*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 06:38:53 -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/t1258724369ku4vxmffl1l7hox.htm/, Retrieved Fri, 20 Nov 2009 14:39:42 +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/t1258724369ku4vxmffl1l7hox.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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2.09 0 2.11 2.05 0 2.09 2.08 0 2.05 2.06 0 2.08 2.06 0 2.06 2.08 0 2.06 2.07 0 2.08 2.06 0 2.07 2.07 0 2.06 2.06 0 2.07 2.09 0 2.06 2.07 0 2.09 2.09 0 2.07 2.28 0 2.09 2.33 0 2.28 2.35 0 2.33 2.52 0 2.35 2.63 0 2.52 2.58 0 2.63 2.70 0 2.58 2.81 0 2.70 2.97 0 2.81 3.04 0 2.97 3.28 0 3.04 3.33 0 3.28 3.50 0 3.33 3.56 0 3.50 3.57 0 3.56 3.69 0 3.57 3.82 0 3.69 3.79 0 3.82 3.96 0 3.79 4.06 0 3.96 4.05 0 4.06 4.03 0 4.05 3.94 0 4.03 4.02 0 3.94 3.88 0 4.02 4.02 0 3.88 4.03 0 4.02 4.09 0 4.03 3.99 0 4.09 4.01 0 3.99 4.01 0 4.01 4.19 0 4.01 4.30 0 4.19 4.27 0 4.30 3.82 0 4.27 3.15 1 3.82 2.49 1 3.15 1.81 1 2.49 1.26 1 1.81 1.06 1 1.26 0.84 1 1.06 0.78 1 0.84 0.70 1 0.78 0.36 1 0.70 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.335653443287973 -0.898644896907781X[t] + 0.828817234617628Y1[t] + 0.00939051102773215t + 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) 0.335653443287973 0.056319 5.9598 0 0 X -0.898644896907781 0.09671 -9.2921 0 0 Y1 0.828817234617628 0.025505 32.4965 0 0 t 0.00939051102773215 0.001831 5.1285 4e-06 2e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.99514204947928 R-squared 0.990307698641821 Adjusted R-squared 0.989779027658647 F-TEST (value) 1873.20229435968 F-TEST (DF numerator) 3 F-TEST (DF denominator) 55 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.115711745031606 Sum Squared Residuals 0.736406436604271 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2.09 2.09384831935891 -0.00384831935890569 2 2.05 2.08666248569428 -0.0366624856942818 3 2.08 2.06290030733731 0.0170996926626892 4 2.06 2.09715533540357 -0.0371553354035694 5 2.06 2.08996950173895 -0.0299695017389490 6 2.08 2.09936001276668 -0.0193600127666811 7 2.07 2.12532686848677 -0.0553268684867661 8 2.06 2.12642920716832 -0.0664292071683215 9 2.07 2.12753154584988 -0.0575315458498778 10 2.06 2.14521022922379 -0.0852102292237858 11 2.09 2.14631256790534 -0.0563125679053421 12 2.07 2.18056759597160 -0.110567595971603 13 2.09 2.17338176230698 -0.0833817623069825 14 2.28 2.19934861802707 0.0806513819729328 15 2.33 2.36621440363215 -0.0362144036321483 16 2.35 2.41704577639076 -0.067045776390762 17 2.52 2.44301263211085 0.0769873678891531 18 2.63 2.59330207302358 0.0366979269764243 19 2.58 2.69386247985925 -0.113862479859247 20 2.7 2.6618121291561 0.0381878708439025 21 2.81 2.77066070833795 0.0393392916620549 22 2.97 2.87122111517362 0.098778884826384 23 3.04 3.01322238374017 0.0267776162598311 24 3.28 3.08063010119113 0.199369898808865 25 3.33 3.2889367485271 0.0410632514729026 26 3.5 3.33976812128571 0.160231878714289 27 3.56 3.49005756219844 0.06994243780156 28 3.57 3.54917710730323 0.0208228926967699 29 3.69 3.56685579067714 0.123144209322862 30 3.82 3.67570436985899 0.144295630141014 31 3.79 3.79284112138701 -0.00284112138700933 32 3.96 3.77736711537621 0.182632884623787 33 4.06 3.92765655628894 0.132343443711058 34 4.05 4.01992879077844 0.0300712092215637 35 4.03 4.02103112945999 0.00896887054000805 36 3.94 4.01384529579537 -0.0738452957953723 37 4.02 3.94864225570752 0.071357744292482 38 3.88 4.02433814550466 -0.144338145504660 39 4.02 3.91769424368592 0.102305756314075 40 4.03 4.04311916756012 -0.0131191675601236 41 4.09 4.06079785093403 0.0292021490659669 42 3.99 4.11991739603882 -0.129917396038822 43 4.01 4.04642618360479 -0.0364261836047924 44 4.01 4.07239303932488 -0.0623930393248766 45 4.19 4.08178355035261 