WS 7: Multiple Regression 3

*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, 20 Nov 2009 03:44:27 -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/Nov/20/t1258713903i6ptbxsajd9fk5a.htm/, Retrieved Fri, 20 Nov 2009 11:45:15 +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/Nov/20/t1258713903i6ptbxsajd9fk5a.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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79.8 109.87 83.4 95.74 113.6 123.06 112.9 123.39 104 120.28 109.9 115.33 99 110.4 106.3 114.49 128.9 132.03 111.1 123.16 102.9 118.82 130 128.32 87 112.24 87.5 104.53 117.6 132.57 103.4 122.52 110.8 131.8 112.6 124.55 102.5 120.96 112.4 122.6 135.6 145.52 105.1 118.57 127.7 134.25 137 136.7 91 121.37 90.5 111.63 122.4 134.42 123.3 137.65 124.3 137.86 120 119.77 118.1 130.69 119 128.28 142.7 147.45 123.6 128.42 129.6 136.9 151.6 143.95 110.4 135.64 99.2 122.48 130.5 136.83 136.2 153.04 129.7 142.71 128 123.46 121.6 144.37 135.8 146.15 143.8 147.61 147.5 158.51 136.2 147.4 156.6 165.05 123.3 154.64 104.5 126.2 139.8 157.36 136.5 154.15 112.1 123.21 118.5 113.07 94.4 110.45 102.3 113.57 111.4 122.44 99.2 114.93 87.8 111.85 115.8 126.04

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 Investgoed[t] = -0.265915380217017 + 1.00324478774887Uitvoer[t] -27.2082055661959M1[t] -17.7491483570902M2[t] -10.7458319520821M3[t] -14.3164901701178M4[t] -13.5402815455929M5[t] + 0.110014736590909M6[t] -14.6658456995007M7[t] -8.21961363494651M8[t] -4.88144820951547M9[t] -9.68048635839078M10[t] -11.2145734937827M11[t] -0.0555664956132794t + 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) -0.265915380217017 7.714885 -0.0345 0.972653 0.486327 Uitvoer 1.00324478774887 0.056436 17.7766 0 0 M1 -27.2082055661959 3.164989 -8.5966 0 0 M2 -17.7491483570902 3.411725 -5.2024 4e-06 2e-06 M3 -10.7458319520821 3.106695 -3.4589 0.00118 0.00059 M4 -14.3164901701178 3.103017 -4.6137 3.2e-05 1.6e-05 M5 -13.5402815455929 3.122402 -4.3365 7.8e-05 3.9e-05 M6 0.110014736590909 3.275754 0.0336 0.973354 0.486677 M7 -14.6658456995007 3.209383 -4.5697 3.7e-05 1.8e-05 M8 -8.21961363494651 3.188231 -2.5781 0.013202 0.006601 M9 -4.88144820951547 3.090582 -1.5795 0.121084 0.060542 M10 -9.68048635839078 3.147703 -3.0754 0.003532 0.001766 M11 -11.2145734937827 3.138524 -3.5732 0.000841 0.000421 t -0.0555664956132794 0.040998 -1.3553 0.181925 0.090962 Multiple Linear Regression - Regression Statistics Multiple R 0.970324156962185 R-squared 0.941528969584375 Adjusted R-squared 0.925004547945176 F-TEST (value) 56.9780286500874 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 4.88345637341356 Sum Squared Residuals 1097.01472294754 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 79.8 82.6968173879417 -2.89681738794174 2 83.4 77.9244592505432 5.47554074945685 3 113.6 112.280856761237 1.31914323876273 4 112.9 108.985702827545 3.91429717245464 5 104 106.586253666558 -2.58625366655799 6 109.9 115.214921753772 -5.3149217537716 7 99 95.4374980184648 3.56250198153516 8 106.3 105.931434769299 0.368565230701424 9 128.9 126.810947276232 2.08905272376843 10 111.1 113.057561364410 -1.95756136441048 11 102.9 107.113825354575 -4.21382535457512 12 130 127.803657836359 2.19634216364111 13 87 84.4077095875478 2.59229041245216 14 87.5 86.0761829874964 1.42381701250359 15 117.6 121.154916745370 -3.55491674536967 16 103.4 107.446081914844 -4.04608191484446 17 110.8 117.476835674066 -6.67683567406565 18 112.6 123.798040749457 -11.1980407494569 19 102.5 