Workshop 7

*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, 27 Nov 2009 15:03:48 -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/27/t1259359477z2nq69dq5osaycc.htm/, Retrieved Fri, 27 Nov 2009 23:04:49 +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/27/t1259359477z2nq69dq5osaycc.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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126.51 0 131.02 0 136.51 0 138.04 0 132.92 0 129.61 0 122.96 0 124.04 0 121.29 0 124.56 0 118.53 0 113.14 0 114.15 0 122.17 0 129.23 0 131.19 0 129.12 0 128.28 0 126.83 0 138.13 0 140.52 0 146.83 0 135.14 0 131.84 0 125.7 0 128.98 0 133.25 0 136.76 0 133.24 0 128.54 0 121.08 0 120.23 0 119.08 0 125.75 0 126.89 0 126.6 0 121.89 0 123.44 0 126.46 0 129.49 0 127.78 0 125.29 0 119.02 0 119.96 0 122.86 0 131.89 0 132.73 0 135.01 0 136.71 1 142.73 1 144.43 1 144.93 1 138.75 1 130.22 1 122.19 1 128.4 1 140.43 1 153.5 1 149.33 1 142.97 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 14 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 128.755416666667 + 12.9404166666667X[t] -5.35740277777782M1[t] -0.641638888888884M2[t] + 3.70612499999999M3[t] + 5.85188888888888M4[t] + 2.17165277777778M5[t] -1.76258333333333M6[t] -7.69481944444444M7[t] -3.91905555555555M8[t] -1.19529166666666M9[t] + 6.51447222222222M10[t] + 2.57223611111111M11[t] -0.0397638888888885t + 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) 128.755416666667 3.691113 34.8825 0 0 X 12.9404166666667 3.034076 4.265 9.8e-05 4.9e-05 M1 -5.35740277777782 4.277103 -1.2526 0.21669 0.108345 M2 -0.641638888888884 4.264529 -0.1505 0.88106 0.44053 M3 3.70612499999999 4.253121 0.8714 0.388067 0.194033 M4 5.85188888888888 4.242888 1.3792 0.174495 0.087248 M5 2.17165277777778 4.233838 0.5129 0.610456 0.305228 M6 -1.76258333333333 4.225979 -0.4171 0.678558 0.339279 M7 -7.69481944444444 4.219318 -1.8237 0.074696 0.037348 M8 -3.91905555555555 4.21386 -0.93 0.357206 0.178603 M9 -1.19529166666666 4.20961 -0.2839 0.777728 0.388864 M10 6.51447222222222 4.206571 1.5486 0.128321 0.064161 M11 2.57223611111111 4.204747 0.6117 0.543718 0.271859 t -0.0397638888888885 0.071514 -0.556 0.580885 0.290442 Multiple Linear Regression - Regression Statistics Multiple R 0.734406514839355 R-squared 0.539352929038487 Adjusted R-squared 0.40917006115806 F-TEST (value) 4.14304076888121 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0.000163786037073121 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 6.64732770487073 Sum Squared Residuals 2032.60041833333 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 126.51 123.358250000000 3.1517499999998 2 131.02 128.03425 2.98575 3 136.51 132.34225 4.16775000000001 4 138.04 134.44825 3.59174999999999 5 132.92 130.72825 2.19175000000001 6 129.61 126.75425 2.85575000000003 7 122.96 120.78225 2.17775000000002 8 124.04 124.51825 -0.478249999999984 9 121.29 127.20225 -5.91224999999999 10 124.56 134.87225 -10.3122500000000 11 118.53 130.89025 -12.3602500000000 12 113.14 128.27825 -15.13825 13 114.15 122.881083333333 -8.73108333333327 14 122.17 127.557083333333 -5.38708333333332 15 129.23 131.865083333333 -2.63508333333333 16 131.19 133.971083333333 -2.78108333333333 17 129.12 130.251083333333 -1.13108333333333 18 128.28 126.277083333333 2.00291666666667 19 126.83 120.305083333333 6.52491666666667 20 138.13 124.041083333333 14.0889166666667 21 140.52 126.725083333333 