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: Sat, 21 Nov 2009 02:45:32 -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/21/t125879687127dmu3lghjvib6d.htm/, Retrieved Sat, 21 Nov 2009 10:48:03 +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/21/t125879687127dmu3lghjvib6d.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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10,9 0 10 0 9,2 0 9,2 0 9,5 0 9,6 0 9,5 0 9,1 0 8,9 0 9 0 10,1 0 10,3 0 10,2 0 9,6 0 9,2 0 9,3 0 9,4 0 9,4 0 9,2 0 9 0 9 0 9 0 9,8 0 10 0 9,8 0 9,3 0 9 0 9 0 9,1 0 9,1 0 9,1 0 9,2 0 8,8 0 8,3 0 8,4 0 8,1 0 7,7 1 7,9 1 7,9 1 8 1 7,9 1 7,6 1 7,1 1 6,8 1 6,5 1 6,9 1 8,2 1 8,7 1 8,3 1 7,9 1 7,5 1 7,8 1 8,3 1 8,4 1 8,2 1 7,7 1 7,2 1 7,3 1 8,1 1 8,5 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] = + 9.32222222222222 -1.55555555555556X[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) 9.32222222222222 0.094219 98.9423 0 0 X -1.55555555555556 0.148973 -10.4419 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.807936981724256 R-squared 0.6527621664377 Adjusted R-squared 0.646775307238351 F-TEST (value) 109.032490109100 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 6.10622663543836e-15 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.565312784271948 Sum Squared Residuals 18.5355555555555 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 10.9 9.32222222222222 1.57777777777778 2 10 9.32222222222222 0.67777777777778 3 9.2 9.32222222222222 -0.122222222222223 4 9.2 9.32222222222222 -0.122222222222223 5 9.5 9.32222222222222 0.177777777777778 6 9.6 9.32222222222222 0.277777777777778 7 9.5 9.32222222222222 0.177777777777778 8 9.1 9.32222222222222 -0.222222222222222 9 8.9 9.32222222222222 -0.422222222222222 10 9 9.32222222222222 -0.322222222222222 11 10.1 9.32222222222222 0.777777777777778 12 10.3 9.32222222222222 0.977777777777779 13 10.2 9.32222222222222 0.877777777777777 14 9.6 9.32222222222222 0.277777777777778 15 9.2 9.32222222222222 -0.122222222222223 16 9.3 9.32222222222222 -0.0222222222222214 17 9.4 9.32222222222222 0.0777777777777783 18 9.4 9.32222222222222 0.0777777777777783 19 9.2 9.32222222222222 -0.122222222222223 20 9 9.32222222222222 -0.322222222222222 21 9 9.32222222222222 -0.322222222222222 22 9 9.32222222222222 -0.322222222222222 23 9.8 9.32222222222222 0.477777777777779 24 10 9.32222222222222 0.677777777777778 25 9.8 9.32222222222222 0.477777777777779 26 9.3 9.32222222222222 -0.0222222222222214 27 9 9.32222222222222 -0.322222222222222 28 9 9.32222222222222 -0.322222222222222 29 9.1 9.32222222222222 -0.222222222222222 30 9.1 9.32222222222222 -0.222222222222222 31 9.1 9.32222222222222 -0.222222222222222 32 9.2 9.32222222222222 -0.122222222222223 33 8.8 9.32222222222222 -0.522222222222221 34 8.3 9.32222222222222 -1.02222222222222 35 8.4 9.32222222222222 -0.922222222222222 36 8.1 9.32222222222222 -1.22222222222222 37 7.7 7.76666666666667 -0.0666666666666666 38 7.9 7.76666666666667 0.133333333333334 39 7.9 7.76666666666667 0.133333333333334 40 8 7.76666666666667 0.233333333333333 41 7.9 7.76666666666667 0.133333333333334 42 7.6 7.76666666666667 -0.166666666666667 43 7.1 7.76666666666667 -0.666666666666667 44 6.8 7.76666666666667 -0.966666666666667 45 6.5 7.76666666666667 -1.26666666666667 46 6.9 7.76666666666667 -0.866666666666666 47 8.2 7.76666666666667 0.433333333333333 48 8.7 7.76666666666667 0.933333333333333 49 8.3 7.76666666666667 0.533333333333334 50 7.9 7.76666666666667 0.133333333333334 