Multiple Regression with 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: Fri, 18 Dec 2009 10:09:32 -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/Dec/18/t1261156382a7axw5tg41lzq41.htm/, Retrieved Fri, 18 Dec 2009 18:13:14 +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/18/t1261156382a7axw5tg41lzq41.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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113 14,3 110 14,2 107 15,9 103 15,3 98 15,5 98 15,1 137 15 148 12,1 147 15,8 139 16,9 130 15,1 128 13,7 127 14,8 123 14,7 118 16 114 15,4 108 15 111 15,5 151 15,1 159 11,7 158 16,3 148 16,7 138 15 137 14,9 136 14,6 133 15,3 126 17,9 120 16,4 114 15,4 116 17,9 153 15,9 162 13,9 161 17,8 149 17,9 139 17,4 135 16,7 130 16 127 16,6 122 19,1 117 17,8 112 17,2 113 18,6 149 16,3 157 15,1 157 19,2 147 17,7 137 19,1 132 18 125 17,5 123 17,8 117 21,1 114 17,2 111 19,4 112 19,8 144 17,6 150 16,2 149 19,5 134 19,9 123 20 116 17,3

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 WK<25j[t] = + 122.790035587189 + 0.422454368040411ExpBE[t] -3.11273102973248M1[t] -6.23101825278383M2[t] -12.3942142119160M3[t] -16.1267363104121M4[t] -21.1605326598553M5[t] -20.1322925037309M6[t] + 17.2591436115257M7[t] + 26.5800941338538M8[t] + 24.1240730111353M9[t] + 13.0818275743313M10[t] + 3.29305475835152M11[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) 122.790035587189 10.176011 12.0666 0 0 ExpBE 0.422454368040411 0.599909 0.7042 0.484784 0.242392 M1 -3.11273102973248 4.497829 -0.6921 0.49231 0.246155 M2 -6.23101825278383 4.485714 -1.3891 0.171355 0.085678 M3 -12.3942142119160 4.619096 -2.6833 0.010034 0.005017 M4 -16.1267363104121 4.482905 -3.5974 0.00077 0.000385 M5 -21.1605326598553 4.485088 -4.718 2.2e-05 1.1e-05 M6 -20.1322925037309 4.542622 -4.4319 5.6e-05 2.8e-05 M7 17.2591436115257 4.480079 3.8524 0.000353 0.000177 M8 26.5800941338538 4.690536 5.6668 1e-06 0 M9 24.1240730111353 4.580979 5.2661 3e-06 2e-06 M10 13.0818275743313 4.593924 2.8476 0.006513 0.003256 M11 3.29305475835152 4.536771 0.7259 0.471525 0.235762 Multiple Linear Regression - Regression Statistics Multiple R 0.930444080910634 R-squared 0.865726187701635 Adjusted R-squared 0.831443512221201 F-TEST (value) 25.2525853239119 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 2.22044604925031e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 7.08238132631051 Sum Squared Residuals 2357.52588680978 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 113 125.718402020434 -12.7184020204338 2 110 122.557869360579 -12.5578693605786 3 107 117.112845827115 -10.1128458271151 4 103 113.126851107795 -10.1268511077947 5 98 108.177545631960 -10.1775456319596 6 98 109.036804040868 -11.0368040408679 7 137 146.385994719320 -9.38599471932042 8 148 154.481827574331 -6.48182757433128 9 147 153.588887613362 -6.58888761336242 10 139 143.011341981403 -4.01134198140281 11 130 132.462151302950 -2.46215130295027 12 128 128.577660429342 -0.577660429342202 13 127 125.929629204454 1.07037079554582 14 123 122.769096544599 0.230903455401219 15 118 117.155091263919 0.84490873608082 16 114 113.169096544599 0.830903455401216 17 108 107.966318447939 0.0336815520606157 18 111 109.205785788084 1.79421421191598 19 151 146.428240156124 4.57175984387558 20 159 154.312845827115 4.68715417288485 21 158 153.800114797383 4.19988520261738 22 148 142.926851107795 5.07314889220525 23 138 132.419905866146 5.58009413385374 24 137 129.084605670991 7.91539432900931 25 136 