workshop 7,4

*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 06:24:51 -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/t1258723654kn7x2al4znkptuw.htm/, Retrieved Fri, 20 Nov 2009 14:27:46 +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/t1258723654kn7x2al4znkptuw.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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No (this computation is public)

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Dataseries X:

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8,3 9,2 8,3 8,6 8,9 8,9 8,3 9,5 8,3 8,3 8,6 8,9 8,4 9,6 8,3 8,3 8,3 8,6 8,5 9,5 8,4 8,3 8,3 8,3 8,4 9,1 8,5 8,4 8,3 8,3 8,6 8,9 8,4 8,5 8,4 8,3 8,5 9 8,6 8,4 8,5 8,4 8,5 10,1 8,5 8,6 8,4 8,5 8,4 10,3 8,5 8,5 8,6 8,4 8,5 10,2 8,4 8,5 8,5 8,6 8,5 9,6 8,5 8,4 8,5 8,5 8,5 9,2 8,5 8,5 8,4 8,5 8,5 9,3 8,5 8,5 8,5 8,4 8,5 9,4 8,5 8,5 8,5 8,5 8,5 9,4 8,5 8,5 8,5 8,5 8,5 9,2 8,5 8,5 8,5 8,5 8,5 9 8,5 8,5 8,5 8,5 8,6 9 8,5 8,5 8,5 8,5 8,4 9 8,6 8,5 8,5 8,5 8,1 9,8 8,4 8,6 8,5 8,5 8,0 10 8,1 8,4 8,6 8,5 8,0 9,8 8,0 8,1 8,4 8,6 8,0 9,3 8,0 8,0 8,1 8,4 8,0 9 8,0 8,0 8,0 8,1 7,9 9 8,0 8,0 8,0 8,0 7,8 9,1 7,9 8,0 8,0 8,0 7,8 9,1 7,8 7,9 8,0 8,0 7,9 9,1 7,8 7,8 7,9 8,0 8,1 9,2 7,9 7,8 7,8 7,9 8,0 8,8 8,1 7,9 7,8 7,8 7,6 8,3 8,0 8,1 7,9 7,8 7,3 8,4 7,6 8,0 8,1 7,9 7,0 8,1 7,3 7,6 8,0 8,1 6,8 7,7 7,0 7,3 7,6 8,0 7,0 7,9 6,8 7,0 7,3 7,6 7,1 7,9 7,0 6,8 7,0 7,3 7,2 8 7,1 7,0 6,8 7,0 7,1 7,9 7,2 7,1 7,0 6,8 6,9 7,6 7,1 7,2 7,1 7,0 6,7 7,1 6,9 7,1 7,2 7,1 6,7 6,8 6,7 6,9 7,1 7,2 6,6 6,5 6,7 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 Y[t] = -0.272223696302566 + 0.161944495338189X[t] + 1.20816388613964Y1[t] -0.4678267601091Y2[t] -0.193917063933410Y3[t] + 0.289842816853047Y4[t] + 0.0948685017759605M1[t] + 0.0220364659106159M2[t] + 0.0138541990473369M3[t] + 0.0880521230767941M4[t] + 0.127829043601995M5[t] + 0.146966512755292M6[t] + 0.0874807673669855M7[t] -0.0184272763256771M8[t] -0.0800409575166678M9[t] -0.104472012932639M10[t] + 0.0192887853678653M11[t] + 0.003684875625332t + 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) -0.272223696302566 0.530892 -0.5128 0.611007 0.305504 X 0.161944495338189 0.069133 2.3425 0.024352 0.012176 Y1 1.20816388613964 0.173889 6.9479 0 0 Y2 -0.4678267601091 0.258642 -1.8088 0.078199 0.0391 Y3 -0.193917063933410 0.256736 -0.7553 0.454596 0.227298 Y4 0.289842816853047 0.147705 1.9623 0.05689 0.028445 M1 0.0948685017759605 0.09642 0.9839 0.331223 0.165612 M2 0.0220364659106159 0.096809 0.2276 0.821124 0.410562 M3 0.0138541990473369 0.097155 0.1426 0.887341 0.443671 M4 0.0880521230767941 0.096567 0.9118 0.367464 0.183732 M5 0.127829043601995 0.097706 1.3083 0.198427 0.099214 M6 0.146966512755292 0.102955 1.4275 0.161399 0.080699 M7 0.0874807673669855 0.102078 0.857 0.396682 0.198341 M8 -0.0184272763256771 0.101686 -0.1812 0.857135 0.428568 M9 -0.0800409575166678 0.107434 -0.745 0.460722 0.230361 M10 -0.104472012932639 0.107614 -0.9708 0.33763 0.168815 M11 0.0192887853678653 0.103038 0.1872 0.852474 0.426237 t 0.003684875625332 0.002479 1.4866 0.145159 0.072579 Multiple Linear Regression - Regression Statistics Multiple R 0.981039838756772 R-squared 0.962439165227914 Adjusted R-squared 0.946066493660594 F-TEST (value) 58.7832695031255 F-TEST (DF numerator) 17 F-TEST (DF denominator) 39 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.141875843766754 Sum Squared Residuals 0.785021446736605 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.3 8.17440835721559 0.125591642784410 2 8.3 8.3523676927898 -0.052367692789801 3 8.4 8.33528702520978 0.06471297479022 4 8.5 8.4308389188888 0.0691610811112 5 8.4 8.48355662950711 -0.0835566295071102 6 8.6 8.28699930419989 0.313000695800113 7 8.5 8.54540091250153 -0.0454009125015326 8 8.5 8.45531093674907 0.0446890632509271 9 8.4 8.40878601178998 -0.00878601178997573 10 8.5 8.3283892636155 0.171610736384496 11 8.5 8.497283023278 0.00271697672200471 12 8.5 8.38951034578262 0.110489654217383 13 8.5 8.45588218463908 0.0441178153609167 14 8.5 8.4319137556182 0.0680862443818058 15 8.5 8.42741636438025 0.072583635619753 16 8.5 8.4729102649674 0.0270897350326019 17 8.5 8.4839831620503 0.0160168379497067 18 8.6 8.50680550682892 0.0931944931710776 19 8.4 8.57182102567991 -0.171821025679911 20 8.1 8.3107380006443 -0.210738000644296 21 8 7.99692257393286 0.00307742606713911 22 8 8.03108682896534 -0.0310868289653364 23 8 8.1245494870424 -0.124549487042402 24 8 7.99280109003584 0.00719890996416227 25 7.9 8.06237018575183 -0.162370185751825 26 7.8 7.88860108643167 -0.0886010864316686 27 7.8 7.81006998259067 -0.0100699825906667 28 7.9 7.9541271646497 -0.0541271646497065 29 8.1 8.12500722365606 -0.0250072236560602 30 8 8.24891758983112 -0.248917589831125 31 7.6 7.87837102536993 -0.278371025369932 32 7.3 7.3460602972901 -0.0460602972900963 33 7 7.14158995108868 -0.141589951088679 34 6.8 6.88254737924166 -0.0825473792416607 35 7 6.88333519547874 0.116664804521259 36 7.1 7.17415168911006 -0.0741516891100646 37 7.2 7.26798112036809 -0.0679811203680868 38 7.1 7.15992124704002 -0.0599212470400195 39 6.9 6.97781829955301 -0.0778182995530084 40 6.7 6.78947132561365 -0.0894713256136493 41 6.7 6.68465833603526 0.0153416639647373 42 6.6 6.76226181533563 -0.162261815335632 43 6.9 6.63123720451004 0.268762795489959 44 7.3 7.09080515886455 0.209194841135449 45 7.5 7.47615783378445 0.0238421662155463 46 7.3 7.3579765281775 -0.0579765281774995 47 7.1 7.09483229420086 0.00516770579913756 48 6.9 6.94353687507148 -0.0435368750714809 49 7.1 7.03935815202541 0.0606418479745848 50 7.5 7.36719621812032 0.132803781879683 51 7.7 7.7494083282663 -0.0494083282662976 52 7.8 7.75265232588045 0.0473476741195537 53 7.8 7.72279464875127 0.0772053512487265 54 7.7 7.69501578380443 0.00498421619556606 55 7.8 7.57316983193858 0.226830168061416 56 7.8 7.79708560645198 0.00291439354801666 57 7.9 7.77654362940403 0.123456370595969 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.643757673305873 0.712484653388254 0.356242326694127 22 0.518421467965326 0.963157064069347 0.481578532034674 23 0.362979864301384 0.725959728602769 0.637020135698616 24 0.355569402950905 0.71113880590181 0.644430597049095 25 0.277627438212345 0.55525487642469 0.722372561787655 26 0.205758330188626 0.411516660377252 0.794241669811374 27 0.196173657676711 0.392347315353422 0.803826342323289 28 0.161081526607222 0.322163053214444 0.838918473392778 29 0.117738835647324 0.235477671294647 0.882261164352676 30 0.158409619724040 0.316819239448079 0.84159038027596 31 0.521636104133476 0.956727791733049 0.478363895866524 32 0.841324047940721 0.317351904118558 0.158675952059279 33 0.977407684972091 0.0451846300558172 0.0225923150279086 34 0.988250556280508 0.0234988874389842 0.0117494437194921 35 0.978504159912164 0.0429916801756727 0.0214958400878364 36 0.936721026339766 0.126557947320467 0.0632789736602336 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 0.1875 NOK 10% type I error level 3 0.1875 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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