*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 09:35:19 -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/t1258648850mdc3ywgv0kke5aa.htm/, Retrieved Thu, 19 Nov 2009 17:41:01 +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/t1258648850mdc3ywgv0kke5aa.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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56.6 0 56 0 54.8 0 52.7 0 50.9 0 50.6 0 52.1 0 53.3 0 53.9 0 54.3 0 54.2 0 54.2 0 53.5 0 51.4 0 50.5 0 50.3 0 49.8 0 50.7 0 52.8 0 55.3 0 57.3 0 57.5 0 56.8 0 56.4 0 56.3 0 56.4 0 57 0 57.9 0 58.9 0 58.8 0 56.5 1 51.9 1 47.4 1 44.9 1 43.9 1 43.4 1 42.9 1 42.6 1 42.2 1 41.2 1 40.2 1 39.3 1 38.5 1 38.3 1 37.9 1 37.6 1 37.3 1 36 1 34.5 1 33.5 1 32.9 1 32.9 1 32.8 1 31.9 1 30.5 1 29.2 1 28.7 1 28.4 1 28 1 27.4 1 26.9 1

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] = + 59.5264480874317 -6.82688172043011X[t] -0.332459016393442t + 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) 59.5264480874317 1.374036 43.3223 0 0 X -6.82688172043011 2.426145 -2.8139 0.006672 0.003336 t -0.332459016393442 0.068889 -4.826 1.1e-05 5e-06 Multiple Linear Regression - Regression Statistics Multiple R 0.889129937195268 R-squared 0.790552045216862 Adjusted R-squared 0.783329701948478 F-TEST (value) 109.459217852122 F-TEST (DF numerator) 2 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 4.73656273647105 Sum Squared Residuals 1301.23154027851 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 56.6 59.1939890710383 -2.59398907103832 2 56 58.8615300546448 -2.86153005464482 3 54.8 58.5290710382514 -3.72907103825137 4 52.7 58.1966120218579 -5.49661202185791 5 50.9 57.8641530054645 -6.96415300546448 6 50.6 57.531693989071 -6.93169398907103 7 52.1 57.1992349726776 -5.09923497267759 8 53.3 56.8667759562841 -3.56677595628415 9 53.9 56.5343169398907 -2.63431693989071 10 54.3 56.2018579234973 -1.90185792349727 11 54.2 55.8693989071038 -1.66939890710382 12 54.2 55.5369398907104 -1.33693989071037 13 53.5 55.204480874317 -1.70448087431694 14 51.4 54.8720218579235 -3.47202185792349 15 50.5 54.5395628415300 -4.03956284153005 16 50.3 54.2071038251366 -3.90710382513661 17 49.8 53.8746448087432 -4.07464480874317 18 50.7 53.5421857923497 -2.84218579234972 19 52.8 53.2097267759563 -0.409726775956284 20 55.3 52.8772677595628 2.42273224043716 21 57.3 52.5448087431694 4.7551912568306 22 57.5 52.212349726776 5.28765027322405 23 56.8 51.8798907103825 4.92010928961749 24 56.4 51.5474316939891 4.85256830601093 25 56.3 51.2149726775956 5.08502732240437 26 56.4 50.8825136612022 5.51748633879781 27 57 50.5500546448087 6.44994535519126 28 57.9 50.2175956284153 7.6824043715847 29 58.9 49.8851366120219 9.01486338797814 30 58.8 49.5526775956284 9.24732240437158 31 56.5 42.3933368588049 14.1066631411951 32 51.9 42.0608778424114 9.83912215758858 33 47.4 41.728418826018 5.67158117398202 34 44.9 41.3959598096245 3.50404019037546 35 43.9 41.0635007932311 2.83649920676891 36 43.4 40.7310417768376 2.66895822316235 37 42.9 40.3985827604442 2.50141723955579 38 42.6 40.0661237440508 2.53387625594924 39 42.2 39.7336647276573 2.46633527234268 40 41.2 39.4012057112639 1.79879428873612 41 40.2 39.0687466948704 1.13125330512957 42 39.3 38.736287678477 0.563712321523002 43 38.5 38.4038286620836 0.0961713379164476 44 38.3 38.0713696456901 0.228630354309887 45 37.9 37.7389106292967 0.161089370703331 46 37.6 37.4064516129032 0.193548387096776 47 37.3 37.0739925965098 0.226007403490214 48 36 36.7415335801163 -0.74153358011634 49 34.5 36.4090745637229 -1.90907456372290 50 33.5 36.0766155473295 -2.57661554732946 51 32.9 35.744156530936 -2.84415653093601 52 32.9 35.4116975145426 -2.51169751454257 53 32.8 35.0792384981491 -2.27923849814913 