WS 7: 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: Thu, 26 Nov 2009 09:36:09 -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/26/t12592535217p1ydwtdj23oitw.htm/, Retrieved Thu, 26 Nov 2009 17:38:54 +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/26/t12592535217p1ydwtdj23oitw.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:

Seasonal Dummis

Dataseries X:

» Textbox « » Textfile « » CSV «

8.1 1.3 7.7 1.3 7.5 1.2 7.6 1.1 7.8 1.4 7.8 1.2 7.8 1.5 7.5 1.1 7.5 1.3 7.1 1.5 7.5 1.1 7.5 1.4 7.6 1.3 7.7 1.5 7.7 1.6 7.9 1.7 8.1 1.1 8.2 1.6 8.2 1.3 8.2 1.7 7.9 1.6 7.3 1.7 6.9 1.9 6.6 1.8 6.7 1.9 6.9 1.6 7.0 1.5 7.1 1.6 7.2 1.6 7.1 1.7 6.9 2.0 7.0 2.0 6.8 1.9 6.4 1.7 6.7 1.8 6.6 1.9 6.4 1.7 6.3 2.0 6.2 2.1 6.5 2.4 6.8 2.5 6.8 2.5 6.4 2.6 6.1 2.2 5.8 2.5 6.1 2.8 7.2 2.8 7.3 2.9 6.9 3.0 6.1 3.1 5.8 2.9 6.2 2.7 7.1 2.2 7.7 2.5 7.9 2.3 7.7 2.6 7.4 2.3 7.5 2.2 8.0 1.8 8.1 1.8

