*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: Tue, 17 Nov 2009 11:05:50 -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/17/t1258481279t7yyzi75yj9paza.htm/, Retrieved Tue, 17 Nov 2009 19:08:11 +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/17/t1258481279t7yyzi75yj9paza.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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User-defined keywords:

Dataseries X:

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8.9 1.4 8.8 1.2 8.3 1 7.5 1.7 7.2 2.4 7.4 2 8.8 2.1 9.3 2 9.3 1.8 8.7 2.7 8.2 2.3 8.3 1.9 8.5 2 8.6 2.3 8.5 2.8 8.2 2.4 8.1 2.3 7.9 2.7 8.6 2.7 8.7 2.9 8.7 3 8.5 2.2 8.4 2.3 8.5 2.8 8.7 2.8 8.7 2.8 8.6 2.2 8.5 2.6 8.3 2.8 8 2.5 8.2 2.4 8.1 2.3 8.1 1.9 8 1.7 7.9 2 7.9 2.1 8 1.7 8 1.8 7.9 1.8 8 1.8 7.7 1.3 7.2 1.3 7.5 1.3 7.3 1.2 7 1.4 7 2.2 7 2.9 7.2 3.1 7.3 3.5 7.1 3.6 6.8 4.4 6.4 4.1 6.1 5.1 6.5 5.8 7.7 5.9 7.9 5.4 7.5 5.5 6.9 4.8 6.6 3.2 6.9 2.7