54 31.9 34.7467794817557 -2.84677948175569 55 30.5 34.4143204653622 -3.91432046536224 56 29.2 34.0818614489688 -4.8818614489688 57 28.7 33.7494024325754 -5.04940243257536 58 28.4 33.4169434161819 -5.01694341618192 59 28 33.0844843997885 -5.08448439978848 60 27.4 32.7520253833950 -5.35202538339503 61 26.9 32.4195663670016 -5.51956636700159 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.00377096022939169 0.00754192045878337 0.996229039770608 7 0.0138452218493712 0.0276904436987424 0.986154778150629 8 0.0261988986077832 0.0523977972155665 0.973801101392217 9 0.028903627041348 0.057807254082696 0.971096372958652 10 0.0249323656331431 0.0498647312662862 0.975067634366857 11 0.0167079535486792 0.0334159070973584 0.98329204645132 12 0.0101089694240895 0.0202179388481790 0.98989103057591 13 0.00540460417068657 0.0108092083413731 0.994595395829313 14 0.00434926065613839 0.00869852131227678 0.995650739343862 15 0.00508673706585306 0.0101734741317061 0.994913262934147 16 0.00684055219726758 0.0136811043945352 0.993159447802732 17 0.0146255397083463 0.0292510794166926 0.985374460291654 18 0.0351390009741941 0.0702780019483882 0.964860999025806 19 0.107204950931757 0.214409901863514 0.892795049068243 20 0.357239695961125 0.71447939192225 0.642760304038875 21 0.688163865287569 0.623672269424863 0.311836134712431 22 0.831571325060838 0.336857349878324 0.168428674939162 23 0.882565816154924 0.234868367690152 0.117434183845076 24 0.910361636012744 0.179276727974511 0.0896383639872557 25 0.92934467961954 0.141310640760919 0.0706553203804593 26 0.942952190438796 0.114095619122408 0.0570478095612038 27 0.948548396850819 0.102903206298362 0.0514516031491811 28 0.94664130894065 0.106717382118699 0.0533586910593494 29 0.940742134286733 0.118515731426534 0.0592578657132672 30 0.926111161892818 0.147777676214365 0.0738888381071825 31 0.999943847260043 0.000112305479913166 5.61527399565828e-05 32 0.999999999956288 8.74234017493661e-11 4.37117008746830e-11 33 0.999999999999357 1.28592547458165e-12 6.42962737290823e-13 34 0.999999999999593 8.13360907797723e-13 4.06680453898861e-13 35 0.99999999999972 5.58146056209201e-13 2.79073028104601e-13 36 0.999999999999616 7.67643050327044e-13 3.83821525163522e-13 37 0.999999999999166 1.66772828386272e-12 8.33864141931358e-13 38 0.999999999997258 5.48321706618679e-12 2.74160853309340e-12 39 0.999999999992086 1.58281732782637e-11 7.91408663913185e-12 40 0.999999999973194 5.36121284965216e-11 2.68060642482608e-11 41 0.999999999917361 1.65277732924871e-10 8.26388664624354e-11 42 0.999999999808153 3.83694812259176e-10 1.91847406129588e-10 43 0.99999999968975 6.20497878330422e-10 3.10248939165211e-10 44 0.999999998863029 2.27394251997572e-09 1.13697125998786e-09 45 0.99999999479187 1.04162583974537e-08 5.20812919872685e-09 46 0.99999998069463 3.86107393863849e-08 1.93053696931924e-08 47 0.999999981185573 3.76288546090544e-08 1.88144273045272e-08 48 0.999999953324416 9.33511676786104e-08 4.66755838393052e-08 49 0.999999730413843 5.39172314394677e-07 2.69586157197338e-07 50 0.999998899345238 2.20130952438051e-06 1.10065476219025e-06 51 0.999995943193881 8.1136122375806e-06 4.0568061187903e-06 52 0.999971270239141 5.7459521717566e-05 2.8729760858783e-05 53 0.999920402852088 0.000159194295824794 7.95971479123969e-05 54 0.99994671657612 0.000106566847758278 5.32834238791391e-05 55 0.999981293268923 3.74134621533131e-05 1.87067310766566e-05 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 27 0.54 NOK 5% type I error level 35 0.7 NOK 10% type I error level 38 0.76 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; 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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