13.7949166666667 22 146.83 134.395083333333 12.4349166666667 23 135.14 130.413083333333 4.72691666666666 24 131.84 127.801083333333 4.03891666666667 25 125.7 122.403916666667 3.29608333333338 26 128.98 127.079916666667 1.90008333333333 27 133.25 131.387916666667 1.86208333333334 28 136.76 133.493916666667 3.26608333333333 29 133.24 129.773916666667 3.46608333333334 30 128.54 125.799916666667 2.74008333333332 31 121.08 119.827916666667 1.25208333333333 32 120.23 123.563916666667 -3.33391666666666 33 119.08 126.247916666667 -7.16791666666667 34 125.75 133.917916666667 -8.16791666666666 35 126.89 129.935916666667 -3.04591666666666 36 126.6 127.323916666667 -0.723916666666674 37 121.89 121.92675 -0.0367499999999588 38 123.44 126.60275 -3.16275 39 126.46 130.91075 -4.45075000000001 40 129.49 133.01675 -3.52674999999999 41 127.78 129.29675 -1.51675000000001 42 125.29 125.32275 -0.0327500000000041 43 119.02 119.35075 -0.330750000000012 44 119.96 123.08675 -3.12675000000001 45 122.86 125.77075 -2.91075000000001 46 131.89 133.44075 -1.55075000000002 47 132.73 129.45875 3.27124999999998 48 135.01 126.84675 8.16324999999999 49 136.71 134.39 2.32000000000005 50 142.73 139.066 3.66399999999999 51 144.43 143.374 1.05600000000000 52 144.93 145.48 -0.549999999999996 53 138.75 141.76 -3.01000000000001 54 130.22 137.786 -7.56600000000001 55 122.19 131.814 -9.62400000000001 56 128.4 135.55 -7.15 57 140.43 138.234 2.196 58 153.5 145.904 7.596 59 149.33 141.922 7.408 60 142.97 139.31 3.65999999999999 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.116948266371262 0.233896532742524 0.883051733628738 18 0.125961692498477 0.251923384996954 0.874038307501523 19 0.256280376398446 0.512560752796891 0.743719623601554 20 0.78092591987831 0.438148160243379 0.219074080121689 21 0.972091337643814 0.0558173247123718 0.0279086623561859 22 0.996966342313464 0.00606731537307173 0.00303365768653586 23 0.996902287756044 0.00619542448791248 0.00309771224395624 24 0.9967561021174 0.00648779576519867 0.00324389788259934 25 0.993332425268704 0.0133351494625916 0.00666757473129582 26 0.987584541974307 0.024830916051385 0.0124154580256925 27 0.9805965092332 0.0388069815336011 0.0194034907668006 28 0.973458804843195 0.0530823903136104 0.0265411951568052 29 0.968804126751367 0.0623917464972652 0.0311958732486326 30 0.974786252718018 0.0504274945639638 0.0252137472819819 31 0.988738562956903 0.0225228740861935 0.0112614370430967 32 0.995459934898556 0.0090801302028882 0.0045400651014441 33 0.993981182185392 0.0120376356292158 0.00601881781460791 34 0.991016089062122 0.0179678218757567 0.00898391093787835 35 0.981335363768336 0.037329272463328 0.018664636231664 36 0.963617040720655 0.0727659185586891 0.0363829592793445 37 0.93293827600066 0.134123447998680 0.0670617239993402 38 0.91324912848744 0.173501743025119 0.0867508715125594 39 0.884499205513813 0.231001588972374 0.115500794486187 40 0.82247787696611 0.355044246067781 0.177522123033891 41 0.710815088744368 0.578369822511263 0.289184911255632 42 0.644483397320951 0.711033205358098 0.355516602679049 43 0.685509707813074 0.628980584373852 0.314490292186926 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 4 0.148148148148148 NOK 5% type I error level 11 0.407407407407407 NOK 10% type I error level 16 0.592592592592593 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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