Model 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: Thu, 19 Nov 2009 13:41:09 -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/t1258663487fwbv92mcd9banm7.htm/, Retrieved Thu, 19 Nov 2009 21:44:59 +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/t1258663487fwbv92mcd9banm7.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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3 101.2 3.21 101.1 3.37 100.7 3.51 100.1 3.75 99.9 4.11 99.7 4.25 99.5 4.25 99.2 4.5 99 4.7 99 4.75 99.3 4.75 99.5 4.75 99.7 4.75 100 4.75 100.4 4.75 100.6 4.58 100.7 4.5 100.7 4.5 100.6 4.49 100.5 4.03 100.6 3.75 100.5 3.39 100.4 3.25 100.3 3.25 100.4 3.25 100.4 3.25 100.4 3.25 100.4 3.25 100.4 3.25 100.5 3.25 100.6 3.25 100.6 3.25 100.5 3.25 100.5 3.25 100.7 2.85 101.1 2.75 101.5 2.75 101.9 2.55 102.1 2.5 102.1 2.5 102.1 2.1 102.4 2 102.8 2 103.1 2 103.1 2 102.9 2 102.4 2 101.9 2 101.3 2 100.7 2 100.6 2 101 2 101.5 2 101.9 2 102.1 2 102.3 2 102.5 2 102.9 2 103.6 2 104.3

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 Tprod[t] = + 102.226427377846 -0.601785698427419Rente[t] + 0.0235445632537601t + 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) 102.226427377846 0.828026 123.458 0 0 Rente -0.601785698427419 0.173837 -3.4618 0.001024 0.000512 t 0.0235445632537601 0.009938 2.3692 0.021236 0.010618 Multiple Linear Regression - Regression Statistics Multiple R 0.821645845867635 R-squared 0.675101896031541 Adjusted R-squared 0.663701962558964 F-TEST (value) 59.2198101555151 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 1.22124532708767e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.689015006741784 Sum Squared Residuals 27.0602757323767 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 101.2 100.444614845818 0.755385154181666 2 101.1 100.341784412402 0.758215587598025 3 100.7 100.269043263907 0.430956736092665 4 100.1 100.208337829381 -0.108337829381265 5 99.9 100.087453825012 -0.187453825012433 6 99.7 99.8943555368323 -0.194355536832326 7 99.5 99.8336501023063 -0.33365010230625 8 99.2 99.85719466556 -0.657194665560007 9 99 99.730292804207 -0.730292804206915 10 99 99.6334802277752 -0.633480227775192 11 99.3 99.6269355061076 -0.326935506107584 12 99.5 99.6504800693613 -0.150480069361341 13 99.7 99.6740246326151 0.0259753673849017 14 100 99.6975691958689 0.302430804131139 15 100.4 99.7211137591226 0.678886240877384 16 100.6 99.7446583223764 0.855341677623613 17 100.7 99.8705064543628 0.8294935456372 18 100.7 99.9421938734908 0.757806126509247 19 100.6 99.9657384367445 0.634261563255478 20 100.5 99.9953008569826 0.50469914301745 21 100.6 100.295666841513 0.304333158487071 22 100.5 100.487711400326 0.0122885996736398 23 100.4 100.727898815014 -0.327898815013985 24 100.3 100.835693376048 -0.535693376047592 25 100.4 100.859237939301 -0.459237939301344 26 100.4 100.882782502555 -0.482782502555104 27 100.4 100.906327065809 -0.506327065808864 28 100.4 100.929871629063 -0.529871629062624 29 100.4 100.953416192316 -0.553416192316384 30 100.5 100.97696075557 -0.47696075557015 31 100.6 101.000505318824 -0.400505318823916 32 100.6 101.024049882078 -0.424049882077676 33 100.5 101.047594445331 -0.54759444533143 34 100.5 101.071139008585 -0.57113900858519 35 100.7 101.094683571839 -0.394683571838948 36 101.1 101.358942414464 -0.258942414463683 37 101.5 101.442665547560 0.0573344524398202 38 101.9 101.466210110814 0.433789889186066 39 102.1 101.610111813753 0.489888186246811 40 102.1 101.663745661928 0.43625433807168 41 102.1 101.687290225182 0.41270977481792 42 102.4 101.951549067807 0.448450932193204 43 102.8 102.035272200903 0.764727799096693 44 103.1 102.058816764157 1.04118323584293 45 103.1 102.082361327411 1.01763867258917 46 102.9 102.105905890665 0.794094109335421 47 102.4 102.129450453918 0.270549546081661 48 101.9 102.152995017172 -0.252995017172099 49 101.3 102.176539580426 -0.876539580425867 50 100.7 102.200084143680 -1.50008414367962 51 100.6 102.223628706933 -1.62362870693339 52 101 102.247173270187 -1.24717327018714 53 101.5 102.270717833441 -0.770717833440905 54 101.9 102.294262396695 -0.394262396694659 55 102.1 102.317806959948 -0.217806959948431 56 102.3 102.341351523202 -0.0413515232021879 57 102.5 102.364896086456 0.135103913544055 58 102.9 102.388440649710 0.5115593502903 59 103.6 102.411985212963 1.18801478703653 60 104.3 102.435529776217 1.86447022378277 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.00808250383865616 0.0161650076773123 0.991917496161344 7 0.00226721635627179 0.00453443271254359 0.997732783643728 8 0.00131569383772138 0.00263138767544276 0.998684306162279 9 0.000434651757135615 0.00086930351427123 0.999565348242864 10 0.00061029236718969 0.00122058473437938 0.99938970763281 11 0.00396572816705542 0.00793145633411084 0.996034271832945 12 0.00308745799923273 0.00617491599846547 0.996912542000767 13 0.00124667684849381 0.00249335369698761 0.998753323151506 14 0.000451043533484907 0.000902087066969814 0.999548956466515 15 0.000172273225088986 0.000344546450177973 0.99982772677491 16 5.94591794488002e-05 0.000118918358897600 0.999940540820551 17 0.000112262175233024 0.000224524350466049 0.999887737824767 18 0.000322503011659071 0.000645006023318142 0.99967749698834 19 0.000695812827501956 0.00139162565500391 0.999304187172498 20 0.00149893655288699 0.00299787310577397 0.998501063447113 21 0.0108022481718477 0.0216044963436955 0.989197751828152 22 0.0329522599627337 0.0659045199254674 0.967047740037266 23 0.0618725545397259 0.123745109079452 0.938127445460274 24 0.0748471765074719 0.149694353014944 0.925152823492528 25 0.0628209662961335 0.125641932592267 0.937179033703866 26 0.0480901840427665 0.096180368085533 0.951909815957233 27 0.0347449902525305 0.0694899805050609 0.96525500974747 28 0.0240519115555437 0.0481038231110874 0.975948088444456 29 0.0160985312168022 0.0321970624336044 0.983901468783198 30 0.00996304871308032 0.0199260974261606 0.99003695128692 31 0.00582479603999258 0.0116495920799852 0.994175203960007 32 0.00331583488326872 0.00663166976653745 0.996684165116731 33 0.00194730565383747 0.00389461130767494 0.998052694346163 34 0.00118064045659334 0.00236128091318668 0.998819359543407 35 0.000710970426036025 0.00142194085207205 0.999289029573964 36 0.000469341395018991 0.000938682790037982 0.99953065860498 37 0.000369508954605003 0.000739017909210005 0.999630491045395 38 0.000423239054801883 0.000846478109603767 0.999576760945198 39 0.000432152889863948 0.000864305779727895 0.999567847110136 40 0.000340362585049752 0.000680725170099504 0.99965963741495 41 0.000249563419334772 0.000499126838669544 0.999750436580665 42 0.000160499889555568 0.000320999779111136 0.999839500110444 43 0.000219969511172204 0.000439939022344408 0.999780030488828 44 0.000796456942773482 0.00159291388554696 0.999203543057227 45 0.00475524190305482 0.00951048380610965 0.995244758096945 46 0.0401257256510328 0.0802514513020656 0.959874274348967 47 0.240874337226897 0.481748674453793 0.759125662773103 48 0.7707594591131 0.4584810817738 0.2292405408869 49 0.9858237181948 0.0283525636103984 0.0141762818051992 50 0.991085683126021 0.0178286337479575 0.00891431687397873 51 0.98349506915711 0.0330098616857813 0.0165049308428907 52 0.96212658574448 0.0757468285110397 0.0378734142555198 53 0.921843134745118 0.156313730509765 0.0781568652548823 54 0.891041274078858 0.217917451842283 0.108958725921142 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 28 0.571428571428571 NOK 5% type I error level 37 0.755102040816326 NOK 10% type I error level 42 0.857142857142857 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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