Multiple Linear Regression Model 3

*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, 20 Nov 2009 12:13:31 -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/20/t1258744454h3b5ualk8tpkbzj.htm/, Retrieved Fri, 20 Nov 2009 20:14:26 +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/20/t1258744454h3b5ualk8tpkbzj.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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No (this computation is public)

User-defined keywords:

Dataseries X:

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9.3 104.1 8.7 90.2 8.2 99.2 8.3 116.5 8.5 98.4 8.6 90.6 8.5 130.5 8.2 107.4 8.1 106 7.9 196.5 8.6 107.8 8.7 90.5 8.7 123.8 8.5 114.7 8.4 115.3 8.5 197 8.7 88.4 8.7 93.8 8.6 111.3 8.5 105.9 8.3 123.6 8 171 8.2 97 8.1 99.2 8.1 126.6 8 103.4 7.9 121.3 7.9 129.6 8 110.8 8 98.9 7.9 122.8 8 120.9 7.7 133.1 7.2 203.1 7.5 110.2 7.3 119.5 7 135.1 7 113.9 7 137.4 7.2 157.1 7.3 126.4 7.1 112.2 6.8 128.8 6.4 136.8 6.1 156.5 6.5 215.2 7.7 146.7 7.9 130.8 7.5 133.1 6.9 153.4 6.6 159.9 6.9 174.6 7.7 145 8 112.9 8 137.8 7.7 150.6 7.3 162.1 7.4 226.4 8.1 112.3 8.3 126.3

