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: Thu, 19 Nov 2009 14:38:11 -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/t1258666753evqowb8r8vkajna.htm/, Retrieved Thu, 19 Nov 2009 22:39:26 +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/t1258666753evqowb8r8vkajna.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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106.1 97.89 106 98.69 105.9 99.01 105.8 99.18 105.7 98.45 105.6 98.13 105.4 98.29 105.4 99.1 105.5 99.26 105.6 98.85 105.7 98.05 105.9 98.53 106.1 99.34 106 100.14 105.8 100.3 105.8 100.22 105.7 99.9 105.5 99.58 105.3 99.9 105.2 100.78 105.2 100.78 105 100.46 105.1 100.06 105.1 100.28 105.2 100.78 104.9 101.58 104.8 102.06 104.5 102.02 104.5 101.68 104.4 101.32 104.4 101.81 104.2 102.3 104.1 102.12 103.9 102.1 103.8 101.75 103.9 101.5 104.2 102.16 104.1 103.47 103.8 104.05 103.6 104.09 103.7 103.55 103.5 102.77 103.4 102.89 103.1 103.6 103.1 103.76 103.1 103.92 103.2 103.35 103.3 103.32 103.5 104.2 103.6 105.44 103.5 105.81 103.3 106.25 103.2 105.94 103.1 105.82 103.2 105.96 103 106.49 103 106.32 103.1 105.88 103.4 105.07

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 Werkl[t] = + 143.751260601764 -0.385392472173445Infl[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) 143.751260601764 2.15361 66.749 0 0 Infl -0.385392472173445 0.021134 -18.2352 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.923941333344126 R-squared 0.853667587461722 Adjusted R-squared 0.851100352154033 F-TEST (value) 332.524091151642 F-TEST (DF numerator) 1 F-TEST (DF denominator) 57 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.406752304805825 Sum Squared Residuals 9.43050393549652 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 106.1 106.025191500705 0.0748084992951664 2 106 105.716877522966 0.283122477033612 3 105.9 105.593551931871 0.306448068129123 4 105.8 105.528035211601 0.271964788398600 5 105.7 105.809371716288 -0.109371716288011 6 105.6 105.932697307384 -0.332697307383524 7 105.4 105.871034511836 -0.471034511835758 8 105.4 105.558866609375 -0.158866609375272 9 105.5 105.497203813828 0.00279618617247816 10 105.6 105.655214727419 -0.0552147274186441 11 105.7 105.963528705157 -0.263528705157391 12 105.9 105.778540318514 0.121459681485867 13 106.1 105.466372416054 0.633627583946348 14 106 105.158058438315 0.841941561685108 15 105.8 105.096395642767 0.703604357232855 16 105.8 105.127227040541 0.67277295945898 17 105.7 105.250552631637 0.449447368363486 18 105.5 105.373878222732 0.126121777267978 19 105.3 105.250552631637 0.0494473683634804 20 105.2 104.911407256124 0.288592743876116 21 105.2 104.911407256124 0.288592743876116 22 105 105.034732847219 -0.034732847219392 23 105.1 105.188889836089 -0.0888898360887725 24 105.1 105.104103492211 -0.00410349221061495 25 105.2 104.911407256124 0.288592743876116 26 104.9 104.603093278385 0.296906721614874 27 104.8 104.418104891742 0.381895108258121 28 104.5 104.433520590629 0.0664794093711834 29 104.5 104.564554031168 -0.0645540311677838 30 104.4 104.703295321150 -0.303295321150224 31 104.4 104.514453009785 -0.114453009785232 32 104.2 104.325610698420 -0.125610698420249 33 104.1 104.394981343411 -0.294981343411474 34 103.9 104.402689192855 -0.502689192854936 35 103.8 104.537576558116 -0.737576558115648 36 103.9 104.633924676159 -0.733924676159 37 104.2 104.379565644525 -0.179565644524531 38 104.1 103.874701505977 0.225298494022674 39 103.8 103.651173872117 0.148826127883275 40 103.6 103.635758173230 -0.0357581732297879 41 103.7 103.843870108203 -0.143870108203442 42 103.5 104.144476236499 -0.644476236498733 43 103.4 104.098229139838 -0.698229139837912 44 103.1 103.824600484595 -0.72460048459478 45 103.1 103.762937689047 -0.662937689047024 46 103.1 103.701274893499 -0.601274893499274 