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: Wed, 22 Dec 2010 07:54:08 +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/t1293004371muhucfzyo4d1fkt.htm/, Retrieved Wed, 22 Dec 2010 08:52:54 +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/2010/Dec/22/t1293004371muhucfzyo4d1fkt.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:

Workshop 7

Dataseries X:

» Textbox « » Textfile « » CSV «

-5 -6 33 5 15 -1 -3 24 6 17 -2 -4 24 6 13 -5 -7 31 5 12 -4 -7 25 5 13 -6 -7 28 3 10 -2 -3 24 5 14 -2 0 25 5 13 -2 -5 16 5 10 -2 -3 17 3 11 2 3 11 6 12 1 2 12 6 7 -8 -7 39 4 11 -1 -1 19 6 9 1 0 14 5 13 -1 -3 15 4 12 2 4 7 5 5 2 2 12 5 13 1 3 12 4 11 -1 0 14 3 8 -2 -10 9 2 8 -2 -10 8 3 8 -1 -9 4 2 8 -8 -22 7 -1 0 -4 -16 3 0 3 -6 -18 5 -2 0 -3 -14 0 1 -1 -3 -12 -2 -2 -1 -7 -17 6 -2 -4 -9 -23 11 -2 1 -11 -28 9 -6 -1 -13 -31 17 -4 0 -11 -21 21 -2 -1 -9 -19 21 0 6 -17 -22 41 -5 0 -22 -22 57 -4 -3 -25 -25 65 -5 -3 -20 -16 68 -1 4 -24 -22 73 -2 1 -24 -21 71 -4 0 -22 -10 71 -1 -4 -19 -7 70 1 -2 -18 -5 69 1 3 -17 -4 65 -2 2 -11 7 57 1 5 -11 6 57 1 6 -12 3 57 3 6 -10 10 55 3 3 -15 0 65 1 4 -15 -2 65 1 7 -15 -1 64 0 5 -13 2 60 2 6 -8 8 43 2 1 -13 -6 47 -1 3 -9 -4 40 1 6 -7 4 31 0 0 -4 7 27 1 3 -4 3 24 1 4 -2 3 23 3 7 0 8 17 2 6

