*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: Thu, 19 Nov 2009 08:36:14 -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/19/t12586462321oraq5r0jvhjzwh.htm/, Retrieved Thu, 19 Nov 2009 16:57:24 +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/19/t12586462321oraq5r0jvhjzwh.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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2.1 0 2.0 2.4 2.0 0 2.1 2.0 1.8 0 2.0 2.1 2.7 0 1.8 2.0 2.3 0 2.7 1.8 1.9 0 2.3 2.7 2.0 0 1.9 2.3 2.3 0 2.0 1.9 2.8 0 2.3 2.0 2.4 0 2.8 2.3 2.3 0 2.4 2.8 2.7 0 2.3 2.4 2.7 0 2.7 2.3 2.9 0 2.7 2.7 3.0 0 2.9 2.7 2.2 0 3.0 2.9 2.3 0 2.2 3.0 2.8 0 2.3 2.2 2.8 0 2.8 2.3 2.8 0 2.8 2.8 2.2 0 2.8 2.8 2.6 0 2.2 2.8 2.8 0 2.6 2.2 2.5 0 2.8 2.6 2.4 0 2.5 2.8 2.3 0 2.4 2.5 1.9 0 2.3 2.4 1.7 0 1.9 2.3 2.0 0 1.7 1.9 2.1 0 2.0 1.7 1.7 0 2.1 2.0 1.8 0 1.7 2.1 1.8 0 1.8 1.7 1.8 0 1.8 1.8 1.3 0 1.8 1.8 1.3 0 1.3 1.8 1.3 0 1.3 1.3 1.2 0 1.3 1.3 1.4 0 1.2 1.3 2.2 1 1.4 1.2 2.9 1 2.2 1.4 3.1 1 2.9 2.2 3.5 1 3.1 2.9 3.6 1 3.5 3.1 4.4 1 3.6 3.5 4.1 1 4.4 3.6 5.1 1 4.1 4.4 5.8 1 5.1 4.1 5.9 1 5.8 5.1 5.4 1 5.9 5.8 5.5 1 5.4 5.9 4.8 1 5.5 5.4 3.2 1 4.8 5.5 2.7 1 3.2 4.8 2.1 1 2.7 3.2 1.9 1 2.1 2.7 0.6 1 1.9 2.1

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.918381042450223 + 0.654697232943884X[t] + 1.08057771183516Y1[t] -0.255818518025034Y2[t] -0.201421589282912M1[t] -0.308680635057811M2[t] -0.200007912257289M3[t] -0.304535445515861M4[t] -0.48088115921706M5[t] -0.292490144066929M6[t] -0.403038896054930M7[t] -0.226210468212112M8[t] -0.422739955704922M9[t] -0.322312577454471M10[t] -0.0326139813926506M11[t] -0.0138870270198022t + 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) 0.918381042450223 0.37768 2.4316 0.019487 0.009743 X 0.654697232943884 0.305291 2.1445 0.037969 0.018985 Y1 1.08057771183516 0.164152 6.5828 0 0 Y2 -0.255818518025034 0.155223 -1.6481 0.106982 0.053491 M1 -0.201421589282912 0.336739 -0.5982 0.553028 0.276514 M2 -0.308680635057811 0.336817 -0.9165 0.364783 0.182391 M3 -0.200007912257289 0.338412 -0.591 0.557752 0.278876 M4 -0.304535445515861 0.343353 -0.8869 0.380281 0.190141 M5 -0.48088115921706 0.341655 -1.4075 0.166814 0.083407 M6 -0.292490144066929 0.345638 -0.8462 0.402335 0.201167 M7 -0.403038896054930 0.34213 -1.178 0.245577 0.122789 M8 -0.226210468212112 0.34373 -0.6581 0.514149 0.257075 M9 -0.422739955704922 0.340734 -1.2407 0.221778 0.110889 M10 -0.322312577454471 0.353886 -0.9108 0.367734 0.183867 M11 -0.0326139813926506 0.35551 -0.0917 0.927353 0.463676 t -0.0138870270198022 0.007196 -1.9297 0.060583 0.030291 Multiple Linear Regression - Regression Statistics Multiple R 0.931744212410473 R-squared 0.868147277360413 Adjusted R-squared 0.81990847639471 F-TEST (value) 17.9968668370853 F-TEST (DF numerator) 15 F-TEST (DF denominator) 41 p-value 2.00950367457153e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.499925504351956 Sum Squared Residuals 10.2469459059638 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2.1 2.25026340655775 -0.150263406557748 2 2 2.33950251215658 -0.339502512156579 3 1.8 2.30064858495128 -0.500648584951281 4 2.7 1.99170033410838 0.708299665891623 5 2.3 2.82515123764403 -0.525151237644028 6 1.9 2.33718747481776 -0.437187474817762 7 2 1.88284801828591 0.117151981714092 8 2.3 2.25617459750245 0.0438254024975471 9 2.8 2.34434954473788 