Paper, Multiple Regression 4

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Title produced by software: Multiple Regression

Date of computation: Sun, 27 Dec 2009 04:45:56 -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/t1261914499yfttb38cqyxkm7s.htm/, Retrieved Sun, 27 Dec 2009 12:48:31 +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/t1261914499yfttb38cqyxkm7s.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 97 93,8 95,1 96,9 98,6 112,7 107,6 97 95,1 96,9 102,9 101 112,7 97 95,1 97,4 95,4 102,9 112,7 97 111,4 96,5 97,4 102,9 112,7 87,4 89,2 111,4 97,4 102,9 96,8 87,1 87,4 111,4 97,4 114,1 110,5 96,8 87,4 111,4 110,3 110,8 114,1 96,8 87,4 103,9 104,2 110,3 114,1 96,8 101,6 88,9 103,9 110,3 114,1 94,6 89,8 101,6 103,9 110,3 95,9 90 94,6 101,6 103,9 104,7 93,9 95,9 94,6 101,6 102,8 91,3 104,7 95,9 94,6 98,1 87,8 102,8 104,7 95,9 113,9 99,7 98,1 102,8 104,7 80,9 73,5 113,9 98,1 102,8 95,7 79,2 80,9 113,9 98,1 113,2 96,9 95,7 80,9 113,9 105,9 95,2 113,2 95,7 80,9 108,8 95,6 105,9 113,2 95,7 102,3 89,7 108,8 105,9 113,2 99 92,8 102,3 108,8 105,9 100,7 88 99 102,3 108,8 115,5 101,1 100,7 99 102,3 100,7 92,7 115,5 100,7 99 109,9 95,8 100,7 115,5 100,7 114,6 103,8 109,9 100,7 115,5 85,4 81,8 114,6 109,9 100,7 100,5 87,1 85,4 114,6 109,9 114,8 105,9 100,5 85,4 114,6 116,5 108,1 114,8 100,5 85,4 112,9 102,6 116,5 114,8 100,5 102 93,7 112,9 116,5 114,8 106 103,5 1 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] = + 38.1816714317873 + 0.311752284546608`X(t)`[t] -0.126216251824039`Y(t-1)`[t] + 0.0510552001571137`Y(t-2)`[t] + 0.363728019129759`Y(t-3)`[t] -1.66302616229589M1[t] + 2.51624326560834M2[t] + 12.0154766461194M3[t] + 8.22541372349253M4[t] + 5.88813520404393M5[t] + 10.2699498878386M6[t] -6.51336304862936M7[t] + 0.417995413824675M8[t] + 8.82630473985045M9[t] + 18.9912785329026M10[t] + 10.5108024928395M11[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) 38.1816714317873 16.483384 2.3164 0.025257 0.012629 `X(t)` 0.311752284546608 0.087036 3.5819 0.000848 0.000424 `Y(t-1)` -0.126216251824039 0.116877 -1.0799 0.286066 0.143033 `Y(t-2)` 0.0510552001571137 0.11773 0.4337 0.666652 0.333326 `Y(t-3)` 0.363728019129759 0.125781 2.8918 0.005933 0.002966 M1 -1.66302616229589 2.766841 -0.6011 0.550887 0.275443 M2 2.51624326560834 3.250137 0.7742 0.442955 0.221478 M3 12.0154766461194 4.179288 2.875 0.006203 0.003102 M4 8.22541372349253 3.748893 2.1941 0.033554 0.016777 M5 5.88813520404393 3.071497 1.917 0.061743 0.030871 M6 10.2699498878386 3.153411 3.2568 0.002173 0.001087 M7 -6.51336304862936 2.852894 -2.2831 0.027313 0.013656 M8 0.417995413824675 3.232266 0.1293 0.897694 0.448847 M9 8.82630473985045 4.748572 1.8587 0.069762 0.034881 M10 18.9912785329026 5.509525 3.447 0.001259 0.00063 M11 10.5108024928395 3.411471 3.081 0.003551 0.001776 Multiple Linear Regression - Regression Statistics Multiple R 0.940717446934413 R-squared 0.8849493149668 Adjusted R-squared 0.845727490523663 F-TEST (value) 22.5626759471577 F-TEST (DF numerator) 15 F-TEST (DF denominator) 44 p-value 7.7715611723761e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.53485642588882 Sum Squared Residuals 549.78923787249 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 95.1 99.1931172304992 -4.09311723049916 2 97 98.7479450208199 -1.74794502081988 3 112.7 111.599312056805 1.10068794319494 4 102.9 103.212383348398 -0.312383348398138 5 97.4 101.858861182177 -4.45886118217734 6 111.4 112.487981702803 -1.08798170280302 7 87.4 87.8165113752725 -0.416511375272481 8 96.8 95.8366487809415 0.963351219058486 9 114.1 114.220396262258 -0.120396262257859 10 110.3 114.045801006481 -3.74580100648082 11 103.9 108.289679987879 -4.38967998787919 12 101.6 