WS 7 Multiple Regression (model 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: Sat, 21 Nov 2009 04:22:15 -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/21/t12588026127vxvmqbomhb8blb.htm/, Retrieved Sat, 21 Nov 2009 12:23:44 +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/21/t12588026127vxvmqbomhb8blb.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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109.9 104 112.9 113.6 83.4 99 109.9 104 112.9 113.6 106.3 99 109.9 104 112.9 128.9 106.3 99 109.9 104 111.1 128.9 106.3 99 109.9 102.9 111.1 128.9 106.3 99 130 102.9 111.1 128.9 106.3 87 130 102.9 111.1 128.9 87.5 87 130 102.9 111.1 117.6 87.5 87 130 102.9 103.4 117.6 87.5 87 130 110.8 103.4 117.6 87.5 87 112.6 110.8 103.4 117.6 87.5 102.5 112.6 110.8 103.4 117.6 112.4 102.5 112.6 110.8 103.4 135.6 112.4 102.5 112.6 110.8 105.1 135.6 112.4 102.5 112.6 127.7 105.1 135.6 112.4 102.5 137 127.7 105.1 135.6 112.4 91 137 127.7 105.1 135.6 90.5 91 137 127.7 105.1 122.4 90.5 91 137 127.7 123.3 122.4 90.5 91 137 124.3 123.3 122.4 90.5 91 120 124.3 123.3 122.4 90.5 118.1 120 124.3 123.3 122.4 119 118.1 120 124.3 123.3 142.7 119 118.1 120 124.3 123.6 142.7 119 118.1 120 129.6 123.6 142.7 119 118.1 151.6 129.6 123.6 142.7 119 110.4 151.6 129.6 123.6 142.7 99.2 110.4 151.6 129.6 123.6 130.5 99.2 110.4 151.6 129.6 136.2 130.5 99.2 110.4 151.6 129.7 136.2 130.5 99.2 110.4 128 129.7 136.2 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Yt[t] = + 3.44967554333794 + 0.345771119866665`Yt-1`[t] + 0.572393185503651`Yt-2`[t] + 0.329280609238653`Yt-3`[t] -0.284562624705873`Yt-4`[t] -8.40904074349483M1[t] -6.16791229568818M2[t] + 6.13035172360855M3[t] + 24.5480317538756M4[t] + 2.83995158796344M5[t] -7.88662923258578M6[t] + 18.9911058720556M7[t] -18.0391515918807M8[t] -29.8249609435041M9[t] + 22.5604543650976M10[t] + 32.1620988449328M11[t] -0.0850084829355323t + 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) 3.44967554333794 11.4355 0.3017 0.764511 0.382256 `Yt-1` 0.345771119866665 0.160954 2.1483 0.037961 0.018981 `Yt-2` 0.572393185503651 0.158911 3.602 0.000882 0.000441 `Yt-3` 0.329280609238653 0.174203 1.8902 0.066177 0.033089 `Yt-4` -0.284562624705873 0.184055 -1.5461 0.130164 0.065082 M1 -8.40904074349483 8.040222 -1.0459 0.302059 0.15103 M2 -6.16791229568818 9.297597 -0.6634 0.510985 0.255492 M3 6.13035172360855 9.17902 0.6679 0.508153 0.254076 M4 24.5480317538756 8.843882 2.7757 0.008416 0.004208 M5 2.83995158796344 6.663897 0.4262 0.672327 0.336164 M6 -7.88662923258578 6.466877 -1.2195 0.229963 0.114981 M7 18.9911058720556 9.53939 1.9908 0.053542 0.026771 M8 -18.0391515918807 8.38865 -2.1504 0.037779 0.01889 M9 -29.8249609435041 9.960682 -2.9943 0.004759 0.002379 M10 22.5604543650976 14.949974 1.5091 0.139342 0.069671 M11 32.1620988449328 12.126634 2.6522 0.011504 0.005752 t -0.0850084829355323 0.08498 -1.0003 0.323314 0.161657 Multiple Linear Regression - Regression Statistics Multiple R 0.937994052362038 R-squared 0.879832842266557 Adjusted R-squared 0.830533495504119 F-TEST (value) 17.8467444306364 F-TEST (DF numerator) 16 F-TEST (DF denominator) 39 p-value 3.72590847064203e-13 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 7.32886552923144 Sum Squared Residuals 2094.77852787672 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 109.9 109.212767735444 0.68723226455576 2 99 99.4903502639618 -0.490350263961782 3 106.3 108.580416803318 -2.28041680331797 4 128.9 