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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 114.594644234140 -3.37758499824748X[t] + 19.3930884157026M1[t] + 8.24068261479143M2[t] + 18.3460353137049M3[t] + 46.4397669120224M4[t] + 5.50636041009466M5[t] -7.19766649141254M6[t] + 16.2378930073607M7[t] + 12.9232457062741M8[t] + 23.2659433052927M9[t] + 88.3890545040308M10[t] + 2.12402690150718M11[t] + 0.719130301437084t + 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) 114.594644234140 37.364412 3.0669 0.003616 0.001808 X -3.37758499824748 4.051767 -0.8336 0.40881 0.204405 M1 19.3930884157026 8.787307 2.2069 0.032347 0.016174 M2 8.24068261479143 8.965368 0.9192 0.362802 0.181401 M3 18.3460353137049 9.142056 2.0068 0.050672 0.025336 M4 46.4397669120224 8.948323 5.1898 5e-06 2e-06 M5 5.50636041009466 8.729952 0.6307 0.53133 0.265665 M6 -7.19766649141254 8.699136 -0.8274 0.412282 0.206141 M7 16.2378930073607 8.725626 1.8609 0.069151 0.034575 M8 12.9232457062741 8.824547 1.4645 0.149868 0.074934 M9 23.2659433052927 9.045111 2.5722 0.013399 0.0067 M10 88.3890545040308 9.131021 9.6801 0 0 M11 2.12402690150718 8.661705 0.2452 0.807376 0.403688 t 0.719130301437084 0.152503 4.7155 2.3e-05 1.1e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.924217732216523 R-squared 0.854178416543453 Adjusted R-squared 0.812967969044863 F-TEST (value) 20.7272298261913 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 5.55111512312578e-15 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 13.6875386859959 Sum Squared Residuals 8618.04090290921 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 104.1 103.295322467578 0.804677532421608 2 90.2 94.8885979670522 -4.68859796705217 3 99.2 107.401873466526 -8.2018734665264 4 116.5 135.876976866456 -19.3769768664563 5 98.4 94.9871836663162 3.41281633368384 6 90.6 82.6645285664213 7.9354714335787 7 130.5 107.156976866456 23.3430231335436 8 107.4 105.574735366281 1.82526463371890 9 106 116.974321766562 -10.9743217665615 10 196.5 183.492080266386 13.0079197336138 11 107.8 95.5818734665264 12.2181265334735 12 90.5 93.8392183666316 -3.33921836663156 13 123.8 113.951437083771 9.84856291622866 14 114.7 104.193678583947 10.5063214160533 15 115.3 115.355920084122 -0.0559200841219734 16 197 143.831023484052 53.1689765159481 17 88.4 102.941230283912 -14.5412302839116 18 93.8 90.9563336838416 2.84366631615844 19 111.3 115.448781983877 -4.14878198387660 20 105.9 113.191023484052 -7.29102348405185 21 123.6 124.928368384157 -1.32836838415701 22 171 191.783885383807 -20.7838853838065 23 97 105.562471083070 -8.56247108307045 24 99.2 104.495332982825 -5.2953329828251 25 126.6 124.607551699965 1.99244830003516 26 103.4 114.512034700315 -11.1120347003155 27 121.3 125.674276200491 -4.37427620049072 28 129.6 154.487138100245 -24.8871381002454 29 110.8 113.935103399930 -3.1351033999299 30 98.9 101.950206799860 -3.0502067998598 31 122.8 126.442655099895 -3.64265509989485 32 120.9 123.509379600421 -2.60937960042059 33 133.1 135.584483000351 -2.48448300035051 34 203.1 203.115516999649 -0.0155169996494948 35 110.2 116.556344199089 -6.35634419908867 36 119.5 115.826964598668 3.67303540133192 37 135.1 136.952458815282 -1.85245881528208 38 113.9 126.519183315808 -12.6191833158080 39 137.4 137.343666316158 0.0563336838415397 40 157.1 165.481011216264 -8.38101121626362 41 126.4 124.928976515948 1.47102348405187 42 112.2 113.619596915528 -1.41959691552754 43 128.8 138.787562215212 -9.98756221521207 44 136.8 137.543079214862 -0.74307921486157 45 156.5 149.618182614791 6.88181738520853 46 215.2 214.109390115668 1.09060988433224 47 146.7 124.510390816684 22.1896091833158 48 130.8 122.429977216965 8.37002278303542 49 133.1 143.893229933403 -10.7932299334033 50 153.4 135.486505432878 17.9134945671223 51 159.9 147.324263932702 12.5757360672975 52 174.6 175.123850332983 -0.523850332982872 53 145 132.207506133894 12.7924938661058 54 112.9 119.209334034350 -6.3093340343498 55 137.8 143.36402383456 -5.5640238345601 56 150.6 141.781782334385 8.81821766561513 57 162.1 154.194644234140 7.9053557658605 58 226.4 219.69912723449 6.70087276551 59 112.3 131.788920434630 -19.4889204346302 60 126.3 129.708506834911 -3.4085068349106 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.999618847419858 0.000762305160283137 0.000381152580141569 18 0.99958555644055 0.00082888711889816 0.00041444355944908 19 0.99985573752144 0.000288524957118729 0.000144262478559364 20 0.99966363981492 0.000672720370161743 0.000336360185080872 21 0.99918570367089 0.00162859265821895 0.000814296329109474 22 0.999616692418242 0.000766615163516541 0.000383307581758271 23 0.99933038011247 0.00133923977505964 0.000669619887529818 24 0.998380241037703 0.00323951792459501 0.00161975896229750 25 0.99787632070473 0.00424735859054138 0.00212367929527069 26 0.995839830735828 0.00832033852834286 0.00416016926417143 27 0.991413653250918 0.0171726934981637 0.00858634674908183 28 0.9948617924418 0.0102764151163989 0.00513820755819944 29 0.990856626076422 0.0182867478471555 0.00914337392357775 30 0.98298102604048 0.0340379479190416 0.0170189739595208 31 0.973522856677194 0.0529542866456121 0.0264771433228061 32 0.95393786258523 0.0921242748295378 0.0460621374147689 33 0.925210216701095 0.149579566597810 0.0747897832989052 34 0.886420693410233 0.227158613179535 0.113579306589767 35 0.826361590979276 0.347276818041448 0.173638409020724 36 0.762346981435996 0.475306037128008 0.237653018564004 37 0.686670022293673 0.626659955412654 0.313329977706327 38 0.738298716916327 0.523402566167345 0.261701283083673 39 0.701863181357667 0.596273637284667 0.298136818642333 40 0.751805375567711 0.496389248864578 0.248194624432289 41 0.91947101901493 0.161057961970139 0.0805289809850693 42 0.83215515953407 0.335689680931858 0.167844840465929 43 0.688813454480907 0.622373091038185 0.311186545519092 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 10 0.370370370370370 NOK 5% type I error level 14 0.518518518518518 NOK 10% type I error level 16 0.592592592592593 NOK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; 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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