*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, 22 Dec 2010 07:21:41 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/22/t1293002438s3g3vgw9zlizok7.htm/, Retrieved Wed, 22 Dec 2010 08:20:38 +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/2010/Dec/22/t1293002438s3g3vgw9zlizok7.htm/}, year = {2010}, } @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 = {2010}, 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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-999.00 3.823082797 3.00 6.30 0 3.00 -999.00 0.529558673 1.00 -999.00 -0.036212173 3.00 2.10 3.406028945 4.00 9.10 1.02325246 4.00 15.80 -1.638272164 1.00 5.20 2.204119983 4.00 10.90 0.51851394 1.00 8.30 1.717337583 1.00 11.00 -0.37161107 4.00 3.20 2.667452953 5.00 7.60 -0.259637311 2.00 -999.00 2.272073788 5.00 6.30 -1.124938737 1.00 8.60 0.477121255 2.00 6.60 -0.105130343 2.00 9.50 -0.698970004 2.00 4.80 0.149219113 1.00 12.00 1.77815125 1.00 -999.00 2.723455672 5.00 3.30 1.441852176 5.00 11.00 -0.920818754 2.00 -999.00 2.315970345 1.00 4.70 1.929418926 1.00 -999.00 1.560265398 1.00 10.40 -0.995678626 3.00 7.40 0.017033339 4.00 2.10 2.716837723 5.00 -999.00 2 1.00 -999.00 1.544068044 4.00 7.70 -2.301029996 4.00 17.90 -2 1.00 6.10 1.792391689 1.00 8.20 -0.913640169 1.00 8.40 0.130333768 3.00 11.90 -1.638272164 3.00 10.80 -1.318758763 3.00 13.80 0.230448921 1.00 14.30 0.544068044 1.00 -999.00 2.397940009 5.00 15.20 -0.318758763 2.00 10.00 1 4.00 11.90 0.209515015 2.0 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 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation SWS[t] = -193.772904906513 -129.452856414406logWb[t] + 19.1756436818691D[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) -193.772904906513 105.217683 -1.8416 0.070556 0.035278 logWb -129.452856414406 39.532938 -3.2746 0.001773 0.000887 D 19.1756436818691 37.204302 0.5154 0.608189 0.304095 Multiple Linear Regression - Regression Statistics Multiple R 0.397167878821118 R-squared 0.157742323967267 Adjusted R-squared 0.129191216305140 F-TEST (value) 5.52491083127097 F-TEST (DF numerator) 2 F-TEST (DF denominator) 59 p-value 0.00631868697878335 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 396.38510498693 Sum Squared Residuals 9270147.93587446 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 -999 -631.15496224133 -367.845037758669 2 6.3 -136.245973860906 142.545973860906 3 -999 -243.150144083516 -755.849855916484 4 -999 -131.558204629083 -867.441795370917 5 2.1 -557.990506139431 560.090506139431 6 9.1 -249.533283959104 258.633283959104 7 15.8 37.4817499893659 -21.6817499893659 8 5.2 -402.399957858458 407.599957858458 9 10.9 -241.720371848331 252.620371848331 10 8.3 -396.911516771805 405.211516771805 11 11 -68.9642156923227 79.9642156923227 12 3.2 -443.204090614059 446.404090614059 13 7.6 -121.810826002069 129.410826002069 14 -999 -392.021128338066 -606.978871661934 15 6.3 -28.9707284287799 35.2707284287799 16 8.6 -217.186326858551 225.786326858551 17 6.6 -141.812194345598 148.412194345598 18 9.5 -64.9379539769861 74.437953976986 19 4.8 -193.914101634118 198.714101634118 20 12 -404.784019673990 416.784019673990 21 -999 -450.453802555582 -548.546197444418 22 3.3 -284.546569207694 287.846569207694 23 11 -36.2189995975207 47.2189995975207 24 -999 -474.40623775595 -524.59376224405 25 4.7 -424.366052415358 429.066052415358 26 -999 -376.578073760303 -622.421926239697 27 10.4 -7.35253165443473 17.7525316544347 28 7.4 -119.275344566861 126.675344566861 29 2.1 -449.597090153927 451.697090153927 30 -999 -433.502974053455 -565.497025946545 31 -999 -316.954348973041 -682.04565102696 32 7.7 180.804575498392 -173.104575498392 33 17.9 84.3084516041676 -66.4084516041676 34 6.1 -406.627485179135 412.727485179135 35 8.2 -56.3239316126533 64.5239316126533 36 8.4 -153.118052415758 161.518052415758 37 11.9 75.8330373531041 -63.9330373531041 38 10.8 34.4711149309727 -23.6711149309727 39 13.8 -204.429532305711 218.229532305711 40 14.3 -245.028423604242 259.328423604242 