JJ Workshop 7, Multiple Regression (MD, LT & Wegwerken autocorrelatie)

*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, 20 Nov 2009 12:35:59 -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/20/t1258745997vs3mo909xxh400d.htm/, Retrieved Fri, 20 Nov 2009 20:40:09 +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/20/t1258745997vs3mo909xxh400d.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:

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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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation TIA[t] = + 49.3851818122445 + 0.328226028431917IAidM[t] -0.0569153192027402`TIA(t-1)`[t] + 0.0550408037582717`TIA(t-2)`[t] + 0.315315051430443`TIA(t-3)`[t] -0.159716545463239`TIA(t-4)`[t] + 1.07935611922313M1[t] + 5.04024362561899M2[t] + 12.9910144316273M3[t] + 7.48173555248839M4[t] + 5.69904974515526M5[t] + 10.7442308754480M6[t] -5.15002616864018M7[t] + 2.4269950608123M8[t] + 10.1458174187584M9[t] + 19.4896364076166M10[t] + 7.43512929557958M11[t] + 0.0224084944263686t + 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) 49.3851818122445 34.912994 1.4145 0.16458 0.08229 IAidM 0.328226028431917 0.091988 3.5681 0.000915 0.000458 `TIA(t-1)` -0.0569153192027402 0.146032 -0.3897 0.698695 0.349347 `TIA(t-2)` 0.0550408037582717 0.160799 0.3423 0.733835 0.366918 `TIA(t-3)` 0.315315051430443 0.174454 1.8074 0.07786 0.03893 `TIA(t-4)` -0.159716545463239 0.168604 -0.9473 0.348911 0.174456 M1 1.07935611922313 4.004491 0.2695 0.788837 0.394418 M2 5.04024362561899 4.504958 1.1188 0.269575 0.134787 M3 12.9910144316273 5.105735 2.5444 0.014712 0.007356 M4 7.48173555248839 4.445992 1.6828 0.099834 0.049917 M5 5.69904974515526 3.365714 1.6933 0.097811 0.048905 M6 10.7442308754480 3.410141 3.1507 0.003 0.0015 M7 -5.15002616864018 3.331084 -1.5461 0.129595 0.064797 M8 2.4269950608123 4.135388 0.5869 0.560423 0.280211 M9 10.1458174187584 5.901105 1.7193 0.092921 0.046461 M10 19.4896364076166 6.832734 2.8524 0.006705 0.003352 M11 7.43512929557958 5.026506 1.4792 0.146553 0.073276 t 0.0224084944263686 0.067817 0.3304 0.742721 0.371361 Multiple Linear Regression - Regression Statistics Multiple R 0.94203267354235 R-squared 0.887425558021349 Adjusted R-squared 0.841859712458562 F-TEST (value) 19.4756740945041 F-TEST (DF numerator) 17 F-TEST (DF denominator) 42 p-value 1.17683640610267e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 3.57889345058774 Sum Squared Residuals 537.956089887712 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 95.1 98.8702912655445 -3.77029126554452 2 97 98.4283768644238 -1.4283768644238 3 112.7 112.280113962106 0.41988603789447 4 102.9 103.541909840113 -0.641909840112737 5 97.4 102.252065894731 -4.85206589473064 6 111.4 112.101323400587 -0.701323400586641 7 87.4 87.1362486860718 0.263751313928171 8 96.8 96.0139320263975 0.786067973602462 9 114.1 114.872520373447 -0.772520373446601 10 110.3 114.066371327565 -3.76637132756515 11 103.9 107.833623614856 -3.93362361485621 12 101.6 99.507762429703 2.09223757029699 13 94.6 96.7222811271049 -2.12228112710488 14 95.9 99.6319422629941 -3.73194226299409 15 104.7 108.722888805716 -4.02288880571649 16 102.8 100.113471677535 2.68652832246503 17 98.1 99.3248268297767 -1.22482682977674 18 113.9 111.028371609433 2.87162839056653 19 80.9 83.394443095994 -2.49444309599395 20 95.7 94.434092109663 1.26590789033707 21 113.2 111.058875993335 2.14112400666497 22 105.9 106.756786922336 -0.856786922336051 23 108.8 106.171983373505 2.62801662649468 24 102.3 99.4100852386573 2.88991476134270 25 99 96.9620790251136 2.03792097488638 26 100.7 101.280289849433 -0.580289849432837 27 115.5 110.762113611137 4.7378863888626 28 100.7 101.766985105576 -1.06698510557587 29 109.9 103.744259288091 6.15574071190854 30 114.6 114.494576941861 0.105423058138700 31 85.4 84.610161526994 0.789838473005926 32 100.5 101.134511645962 -0.634511645961985 33 114.8 112.592367566613 2.20763243338713 34 116.5 112.740052119152 3.75994788084775 35 112.9 109.018018200391 3.88198179960910 36 102 101.079835660673 0.920164339327007 37 106 104.072534426043 1.92746557395719 38 105.3 104.869716594199 0.430283405800751 39 118.8 114.40941589727 4.3905841027299 40 106.1 107.540166982053 -1.44016698205297 41 109.3 106.287710547374 3.01228945262617 42 117.2 116.418192655523 0.781807344477457 43 92.5 87.6789389819997 4.82106101800034 44 104.2 102.060118696742 2.13988130325779 45 112.5 115.007444877342 -2.50744487734211 46 122.4 119.171341654162 3.22865834583831 47 113.3 109.612123984464 3.68787601553581 48 100 99.973487740503 0.0265122594969877 49 110.7 108.772814156194 1.92718584380584 50 112.8 107.48967442895 5.31032557104998 51 109.8 115.325467723770 -5.52546772377049 52 117.3 116.837466394723 0.462533605276545 53 109.1 112.191137440027 -3.09113744002732 54 115.9 118.957535392596 -3.05753539259604 55 96 99.3802077089405 -3.38020770894048 56 99.8 103.357345521235 -3.55734552123534 57 116.8 117.868791189263 -1.06879118926339 58 115.7 118.065447976785 -2.36544797678486 59 99.4 105.664250826783 -6.26425082678338 60 94.3 100.228828930464 -5.92882893046369 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.25336744939811 0.50673489879622 0.74663255060189 22 0.139869474815040 0.279738949630080 0.86013052518496 23 0.219910885490772 0.439821770981545 0.780089114509228 24 0.132971905185575 0.265943810371149 0.867028094814425 25 0.0967922759389224 0.193584551877845 0.903207724061078 26 0.104342634952460 0.208685269904921 0.89565736504754 27 0.131256812828266 0.262513625656532 0.868743187171734 28 0.116413776013051 0.232827552026101 0.88358622398695 29 0.123201673994415 0.24640334798883 0.876798326005585 30 0.085344244058933 0.170688488117866 0.914655755941067 31 0.0656565757911339 0.131313151582268 0.934343424208866 32 0.0840582787059347 0.168116557411869 0.915941721294065 33 0.0567722464778919 0.113544492955784 0.943227753522108 34 0.0308638752619234 0.0617277505238468 0.969136124738077 35 0.0224511424481125 0.044902284896225 0.977548857551887 36 0.0218944192770318 0.0437888385540636 0.978105580722968 37 0.020557612429238 0.041115224858476 0.979442387570762 38 0.0744231257863276 0.148846251572655 0.925576874213672 39 0.0402451351879862 0.0804902703759725 0.959754864812014 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 3 0.157894736842105 NOK 10% type I error level 5 0.263157894736842 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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