243907 -2.7092052439074 32 116.5 118.329903570118 -1.82990357011808 33 112.9 111.928917489317 0.971082510682531 34 102 103.667474988803 -1.66747498880260 35 106 103.965285087797 2.03471491220339 36 105.3 104.317263537301 0.98273646269903 37 118.8 114.004891833963 4.79510816603694 38 106.1 108.362543001601 -2.26254300160095 39 109.3 108.676831412264 0.623168587735602 40 117.2 117.628972080892 -0.42897208089191 41 92.5 90.8642451424365 1.63575485756349 42 104.2 100.712556991323 3.48744300867723 43 112.5 115.505400704598 -3.00540070459785 44 122.4 119.848303207327 2.55169679267309 45 113.3 110.649083910123 2.65091608987714 46 100 100.292301696722 -0.292301696722308 47 110.7 108.473268158366 2.22673184163417 48 112.8 107.899182331787 4.9008176682134 49 109.8 115.014440498259 -5.21444049825893 50 117.3 113.366343137419 3.93365686258098 51 109.1 109.846651257284 -0.746651257283766 52 115.9 119.265808403015 -3.36580840301503 53 96 95.960648109351 0.0393518906489888 54 99.8 102.755768902613 -2.95576890261287 55 116.8 116.002700163854 0.79729983614588 56 115.7 110.765067021793 4.93493297820747 57 99.4 101.087706241536 -1.68770624153628 58 94.3 96.085191234137 -1.78519123413703 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.491741105069132 0.983482210138265 0.508258894930868 20 0.392137731264341 0.784275462528682 0.607862268735659 21 0.311109215558408 0.622218431116816 0.688890784441592 22 0.596917888297091 0.806164223405818 0.403082111702909 23 0.66120530879561 0.677589382408781 0.338794691204390 24 0.611769918290068 0.776460163419865 0.388230081709933 25 0.592499906671537 0.815000186656926 0.407500093328463 26 0.682080200516244 0.635839598967513 0.317919799483756 27 0.654918063982162 0.690163872035677 0.345081936017838 28 0.574428749428025 0.85114250114395 0.425571250571975 29 0.60943473315636 0.781130533687279 0.390565266843639 30 0.551125658459404 0.897748683081193 0.448874341540596 31 0.475961593500557 0.951923187001114 0.524038406499443 32 0.553210085264606 0.893579829470787 0.446789914735394 33 0.476229550365818 0.952459100731637 0.523770449634182 34 0.407389853212567 0.814779706425133 0.592610146787433 35 0.340917163510232 0.681834327020464 0.659082836489768 36 0.350286422301174 0.700572844602347 0.649713577698826 37 0.422697301717864 0.845394603435729 0.577302698282136 38 0.514856094099313 0.970287811801374 0.485143905900687 39 0.344308351009028 0.688616702018056 0.655691648990972 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 0 0 OK

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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