*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: Tue, 15 Dec 2009 03:41:15 -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/15/t1260873767upm5swg4yrrfuru.htm/, Retrieved Tue, 15 Dec 2009 11:43:00 +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/15/t1260873767upm5swg4yrrfuru.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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Dataseries X:

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89.1 0 90.3 94.1 82.6 0 89.1 90.3 102.7 0 82.6 89.1 91.8 0 102.7 82.6 94.1 0 91.8 102.7 103.1 0 94.1 91.8 93.2 0 103.1 94.1 91 0 93.2 103.1 94.3 0 91 93.2 99.4 0 94.3 91 115.7 0 99.4 94.3 116.8 0 115.7 99.4 99.8 0 116.8 115.7 96 0 99.8 116.8 115.9 0 96 99.8 109.1 0 115.9 96 117.3 0 109.1 115.9 109.8 0 117.3 109.1 112.8 0 109.8 117.3 110.7 0 112.8 109.8 100 0 110.7 112.8 113.3 0 100 110.7 122.4 0 113.3 100 112.5 0 122.4 113.3 104.2 0 112.5 122.4 92.5 0 104.2 112.5 117.2 0 92.5 104.2 109.3 0 117.2 92.5 106.1 0 109.3 117.2 118.8 0 106.1 109.3 105.3 0 118.8 106.1 106 0 105.3 118.8 102 0 106 105.3 112.9 0 102 106 116.5 0 112.9 102 114.8 0 116.5 112.9 100.5 0 114.8 116.5 85.4 0 100.5 114.8 114.6 0 85.4 100.5 109.9 0 114.6 85.4 100.7 0 109.9 114.6 115.5 0 100.7 109.9 100.7 1 115.5 100.7 99 1 100.7 115.5 102.3 1 99 100.7 108.8 1 102.3 99 105.9 1 108.8 102.3 113.2 1 105.9 108.8 95.7 1 113.2 105.9 80.9 1 95.7 113.2 113.9 1 80.9 95.7 98.1 1 113.9 80.9 102.8 1 98.1 113. 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 52.256193984698 -4.11718487866997X[t] + 0.220811068130770Y1[t] + 0.336607809560939Y2[t] -16.6367537407777M1[t] -24.0326149655328M2[t] + 7.5024817685998M3[t] -3.88948328986188M4[t] -9.8984806608981M5[t] -0.800380067444774M6[t] -10.3918250786584M7[t] -11.9106931547395M8[t] -8.9388430332159M9[t] -0.887711502151985M10[t] + 3.94199266424966M11[t] + 0.0617692125452959t + 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) 52.256193984698 13.757349 3.7984 0.000443 0.000222 X -4.11718487866997 2.827576 -1.4561 0.152469 0.076234 Y1 0.220811068130770 0.130152 1.6966 0.096845 0.048423 Y2 0.336607809560939 0.129791 2.5935 0.012854 0.006427 M1 -16.6367537407777 3.207553 -5.1867 5e-06 3e-06 M2 -24.0326149655328 3.799676 -6.3249 0 0 M3 7.5024817685998 4.293143 1.7475 0.087518 0.043759 M4 -3.88948328986188 3.952901 -0.984 0.330517 0.165258 M5 -9.8984806608981 3.556994 -2.7828 0.00791 0.003955 M6 -0.800380067444774 3.275642 -0.2443 0.808101 0.404051 M7 -10.3918250786584 3.064577 -3.3909 0.001481 0.00074 M8 -11.9106931547395 3.550716 -3.3544 0.001645 0.000823 M9 -8.9388430332159 3.383603 -2.6418 0.011375 0.005688 M10 -0.887711502151985 3.400153 -0.2611 0.795249 0.397625 M11 3.94199266424966 3.163533 1.2461 0.219332 0.109666 t 0.0617692125452959 0.074454 0.8296 0.411226 0.205613 Multiple Linear Regression - Regression Statistics Multiple R 0.902332302776116 R-squared 0.814203584633248 Adjusted R-squared 0.7508638975764 F-TEST (value) 12.8545564789813 F-TEST (DF numerator) 15 F-TEST (DF denominator) 44 p-value 1.81019643719083e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 4.80445749036213 Sum Squared Residuals 1015.64371817466 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 89.1 87.2952437883586 1.80475621164137 2 82.6 78.4170688180601 4.18293118193987 3 102.7 108.174733450415 -5.47473345041491 4 91.8 99.094889311781 -7.2948893117809 5 94.1 97.5066374828394 -3.40663748283945 6 103.1 103.505347621325 -0.405347621324616 7 93.2 96.7371693978234 -3.53716939782339 8 91 96.1235112458414 -5.1235112458414 9 94.3 95.3389289153693 -1.03892891536929 10 99.4 103.439969002776 -4.03996900277598 11 115.7 110.568384600741 5.13161539925906 12 116.8 112.004081388329 4.79591861167107 13 99.8 