41 -999 -408.314870172603 -590.685129827397 42 15.2 -114.157385165302 129.357385165302 43 10 -246.523186593442 256.523186593442 44 11.9 -182.543934696232 194.443934696232 45 6.5 -412.650196327609 419.150196327609 46 7.5 -149.409157343792 156.909157343792 47 -999 -237.268804407078 -761.731195592922 48 10.6 -64.6790018280911 75.2790018280911 49 7.4 -255.745226349519 263.145226349519 50 8.4 -263.192274321076 271.592274321076 51 5.7 -139.247941161316 144.947941161316 52 4.9 -208.260921645833 213.160921645833 53 -999 -249.502253758572 -749.497746241428 54 3.2 -323.698395570122 326.898395570122 55 -999 -174.338309205202 -824.661690794798 56 8.1 2.75019325109646 5.34980674890354 57 11 -149.498179630468 160.498179630468 58 4.9 -175.215166709523 180.115166709523 59 13.2 -28.1737755161938 41.3737755161938 60 9.7 -197.617712757485 207.317712757485 61 12.8 -245.028423604242 257.828423604242 62 -999 -253.234049095272 -745.765950904728 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.842180597548884 0.315638804902231 0.157819402451116 7 0.916399046178063 0.167201907643874 0.0836009538219372 8 0.878415280394344 0.243169439211313 0.121584719605656 9 0.922370717364406 0.155258565271187 0.0776292826355936 10 0.933357462085711 0.133285075828577 0.0666425379142885 11 0.893126991010116 0.213746017979768 0.106873008989884 12 0.866099238553136 0.267801522893728 0.133900761446864 13 0.819353864363042 0.361292271273917 0.180646135636958 14 0.909159135437013 0.181681729125974 0.090840864562987 15 0.869802364464655 0.26039527107069 0.130197635535345 16 0.833349310147752 0.333301379704496 0.166650689852248 17 0.781921645200674 0.436156709598653 0.218078354799326 18 0.716646138612042 0.566707722775916 0.283353861387958 19 0.656319050937958 0.687361898124084 0.343680949062042 20 0.63828167451151 0.72343665097698 0.36171832548849 21 0.69634988540068 0.607300229198639 0.303650114599320 22 0.668737105712832 0.662525788574337 0.331262894287168 23 0.594275073783752 0.811449852432495 0.405724926216248 24 0.656792833572915 0.686414332854171 0.343207166427085 25 0.660468972164636 0.679062055670728 0.339531027835364 26 0.746297746603459 0.507404506793082 0.253702253396541 27 0.680707039670494 0.638585920659012 0.319292960329506 28 0.615468927696285 0.769062144607431 0.384531072303715 29 0.635456460023267 0.729087079953465 0.364543539976733 30 0.697732824072753 0.604534351854493 0.302267175927247 31 0.808116041783644 0.383767916432711 0.191883958216356 32 0.7598789385911 0.480242122817799 0.240121061408900 33 0.698181357915564 0.603637284168872 0.301818642084436 34 0.695113834282298 0.609772331435404 0.304886165717702 35 0.625654780345876 0.748690439308249 0.374345219654124 36 0.561708664521901 0.876582670956198 0.438291335478099 37 0.487119481854069 0.974238963708138 0.512880518145931 38 0.410956046088235 0.82191209217647 0.589043953911765 39 0.356209916776718 0.712419833553436 0.643790083223282 40 0.317339206637268 0.634678413274537 0.682660793362732 41 0.398523990759491 0.797047981518983 0.601476009240509 42 0.331311603510072 0.662623207020144 0.668688396489928 43 0.280149543882652 0.560299087765303 0.719850456117348 44 0.230917435719027 0.461834871438054 0.769082564280973 45 0.235562099797721 0.471124199595441 0.76443790020228 46 0.179389009680566 0.358778019361132 0.820610990319434 47 0.312765170590895 0.62553034118179 0.687234829409105 48 0.237545337558102 0.475090675116203 0.762454662441898 49 0.201290427532035 0.40258085506407 0.798709572467965 50 0.181274813294906 0.362549626589811 0.818725186705094 51 0.132903148163837 0.265806296327673 0.867096851836163 52 0.102908588840534 0.205817177681068 0.897091411159466 53 0.324493539517942 0.648987079035884 0.675506460482058 54 0.227609550055118 0.455219100110236 0.772390449944882 55 0.515950388493818 0.968099223012363 0.484049611506182 56 0.358248913480572 0.716497826961144 0.641751086519428 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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