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: Mon, 14 Dec 2009 11:53:52 -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/14/t1260816997vpci6yfr9ynwfjv.htm/, Retrieved Mon, 14 Dec 2009 19:56:49 +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/14/t1260816997vpci6yfr9ynwfjv.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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3.1 6.3 3.5 7.1 6 7.5 5.7 7.4 4.7 7.1 4.2 6.8 3.6 6.9 4.4 7.2 2.5 7.4 -0.6 7.3 -1.9 6.9 -1.9 6.9 0.7 6.8 -0.9 7.1 -1.7 7.2 -3.1 7.1 -2.1 7 0.2 6.9 1.2 7.1 3.8 7.3 4 7.5 6.6 7.5 5.3 7.5 7.6 7.3 4.7 7 6.6 6.7 4.4 6.5 4.6 6.5 6 6.5 4.8 6.6 4 6.8 2.7 6.9 3 6.9 4.1 6.8 4 6.8 2.7 6.5 2.6 6.1 3.1 6.1 4.4 5.9 3 5.7 2 5.9 1.3 5.9 1.5 6.1 1.3 6.3 3.2 6.2 1.8 5.9 3.3 5.7 1 5.4 2.4 5.6 0.4 6.2 -0.1 6.3 1.3 6 -1.1 5.6 -4.4 5.5 -7.5 5.9 -12.2 6.5 -14.5 6.8 -16 6.8 -16.7 6.5 -16.3 6.2 -16.9 6.2

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 ip[t] = -13.1395991502116 + 2.07226494952323wklh[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) -13.1395991502116 8.862386 -1.4826 0.143496 0.071748 wklh 2.07226494952323 1.336795 1.5502 0.126448 0.063224 Multiple Linear Regression - Regression Statistics Multiple R 0.19782701630127 R-squared 0.039135528378663 Adjusted R-squared 0.0228496898766064 F-TEST (value) 2.40304043133677 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 0.126447615361309 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.96464060027277 Sum Squared Residuals 2099.03931193492 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 3.1 -0.0843299682152474 3.18432996821525 2 3.5 1.57348199140331 1.92651800859669 3 6 2.4023879712126 3.5976120287874 4 5.7 2.19516147626028 3.50483852373972 5 4.7 1.57348199140330 3.12651800859670 6 4.2 0.951802506546335 3.24819749345367 7 3.6 1.15902900149866 2.44097099850134 8 4.4 1.78070848635563 2.61929151364437 9 2.5 2.19516147626028 0.304838523739724 10 -0.6 1.98793498130795 -2.58793498130795 11 -1.9 1.15902900149866 -3.05902900149866 12 -1.9 1.15902900149866 -3.05902900149866 13 0.7 0.951802506546335 -0.251802506546335 14 -0.9 1.57348199140330 -2.47348199140330 15 -1.7 1.78070848635563 -3.48070848635563 16 -3.1 1.57348199140330 -4.67348199140331 17 -2.1 1.36625549645098 -3.46625549645098 18 0.2 1.15902900149866 -0.95902900149866 19 1.2 1.57348199140330 -0.373481991403304 20 3.8 1.98793498130795 1.81206501869205 21 4 2.4023879712126 1.59761202878740 22 6.6 2.4023879712126 4.1976120287874 23 5.3 2.4023879712126 2.8976120287874 24 7.6 1.98793498130795 5.61206501869205 25 4.7 1.36625549645098 3.33374450354902 26 6.6 0.744576011594013 5.85542398840599 27 4.4 0.330123021689366 4.06987697831063 28 4.6 0.330123021689366 4.26987697831063 29 6 0.330123021689366 5.66987697831063 30 4.8 0.537349516641688 4.26265048335831 31 4 0.951802506546335 3.04819749345367 32 2.7 1.15902900149866 1.54097099850134 33 3 1.15902900149866 1.84097099850134 34 4.1 0.951802506546335 3.14819749345366 35 4 0.951802506546335 3.04819749345367 36 2.7 0.330123021689366 2.36987697831063 37 2.6 -0.498782958119928 3.09878295811993 38 3.1 -0.498782958119929 3.59878295811993 39 4.4 -0.913235948024572 5.31323594802457 40 3 -1.32768893792922 4.32768893792922 41 2 -0.913235948024573 2.91323594802457 42 1.3 -0.913235948024573 2.21323594802457 43 1.5 -0.498782958119928 1.99878295811993 44 1.3 -0.0843299682152812 1.38432996821528 45 3.2 -0.291556463167603 3.4915564631676 46 1.8 -0.913235948024573 2.71323594802457 47 3.3 -1.32768893792922 4.62768893792922 48 1 -1.94936842278619 2.94936842278619 49 2.4 -1.53491543288154 3.93491543288154 50 0.4 -0.291556463167604 0.691556463167604 51 -0.1 -0.0843299682152812 -0.0156700317847188 52 1.3 -0.70600945307225 2.00600945307225 53 -1.1 -1.53491543288154 0.434915432881544 54 -4.4 -1.74214192783387 -2.65785807216613 55 -7.5 -0.913235948024572 -6.58676405197543 56 -12.2 0.330123021689365 -12.5301230216894 57 -14.5 0.951802506546336 -15.4518025065463 58 -16 0.951802506546333 -16.9518025065463 59 -16.7 0.330123021689364 -17.0301230216894 60 -16.3 -0.291556463167606 -16.0084435368324 61 -16.9 -0.291556463167606 -16.6084435368324 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.00258004387475797 0.00516008774951593 0.997419956125242 6 0.000273734746124935 0.00054746949224987 0.999726265253875 7 3.82326939519149e-05 7.64653879038297e-05 0.999961767306048 8 4.36479011406359e-06 8.72958022812719e-06 0.999995635209886 9 3.17913307477928e-05 6.35826614955856e-05 0.999968208669252 10 0.000760829561478636 0.00152165912295727 0.999239170438521 11 0.0025543714647826 0.0051087429295652 0.997445628535217 12 0.0033949723472136 0.0067899446944272 0.996605027652786 13 0.00148044395388511 0.00296088790777023 0.998519556046115 14 0.00113838045747458 0.00227676091494915 0.998861619542526 15 0.00113439039331077 0.00226878078662154 0.99886560960669 16 0.00154315989974042 0.00308631979948084 0.99845684010026 17 0.00116496816554407 0.00232993633108814 0.998835031834456 18 0.000520920052408494 0.00104184010481699 0.999479079947591 19 0.000215879384779756 0.000431758769559512 0.99978412061522 20 0.000100684342686815 0.000201368685373629 0.999899315657313 21 4.3014876017004e-05 8.6029752034008e-05 0.999956985123983 22 3.34151240394613e-05 6.68302480789226e-05 0.99996658487596 23 1.64098802479321e-05 3.28197604958641e-05 0.999983590119752 24 2.33638936271725e-05 4.67277872543451e-05 0.999976636106373 25 1.55787542643212e-05 3.11575085286423e-05 0.999984421245736 26 3.10981765459336e-05 6.21963530918672e-05 0.999968901823454 27 2.32693096179985e-05 4.65386192359970e-05 0.999976730690382 28 1.65806932031465e-05 3.31613864062930e-05 0.999983419306797 29 1.73010744044595e-05 3.46021488089190e-05 0.999982698925596 30 1.21704787456854e-05 2.43409574913708e-05 0.999987829521254 31 8.02984411055284e-06 1.60596882211057e-05 0.99999197015589 32 5.1507330287465e-06 1.0301466057493e-05 0.999994849266971 33 4.26554773702543e-06 8.53109547405085e-06 0.999995734452263 34 6.02982027085311e-06 1.20596405417062e-05 0.99999397017973 35 1.68589884482400e-05 3.37179768964799e-05 0.999983141011552 36 2.71840960096969e-05 5.43681920193937e-05 0.99997281590399 37 1.93230363127247e-05 3.86460726254494e-05 0.999980676963687 38 1.62784229567750e-05 3.25568459135501e-05 0.999983721577043 39 1.29498249839165e-05 2.58996499678330e-05 0.999987050175016 40 5.91849510829651e-06 1.18369902165930e-05 0.999994081504892 41 3.27148450728741e-06 6.54296901457482e-06 0.999996728515493 42 1.77864921067691e-06 3.55729842135382e-06 0.99999822135079 43 1.55681335333716e-06 3.11362670667432e-06 0.999998443186647 44 3.39152214363053e-06 6.78304428726106e-06 0.999996608477856 45 1.58261018553358e-05 3.16522037106715e-05 0.999984173898145 46 1.51278675584226e-05 3.02557351168453e-05 0.999984872132442 47 1.32218301775317e-05 2.64436603550635e-05 0.999986778169822 48 5.50016829040213e-06 1.10003365808043e-05 0.99999449983171 49 3.22927480511953e-06 6.45854961023906e-06 0.999996770725195 50 2.42498846654192e-05 4.84997693308383e-05 0.999975750115335 51 0.00107886418020569 0.00215772836041137 0.998921135819794 52 0.0484497260810852 0.0968994521621704 0.951550273918915 53 0.115363691707691 0.230727383415381 0.884636308292309 54 0.168715008515886 0.337430017031771 0.831284991484114 55 0.774110774753551 0.451778450492897 0.225889225246449 56 0.984472702835704 0.0310545943285929 0.0155272971642964 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 47 0.903846153846154 NOK 5% type I error level 48 0.923076923076923 NOK 10% type I error level 49 0.942307692307692 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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