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 Y[t] = + 8.6158773356344 -0.271381633533570X[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) 8.6158773356344 0.216253 39.8417 0 0 X -0.271381633533570 0.075663 -3.5867 0.000688 0.000344 Multiple Linear Regression - Regression Statistics Multiple R 0.426069605833812 R-squared 0.18153530901538 Adjusted R-squared 0.16742384882599 F-TEST (value) 12.8643887010269 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.000688219457232186 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.678230028540176 Sum Squared Residuals 26.6797663535893 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.9 8.23594304868742 0.664056951312576 2 8.8 8.2902193753941 0.509780624605889 3 8.3 8.34449570210083 -0.0444957021008247 4 7.5 8.15452855862733 -0.654528558627327 5 7.2 7.96456141515383 -0.764561415153828 6 7.4 8.07311406856726 -0.673114068567255 7 8.8 8.0459759052139 0.754024094786102 8 9.3 8.07311406856726 1.22688593143274 9 9.3 8.12739039527397 1.17260960472603 10 8.7 7.88314692509376 0.816853074906242 11 8.2 7.99169957850718 0.208300421492814 12 8.3 8.10025223192061 0.199747768079388 13 8.5 8.07311406856726 0.426885931432744 14 8.6 7.99169957850718 0.608300421492815 15 8.5 7.8560087617404 0.6439912382596 16 8.2 7.96456141515383 0.235438584846171 17 8.1 7.99169957850718 0.108300421492815 18 7.9 7.88314692509376 0.0168530749062435 19 8.6 7.88314692509376 0.716853074906243 20 8.7 7.82887059838704 0.871129401612956 21 8.7 7.80173243503369 0.898267564966313 22 8.5 8.01883774186054 0.481162258139458 23 8.4 7.99169957850718 0.408300421492816 24 8.5 7.8560087617404 0.6439912382596 25 8.7 7.8560087617404 0.843991238259599 26 8.7 7.8560087617404 0.843991238259599 27 8.6 8.01883774186054 0.581162258139458 28 8.5 7.91028508844711 0.589714911552886 29 8.3 7.8560087617404 0.443991238259601 30 8 7.93742325180047 0.0625767481995292 31 8.2 7.96456141515383 0.235438584846171 32 8.1 7.99169957850718 0.108300421492815 33 8.1 8.10025223192061 -0.000252231920613064 34 8 8.15452855862733 -0.154528558627327 35 7.9 8.07311406856726 -0.173114068567255 36 7.9 8.0459759052139 -0.145975905213898 37 8 8.15452855862733 -0.154528558627327 38 8 8.12739039527397 -0.127390395273970 39 7.9 8.12739039527397 -0.227390395273969 40 8 8.12739039527397 -0.127390395273970 41 7.7 8.26308121204075 -0.563081212040754 42 7.2 8.26308121204075 -1.06308121204075 43 7.5 8.26308121204075 -0.763081212040755 44 7.3 8.2902193753941 -0.990219375394112 45 7 8.2359430486874 -1.23594304868740 46 7 8.01883774186054 -1.01883774186054 47 7 7.82887059838704 -0.828870598387043 48 7.2 7.77459427168033 -0.574594271680329 49 7.3 7.6660416182669 -0.366041618266901 50 7.1 7.63890345491354 -0.538903454913545 51 6.8 7.42179814808669 -0.621798148086688 52 6.4 7.50321263814676 -1.10321263814676 53 6.1 7.23183100461319 -1.13183100461319 54 6.5 7.04186386113969 -0.541863861139691 55 7.7 7.01472569778633 0.685274302213667 56 7.9 7.15041651455312 0.749583485446882 57 7.5 7.12327835119976 0.376721648800239 58 6.9 7.31324549467326 -0.41324549467326 59 6.6 7.74745610832697 -1.14745610832697 60 6.9 7.88314692509376 -0.983146925093757 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.384369639251134 0.768739278502267 0.615630360748866 6 0.24426665329543 0.48853330659086 0.75573334670457 7 0.550337748382427 0.899324503235146 0.449662251617573 8 0.792856255386748 0.414287489226505 0.207143744613252 9 0.860327272856524 0.279345454286953 0.139672727143476 10 0.851729354694002 0.296541290611997 0.148270645305998 11 0.78692097970441 0.42615804059118 0.21307902029559 12 0.711168095252503 0.577663809494994 0.288831904747497 13 0.6371329627255 0.725734074549 0.3628670372745 14 0.580890792415066 0.838218415169869 0.419109207584934 15 0.524278667037657 0.951442665924685 0.475721332962343 16 0.444140888586676 0.888281777173352 0.555859111413324 17 0.37040881951308 0.74081763902616 0.62959118048692 18 0.307334055815204 0.614668111630408 0.692665944184796 19 0.284949845783743 0.569899691567487 0.715050154216257 20 0.289042074359582 0.578084148719163 0.710957925640418 21 0.297389847889067 0.594779695778133 0.702610152110933 22 0.259258270789634 0.518516541579268 0.740741729210366 23 0.220527660509166 0.441055321018332 0.779472339490834 24 0.206024975331234 0.412049950662468 0.793975024668766 25 0.231716723646915 0.463433447293831 0.768283276353085 26 0.273881457033209 0.547762914066417 0.726118542966791 27 0.292190123762764 0.584380247525528 0.707809876237236 28 0.319152998472568 0.638305996945135 0.680847001527432 29 0.332428519214593 0.664857038429187 0.667571480785407 30 0.321443490526722 0.642886981053445 0.678556509473278 31 0.324623882368913 0.649247764737826 0.675376117631087 32 0.322975022810484 0.645950045620969 0.677024977189516 33 0.318042318723658 0.636084637447315 0.681957681276342 34 0.307165231513706 0.614330463027411 0.692834768486295 35 0.302035567193821 0.604071134387643 0.697964432806179 36 0.301936592281298 0.603873184562596 0.698063407718702 37 0.306851787916932 0.613703575833864 0.693148212083068 38 0.328714991291537 0.657429982583075 0.671285008708463 39 0.35233906349714 0.70467812699428 0.64766093650286 40 0.427999404095852 0.855998808191704 0.572000595904148 41 0.45490927551572 0.90981855103144 0.54509072448428 42 0.468300874305901 0.936601748611802 0.531699125694099 43 0.47236153965512 0.94472307931024 0.52763846034488 44 0.466063834950293 0.932127669900586 0.533936165049707 45 0.464494906843732 0.928989813687463 0.535505093156268 46 0.49335220739229 0.98670441478458 0.50664779260771 47 0.531315682254885 0.93736863549023 0.468684317745115 48 0.540026942347399 0.919946115305201 0.459973057652601 49 0.549524370280809 0.900951259438382 0.450475629719191 50 0.523807776969107 0.952384446061786 0.476192223030893 51 0.461850963978485 0.92370192795697 0.538149036021515 52 0.451670948954608 0.903341897909215 0.548329051045392 53 0.671675199044687 0.656649601910626 0.328324800955313 54 0.908851993031606 0.182296013936789 0.0911480069683943 55 0.80513665110694 0.38972669778612 0.19486334889306 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 = 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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