*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: Wed, 18 Nov 2009 10:51:43 -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/18/t1258566820axb2jbv1tw1fzdg.htm/, Retrieved Wed, 18 Nov 2009 18:53:52 +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/18/t1258566820axb2jbv1tw1fzdg.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:

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Dataseries X:

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1.5 7.2 1.5 1.6 1.8 1.6 1.3 7.4 1.5 1.5 1.6 1.8 1.4 8.8 1.3 1.5 1.5 1.6 1.4 9.3 1.4 1.3 1.5 1.5 1.3 9.3 1.4 1.4 1.3 1.5 1.3 8.7 1.3 1.4 1.4 1.3 1.2 8.2 1.3 1.3 1.4 1.4 1.1 8.3 1.2 1.3 1.3 1.4 1.4 8.5 1.1 1.2 1.3 1.3 1.2 8.6 1.4 1.1 1.2 1.3 1.5 8.5 1.2 1.4 1.1 1.2 1.1 8.2 1.5 1.2 1.4 1.1 1.3 8.1 1.1 1.5 1.2 1.4 1.5 7.9 1.3 1.1 1.5 1.2 1.1 8.6 1.5 1.3 1.1 1.5 1.4 8.7 1.1 1.5 1.3 1.1 1.3 8.7 1.4 1.1 1.5 1.3 1.5 8.5 1.3 1.4 1.1 1.5 1.6 8.4 1.5 1.3 1.4 1.1 1.7 8.5 1.6 1.5 1.3 1.4 1.1 8.7 1.7 1.6 1.5 1.3 1.6 8.7 1.1 1.7 1.6 1.5 1.3 8.6 1.6 1.1 1.7 1.6 1.7 8.5 1.3 1.6 1.1 1.7 1.6 8.3 1.7 1.3 1.6 1.1 1.7 8 1.6 1.7 1.3 1.6 1.9 8.2 1.7 1.6 1.7 1.3 1.8 8.1 1.9 1.7 1.6 1.7 1.9 8.1 1.8 1.9 1.7 1.6 1.6 8 1.9 1.8 1.9 1.7 1.5 7.9 1.6 1.9 1.8 1.9 1.6 7.9 1.5 1.6 1.9 1.8 1.6 8 1.6 1.5 1.6 1.9 1.7 8 1.6 1.6 1.5 1.6 2 7.9 1.7 1.6 1.6 1.5 2 8 2 1.7 1.6 1.6 1.9 7.7 2 2 1.7 1.6 1.7 7.2 1.9 2 2 1.7 1.8 7.5 1.7 1.9 2 2 1.9 7.3 1.8 1.7 1.9 2 1.7 7 1.9 1.8 1.7 1.9 2 7 1.7 1.9 1.8 1.7 2.1 7 2 1. 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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.184356379334972 + 0.00818805553395265X[t] + 0.379584054171523Y1[t] + 0.37974171622158Y2[t] -0.00603062858926295Y3[t] -0.034905189738258Y4[t] + 0.00996622977915651t + 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) 0.184356379334972 0.593353 0.3107 0.757344 0.378672 X 0.00818805553395265 0.060502 0.1353 0.8929 0.44645 Y1 0.379584054171523 0.147719 2.5696 0.013274 0.006637 Y2 0.37974171622158 0.155574 2.4409 0.018311 0.009155 Y3 -0.00603062858926295 0.159721 -0.0378 0.970035 0.485017 Y4 -0.034905189738258 0.145599 -0.2397 0.811536 0.405768 t 0.00996622977915651 0.003656 2.726 0.008866 0.004433 Multiple Linear Regression - Regression Statistics Multiple R 0.942808386330887 R-squared 0.888887653335852 Adjusted R-squared 0.875282059866773 F-TEST (value) 65.3325160240878 F-TEST (DF numerator) 6 F-TEST (DF denominator) 49 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.195155165199376 Sum Squared Residuals 1.86619138669579 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.5 1.36353600112851 0.136463998871487 2 1.3 1.33139075816250 -0.031390758162503 3 1.4 1.28448755566147 0.115512444338533 4 1.4 1.26404839435426 0.135951605645738 5 1.3 1.31319492147343 -0.0131949214734296 6 1.3 1.28666788760379 0.0133321123962128 7 1.2 1.25107539901998 -0.0510753990199833 8 1.1 1.22450509179431 -0.124505091794309 9 1.4 1.16366687461477 0.236333125385228 10 1.2 1.25095601743555 -0.0509560174355492 11 1.5 1.30220272752623 0.197797272473768 12 1.1 1.34932074404939 -0.24932074404939 13 1.3 1.31129163026939 -0.0112916302693913 14 1.5 1.24881222265830 0.251187777341697 15 1.1 1.40831593990407 -0.308315939904075 16 1.4 1.35597164698978 0.0440283530102162 17 1.3 1.31972924286626 -0.0197292428662611 18 1.5 1.39945318447600 0.100546815523998 19 1.6 1.45869613523243 0.141303864767566 20 1.7 1.57351942516390 0.126480574836096 21 1.1 1.66334023634513 -0.563340236345134 22 1.6 1.47594610443696 0.124053895563043 23 1.3 1.44294694418278 -0.142946944182780 24 1.7 1.52821786844761 0.171782131552394 25 1.6 1.59238539347043 0.00761460652956964 26 1.7 1.69819008136853 0.00180991863146940 