99.9153365234982 1.68466347650176 13 94.6 97.114265042791 -2.51426504279097 14 95.9 99.794112407581 -3.89411240758097 15 104.7 109.151137725354 -4.45113772535427 16 102.8 100.960091473151 1.83990852684935 17 98.1 98.693623022506 -0.593623022505887 18 113.9 110.482307964022 3.41769203597844 19 80.9 82.6058257165276 -1.70582571652763 20 95.7 94.5764589836632 1.12354101633684 21 113.2 110.696864316234 2.50313568376643 22 105.9 106.875667149679 -0.97566714967909 23 108.8 105.71791134762 3.08208865238 24 102.3 98.9942806192897 3.30571938071035 25 99 96.610937716753 2.38906228324711 26 100.7 100.433262264308 0.266737735692224 27 115.5 111.269168659417 4.2308313405834 28 100.7 101.878877396741 -1.17887739674137 29 109.9 103.749986081229 6.1500139187711 30 114.6 114.092187245410 0.507812754589537 31 85.4 84.9436408236691 0.45635917633089 32 100.5 100.799058164214 -0.299058164214328 33 114.8 113.381154882496 1.41884511750449 34 116.5 112.57716666425 3.92283333575005 35 112.9 108.389867882186 4.51013211781433 36 102 100.846953077271 1.15304692272946 37 106 104.049395360368 1.95060463963158 38 105.3 104.953795605212 0.346204394788341 39 118.8 114.740219767856 4.05978023214448 40 106.1 107.267310980455 -1.16731098045519 41 109.3 106.874088762538 2.42591123746191 42 117.2 116.610349622576 0.589650377424014 43 92.5 88.2636143171394 4.23638568286061 44 104.2 101.687943192474 2.51205680752606 45 112.5 115.219946832149 -2.71994683214859 46 122.4 119.442215091316 2.95778490868354 47 113.3 109.590588961300 3.7094110387002 48 100 99.9181903004682 0.0818096995318184 49 110.7 108.432284649589 2.26771535041144 50 112.8 107.770884702080 5.02911529792028 51 109.8 114.740161790569 -4.94016179056855 52 117.3 116.481336801255 0.818663198745358 53 109.1 112.623440951550 -3.52344095154978 54 115.9 119.327173465189 -3.42717346518897 55 96 98.5704077673914 -2.57040776739138 56 99.8 104.099890878707 -4.29989087870706 57 116.8 117.881637706864 -1.08163770686447 58 115.7 117.859150088274 -2.15915008827368 59 99.4 106.311951821015 -6.91195182101534 60 94.3 100.525239479473 -6.2252394794734 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.155501098238017 0.311002196476035 0.844498901761983 20 0.0626627005496948 0.125325401099390 0.937337299450305 21 0.0457355802243054 0.0914711604486108 0.954264419775695 22 0.0235733745769818 0.0471467491539637 0.976426625423018 23 0.0673127457144268 0.134625491428854 0.932687254285573 24 0.0349122358434295 0.0698244716868589 0.96508776415657 25 0.0414541815679572 0.0829083631359144 0.958545818432043 26 0.125923256627765 0.25184651325553 0.874076743372235 27 0.169073931160205 0.33814786232041 0.830926068839795 28 0.140038198954866 0.280076397909732 0.859961801045134 29 0.276815389891599 0.553630779783199 0.7231846101084 30 0.201227829813307 0.402455659626615 0.798772170186693 31 0.151435938416461 0.302871876832923 0.848564061583539 32 0.0957952426143486 0.191590485228697 0.904204757385651 33 0.056851437055385 0.11370287411077 0.943148562944615 34 0.0502544322241367 0.100508864448273 0.949745567775863 35 0.0553125984967839 0.110625196993568 0.944687401503216 36 0.044101648158915 0.08820329631783 0.955898351841085 37 0.0246312181559398 0.0492624363118796 0.97536878184406 38 0.0177972795947675 0.0355945591895349 0.982202720405233 39 0.0337314575735064 0.0674629151470127 0.966268542426494 40 0.0230189749113778 0.0460379498227556 0.976981025088622 41 0.0146449221843319 0.0292898443686638 0.985355077815668 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 5 0.217391304347826 NOK 10% type I error level 10 0.434782608695652 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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