105.364965029734 -2.86496502973356 20 112.4 113.400952050583 -1.00095205058257 21 135.6 139.677921515605 -4.07792151560453 22 105.1 107.785869841284 -2.6858698412838 23 127.7 121.927094482181 5.77290551781912 24 137 135.544051210335 1.45594878966492 25 91 92.9005365523357 -1.90053655233570 26 90.5 92.532423033154 -2.03242303315405 27 122.4 122.344121655346 0.0558783446542843 28 123.3 121.958377606126 1.34162239387443 29 124.3 122.889701140464 1.41029885953554 30 120 118.335732716658 1.6642672833421 31 118.1 114.459738867171 3.64026113282925 32 119 118.432584497637 0.567415502363172 33 142.7 140.947386008600 1.75261399139952 34 123.6 117.001033053251 6.59896694674917 35 129.6 123.918895222356 5.68110477764394 36 151.6 142.150777974155 9.44922202584495 37 110.4 106.550041726153 3.84995827384726 38 99.2 102.75083103287 -3.55083103286996 39 130.5 124.095143646461 6.40485635353882 40 136.2 136.731516942221 -0.531516942221357 41 129.7 127.088640413687 2.61135958631285 42 128 121.370908036092 6.62909196390811 43 121.6 127.517329616216 -5.91732961621599 44 135.8 135.693770907350 0.106229092650180 45 143.8 140.441107227281 3.35889277271905 46 147.5 146.521870769255 0.978129230744951 47 136.2 133.786167546360 2.41383245364013 48 156.6 162.652445048297 -6.05244504829693 49 123.3 124.944894746022 -1.64489474602197 50 104.5 105.816103695936 -1.31610369593642 51 139.8 144.024961191586 -4.22496119158616 52 136.5 137.178320709263 -0.678320709263254 53 112.1 106.858569105225 5.24143089477525 54 118.5 110.280396744022 8.21960325597826 55 94.4 92.8204684684149 1.57953153158514 56 102.3 102.341257775132 -0.0412577751322 57 111.4 114.522637972282 -3.12263797228246 58 99.2 102.133664971800 -2.93366497179984 59 87.8 97.454017394528 -9.65401739452808 60 115.8 122.849067930854 -7.04906793085405 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.381049082288669 0.762098164577338 0.618950917711331 18 0.369123833177892 0.738247666355783 0.630876166822108 19 0.251906262571093 0.503812525142187 0.748093737428907 20 0.170427966114207 0.340855932228414 0.829572033885793 21 0.118209646833454 0.236419293666909 0.881790353166546 22 0.0807213702340897 0.161442740468179 0.91927862976591 23 0.417635251993604 0.835270503987207 0.582364748006396 24 0.315378026480575 0.63075605296115 0.684621973519425 25 0.259764357512491 0.519528715024982 0.740235642487509 26 0.197570456922592 0.395140913845185 0.802429543077408 27 0.186049081673888 0.372098163347777 0.813950918326112 28 0.151872323252713 0.303744646505427 0.848127676747287 29 0.227524039792588 0.455048079585177 0.772475960207412 30 0.500808962188126 0.998382075623748 0.499191037811874 31 0.431142637560337 0.862285275120674 0.568857362439663 32 0.406336970121493 0.812673940242987 0.593663029878507 33 0.359274336798357 0.718548673596714 0.640725663201643 34 0.35418731325533 0.70837462651066 0.64581268674467 35 0.287547016503121 0.575094033006242 0.71245298349688 36 0.456178511935351 0.912357023870702 0.543821488064649 37 0.372641086963504 0.745282173927009 0.627358913036496 38 0.375219673292545 0.75043934658509 0.624780326707455 39 0.45916667746083 0.91833335492166 0.54083332253917 40 0.348197508518478 0.696395017036957 0.651802491481522 41 0.256470415458636 0.512940830917271 0.743529584541365 42 0.187638296963563 0.375276593927126 0.812361703036437 43 0.457723492396732 0.915446984793463 0.542276507603268 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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