0.108216449647392 46 4.3 4.24036116361151 0.0596388363884856 47 4.27 4.34092157044719 -0.0709215704471853 48 3.82 4.32544756443639 -0.505447564436388 49 3.15 3.06322542297841 0.0867745770215936 50 2.49 2.51730838681233 -0.0273083868123280 51 1.81 1.97967952299243 -0.169679522992426 52 1.26 1.42547431448017 -0.165474314480172 53 1.06 0.979015346468209 0.0809846535317915 54 0.84 0.822642410572415 0.0173575894275846 55 0.78 0.649693129984269 0.130306870015731 56 0.7 0.609354606934944 0.0906453930650561 57 0.36 0.552439739193266 -0.192439739193266 58 0.35 0.280032390451004 0.0699676095489957 59 0.36 0.281134729132560 0.0788652708674396 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.00578386083578261 0.0115677216715652 0.994216139164217 8 0.000798841662070364 0.00159768332414073 0.99920115833793 9 0.000102979187006964 0.000205958374013927 0.999897020812993 10 1.23242226228452e-05 2.46484452456903e-05 0.999987675777377 11 4.55603627608217e-06 9.11207255216434e-06 0.999995443963724 12 6.20208765365553e-07 1.24041753073111e-06 0.999999379791235 13 1.50053060105216e-07 3.00106120210431e-07 0.99999984994694 14 0.00358020490201834 0.00716040980403668 0.996419795097982 15 0.00158266370220128 0.00316532740440255 0.998417336297799 16 0.000831805994478731 0.00166361198895746 0.999168194005521 17 0.00150911289116506 0.00301822578233011 0.998490887108835 18 0.000648323610282255 0.00129664722056451 0.999351676389718 19 0.00322619541839445 0.0064523908367889 0.996773804581605 20 0.00235509803176037 0.00471019606352074 0.99764490196824 21 0.00146006359798777 0.00292012719597554 0.998539936402012 22 0.00108015183958393 0.00216030367916787 0.998919848160416 23 0.000731686797134811 0.00146337359426962 0.999268313202865 24 0.0012641940261311 0.0025283880522622 0.998735805973869 25 0.00130564517631345 0.00261129035262689 0.998694354823687 26 0.000718781410480645 0.00143756282096129 0.99928121858952 27 0.00053851712292914 0.00107703424585828 0.99946148287707 28 0.000719277071258333 0.00143855414251667 0.999280722928742 29 0.000347825859534004 0.000695651719068009 0.999652174140466 30 0.000165204881148222 0.000330409762296444 0.999834795118852 31 0.000359785601771594 0.000719571203543188 0.999640214398228 32 0.000257537082481048 0.000515074164962096 0.999742462917519 33 0.000130076070487711 0.000260152140975423 0.999869923929512 34 0.000130018680280709 0.000260037360561419 0.99986998131972 35 0.000134701019315864 0.000269402038631728 0.999865298980684 36 0.000483301775399937 0.000966603550799873 0.9995166982246 37 0.000236880525054706 0.000473761050109412 0.999763119474945 38 0.00224997989952375 0.0044999597990475 0.997750020100476 39 0.00126129193045819 0.00252258386091639 0.998738708069542 40 0.000829644787256695 0.00165928957451339 0.999170355212743 41 0.000416677814510844 0.000833355629021688 0.99958332218549 42 0.000959726863065056 0.00191945372613011 0.999040273136935 43 0.000572214699302067 0.00114442939860413 0.999427785300698 44 0.000380971750554365 0.00076194350110873 0.999619028249446 45 0.000414391973779885 0.00082878394755977 0.99958560802622 46 0.00149724438383848 0.00299448876767696 0.998502755616162 47 0.0270836339929127 0.0541672679858254 0.972916366007087 48 0.170700043168993 0.341400086337986 0.829299956831007 49 0.225560096027178 0.451120192054355 0.774439903972822 50 0.372970811082909 0.745941622165817 0.627029188917091 51 0.428971015106453 0.857942030212906 0.571028984893547 52 0.31470519159447 0.62941038318894 0.68529480840553 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 39 0.847826086956522 NOK 5% type I error level 40 0.869565217391304 NOK 10% type I error level 41 0.891304347826087 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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