0.455650455262116 10 2.4 2.89443319647860 -0.494433196478605 11 2.3 2.61010442177404 -0.310104421774041 12 2.7 2.62310101217339 0.0768989878266132 13 2.7 2.86560533240724 -0.165605332407241 14 2.9 2.64213185240253 0.257868147597473 15 3 2.95303309055028 0.0469669094497221 16 2.2 2.89151259785041 -0.691512597850414 17 2.3 1.81123583585878 0.48876416414122 18 2.8 2.29845240959265 0.501547590407348 19 2.8 2.68872363469993 0.111276365300074 20 2.8 2.72375577651042 0.0762442234895748 21 2.2 2.51333926199781 -0.313339261997812 22 2.6 1.95153298612736 0.648467013872636 23 2.8 2.81306675071847 -0.0130667507184674 24 2.5 2.94558184024833 -0.445581840248334 25 2.4 2.35493620679006 0.0450637932099351 26 2.3 2.20247791821936 0.0975220817806422 27 1.9 2.21478769461906 -0.314787694619064 28 1.7 1.68972390140913 0.0102760985908709 29 2 1.38570302553111 0.614296974468891 30 2.1 1.93554403081699 0.164455969183007 31 1.7 1.84242046758520 -0.142420467585196 32 1.8 1.54754893187164 0.252451068128356 33 1.8 1.54751759575256 0.252482404247439 34 1.8 1.60847609518071 0.191523904819294 35 1.3 1.88428766422272 -0.584287664222725 36 1.3 1.36272576267799 -0.0627257626779923 37 1.3 1.27532640538779 0.0246735946122057 38 1.2 1.15418033259309 0.0458196674069067 39 1.4 1.14090825719030 0.259091742809703 40 2.2 1.91888832402534 0.281111675974658 41 2.9 2.54195404916746 0.358045950832536 42 3.1 3.26820762116238 -0.168207621162379 43 3.5 3.18081442190409 0.319185578095915 44 3.6 3.72482320385616 -0.124823203856159 45 4.4 3.52013705331705 0.879862946682951 46 4.1 4.44555772221332 -0.345557722213324 47 5.1 4.19254116328477 0.907458836715234 48 5.8 5.36859138490029 0.431408615099713 49 5.9 5.65386864885715 0.246131351142848 50 5.4 5.46170738462844 -0.0617073846284438 51 5.5 4.99062237268908 0.50937762731092 52 4.8 5.10817484260674 -0.308174842606738 53 3.2 4.13595585179862 -0.93595585179862 54 2.7 2.76060846361021 -0.0606084636102141 55 2.1 2.50519345752488 -0.405193457524884 56 1.9 2.14769749025932 -0.247697490259319 57 0.6 1.87465654419469 -1.27465654419469 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.545195235092514 0.909609529814973 0.454804764907486 20 0.376182279428507 0.752364558857014 0.623817720571493 21 0.447594288741657 0.895188577483314 0.552405711258343 22 0.319108602978687 0.638217205957373 0.680891397021313 23 0.247362030100271 0.494724060200543 0.752637969899729 24 0.28189039528837 0.56378079057674 0.71810960471163 25 0.256230594218741 0.512461188437483 0.743769405781259 26 0.241597701303076 0.483195402606152 0.758402298696924 27 0.499736242217337 0.999472484434675 0.500263757782663 28 0.507493646068328 0.985012707863344 0.492506353931672 29 0.435017231678514 0.870034463357029 0.564982768321486 30 0.348481267722258 0.696962535444515 0.651518732277742 31 0.303593934121942 0.607187868243883 0.696406065878058 32 0.218856992319492 0.437713984638984 0.781143007680508 33 0.152177263530724 0.304354527061448 0.847822736469276 34 0.134780565846445 0.269561131692889 0.865219434153555 35 0.194034740653128 0.388069481306255 0.805965259346872 36 0.186383932350750 0.372767864701499 0.81361606764925 37 0.126815856133369 0.253631712266737 0.873184143866631 38 0.063261094160008 0.126522188320016 0.936738905839992 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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