47 103.2 103.920948602638 -0.720948602638132 48 103.3 103.932510376803 -0.632510376803342 49 103.5 103.593365001291 -0.0933650012907036 50 103.6 103.115478335796 0.484521664204361 51 103.5 102.972883121091 0.527116878908543 52 103.3 102.803310433335 0.496689566664855 53 103.2 102.922782099709 0.277217900291092 54 103.1 102.969029196370 0.130970803630268 55 103.2 102.915074250265 0.284925749734560 56 103 102.710816240014 0.289183759986483 57 103 102.776332960283 0.223667039716997 58 103.1 102.945905648039 0.154094351960676 59 103.4 103.258073550500 0.141926449500196 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0657043912428305 0.131408782485661 0.93429560875717 6 0.0889452644193762 0.177890528838752 0.911054735580624 7 0.146880293943616 0.293760587887231 0.853119706056384 8 0.148187746719477 0.296375493438954 0.851812253280523 9 0.0930978776883454 0.186195755376691 0.906902122311655 10 0.0516104980930191 0.103220996186038 0.94838950190698 11 0.0281123004990112 0.0562246009980225 0.971887699500989 12 0.0171703474984610 0.0343406949969221 0.98282965250154 13 0.0292404902431319 0.0584809804862639 0.970759509756868 14 0.0287512808073009 0.0575025616146019 0.9712487191927 15 0.0225811866843572 0.0451623733687144 0.977418813315643 16 0.0190712956310427 0.0381425912620854 0.980928704368957 17 0.0152832464742233 0.0305664929484466 0.984716753525777 18 0.0149056651691330 0.0298113303382660 0.985094334830867 19 0.0257394261855862 0.0514788523711724 0.974260573814414 20 0.0444340514632611 0.0888681029265221 0.955565948536739 21 0.0573603970954792 0.114720794190958 0.94263960290452 22 0.0898840696953125 0.179768139390625 0.910115930304688 23 0.105568827119153 0.211137654238306 0.894431172880847 24 0.11615861273552 0.23231722547104 0.88384138726448 25 0.144949624674705 0.289899249349410 0.855050375325295 26 0.198840540687441 0.397681081374882 0.801159459312559 27 0.302267900601939 0.604535801203878 0.697732099398061 28 0.403034447267361 0.806068894534723 0.596965552732639 29 0.509027696449989 0.981944607100021 0.490972303550011 30 0.627318536595967 0.745362926808065 0.372681463404033 31 0.713853251737293 0.572293496525414 0.286146748262707 32 0.76748647062328 0.465027058753439 0.232513529376720 33 0.81051409478064 0.378971810438721 0.189485905219361 34 0.843316677384611 0.313366645230777 0.156683322615389 35 0.887095563361121 0.225808873277758 0.112904436638879 36 0.90705627451035 0.185887450979298 0.0929437254896492 37 0.943297341635426 0.113405316729147 0.0567026583645735 38 0.987305741602801 0.0253885167943971 0.0126942583971985 39 0.994011958559478 0.0119760828810436 0.00598804144052182 40 0.993655045565876 0.0126899088682481 0.00634495443412403 41 0.996477870532076 0.0070442589358482 0.0035221294679241 42 0.99655303222439 0.00689393555122 0.00344696777561 43 0.995721062587448 0.00855787482510414 0.00427893741255207 44 0.994720803254726 0.0105583934905484 0.00527919674527419 45 0.993357916272452 0.0132841674550970 0.00664208372754852 46 0.992279352634354 0.0154412947312926 0.0077206473656463 47 0.99204320675749 0.0159135864850223 0.00795679324251115 48 0.99552753917083 0.00894492165834016 0.00447246082917008 49 0.993550561497449 0.0128988770051021 0.00644943850255107 50 0.995299009159069 0.00940198168186223 0.00470099084093112 51 0.998682338430718 0.00263532313856470 0.00131766156928235 52 0.999814899465275 0.000370201069450011 0.000185100534725006 53 0.999235784661285 0.00152843067742943 0.000764215338714716 54 0.996915276331129 0.00616944733774284 0.00308472366887142 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 9 0.18 NOK 5% type I error level 22 0.44 NOK 10% type I error level 27 0.54 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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