WS 7 Multiple regression 2

*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: Thu, 19 Nov 2009 12:11:30 -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/19/t12586579245zlflrj8daehne3.htm/, Retrieved Thu, 19 Nov 2009 20:12:16 +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/19/t12586579245zlflrj8daehne3.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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108.01 102.9 101.21 97.4 119.93 111.4 94.76 87.4 95.26 96.8 117.96 114.1 115.86 110.3 111.44 103.9 108.16 101.6 108.77 94.6 109.45 95.9 124.83 104.7 115.31 102.8 109.49 98.1 124.24 113.9 92.85 80.9 98.42 95.7 120.88 113.2 111.72 105.9 116.1 108.8 109.37 102.3 111.65 99 114.29 100.7 133.68 115.5 114.27 100.7 126.49 109.9 131 114.6 104 85.4 108.88 100.5 128.48 114.8 132.44 116.5 128.04 112.9 116.35 102 120.93 106 118.59 105.3 133.1 118.8 121.05 106.1 127.62 109.3 135.44 117.2 114.88 92.5 114.34 104.2 128.85 112.5 138.9 122.4 129.44 113.3 114.96 100 127.98 110.7 127.03 112.8 128.75 109.8 137.91 117.3 128.37 109.1 135.9 115.9 122.19 96 113.08 99.8 136.2 116.8 138 115.7 115.24 99.4 110.95 94.3 99.23 91 102.39 93.2 112.67 103.1

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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y(omzet)[t] = -37.7533688137505 + 1.48903215087652`X(prod)`[t] -0.714477893125787M1[t] + 0.398360687926065M2[t] -3.58771567669893M3[t] + 11.7993653902309M4[t] -4.26042698337578M5[t] -5.93922538841844M6[t] -4.85054153031326M7[t] -2.50383254961587M8[t] + 0.748592440063247M9[t] + 2.17500536687042M10[t] + 0.847482927713403M11[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) -37.7533688137505 13.371138 -2.8235 0.006946 0.003473 `X(prod)` 1.48903215087652 0.119588 12.4514 0 0 M1 -0.714477893125787 3.06093 -0.2334 0.81645 0.408225 M2 0.398360687926065 3.088947 0.129 0.897937 0.448969 M3 -3.58771567669893 3.056891 -1.1736 0.246449 0.123225 M4 11.7993653902309 3.996747 2.9522 0.004911 0.002455 M5 -4.26042698337578 3.288474 -1.2956 0.201453 0.100726 M6 -5.93922538841844 3.050807 -1.9468 0.057552 0.028776 M7 -4.85054153031326 3.048646 -1.591 0.118303 0.059152 M8 -2.50383254961587 3.032442 -0.8257 0.413156 0.206578 M9 0.748592440063247 3.258669 0.2297 0.819304 0.409652 M10 2.17500536687042 3.248777 0.6695 0.506462 0.253231 M11 0.847482927713403 3.193335 0.2654 0.791869 0.395934 Multiple Linear Regression - Regression Statistics Multiple R 0.932939088464168 R-squared 0.870375342784353 Adjusted R-squared 0.837279685622912 F-TEST (value) 26.2987780704469 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 4.76704844747167 Sum Squared Residuals 1068.06329232548 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 108.01 114.753561618318 -6.74356161831793 2 101.21 107.676723369549 -6.46672336954877 3 119.93 124.537097117195 -4.60709711719513 4 94.76 104.187406563088 -9.4274065630884 5 95.26 102.124516407721 -6.86451640772103 6 117.96 126.205974212842 -8.24597421284223 7 115.86 121.636335897617 -5.77633589761661 8 111.44 114.453239112704 -3.01323911270428 9 108.16 114.280890155367 -6.12089015536737 10 108.77 105.284078026039 3.48592197396112 11 109.45 105.892297383021 3.55770261697864 12 124.83 118.148297383021 6.68170261697865 13 115.31 114.604658403230 0.705341596769839 14 109.49 108.719045875162 0.770954124837636 15 124.24 128.259677494386 -4.01967749438644 16 92.85 94.508697582391 -1.65869758239103 17 98.42 100.486581041757 -2.06658104175687 18 120.88 124.865845277053 -3.98584527705336 19 111.72 115.08459443376 -3.36459443375993 20 116.1 121.749496651999 -5.64949665199923 21 109.37 115.323212660981 -5.95321266098093 22 111.65 111.835819489896 -0.185819489895579 23 114.29 113.039651707229 1.25034829277134 24 133.68 134.229844612488 -0.549844612487789 25 114.27 111.477690886389 2.79230911361052 26 126.49 126.289625255505 0.200374744494649 27 131 129.302 1.69800000000001 28 104 101.209342261335 2.79065773866463 29 108.88 107.633935365964 1.24606463403582 30 128.48 127.248296718456 1.23170328154421 31 132.44 130.868335233051 1.57166476694894 32 128.04 127.854528470593 0.185471529407012 33 116.35 114.876503015718 1.47349698428201 34 120.93 122.259044546031 -1.32904454603124 35 118.59 119.889199601261 -1.29919960126066 36 133.1 139.143650710380 -6.04365071038032 37 121.05 119.518464501123 1.53153549887731 38 127.62 125.396205964979 2.22379403502059 39 135.44 133.173483592279 2.26651640772104 40 114.88 111.781470532559 3.09852946744132 41 114.34 113.143354324207 1.19664567579269 42 128.85 123.823522771440 5.02647722856021 43 138.9 139.653624923223 -0.753624923222548 44 129.44 128.450141330944 0.989858669056423 45 114.96 111.898438713965 3.06156128603505 46 127.98 129.257495655151 -1.2774956551509 47 127.03 131.056940732835 -4.02694073283458 48 128.75 125.742361352492 3.00763864750839 49 137.91 136.195624590940 1.71437540906025 50 128.37 125.098399534804 3.27160046519589 51 135.9 131.237741796139 4.66225820386052 52 122.19 116.993083060627 5.19691693937349 53 113.08 106.591612860351 6.48838713964939 54 136.2 130.226361020209 5.97363897979116 55 138 129.677109512350 8.32289048765015 56 115.24 107.75259443376 7.48740556624007 57 110.95 103.410955453969 7.53904454603125 58 99.23 99.9235622828834 -0.693562282883395 59 102.39 101.871910575655 0.518089424345254 60 112.67 115.765845941619 -3.09584594161890 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.78939471729086 0.421210565418281 0.210605282709140 17 0.731869941789105 0.53626011642179 0.268130058210895 18 0.715313485861854 0.569373028276291 0.284686514138146 19 0.714348919419678 0.571302161160644 0.285651080580322 20 0.739158298742416 0.521683402515169 0.260841701257584 21 0.828318734219558 0.343362531560885 0.171681265780442 22 0.747452500914341 0.505094998171319 0.252547499085659 23 0.677872149063013 0.644255701873975 0.322127850936987 24 0.601753640537746 0.796492718924509 0.398246359462254 25 0.607248502458548 0.785502995082905 0.392751497541452 26 0.84947542857768 0.301049142844641 0.150524571422321 27 0.882023117800227 0.235953764399546 0.117976882199773 28 0.938706196651391 0.122587606697217 0.0612938033486086 29 0.947523412823658 0.104953174352684 0.0524765871763418 30 0.965881391775526 0.068237216448948 0.034118608224474 31 0.967768539294364 0.0644629214112721 0.0322314607056361 32 0.958146282486213 0.0837074350275734 0.0418537175137867 33 0.95924938217724 0.0815012356455206 0.0407506178227603 34 0.935288792448867 0.129422415102266 0.064711207551133 35 0.902359043703227 0.195281912593545 0.0976409562967727 36 0.908238260180532 0.183523479638937 0.0917617398194683 37 0.870573853729185 0.258852292541630 0.129426146270815 38 0.817954540849281 0.364090918301437 0.182045459150719 39 0.758210497196583 0.483579005606834 0.241789502803417 40 0.70478511739159 0.59042976521682 0.29521488260841 41 0.654065236832393 0.691869526335213 0.345934763167607 42 0.584096099323823 0.831807801352354 0.415903900676177 43 0.672066966371459 0.655866067257082 0.327933033628541 44 0.61127830511023 0.777443389779538 0.388721694889769 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 4 0.137931034482759 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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