CVM Paper: Multiple Linear Regression (seasonal dummies)

*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, 17 Dec 2009 04:28:53 -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/Dec/17/t1261049376zwrcmijtb565bni.htm/, Retrieved Thu, 17 Dec 2009 12:29:48 +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/Dec/17/t1261049376zwrcmijtb565bni.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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25.6 7.4 1.8 23.7 7.1 2.7 22 6.8 2.3 21.3 6.9 1.9 20.7 7.2 2 20.4 7.4 2.3 20.3 7.3 2.8 20.4 6.9 2.4 19.8 6.9 2.3 19.5 6.8 2.7 23.1 7.1 2.7 23.5 7.2 2.9 23.5 7.1 3 22.9 7 2.2 21.9 6.9 2.3 21.5 7.1 2.8 20.5 7.3 2.8 20.2 7.5 2.8 19.4 7.5 2.2 19.2 7.5 2.6 18.8 7.3 2.8 18.8 7 2.5 22.6 6.7 2.4 23.3 6.5 2.3 23 6.5 1.9 21.4 6.5 1.7 19.9 6.6 2 18.8 6.8 2.1 18.6 6.9 1.7 18.4 6.9 1.8 18.6 6.8 1.8 19.9 6.8 1.8 19.2 6.5 1.3 18.4 6.1 1.3 21.1 6.1 1.3 20.5 5.9 1.2 19.1 5.7 1.4 18.1 5.9 2.2 17 5.9 2.9 17.1 6.1 3.1 17.4 6.3 3.5 16.8 6.2 3.6 15.3 5.9 4.4 14.3 5.7 4.1 13.4 5.4 5.1 15.3 5.6 5.8 22.1 6.2 5.9 23.7 6.3 5.4 22.2 6 5.5 19.5 5.6 4.8 16.6 5.5 3.2 17.3 5.9 2.7 19.8 6.5 2.1 21.2 6.8 1.9 21.5 6.8 0.6 20.6 6.5 0.7 19.1 6.2 -0.2 19.6 6.2 -1 23.5 6.5 -1.7 24 6.7 -0.7

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 W<25j[t] = + 6.0283260818606 + 2.69535910941759`W>25j`[t] -0.271201565433912Inflatie[t] -0.238306399471402M1[t] -1.47486330634128M2[t] -2.94805085936598M3[t] -3.82645389474653M4[t] -4.40827460192685M5[t] -4.71544560113093M6[t] -4.85845387804123M7[t] -4.52413730096343M8[t] -4.76743039081759M9[t] -4.18398729768748M10[t] -0.54712015654339M11[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) 6.0283260818606 2.09838 2.8728 0.006136 0.003068 `W>25j` 2.69535910941759 0.299566 8.9976 0 0 Inflatie -0.271201565433912 0.107819 -2.5153 0.015448 0.007724 M1 -0.238306399471402 0.740577 -0.3218 0.749073 0.374537 M2 -1.47486330634128 0.740435 -1.9919 0.052339 0.02617 M3 -2.94805085936598 0.740542 -3.9809 0.000242 0.000121 M4 -3.82645389474653 0.739417 -5.175 5e-06 2e-06 M5 -4.40827460192685 0.745741 -5.9113 0 0 M6 -4.71544560113093 0.75202 -6.2704 0 0 M7 -4.85845387804123 0.74618 -6.5111 0 0 M8 -4.52413730096343 0.740292 -6.1113 0 0 M9 -4.76743039081759 0.738669 -6.4541 0 0 M10 -4.18398729768748 0.740359 -5.6513 1e-06 0 M11 -0.54712015654339 0.738546 -0.7408 0.462575 0.231288 Multiple Linear Regression - Regression Statistics Multiple R 0.913680740314576 R-squared 0.834812495221791 Adjusted R-squared 0.788129069958384 F-TEST (value) 17.8824173785758 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 8.79296635503124e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.16761859153805 Sum Squared Residuals 62.7133260640438 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 25.6 25.2475142742984 0.352485725701641 2 23.7 22.9582682257126 0.741731774287357 3 22 20.7849535660362 1.21504643396377 4 21.3 20.284567067771 1.01543293222899 5 20.7 20.4842339368726 0.215766063127420 6 20.4 20.6347742899218 -0.234774289921848 7 20.3 20.0866293193528 0.213370680647179 8 20.4 19.4512828788372 0.948717121162844 9 19.8 19.2351099455264 0.564890054473616 10 19.5 19.4405365015412 0.0594634984588263 11 23.1 23.8860113755105 -0.78601137551053 12 23.5 24.6484271299089 -1.14842712990890 13 23.5 24.1134646629523 -0.613464662952346 14 22.9 22.8243330974878 0.0756669025121628 15 21.9 21.054489476978 0.845510523022009 16 21.5 20.579557480764 0.920442519235997 17 20.5 20.5368085954672 -0.0368085954672038 18 20.2 20.7687094181466 -0.568709418146646 19 19.4 20.7884220804967 -1.38842208049668 20 