WS 7: Multiple Regression 4

*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 03:53:05 -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/t1258714672b1nv4zrvjt7lhkn.htm/, Retrieved Fri, 20 Nov 2009 11:58:26 +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/t1258714672b1nv4zrvjt7lhkn.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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104 120.28 112.9 113.6 83.4 79.8 109.9 115.33 104 112.9 113.6 83.4 99 110.4 109.9 104 112.9 113.6 106.3 114.49 99 109.9 104 112.9 128.9 132.03 106.3 99 109.9 104 111.1 123.16 128.9 106.3 99 109.9 102.9 118.82 111.1 128.9 106.3 99 130 128.32 102.9 111.1 128.9 106.3 87 112.24 130 102.9 111.1 128.9 87.5 104.53 87 130 102.9 111.1 117.6 132.57 87.5 87 130 102.9 103.4 122.52 117.6 87.5 87 130 110.8 131.8 103.4 117.6 87.5 87 112.6 124.55 110.8 103.4 117.6 87.5 102.5 120.96 112.6 110.8 103.4 117.6 112.4 122.6 102.5 112.6 110.8 103.4 135.6 145.52 112.4 102.5 112.6 110.8 105.1 118.57 135.6 112.4 102.5 112.6 127.7 134.25 105.1 135.6 112.4 102.5 137 136.7 127.7 105.1 135.6 112.4 91 121.37 137 127.7 105.1 135.6 90.5 111.63 91 137 127.7 105.1 122.4 134.42 90.5 91 137 127.7 123.3 137.65 122.4 90.5 91 137 124.3 137.86 123.3 122.4 90.5 91 120 119.77 124.3 123.3 122.4 90.5 118.1 130.69 120 124.3 123.3 122.4 119 128.28 118.1 120 124.3 123.3 142.7 147.45 119 118.1 120 124.3 123.6 128.42 142.7 119 118 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 I[t] = -25.1447875446987 + 0.730220332179792U[t] + 0.0643404999581819m1[t] + 0.194820257073925m2[t] + 0.170133757972612m3[t] + 0.046402945892129m4[t] -3.23374546523884M1[t] + 2.95184473503329M2[t] -10.5575350599474M3[t] -2.0182951502947M4[t] + 6.81571807814321M5[t] -1.63565124828532M6[t] -6.03568977932192M7[t] + 7.71868122050343M8[t] -24.0107549608620M9[t] -21.4349262435542M10[t] -1.73157334236155M11[t] -0.152079578124883t + 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) -25.1447875446987 10.733265 -2.3427 0.024483 0.012241 U 0.730220332179792 0.078983 9.2453 0 0 m1 0.0643404999581819 0.094311 0.6822 0.49924 0.24962 m2 0.194820257073925 0.099556 1.9569 0.057736 0.028868 m3 0.170133757972612 0.098832 1.7214 0.093303 0.046652 m4 0.046402945892129 0.108066 0.4294 0.670062 0.335031 M1 -3.23374546523884 7.362942 -0.4392 0.66301 0.331505 M2 2.95184473503329 8.497245 0.3474 0.730216 0.365108 M3 -10.5575350599474 5.519921 -1.9126 0.063355 0.031678 M4 -2.0182951502947 5.641557 -0.3578 0.722506 0.361253 M5 6.81571807814321 4.867396 1.4003 0.169542 0.084771 M6 -1.63565124828532 5.41332 -0.3022 0.764183 0.382092 M7 -6.03568977932192 7.446678 -0.8105 0.422688 0.211344 M8 7.71868122050343 6.132789 1.2586 0.215855 0.107928 M9 -24.0107549608620 4.966142 -4.8349 2.2e-05 1.1e-05 M10 -21.4349262435542 8.439853 -2.5397 0.015306 0.007653 M11 -1.73157334236155 6.847911 -0.2529 0.801738 0.400869 t -0.152079578124883 0.053864 -2.8234 0.007524 0.003762 Multiple Linear Regression - Regression Statistics Multiple R 0.980509202308032 R-squared 0.961398295810733 Adjusted R-squared 0.94412911235764 F-TEST (value) 55.6713233385993 F-TEST (DF numerator) 17 F-TEST (DF denominator) 38 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 4.05308389685655 Sum Squared Residuals 624.244584848399 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 104 106.588023112507 -2.5880231125075 2 109.9 113.603028556769 -3.70302855676944 3 99 