127.673494758077 1.22650524192338 5 111.1 112.605225545926 -1.5052255459263 6 102.9 114.080477357934 -11.1804773579336 7 130 133.213716703208 -3.21371670320849 8 87 88.4829138207926 -1.48291382079260 9 87.5 79.6209068831236 7.87909311687642 10 117.6 118.738210325022 -1.13821032502169 11 103.4 117.078040295869 -13.6780402958686 12 110.8 109.550851116525 1.24914888347462 13 112.6 105.256589968687 7.34341003131297 14 102.5 99.0296878672095 3.47031213279049 15 112.4 115.258428606013 -2.85842860601337 16 135.6 129.720004740244 5.87999525975588 17 105.1 117.777551731008 -12.6775517310083 18 127.7 115.833425716267 11.8665742837331 19 137 137.804727638847 -0.80472763884663 20 91 100.196307624162 -9.19630762416215 21 90.5 93.8641767232433 -3.36417672324327 22 122.4 116.292805803375 6.10719419662516 23 123.3 118.760003496527 4.53999650347329 24 124.3 118.008673225956 6.29132677404438 25 120 121.021881733411 -1.02188173341119 26 118.1 113.482383888557 4.61761611144324 27 119 122.650557846509 -3.65055784650895 28 142.7 138.506407104831 4.19359289516854 29 123.6 126.021233992459 -2.42123399245884 30 129.6 123.008156331213 6.59184366878675 31 151.6 148.49064390572 3.10935609427987 32 110.4 109.383307866949 1.01669213305060 33 99.2 103.270199762278 -4.07019976227828 34 130.5 133.652168457702 -3.15216845770247 35 136.2 127.753897984626 8.44610201537378 36 129.7 123.429630060671 6.27036993932885 37 128 129.444294178354 -1.44429417835389 38 121.6 120.262136853044 1.33786314695553 39 135.8 125.527057886028 10.2729421139719 40 143.8 146.396242973125 -2.59624297312532 41 147.5 133.873667080235 13.6263329197646 42 136.2 135.417561853593 0.782438146406968 43 156.6 159.014385210255 -2.41438521025487 44 123.3 121.426644369008 1.87335563099220 45 104.5 104.944716631355 -0.444716631354871 46 139.8 141.616815413901 -1.816815413901 47 136.5 135.808058222978 0.691941777021569 48 112.1 125.910845596848 -13.8108455968478 49 118.5 124.064466384104 -5.56446638410365 50 94.4 103.335441127227 -8.93544112722747 51 102.3 103.783538858132 -1.48353885813164 52 111.4 120.103850423722 -8.70385042372248 53 99.2 96.2223216503712 2.97767834962882 54 87.8 95.8603787409932 -8.06037874099323 55 115.8 112.47652654197 3.32347345803011 56 79.7 71.910826319088 7.78917368091195 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 20 0.710064083303298 0.579871833393404 0.289935916696702 21 0.64395612483201 0.71208775033598 0.35604387516799 22 0.587317560642499 0.825364878715003 0.412682439357501 23 0.646064363175146 0.707871273649709 0.353935636824854 24 0.632851767946538 0.734296464106925 0.367148232053462 25 0.504429895369259 0.991140209261482 0.495570104630741 26 0.406311409239684 0.812622818479367 0.593688590760316 27 0.390006227417301 0.780012454834602 0.609993772582699 28 0.303784700402278 0.607569400804557 0.696215299597722 29 0.421941090899175 0.843882181798349 0.578058909100825 30 0.51025634677933 0.97948730644134 0.48974365322067 31 0.466325422073429 0.932650844146858 0.533674577926571 32 0.412297689936485 0.82459537987297 0.587702310063515 33 0.562846500250611 0.874306999498777 0.437153499749389 34 0.67166805795861 0.656663884082779 0.328331942041389 35 0.545693584745583 0.908612830508835 0.454306415254417 36 0.620744556270439 0.758510887459122 0.379255443729561 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 0 0 OK

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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