27 1.9 1.71783746153524 0.182162538464756 28 1.8 1.82751685518109 -0.0275168551810911 29 1.9 1.87836047890231 0.0216395210976891 30 1.6 1.88279549223139 -0.282795492231388 31 1.5 1.80966389673912 -0.309663896739125 32 1.6 1.67063666234955 -0.0706366623495547 33 1.6 1.67972460108005 -0.079724601080054 34 1.7 1.73873962226177 -0.0387396222617723 35 2 1.78873290801959 0.211267091980415 36 2 1.94787681225193 0.0521231877480738 37 1.9 2.06870607737844 -0.168706077378445 38 1.7 2.03132016642287 -0.331320166422868 39 1.8 1.91938027348427 -0.119380273484270 40 1.9 1.8903220171884 0.00967798281160137 41 1.7 1.97846105203836 -0.278461052038358 42 2 1.95686261769409 0.0431373823059071 43 2.1 2.00066213864764 0.0993378613523616 44 2.4 2.16186250656124 0.238137493438761 45 2.5 2.32966877913828 0.170331220861722 46 2.5 2.47880369831387 0.0211963016861333 47 2.6 2.51898797550439 0.0810120244956094 48 2.2 2.55256276870671 -0.352562768706714 49 2.5 2.44272261280541 0.0572773871945918 50 2.8 2.41733953170204 0.382660468297955 51 2.8 2.66385089170175 0.136149108298245 52 2.9 2.8015301347727 0.0984698652272998 53 3 2.83389880225717 0.166101197742828 54 3.1 2.90441321883379 0.195586781166211 55 2.9 2.98725254613314 -0.0872525461331443 56 2.7 2.95763897152759 -0.257638971527587 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 10 0.170074567260247 0.340149134520493 0.829925432739753 11 0.199093094032655 0.398186188065311 0.800906905967345 12 0.254573486923100 0.509146973846200 0.7454265130769 13 0.153344099691017 0.306688199382034 0.846655900308983 14 0.270035924438452 0.540071848876904 0.729964075561548 15 0.191065912108966 0.382131824217932 0.808934087891034 16 0.134325510044371 0.268651020088742 0.865674489955629 17 0.0824349499433786 0.164869899886757 0.917565050056621 18 0.113276541717024 0.226553083434048 0.886723458282976 19 0.143887576373220 0.287775152746440 0.85611242362678 20 0.157555204993441 0.315110409986881 0.84244479500656 21 0.584319919946329 0.831360160107342 0.415680080053671 22 0.563367858713044 0.873264282573911 0.436632141286956 23 0.498288479912898 0.996576959825797 0.501711520087102 24 0.569402140857322 0.861195718285356 0.430597859142678 25 0.537541246133993 0.924917507732014 0.462458753866007 26 0.52534369227134 0.94931261545732 0.47465630772866 27 0.60605566333936 0.78788867332128 0.39394433666064 28 0.5619510998765 0.876097800247 0.4380489001235 29 0.659266758797645 0.68146648240471 0.340733241202355 30 0.639948900726864 0.720102198546273 0.360051099273136 31 0.666777081962525 0.66644583607495 0.333222918037475 32 0.59748759812209 0.80502480375582 0.40251240187791 33 0.509116926765446 0.981766146469108 0.490883073234554 34 0.421082291683322 0.842164583366643 0.578917708316678 35 0.514966717928441 0.970066564143117 0.485033282071559 36 0.502418760742761 0.995162478514477 0.497581239257239 37 0.444851871801673 0.889703743603346 0.555148128198327 38 0.399007451058808 0.798014902117617 0.600992548941192 39 0.324614320954878 0.649228641909756 0.675385679045122 40 0.312524592014604 0.625049184029208 0.687475407985396 41 0.327581205546785 0.655162411093571 0.672418794453215 42 0.248108086766593 0.496216173533186 0.751891913233407 43 0.320306068965422 0.640612137930845 0.679693931034578 44 0.270107117154925 0.54021423430985 0.729892882845075 45 0.226380959215962 0.452761918431925 0.773619040784038 46 0.129441102014383 0.258882204028766 0.870558897985617 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 0 0 OK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; 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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