96.2695669431744 2.73043305682563 4 106.3 106.544783992429 -0.244783992428747 5 128.9 126.971730070363 1.92826992963728 6 111.1 113.186829413931 -2.08682941393122 7 102.9 109.459416296700 -6.55941629670036 8 130 130.187172633729 -0.187172633729466 9 87 84.7301410588982 2.26985894110176 10 87.5 81.8158096532213 5.68419034677873 11 117.6 117.727480971153 -0.127480971152990 12 103.4 107.944087815115 -4.54408781511537 13 110.8 114.374902298524 -3.57490229852384 14 112.6 117.96821514953 -5.3682151495301 15 102.5 102.223576894313 0.27642310568715 16 112.4 112.1092039611 0.290796038899986 17 135.6 136.846696542066 -1.24669654206571 18 105.1 110.350404176411 -5.25040417641103 19 127.7 121.021240041638 6.67875995836246 20 137 136.331141084775 0.668858915224647 21 91 94.1441208189833 -3.14412081898335 22 90.5 90.7374223959174 -0.237422395917378 23 122.4 120.567465540290 1.83253445970969 24 123.3 119.066017327653 4.23398267234656 25 124.3 119.886608814644 4.41339118535619 26 120 118.354177765364 1.64582223463643 27 118.1 114.218254883049 3.88174511695064 28 119 120.097506567961 -1.09750656796095 29 142.7 141.780339734292 0.919660265707606 30 123.6 120.458219181249 3.14178081875114 31 129.6 125.551660817403 4.04833918259692 32 151.6 145.040914385862 6.55908561413800 33 110.4 107.525875247848 2.8741247521523 34 99.2 102.10964815419 -2.90964815418991 35 130.5 127.413735403811 3.08626459618946 36 136.2 134.6733255033 1.5266744967 37 129.7 126.391639864646 3.30836013535443 38 128 123.866144938476 4.13385506152412 39 121.6 126.520056817208 -4.92005681720789 40 135.8 134.622663068021 1.17733693197917 41 143.8 143.446657320597 0.35334267940286 42 147.5 144.916040627877 2.58395937212268 43 136.2 136.166717044136 0.033282955863508 44 156.6 164.671176525905 -8.0711765259055 45 123.3 125.299862874271 -1.99986287427072 46 104.5 107.037119796671 -2.53711979667143 47 139.8 144.591318084746 -4.79131808474616 48 136.5 137.716569353931 -1.2165693539312 49 112.1 113.658825909679 -1.55882590967928 50 118.5 115.208433589861 3.291566410139 51 94.4 96.3685444622555 -1.96854446225553 52 102.3 102.425842410489 -0.125842410489466 53 111.4 113.354576332682 -1.95457633268204 54 99.2 97.5885066005316 1.61149339946844 55 87.8 92.0009658001225 -4.20096580012253 56 115.8 114.769595369728 1.03040463027231 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.856414727521685 0.287170544956629 0.143585272478315 22 0.900705038209417 0.198589923581166 0.0992949617905828 23 0.860096039398146 0.279807921203708 0.139903960601854 24 0.835143834636266 0.329712330727467 0.164856165363734 25 0.858744623367108 0.282510753265784 0.141255376632892 26 0.94678225993733 0.106435480125339 0.0532177400626696 27 0.956659776393093 0.0866804472138135 0.0433402236069068 28 0.95746143168005 0.085077136639899 0.0425385683199495 29 0.936089062099625 0.127821875800751 0.0639109379003754 30 0.95918263660974 0.081634726780518 0.040817363390259 31 0.92821979931457 0.143560401370859 0.0717802006854293 32 0.878505547338121 0.242988905323758 0.121494452661879 33 0.791774060662984 0.416451878674031 0.208225939337016 34 0.725468147052283 0.549063705895433 0.274531852947717 35 0.554495772308269 0.891008455383463 0.445504227691731 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 0 0 OK 10% type I error level 3 0.2 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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