101.158696330884 -1.35869633088371 14 96 90.4410847509678 5.55891524903223 15 115.9 115.476535876213 0.423464123787176 16 109.1 107.261370609767 1.83862939023281 17 117.3 106.511122598250 10.7888774017503 18 109.8 115.192710057906 -5.39271005790625 19 112.8 106.767135286657 6.0328647133431 20 110.7 103.447911055806 7.25208894419367 21 100 107.027650575483 -7.02765057548341 22 113.3 112.070996490015 1.22900350998458 23 122.4 116.297553512800 6.10244648720046 24 112.5 118.903594648246 -6.40359464824567 25 104.2 103.205711612523 0.994288387476765 26 92.5 90.7064704201747 1.79352957982533 27 117.2 116.926002050367 0.273997949633184 28 109.3 107.111528215417 2.18847178458255 29 106.1 107.734105514849 -1.63410551484863 30 118.8 113.528178207297 5.27182179270263 31 105.3 105.725657983295 -0.425657983294858 32 106 105.562528881418 0.437471118582446 33 102 104.206510534105 -2.20651053410530 34 112.9 111.671792471884 1.22820752811592 35 116.5 117.623675255213 -1.12367525521267 36 114.8 118.207396772993 -3.40739677299331 37 100.5 102.468821543358 -1.96882154335802 38 85.4 91.4048979806245 -6.00489798062453 39 114.6 114.854025121806 -0.254025121806433 40 109.9 104.888734540938 5.01126545906168 41 100.7 107.732642401412 -7.03264240141219 42 115.5 113.278993675671 2.22100632432867 43 100.7 99.8033449587078 0.896655041292176 44 99 100.060037868338 -1.06003786833849 45 102.3 97.7364828050832 4.56351719491683 46 108.8 106.005826797270 2.79417320272967 47 105.9 113.443377890618 -7.54337789061836 48 113.2 111.110753103481 2.08924689651913 49 95.7 95.1715267248764 0.528473275123591 50 80.9 86.4304780301729 -5.5304780301729 51 113.9 108.868703501199 5.03129649880097 52 98.1 99.8434773220961 -1.74347732209615 53 102.8 101.51549200265 1.28450799734997 54 104.7 106.394770437800 -1.69477043780044 55 95.9 98.866692373517 -2.96669237351703 56 94.6 96.1060109485962 -1.50601094859622 57 101.6 95.8904271699588 5.70957283004116 58 103.9 105.111415238054 -1.2114152380542 59 110.3 112.867008740628 -2.56700874062849 60 114.1 111.174174086951 2.92582591304878 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.869075307891764 0.261849384216472 0.130924692108236 20 0.814716826246422 0.370566347507156 0.185283173753578 21 0.85174798323729 0.296504033525421 0.148252016762710 22 0.846989440551556 0.306021118896889 0.153010559448444 23 0.94424845473594 0.111503090528119 0.0557515452640594 24 0.998328137250768 0.00334372549846490 0.00167186274923245 25 0.997274695682457 0.0054506086350856 0.0027253043175428 26 0.997528610878277 0.00494277824344642 0.00247138912172321 27 0.9943443696376 0.0113112607247984 0.00565563036239919 28 0.98822498509394 0.0235500298121193 0.0117750149060597 29 0.984566569245833 0.0308668615083347 0.0154334307541674 30 0.982211590654233 0.0355768186915346 0.0177884093457673 31 0.971811079173001 0.0563778416539975 0.0281889208269987 32 0.965491117540715 0.0690177649185694 0.0345088824592847 33 0.950969035752081 0.0980619284958384 0.0490309642479192 34 0.914274740512324 0.171450518975352 0.0857252594876758 35 0.916394608813004 0.167210782373993 0.0836053911869964 36 0.884027845525573 0.231944308948854 0.115972154474427 37 0.82341976535224 0.353160469295521 0.176580234647761 38 0.770952054052808 0.458095891894384 0.229047945947192 39 0.716746408554063 0.566507182891874 0.283253591445937 40 0.73797755144934 0.52404489710132 0.26202244855066 41 0.797942771577721 0.404114456844557 0.202057228422279 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 3 0.130434782608696 NOK 5% type I error level 7 0.304347826086957 NOK 10% type I error level 10 0.434782608695652 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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