model 1 multiple regression

*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: Sun, 06 Dec 2009 04:43:28 -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/06/t12601035179to9g9c88jbggr8.htm/, Retrieved Sun, 06 Dec 2009 13:45:29 +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/06/t12601035179to9g9c88jbggr8.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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3,9 4,2 3,6 4,5 3,3 4,6 3,2 4,9 3,4 4,9 3,4 4,5 3,5 4,6 3,2 4,7 3,3 4,7 3,3 4,3 3,4 4,2 3,7 4,4 3,9 4 4 3,8 3,7 3,6 3,9 3,6 4,2 3,3 4,4 3,4 4,3 3,4 4,2 3,3 4,3 3,3 4,3 3,2 4,3 3,1 4,5 3,1 5 2,4 5,2 2,4 5,2 2,4 5,4 2,1 5,5 2 5,4 2 5,5 2,1 5,4 2,1 5,7 2 5,7 2 6,1 2 6,5 1,7 6,9 1,3 6,8 1,2 6,7 1,1 6,6 1,4 6,5 1,5 6,4 1,4 6,1 1,1 6,2 1,1 6,3 1 6,4 1,4 6,5 1,3 6,7 1,2 7 1,5 7 1,6 6,8 1,8 6,7 1,5 6,7 1,3 6,5 1,6 6,4 1,6 6,1 1,8 6,2 1,8 6 1,6 6,1 1,8 6,1 2 6,2 1,3

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Werkl[t] = + 7.78924216993382 -0.990087722076194Infl[t] + 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) 7.78924216993382 0.095941 81.1877 0 0 Infl -0.990087722076194 0.033603 -29.4643 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.967658866922448 R-squared 0.936363682733636 Adjusted R-squared 0.935285101085053 F-TEST (value) 868.143532725851 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.324365515382357 Sum Squared Residuals 6.20756626658645 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 3.9 3.63087373721376 0.269126262786235 2 3.6 3.33384742059094 0.266152579409059 3 3.3 3.23483864838332 0.0651613516166811 4 3.2 2.93781233176046 0.262187668239540 5 3.4 2.93781233176046 0.46218766823954 6 3.4 3.33384742059094 0.066152579409062 7 3.5 3.23483864838332 0.265161351616681 8 3.2 3.1358298761757 0.0641701238243012 9 3.3 3.1358298761757 0.164170123824301 10 3.3 3.53186496500618 -0.231864965006177 11 3.4 3.6308737372138 -0.230873737213796 12 3.7 3.43285619279856 0.267143807201443 13 3.9 3.82889128162903 0.071108718370965 14 4 4.02690882604427 -0.026908826044274 15 3.7 4.22492637045951 -0.524926370459512 16 3.9 4.22492637045951 -0.324926370459513 17 4.2 4.52195268708237 -0.321952687082371 18 4.4 4.42294391487475 -0.0229439148747511 19 4.3 4.42294391487475 -0.122943914874752 20 4.2 4.52195268708237 -0.321952687082371 21 4.3 4.52195268708237 -0.221952687082371 22 4.3 4.62096145928999 -0.32096145928999 23 4.3 4.71997023149761 -0.41997023149761 24 4.5 4.71997023149761 -0.219970231497610 25 5 5.41303163695095 -0.413031636950946 26 5.2 5.41303163695095 -0.213031636950946 27 5.2 5.41303163695095 -0.213031636950946 28 5.4 5.7100579535738 -0.310057953573803 29 5.5 5.80906672578142 -0.309066725781423 30 5.4 5.80906672578142 -0.409066725781423 31 5.5 5.7100579535738 -0.210057953573804 32 5.4 5.7100579535738 -0.310057953573803 33 5.7 5.80906672578142 -0.109066725781423 34 5.7 5.80906672578142 -0.109066725781423 35 6.1 5.80906672578142 0.290933274218576 36 6.5 6.10609304240428 0.393906957595718 37 6.9 6.50212813123476 0.397871868765241 38 6.8 6.60113690344238 0.198863096557621 39 6.7 6.70014567565 -0.000145675649997914 40 6.6 6.40311935902714 0.196880640972860 41 6.5 6.30411058681952 0.195889413180479 42 6.4 6.40311935902714 -0.00311935902713965 43 6.1 6.70014567565 -0.600145675649998 44 6.2 6.70014567565 -0.500145675649998 45 6.3 6.79915444785762 -0.499154447857618 46 6.4 6.40311935902714 -0.00311935902713965 47 6.5 6.50212813123476 -0.00212813123475922 48 6.7 6.60113690344238 0.0988630965576215 49 7 6.30411058681952 0.69588941318048 50 7 6.2051018146119 0.794898185388099 51 6.8 6.00708427019666 0.792915729803338 52 6.7 6.30411058681952 0.395889413180480 53 6.7 6.50212813123476 0.197871868765241 54 6.5 6.2051018146119 0.294898185388099 