Workshop 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, 19 Nov 2009 12:11:43 -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/t1258657992rj50h65cpcdxeid.htm/, Retrieved Thu, 19 Nov 2009 20:13:23 +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/t1258657992rj50h65cpcdxeid.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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Dataseries X:

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0,7790 0,0000 0,7775 0,7461 0,7744 0,0520 0,7790 0,7775 0,7905 0,3130 0,7744 0,7790 0,7719 0,3640 0,7905 0,7744 0,7811 0,3630 0,7719 0,7905 0,7557 -0,1550 0,7811 0,7719 0,7637 0,0520 0,7557 0,7811 0,7595 0,5680 0,7637 0,7557 0,7471 0,6680 0,7595 0,7637 0,7615 1,3780 0,7471 0,7595 0,7487 0,2520 0,7615 0,7471 0,7389 -0,4020 0,7487 0,7615 0,7337 -0,0500 0,7389 0,7487 0,7510 0,5550 0,7337 0,7389 0,7382 0,0500 0,7510 0,7337 0,7159 0,1500 0,7382 0,7510 0,7542 0,4500 0,7159 0,7382 0,7636 0,2990 0,7542 0,7159 0,7433 0,1990 0,7636 0,7542 0,7658 0,4960 0,7433 0,7636 0,7627 0,4440 0,7658 0,7433 0,7480 -0,3930 0,7627 0,7658 0,7692 -0,4440 0,7480 0,7627 0,7850 0,1980 0,7692 0,7480 0,7913 0,4940 0,7850 0,7692 0,7720 0,1330 0,7913 0,7850 0,7880 0,3880 0,7720 0,7913 0,8070 0,4840 0,7880 0,7720 0,8268 0,2780 0,8070 0,7880 0,8244 0,3690 0,8268 0,8070 0,8487 0,1650 0,8244 0,8268 0,8572 0,1550 0,8487 0,8244 0,8214 0,0870 0,8572 0,8487 0,8827 0,4140 0,8214 0,8572 0,9216 0,3600 0,8827 0,8214 0,88 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.0933902629630105 + 0.0131973758026373X[t] + 1.12898049003522Y1[t] -0.261041337127442Y2[t] + 0.00904104327854089M1[t] -0.0127712637042683M2[t] + 0.0157127599307248M3[t] -0.00646660452244727M4[t] + 0.0238195808627235M5[t] + 0.00955915069397892M6[t] + 0.00539507651379937M7[t] + 0.0051366371530282M8[t] -0.0216543225624553M9[t] + 0.0078438959252224M10[t] -0.0141773306869561M11[t] + 0.000257008712972165t + 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) 0.0933902629630105 0.050427 1.852 0.071238 0.035619 X 0.0131973758026373 0.013289 0.9931 0.326499 0.163249 Y1 1.12898049003522 0.181057 6.2355 0 0 Y2 -0.261041337127442 0.169251 -1.5423 0.130676 0.065338 M1 0.00904104327854089 0.021523 0.4201 0.676638 0.338319 M2 -0.0127712637042683 0.02203 -0.5797 0.565272 0.282636 M3 0.0157127599307248 0.022069 0.712 0.480516 0.240258 M4 -0.00646660452244727 0.022106 -0.2925 0.771359 0.385679 M5 0.0238195808627235 0.02191 1.0872 0.283312 0.141656 M6 0.00955915069397892 0.022279 0.4291 0.670116 0.335058 M7 0.00539507651379937 0.022383 0.241 0.810732 0.405366 M8 0.0051366371530282 0.022278 0.2306 0.818798 0.409399 M9 -0.0216543225624553 0.02212 -0.979 0.333342 0.166671 M10 0.0078438959252224 0.024009 0.3267 0.74555 0.372775 M11 -0.0141773306869561 0.022671 -0.6254 0.535198 0.267599 t 0.000257008712972165 0.000272 0.9443 0.350542 0.175271 Multiple Linear Regression - Regression Statistics Multiple R 0.938532903991567 R-squared 0.880844011874843 Adjusted R-squared 0.837250357682712 F-TEST (value) 20.2057851813177 F-TEST (DF numerator) 15 F-TEST (DF denominator) 41 p-value 2.75335310107039e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0317207095155677 Sum Squared Residuals 0.041254339899012 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 0.779 0.785707704326124 -0.00670770432612371 2 0.7744 0.758335442347275 0.0160645576527254 3 0.7905 0.784936117519875 0.00556388248012521 4 0.7719 0.783064203985963 -0.0111642039859629 5 0.7811 0.788392398065896 -0.00729239806589634 6 0.7557 0.782794725323252 -0.0270947253232522 7 0.7637 0.750541831898724 0.0131581681012764 8 0.7595 0.773012541048404 -0.0135125410484043 9 0.7471 0.740968278870989 0.00613172112901088 10 0.7615 0.76719065843101 -0.00569065843101006 