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 8.55716459197787 -0.68222683264177X[t] -0.161867219917016M1[t] -0.320933609958506M2[t] -0.448222683264177M3[t] -0.200933609958506M4[t] + 0.0435546334716457M5[t] + 0.259066390041494M6[t] + 0.206355463347164M7[t] + 0.0527109266943289M8[t] -0.167289073305671M9[t] -0.326355463347165M10[t] -0.0145781466113415M11[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) 8.55716459197787 0.365596 23.4061 0 0 X -0.68222683264177 0.135202 -5.046 7e-06 4e-06 M1 -0.161867219917016 0.356564 -0.454 0.651944 0.325972 M2 -0.320933609958506 0.356287 -0.9008 0.372302 0.186151 M3 -0.448222683264177 0.356452 -1.2575 0.2148 0.1074 M4 -0.200933609958506 0.356287 -0.564 0.575459 0.28773 M5 0.0435546334716457 0.35722 0.1219 0.903477 0.451738 M6 0.259066390041494 0.356287 0.7271 0.470754 0.235377 M7 0.206355463347164 0.356205 0.5793 0.565142 0.282571 M8 0.0527109266943289 0.356236 0.148 0.883002 0.441501 M9 -0.167289073305671 0.356236 -0.4696 0.640811 0.320406 M10 -0.326355463347165 0.356205 -0.9162 0.364239 0.182119 M11 -0.0145781466113415 0.356359 -0.0409 0.967542 0.483771 Multiple Linear Regression - Regression Statistics Multiple R 0.647973621712503 R-squared 0.419869814435218 Adjusted R-squared 0.271751469184636 F-TEST (value) 2.83469150107567 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0.00529516321303269 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.563193756674525 Sum Squared Residuals 14.9077987551867 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.1 7.50840248962657 0.591597510373431 2 7.7 7.34933609958506 0.350663900414938 3 7.5 7.29026970954357 0.209730290456432 4 7.6 7.60578146611342 -0.00578146611341634 5 7.8 7.64560165975104 0.154398340248963 6 7.8 7.99755878284924 -0.197558782849239 7 7.8 7.74017980636238 0.0598201936376213 8 7.5 7.85942600276625 -0.359426002766251 9 7.5 7.5029806362379 -0.00298063623789704 10 7.1 7.20746887966805 -0.107468879668050 11 7.5 7.79213692946058 -0.292136929460580 12 7.5 7.60204702627939 -0.102047026279391 13 7.6 7.50840248962655 0.0915975103734475 14 7.7 7.21289073305671 0.487109266943292 15 7.7 7.01737897648686 0.68262102351314 16 7.9 7.19644536652835 0.703554633471647 17 8.1 7.85026970954357 0.249730290456432 18 8.2 7.72466804979253 0.475331950207469 19 8.2 7.87662517289073 0.323374827109267 20 8.2 7.45008990318119 0.74991009681881 21 7.9 7.29831258644537 0.601687413554634 22 7.3 7.0710235131397 0.228976486860305 23 6.9 7.24635546334716 -0.346355463347164 24 6.6 7.32915629322268 -0.729156293222683 25 6.7 7.09906639004149 -0.39906639004149 26 6.9 7.14466804979253 -0.244668049792531 27 7 7.08560165975104 -0.0856016597510371 28 7.1 7.26466804979253 -0.164668049792531 29 7.2 7.50915629322268 -0.309156293222683 30 7.1 7.65644536652835 -0.556445366528354 31 6.9 7.3990663900415 -0.499066390041494 32 7 7.24542185338866 -0.245421853388658 33 6.8 7.09364453665284 -0.293644536652836 34 6.4 7.0710235131397 -0.671023513139695 35 6.7 7.31457814661134 -0.614578146611341 36 6.6 7.2609336099585 -0.660933609958506 37 6.4 7.23551175656984 -0.835511756569844 38 6.3 6.87177731673582 -0.571777316735823 39 6.2 6.67626556016597 -0.476265560165975 40 6.5 6.71888658367912 -0.218886583679115 41 6.8 6.89515214384509 -0.0951521438450902 42 6.8 7.11066390041494 -0.310663900414938 43 6.4 6.98973029045643 -0.589730290456431 44 6.1 7.1089764868603 -1.00897648686030 45 5.8 6.68430843706777 -0.884308437067774 46 6.1 6.32057399723375 -0.220573997233749 47 7.2 6.63235131396957 0.567648686030428 48 7.3 6.57870677731674 0.721293222683264 49 6.9 6.34861687413554 0.551383125864457 50 6.1 6.12132780082988 -0.0213278008298763 51 5.8 6.13048409405256 -0.330484094052560 52 6.2 6.51421853388658 -0.314218533886584 53 7.1 7.09982019363762 0.000179806362378812 54 7.7 7.11066390041494 0.589336099585062 55 7.9 7.19439834024896 0.705601659751037 56 7.7 6.8360857538036 0.863914246196404 57 7.4 6.82075380359613 0.579246196403873 58 7.5 6.72991009681881 0.77008990318119 59 8 7.31457814661134 0.685421853388658 60 8.1 7.32915629322268 0.770843706777317 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.0485142983165755 0.0970285966331509 0.951485701683425 17 0.0330873435801727 0.0661746871603455 0.966912656419827 18 0.0152342514199236 0.0304685028398473 0.984765748580076 19 0.0134188015201041 0.0268376030402081 0.986581198479896 20 0.0117641004694838 0.0235282009389676 0.988235899530516 21 0.00680012348443746 0.0136002469688749 0.993199876515563 22 0.00270587504064089 0.00541175008128177 0.99729412495936 23 0.0116741509858868 0.0233483019717735 0.988325849014113 24 0.0309054120982048 0.0618108241964096 0.969094587901795 25 0.0769453133487094 0.153890626697419 0.92305468665129 26 0.082181163672838 0.164362327345676 0.917818836327162 27 0.0792176718341745 0.158435343668349 0.920782328165826 28 0.0713152189326125 0.142630437865225 0.928684781067387 29 0.0569354637389486 0.113870927477897 0.943064536261051 30 0.0524654116762258 0.104930823352452 0.947534588323774 31 0.0441801173162616 0.0883602346325231 0.955819882683738 32 0.0275844831331629 0.0551689662663259 0.972415516866837 33 0.0184694965281991 0.0369389930563982 0.9815305034718 34 0.0169402631167168 0.0338805262334336 0.983059736883283 35 0.0155394001939312 0.0310788003878624 0.984460599806069 36 0.0203239674259229 0.0406479348518457 0.979676032574077 37 0.0380344466279424 0.0760688932558848 0.961965553372058 38 0.0310947013714862 0.0621894027429723 0.968905298628514 39 0.0192025756735674 0.0384051513471347 0.980797424326433 40 0.00954221353998928 0.0190844270799786 0.99045778646001 41 0.00498148021680104 0.00996296043360208 0.995018519783199 42 0.00363184412360111 0.00726368824720222 0.996368155876399 43 0.00445772774181618 0.00891545548363236 0.995542272258184 44 0.081652540413984 0.163305080827968 0.918347459586016 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.137931034482759 NOK 5% type I error level 15 0.517241379310345 NOK 10% type I error level 22 0.758620689655172 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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