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 6 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk R Framework error message The field 'Names of X columns' contains a hard return which cannot be interpreted. Please, resubmit your request without hard returns in the 'Names of X columns'. Multiple Linear Regression - Estimated Regression Equation CVI[t] = + 0.130074320935853 + 0.252862994160068EconSit[t] -0.252878563273482Werkloos[t] + 0.268135621609646FinSit[t] + 0.224848607252311`Sparen `[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) 0.130074320935853 0.117933 1.103 0.274851 0.137425 EconSit 0.252862994160068 0.00631 40.0754 0 0 Werkloos -0.252878563273482 0.001989 -127.1389 0 0 FinSit 0.268135621609646 0.033069 8.1084 0 0 `Sparen ` 0.224848607252311 0.015009 14.981 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.999189462068346 R-squared 0.99837958110843 Adjusted R-squared 0.998261732461772 F-TEST (value) 8471.71019275742 F-TEST (DF numerator) 4 F-TEST (DF denominator) 55 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.311004478600877 Sum Squared Residuals 5.31980821403917 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 -5 -5.01868901521657 0.0186890152165668 2 -1 -1.26636012716076 0.266360127160759 3 -2 -2.41861755033006 0.418617550330064 4 -5 -5.4403407045866 0.440340704586605 5 -4 -3.6982207176934 -0.301779282306598 6 -6 -5.66767347249007 -0.332326527509928 7 -2 -2.20904157052734 0.209041570527336 8 -2 -1.92817975857292 -0.0718202414270764 9 -2 -1.59113348166886 -0.408866518331139 10 -2 -1.64970869258919 -0.350291307410813 11 2 2.41399612409336 -0.413996124093363 12 1 0.784011530398259 0.215988469601741 13 -8 -7.95635343963642 -0.0436465603635842 14 -1 -1.2950301804917 0.295030180491698 15 1 0.853484437435377 0.146515562564623 16 -1 -0.650967337180266 -0.349032662819734 17 2 1.83629749897154 0.163702501028463 18 2 1.86496755230248 0.135032447697522 19 1 1.39999771034828 -0.399997710348279 20 -1 -0.80702984204547 -0.192970157954531 21 -2 -2.33940258888839 0.339402588888389 22 -2 -1.81838840400526 -0.181611595994739 23 -1 -0.82214677836091 -0.17785322163909 24 -8 -7.47119711510967 -0.528802884890334 25 -4 -3.99982345368875 -0.00017654631124835 26 -6 -6.22212363353208 0.222123633532077 27 -3 -3.36672058294777 0.366720582947768 28 -3 -3.1596443329096 0.159644332909603 29 -7 -7.12153363165473 0.121533631654732 30 -9 -8.778861376721 -0.221138623279003 31 -11 -11.0596589219176 0.0596589219175801 32 -13 -13.080156560114 0.0801565601140388 33 -11 -11.2516182356403 0.251618235640302 34 -9 -8.6356807533347 -0.364319246665302 35 -17 -17.1416107528466 0.141610752846634 36 -22 -21.5940779653696 -0.405922034630368 37 -25 -24.6438310756473 -0.356168924352662 38 -20 -20.4802170808224 0.48021708082241 39 -24 -24.2044693055168 0.204469305516808 40 -24 -24.2069690352814 0.206969035281379 41 -22 -21.5204636637009 -0.47953633629907 42 -19 -19.5230276602233 0.523027660223331 43 -18 -17.6401800723682 -0.359819927631842 44 -17 -17.4050582971954 0.405058297195412 45 -11 -11.1215841686609 0.121584168660938 46 -11 -11.1495985555687 0.149598555568694 47 -12 -11.3719162948296 -0.628083705170392 48 -10 -9.7706640309191 -0.229335969080901 49 -15 -15.1395022412216 0.13950224122158 50 -15 -14.9706824077848 -0.0293175922152171 51 -15 -15.1827736864655 0.182773686465501 52 -13 -12.6515506004198 -0.348449399580232 53 -8 -7.95968009607172 -0.0403199039282814 54 -13 -12.8659859177309 -0.134014082269082 55 -9 -9.37929292152018 0.379292921520184 56 -7 -6.69770916390181 -0.302290836098189 57 -4 -3.9849244849611 -0.0150755150388999 58 -4 -4.01289216452862 0.0128921645286169 59 -2 -2.54919653627891 0.549196536278911 60 0 -0.260594414699635 0.260594414699635 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.549861111267784 0.900277777464432 0.450138888732216 9 0.455771359386105 0.91154271877221 0.544228640613895 10 0.453215841281762 0.906431682563524 0.546784158718238 11 0.432268954272629 0.864537908545258 0.567731045727371 12 0.417430944780833 0.834861889561665 0.582569055219167 13 0.390084748835817 0.780169497671633 0.609915251164183 14 0.321947222340155 0.64389444468031 0.678052777659845 15 0.336953888855423 0.673907777710846 0.663046111144577 16 0.263872591733422 0.527745183466845 0.736127408266578 17 0.239943765185825 0.47988753037165 0.760056234814175 18 0.207706422622355 0.41541284524471 0.792293577377645 19 0.189436749390779 0.378873498781559 0.81056325060922 20 0.158186220997399 0.316372441994797 0.841813779002601 21 0.391521506551332 0.783043013102664 0.608478493448668 22 0.342675088786893 0.685350177573786 0.657324911213107 23 0.280947935617974 0.561895871235949 0.719052064382026 24 0.318197021780682 0.636394043561365 0.681802978219318 25 0.303603176167946 0.607206352335893 0.696396823832054 26 0.389628268253707 0.779256536507414 0.610371731746293 27 0.399546224415517 0.799092448831034 0.600453775584483 28 0.382501639933937 0.765003279867874 0.617498360066063 29 0.328101144554625 0.656202289109249 0.671898855445375 30 0.28480268718677 0.56960537437354 0.71519731281323 31 0.246374358215479 0.492748716430959 0.75362564178452 32 0.188999021897585 0.37799804379517 0.811000978102415 33 0.169523129767837 0.339046259535674 0.830476870232163 34 0.233083026277128 0.466166052554257 0.766916973722872 35 0.177881571498902 0.355763142997804 0.822118428501098 36 0.235651493782901 0.471302987565802 0.764348506217099 37 0.288741552098209 0.577483104196418 0.711258447901791 38 0.345022801618083 0.690045603236165 0.654977198381917 39 0.275566989838129 0.551133979676259 0.72443301016187 40 0.216740076767327 0.433480153534654 0.783259923232673 41 0.311338309609321 0.622676619218642 0.688661690390679 42 0.691661291072412 0.616677417855176 0.308338708927588 43 0.633164567573825 0.73367086485235 0.366835432426175 44 0.703063744178288 0.593872511643423 0.296936255821712 45 0.620466628606177 0.759066742787646 0.379533371393823 46 0.531327195447518 0.937345609104964 0.468672804552482 47 0.864694114103274 0.270611771793451 0.135305885896726 48 0.797558235550036 0.404883528899928 0.202441764449964 49 0.773937072451338 0.452125855097324 0.226062927548662 50 0.662251360338236 0.675497279323527 0.337748639661764 51 0.899409450994253 0.201181098011494 0.100590549005747 52 0.993149177483076 0.0137016450338476 0.00685082251692381 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 1 0.0222222222222222 OK 10% type I error level 1 0.0222222222222222 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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