11 0.7487 0.750060427014922 -0.00136042701492153 12 0.7389 0.737653737112839 0.00124626288716086 13 0.7337 0.743874585699766 -0.0101745856997665 14 0.751 0.72699120634619 0.0240087936538091 15 0.7382 0.769956341344496 -0.0317563413444964 16 0.7159 0.730386757779805 -0.0144867577798046 17 0.7542 0.743054228806185 0.0111457711938155 18 0.7636 0.776119178190505 -0.0125191781905048 19 0.7433 0.771506908537384 -0.0282069085373837 20 0.7658 0.750053005986255 0.015746994013745 21 0.7627 0.753533991611486 0.00916600838851389 22 0.748 0.762869745660852 -0.014869745660852 23 0.7692 0.724645676537289 0.0445543234627115 24 0.785 0.77532442524703 0.00967557475297011 25 0.7913 0.800832715871578 -0.00953271587157835 26 0.772 0.777501288897598 -0.00550128889759753 27 0.788 0.786173768193653 0.00182623180634735 28 0.807 0.788620146177629 0.0183798538223709 29 0.8268 0.83371864877706 -0.00691864877705902 30 0.8244 0.838310216816602 -0.0139102168166024 31 0.8487 0.82383271503445 0.0248672849655508 32 0.8572 0.851760035745586 0.0054399642544144 33 0.8214 0.827581692861597 -0.00618169286159735 34 0.8827 0.819016109040866 0.0636838909591345 35 0.9216 0.875091016756638 0.0465089832433617 36 0.8865 0.925557249371646 -0.0390572493716464 37 0.8816 0.875769428207806 0.00583057179219378 38 0.8884 0.859019242916405 0.0293807570835953 39 0.9466 0.894208943746033 0.0523910562539672 40 0.918 0.939016015103575 -0.0210160151035753 41 0.9337 0.919081957058695 0.0146180429413048 42 0.9559 0.934650840304796 0.0212491596952039 43 0.9626 0.957502440700828 0.00509755929917249 44 0.9434 0.956670178618916 -0.0132701786189158 45 0.8639 0.896918372403417 -0.0330183724034174 46 0.7996 0.842723486867272 -0.0431234868672724 47 0.668 0.757702879691152 -0.0897028796911517 48 0.6572 0.629064588268485 0.0281354117315154 49 0.6928 0.672215565894725 0.0205844341052748 50 0.6438 0.707752819492532 -0.0639528194925323 51 0.6454 0.673424829195943 -0.0280248291959435 52 0.6873 0.659012876953028 0.0282871230469718 53 0.7265 0.738052767292165 -0.0115527672921649 54 0.7912 0.758925039364845 0.0322749606351555 55 0.8114 0.826316103828616 -0.0149161038286159 56 0.8281 0.822504238600839 0.00559576139916069 57 0.8393 0.81539766425251 0.0239023357474900 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.0133254148988905 0.0266508297977811 0.98667458510111 20 0.0440524468220035 0.088104893644007 0.955947553177997 21 0.0363568363064338 0.0727136726128676 0.963643163693566 22 0.0222619275019989 0.0445238550039978 0.977738072498001 23 0.0228613644138852 0.0457227288277703 0.977138635586115 24 0.0097697645828615 0.019539529165723 0.990230235417139 25 0.00474255482362691 0.00948510964725382 0.995257445176373 26 0.00278865897052858 0.00557731794105716 0.997211341029471 27 0.00103794481912466 0.00207588963824932 0.998962055180875 28 0.00104983794224418 0.00209967588448836 0.998950162057756 29 0.000377056607265409 0.000754113214530818 0.999622943392735 30 0.000204126334836595 0.000408252669673191 0.999795873665163 31 0.000119984878958914 0.000239969757917829 0.999880015121041 32 5.00010965545076e-05 0.000100002193109015 0.999949998903445 33 0.000216739244078787 0.000433478488157575 0.999783260755921 34 0.000414566901048388 0.000829133802096777 0.999585433098952 35 0.00446777809872259 0.00893555619744518 0.995532221901277 36 0.00908663666485146 0.0181732733297029 0.990913363335149 37 0.00390498513129657 0.00780997026259313 0.996095014868703 38 0.0109966311728245 0.021993262345649 0.989003368827176 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 12 0.6 NOK 5% type I error level 18 0.9 NOK 10% type I error level 20 1 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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