125.845138330846 10.1548616691539 26 133 123.022569165423 9.97743083457697 27 126 117.957754563196 8.04224543680404 28 120 113.591550912639 6.40844908736081 29 114 108.135300195156 5.86469980484445 30 116 110.219676271381 5.78032372861898 31 153 146.766203650557 6.23379634944324 32 162 155.242245436804 6.75775456319595 33 161 154.433796349443 6.56620365055677 34 149 143.433796349443 5.56620365055676 35 139 133.433796349443 5.56620365055676 36 135 129.845023533463 5.15497646653657 37 130 126.436574446103 3.56342555389733 38 127 123.571759843876 3.42824015612443 39 122 118.464699804844 3.53530019515554 40 117 114.182987027896 2.81701297210423 41 112 108.895718057628 3.10428194237171 42 113 110.515394329009 2.4846056709907 43 149 146.935185397773 2.06481460222707 44 157 155.749190678453 1.25080932154746 45 157 155.025232464700 1.97476753530019 46 147 143.349305475835 3.65069452416485 47 137 134.151968775112 2.84803122488806 48 132 130.394214211916 1.60578578808403 49 125 127.070255998163 -2.07025599816329 50 123 124.078705085524 -1.07870508552406 51 117 119.309608540925 -2.30960854092527 52 114 113.929514407072 0.0704855929284804 53 111 109.825117667317 1.17488233268281 54 112 111.022339570658 0.9776604293422 55 144 147.484376076225 -3.48437607622546 56 150 156.213890483297 -6.21389048329699 57 149 155.151968775112 -6.15196877511193 58 134 144.278705085524 -10.2787050855241 59 123 134.532177706348 -11.5321777063483 60 116 130.098496154288 -14.0984961542877 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.604630663503497 0.790738672993005 0.395369336496503 17 0.980390830612486 0.0392183387750278 0.0196091693875139 18 0.983187697720517 0.0336246045589663 0.0168123022794831 19 0.987236439429576 0.0255271211408487 0.0127635605704244 20 0.996418245743443 0.00716350851311334 0.00358175425655667 21 0.994345709120116 0.0113085817597672 0.00565429087988358 22 0.993846308252414 0.0123073834951727 0.00615369174758633 23 0.99478445856106 0.0104310828778782 0.00521554143893911 24 0.989906921348378 0.0201861573032444 0.0100930786516222 25 0.994116528838714 0.0117669423225719 0.00588347116128594 26 0.99078509623236 0.0184298075352806 0.0092149037676403 27 0.98530852576327 0.0293829484734606 0.0146914742367303 28 0.973468591383175 0.0530628172336493 0.0265314086168247 29 0.978822724376206 0.0423545512475885 0.0211772756237942 30 0.972479015446452 0.055041969107097 0.0275209845535485 31 0.954668733495738 0.090662533008523 0.0453312665042615 32 0.930956419154122 0.138087161691756 0.0690435808458781 33 0.894325096203269 0.211349807593462 0.105674903796731 34 0.860616239818176 0.278767520363648 0.139383760181824 35 0.820746680362464 0.358506639275073 0.179253319637536 36 0.835660067467384 0.328679865065233 0.164339932532616 37 0.758809420094547 0.482381159810907 0.241190579905453 38 0.666511027982846 0.666977944034309 0.333488972017154 39 0.56890200359842 0.862195992803161 0.431097996401581 40 0.478561333758991 0.957122667517981 0.521438666241009 41 0.41347130782955 0.8269426156591 0.58652869217045 42 0.317027457229494 0.634054914458987 0.682972542770506 43 0.203003562849661 0.406007125699323 0.796996437150339 44 0.120869836785123 0.241739673570246 0.879130163214877 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.0344827586206897 NOK 5% type I error level 12 0.413793103448276 NOK 10% type I error level 15 0.517241379310345 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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