Paper, Multiple Regression 4

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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Sun, 27 Dec 2009 04:57:17 -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/27/t1261915139shfleu7xtvohypv.htm/, Retrieved Sun, 27 Dec 2009 12:59:12 +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/27/t1261915139shfleu7xtvohypv.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}, }

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Dataseries X:

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95,1 93,8 96,9 98,6 111,7 109,8 97 93,8 95,1 96,9 98,6 111,7 112,7 107,6 97 95,1 96,9 98,6 102,9 101 112,7 97 95,1 96,9 97,4 95,4 102,9 112,7 97 95,1 111,4 96,5 97,4 102,9 112,7 97 87,4 89,2 111,4 97,4 102,9 112,7 96,8 87,1 87,4 111,4 97,4 102,9 114,1 110,5 96,8 87,4 111,4 97,4 110,3 110,8 114,1 96,8 87,4 111,4 103,9 104,2 110,3 114,1 96,8 87,4 101,6 88,9 103,9 110,3 114,1 96,8 94,6 89,8 101,6 103,9 110,3 114,1 95,9 90 94,6 101,6 103,9 110,3 104,7 93,9 95,9 94,6 101,6 103,9 102,8 91,3 104,7 95,9 94,6 101,6 98,1 87,8 102,8 104,7 95,9 94,6 113,9 99,7 98,1 102,8 104,7 95,9 80,9 73,5 113,9 98,1 102,8 104,7 95,7 79,2 80,9 113,9 98,1 102,8 113,2 96,9 95,7 80,9 113,9 98,1 105,9 95,2 113,2 95,7 80,9 113,9 108,8 95,6 105,9 113,2 95,7 80,9 102,3 89,7 108,8 105,9 113,2 95,7 99 92,8 102,3 108,8 105,9 113,2 100,7 88 99 102,3 108,8 105,9 115,5 101,1 100,7 99 102,3 108,8 100,7 92,7 115,5 100,7 99 102,3 109,9 95,8 100,7 115,5 100,7 99 114,6 103,8 109,9 100,7 115,5 100,7 85,4 81,8 114,6 109,9 100 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 Y(t)[t] = + 39.2600545642095 + 0.32005110995124`X(t)`[t] -0.0508039121660601`Y(t-1)`[t] + 0.088033584837062`Y(t-2)`[t] + 0.354570106951256`Y(t-3)`[t] -0.131980785927691`Y(t-4)`[t] + 0.823294556124404M1[t] + 5.23945758294206M2[t] + 13.7044910853491M3[t] + 8.24670612903364M4[t] + 6.116169736949M5[t] + 11.0269863471651M6[t] -5.04401421530769M7[t] + 2.6309225600948M8[t] + 10.9624252421365M9[t] + 20.6444943432124M10[t] + 8.34460830008787M11[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) 39.2600545642095 16.557228 2.3712 0.022282 0.011141 `X(t)` 0.32005110995124 0.087676 3.6504 0.000705 0.000352 `Y(t-1)` -0.0508039121660601 0.143348 -0.3544 0.724764 0.362382 `Y(t-2)` 0.088033584837062 0.124729 0.7058 0.484118 0.242059 `Y(t-3)` 0.354570106951256 0.126421 2.8047 0.007532 0.003766 `Y(t-4)` -0.131980785927691 0.144702 -0.9121 0.366807 0.183404 M1 0.823294556124404 3.887884 0.2118 0.833296 0.416648 M2 5.23945758294206 4.417946 1.1859 0.242155 0.121077 M3 13.7044910853491 4.578496 2.9932 0.004561 0.002281 M4 8.24670612903364 3.756148 2.1955 0.033571 0.016785 M5 6.116169736949 3.08752 1.9809 0.054018 0.027009 M6 11.0269863471651 3.266656 3.3756 0.001571 0.000786 M7 -5.04401421530769 3.281073 -1.5373 0.131545 0.065773 M8 2.6309225600948 4.046498 0.6502 0.519039 0.25952 M9 10.9624252421365 5.302869 2.0673 0.044762 0.022381 M10 20.6444943432124 5.810047 3.5532 0.000939 0.000469 M11 8.34460830008787 4.162128 2.0049 0.051296 0.025648 Multiple Linear Regression - Regression Statistics Multiple R 0.941877335833354 R-squared 0.887132915756537 Adjusted R-squared 0.845135861154318 F-TEST (value) 21.1236936532608 F-TEST (DF numerator) 16 F-TEST (DF denominator) 43 p-value 2.66453525910038e-15 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.54162796179663 Sum Squared Residuals 539.35453065053 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 95.1 98.9753462613978 -3.87534626139785 2 97 98.4376673415675 -1.43766734156749 3 112.7 112.190597389315 0.509402610684976 4 102.9 103.576258641070 -0.676258641069492 5 97.4 102.444690272304 -5.04469027230441 6 111.4 112.440242674849 -1.04024267484910 7 87.4 