51 7.5 7.76666666666667 -0.266666666666667 52 7.8 7.76666666666667 0.0333333333333331 53 8.3 7.76666666666667 0.533333333333334 54 8.4 7.76666666666667 0.633333333333334 55 8.2 7.76666666666667 0.433333333333333 56 7.7 7.76666666666667 -0.0666666666666666 57 7.2 7.76666666666667 -0.566666666666667 58 7.3 7.76666666666667 -0.466666666666667 59 8.1 7.76666666666667 0.333333333333333 60 8.5 7.76666666666667 0.733333333333333 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.918007370058325 0.163985259883349 0.0819926299416747 6 0.850615930925037 0.298768138149926 0.149384069074963 7 0.768802624743277 0.462394750513446 0.231197375256723 8 0.742163498772428 0.515673002455145 0.257836501227572 9 0.752578985518163 0.494842028963674 0.247421014481837 10 0.718469973703824 0.563060052592351 0.281530026296176 11 0.730513675788857 0.538972648422287 0.269486324211143 12 0.799040434189397 0.401919131621206 0.200959565810603 13 0.831329850961285 0.337340298077430 0.168670149038715 14 0.779918911196258 0.440162177607484 0.220081088803742 15 0.740204373150153 0.519591253699694 0.259795626849847 16 0.683714873465291 0.632570253069418 0.316285126534709 17 0.617232030202464 0.765535939595072 0.382767969797536 18 0.548635559846282 0.902728880307436 0.451364440153718 19 0.492986311944955 0.98597262388991 0.507013688055045 20 0.467373325446833 0.934746650893666 0.532626674553167 21 0.435568136542705 0.87113627308541 0.564431863457295 22 0.399230803195447 0.798461606390893 0.600769196804553 23 0.387604943578951 0.775209887157902 0.612395056421049 24 0.451611970823418 0.903223941646835 0.548388029176582 25 0.475406145450062 0.950812290900123 0.524593854549938 26 0.431741187855749 0.863482375711497 0.568258812144251 27 0.402747688191409 0.805495376382819 0.597252311808591 28 0.371515616959365 0.74303123391873 0.628484383040635 29 0.335058654819736 0.670117309639472 0.664941345180264 30 0.303564267671022 0.607128535342043 0.696435732328978 31 0.280015082675317 0.560030165350635 0.719984917324683 32 0.279968240115542 0.559936480231084 0.720031759884458 33 0.288151654913869 0.576303309827737 0.711848345086131 34 0.360402899637518 0.720805799275037 0.639597100362482 35 0.394630284030537 0.789260568061073 0.605369715969463 36 0.467835387411792 0.935670774823583 0.532164612588208 37 0.391230500761587 0.782461001523174 0.608769499238413 38 0.321439642558603 0.642879285117207 0.678560357441397 39 0.256168773422883 0.512337546845767 0.743831226577117 40 0.203123876603134 0.406247753206267 0.796876123396866 41 0.152661701177379 0.305323402354758 0.84733829882262 42 0.113270912753961 0.226541825507922 0.886729087246039 43 0.122510063488452 0.245020126976903 0.877489936511548 44 0.203822977369796 0.407645954739591 0.796177022630205 45 0.528743550738852 0.942512898522295 0.471256449261148 46 0.734119384192428 0.531761231615144 0.265880615807572 47 0.685326420359947 0.629347159280106 0.314673579640053 48 0.79323281442539 0.413534371149219 0.206767185574610 49 0.764491988975164 0.471016022049673 0.235508011024836 50 0.670523980082705 0.65895203983459 0.329476019917295 51 0.618742544756781 0.762514910486438 0.381257455243219 52 0.503638140737694 0.992723718524612 0.496361859262306 53 0.430081428583591 0.860162857167182 0.569918571416409 54 0.402626270504171 0.805252541008343 0.597373729495829 55 0.313603491866021 0.627206983732042 0.686396508133979 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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