55 6.4 6.2051018146119 0.194898185388099 56 6.1 6.00708427019666 0.0929157298033375 57 6.2 6.00708427019666 0.192915729803338 58 6 6.2051018146119 -0.205101814611901 59 6.1 6.00708427019666 0.0929157298033375 60 6.1 5.80906672578142 0.290933274218576 61 6.2 6.50212813123476 -0.302128131234759 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.12014454293313 0.24028908586626 0.87985545706687 6 0.0778132195760344 0.155626439152069 0.922186780423966 7 0.0332432428804008 0.0664864857608016 0.9667567571196 8 0.0262419436383389 0.0524838872766777 0.973758056361661 9 0.0122498266852527 0.0244996533705055 0.987750173314747 10 0.0331607112328029 0.0663214224656059 0.966839288767197 11 0.0266538592120331 0.0533077184240662 0.973346140787967 12 0.0269350140747991 0.0538700281495982 0.973064985925201 13 0.0201344942895596 0.0402689885791192 0.97986550571044 14 0.0118065059251465 0.0236130118502931 0.988193494074853 15 0.0161120722740568 0.0322241445481136 0.983887927725943 16 0.0084253286312681 0.0168506572625362 0.991574671368732 17 0.00481665195767316 0.00963330391534631 0.995183348042327 18 0.006783612522412 0.013567225044824 0.993216387477588 19 0.0044967848901761 0.0089935697803522 0.995503215109824 20 0.00227889833447751 0.00455779666895503 0.997721101665523 21 0.00117178435359749 0.00234356870719498 0.998828215646403 22 0.00055540537556065 0.0011108107511213 0.99944459462444 23 0.000289928853412957 0.000579857706825913 0.999710071146587 24 0.000167927417471644 0.000335854834943288 0.999832072582528 25 0.00013522319142541 0.00027044638285082 0.999864776808575 26 0.000206479110705970 0.000412958221411939 0.999793520889294 27 0.000220668525945716 0.000441337051891432 0.999779331474054 28 0.000213945224326076 0.000427890448652152 0.999786054775674 29 0.000216253148781410 0.000432506297562819 0.999783746851219 30 0.000247151403579263 0.000494302807158527 0.99975284859642 31 0.000347591982135976 0.000695183964271951 0.999652408017864 32 0.000663322563125216 0.00132664512625043 0.999336677436875 33 0.00163701082758675 0.00327402165517351 0.998362989172413 34 0.00462481975392672 0.00924963950785343 0.995375180246073 35 0.0287141209743449 0.0574282419486899 0.971285879025655 36 0.0974660882299367 0.194932176459873 0.902533911770063 37 0.234136869531709 0.468273739063418 0.765863130468291 38 0.270541221732141 0.541082443464281 0.729458778267859 39 0.240316505274757 0.480633010549514 0.759683494725243 40 0.226057972980809 0.452115945961618 0.773942027019191 41 0.196512167880589 0.393024335761179 0.80348783211941 42 0.147707767754778 0.295415535509556 0.852292232245222 43 0.197394497000085 0.39478899400017 0.802605502999915 44 0.217939404175612 0.435878808351225 0.782060595824388 45 0.251647970375843 0.503295940751687 0.748352029624157 46 0.205738727988319 0.411477455976639 0.79426127201168 47 0.161318918550289 0.322637837100579 0.83868108144971 48 0.121147459380844 0.242294918761688 0.878852540619156 49 0.286490403185382 0.572980806370764 0.713509596814618 50 0.628971160544871 0.742057678910258 0.371028839455129 51 0.896729409766838 0.206541180466324 0.103270590233162 52 0.928623147535896 0.142753704928208 0.0713768524641039 53 0.95167603436268 0.0966479312746396 0.0483239656373198 54 0.977048762275893 0.0459024754482142 0.0229512377241071 55 0.993221959745209 0.0135560805095823 0.00677804025479115 56 0.970348391243274 0.059303217513451 0.0296516087567255 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 17 0.326923076923077 NOK 5% type I error level 25 0.480769230769231 NOK 10% type I error level 33 0.634615384615385 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No 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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