87.2905441356165 0.109455864383466 8 96.8 96.0884137736852 0.711586226314793 9 114.1 115.008625438055 -0.908625438055306 10 110.3 114.377904319294 -4.07790431929433 11 103.9 108.182214702010 -4.28221470201022 12 101.6 99.8248852976868 1.77511470231323 13 94.6 96.8590259048282 -2.25902590482819 14 95.9 99.7246275957104 -3.82462759571044 15 104.7 108.784746031201 -4.08474603120131 16 102.8 99.9837624812144 2.81623751878558 17 98.1 98.9890768245126 -0.88907682451258 18 113.9 110.728658138604 3.17134186139646 19 80.9 83.2207447150793 -2.32074471507925 20 95.7 94.371716549701 1.32828345029891 21 113.2 110.933635061889 2.26636493811105 22 105.9 106.699335921681 -0.79933592168127 23 108.8 106.041930134490 2.75806986550953 24 102.3 99.2707050110153 3.02929498898469 25 99 96.7736552986167 2.22634470138333 26 100.7 101.240720653806 -0.540720653806237 27 115.5 110.834096241556 4.66590375844358 28 100.7 101.773434911407 -1.07343491140675 29 109.9 103.728157691196 6.17184230880433 30 114.6 114.452360380307 0.14763961969306 31 85.4 84.710412777619 0.68958722238106 32 100.5 101.194213135428 -0.694213135427528 33 114.8 112.657213205739 2.14278679426107 34 116.5 112.672449118936 3.82755088106426 35 112.9 108.992643147620 3.90735685238037 36 102 101.209573808881 0.790426191118627 37 106 104.121654922776 1.87834507722428 38 105.3 104.946068286245 0.353931713755042 39 118.8 114.473764626468 4.32623537353181 40 106.1 107.636817242473 -1.53681724247337 41 109.3 106.467806378636 2.83219362136399 42 117.2 116.513352264247 0.68664773575268 43 92.5 87.7649255437922 4.73507445620783 44 104.2 102.057261031152 2.14273896884798 45 112.5 115.119511484541 -2.61951148454149 46 122.4 119.193943638162 3.20605636183815 47 113.3 109.601386189236 3.69861381076446 48 100 100.052754019687 -0.0527540196869052 49 110.7 108.670317612382 2.02968238761843 50 112.8 107.350916122671 5.44908387732913 51 109.8 115.216795711459 -5.41679571145906 52 117.3 116.829726723836 0.470273276164035 53 109.1 112.170268833351 -3.07026883335133 54 115.9 118.865386541993 -2.9653865419931 55 96 99.213372827893 -3.21337282789311 56 99.8 103.288395510034 -3.48839551003415 57 116.8 117.681014809775 -0.881014809775334 58 115.7 117.856367001927 -2.15636700192681 59 99.4 105.481825826644 -6.08182582664414 60 94.3 99.8420818627296 -5.54208186272964 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 20 0.154895454273429 0.309790908546858 0.84510454572657 21 0.106399181785162 0.212798363570324 0.893600818214838 22 0.057564051545779 0.115128103091558 0.94243594845422 23 0.0908779300220735 0.181755860044147 0.909122069977926 24 0.0460741590852886 0.0921483181705771 0.953925840914711 25 0.0640019301856797 0.128003860371359 0.93599806981432 26 0.153815568144891 0.307631136289782 0.846184431855109 27 0.298700728360825 0.59740145672165 0.701299271639175 28 0.248419473857716 0.496838947715431 0.751580526142284 29 0.39153023702845 0.7830604740569 0.60846976297155 30 0.306637452555134 0.613274905110268 0.693362547444866 31 0.247604834394500 0.495209668788999 0.7523951656055 32 0.180453031276551 0.360906062553102 0.819546968723449 33 0.114883462241625 0.22976692448325 0.885116537758375 34 0.111632163087201 0.223264326174402 0.888367836912799 35 0.0902517352497623 0.180503470499525 0.909748264750238 36 0.0586864600623718 0.117372920124744 0.941313539937628 37 0.0371924455810932 0.0743848911621865 0.962807554418907 38 0.0282262658878092 0.0564525317756184 0.97177373411219 39 0.064479969193718 0.128959938387436 0.935520030806282 40 0.217541843140563 0.435083686281126 0.782458156859437 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 0.142857142857143 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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