19.2 21.0142580314009 -1.81425803140093 21 18.8 20.1776528065765 -1.37765280657646 22 18.8 20.0338486365115 -1.23384863651147 23 22.6 22.8892282013737 -0.289228201373672 24 23.3 22.9243966925769 0.375603307423065 25 23 22.7945709192791 0.205429080720904 26 21.4 21.612254325496 -0.212254325495997 27 19.9 20.3272422137829 -0.427242213782885 28 18.8 19.9607908437425 -1.16079084374246 29 18.6 19.7569866736775 -1.15698667367747 30 18.4 19.42269551793 -1.02269551793001 31 18.6 19.0101513300779 -0.410151330077933 32 19.9 19.3444679071557 0.555532092844254 33 19.2 18.4281678671933 0.771832132806741 34 18.4 17.9334673165563 0.466532683443667 35 21.1 21.5703344577004 -0.470334457700419 36 20.5 21.6055029489037 -1.10550294890369 37 19.1 20.7738844144620 -1.67388441446198 38 18.1 19.8594380771285 -1.75943807712849 39 17 18.1964094283001 -1.19640942830005 40 17.1 17.8028379017162 -0.702837901716238 41 17.4 17.6516083902459 -0.251608390245877 42 16.8 17.0477813235566 -0.247781323556649 43 15.3 15.8792040614739 -0.579204061473933 44 14.3 15.7558092862984 -1.45580928629840 45 13.4 14.4327068981850 -1.03270689818505 46 15.3 15.3653807173949 -0.0653807173949337 47 22.1 20.5923431676462 1.50765683235381 48 23.7 21.5446000178483 2.15539998215171 49 22.2 20.4705657290082 1.72943427099178 50 19.5 18.3457062741750 1.15429372582496 51 16.6 17.0369053149028 -0.436905314902841 52 17.3 17.3722467060063 -0.0722467060062862 53 19.8 18.5703624037369 1.22963759626313 54 21.2 19.1260394504448 2.07396054955515 55 21.5 19.3355932085986 2.16440679140137 56 20.6 18.8341818963078 1.76581810369223 57 19.1 18.0263624825188 1.07363751748115 58 19.6 18.8267668279961 0.773233172003911 59 23.5 23.4620827977692 0.0379172022308077 60 24 24.2770732107622 -0.277073210762190 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.0273397186282271 0.0546794372564542 0.972660281371773 18 0.00744337222734536 0.0148867444546907 0.992556627772655 19 0.058242086007885 0.11648417201577 0.941757913992115 20 0.0771353581140986 0.154270716228197 0.922864641885901 21 0.0512416691428479 0.102483338285696 0.948758330857152 22 0.0432686461962748 0.0865372923925496 0.956731353803725 23 0.033097563314059 0.066195126628118 0.96690243668594 24 0.0188847697040868 0.0377695394081736 0.981115230295913 25 0.0272949855333516 0.0545899710667032 0.972705014466648 26 0.0385060487557989 0.0770120975115977 0.96149395124420 27 0.0571927763169398 0.114385552633880 0.94280722368306 28 0.140330079191260 0.280660158382520 0.85966992080874 29 0.193195428098178 0.386390856196355 0.806804571901822 30 0.259133342988957 0.518266685977914 0.740866657011043 31 0.297742086947925 0.59548417389585 0.702257913052075 32 0.318040343268515 0.63608068653703 0.681959656731485 33 0.324588245472652 0.649176490945304 0.675411754527348 34 0.245453242412578 0.490906484825156 0.754546757587422 35 0.175622903943095 0.351245807886191 0.824377096056905 36 0.159436044018570 0.318872088037139 0.84056395598143 37 0.175399429796948 0.350798859593896 0.824600570203052 38 0.45440740900961 0.90881481801922 0.54559259099039 39 0.684639584106719 0.630720831786561 0.315360415893281 40 0.805102449894396 0.389795100211208 0.194897550105604 41 0.793797306097249 0.412405387805502 0.206202693902751 42 0.679369740748603 0.641260518502794 0.320630259251397 43 0.549026737485484 0.901946525029031 0.450973262514516 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 2 0.0740740740740741 NOK 10% type I error level 7 0.259259259259259 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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