paper Bel 20

*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: Sun, 19 Dec 2010 13:25:55 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/19/t1292765051pqi42p2r2w426gb.htm/, Retrieved Sun, 19 Dec 2010 14:24:14 +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/2010/Dec/19/t1292765051pqi42p2r2w426gb.htm/}, year = {2010}, } @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 = {2010}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

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User-defined keywords:

Dataseries X:

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3494,17 0 3667,03 0 3813,06 0 3917,96 0 3895,51 0 3801,06 0 3570,12 0 3701,61 0 3862,27 0 3970,1 0 4138,52 0 4199,75 0 4290,89 0 4443,91 0 4502,64 0 4356,98 0 4591,27 0 4696,96 0 4621,4 0 4562,84 0 4202,52 0 4296,49 0 4435,23 0 4105,18 0 4116,68 0 3844,49 0 3720,98 0 3674,4 0 3857,62 0 3801,06 0 3504,37 0 3032,6 0 3047,03 0 2962,34 1 2197,82 1 2014,45 1 1862,83 1 1905,41 1 1810,99 1 1670,07 1 1864,44 1 2052,02 1 2029,6 1 2070,83 1 2293,41 1 2443,27 1 2513,17 1 2466,92 1 2502,66 1 2539,91 1 2482,6 1 2626,15 1 2656,32 1 2446,66 1 2467,38 1 2462,32 1 2504,58 1 2579,39 1 2649,24 1 2636,87 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 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation Y[t] = + 3992.02121212121 -1669.36750841751X[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) 3992.02121212121 66.550507 59.9848 0 0 X -1669.36750841751 99.207638 -16.827 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.91103560619018 R-squared 0.829985875746308 Adjusted R-squared 0.827054597741934 F-TEST (value) 283.148126690071 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 382.303555342167 Sum Squared Residuals 8477048.48878115 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 3494.17 3992.02121212122 -497.851212121218 2 3667.03 3992.02121212121 -324.991212121213 3 3813.06 3992.02121212121 -178.961212121212 4 3917.96 3992.02121212121 -74.0612121212119 5 3895.51 3992.02121212121 -96.5112121212117 6 3801.06 3992.02121212121 -190.961212121212 7 3570.12 3992.02121212121 -421.901212121212 8 3701.61 3992.02121212121 -290.411212121212 9 3862.27 3992.02121212121 -129.751212121212 10 3970.1 3992.02121212121 -21.9212121212121 11 4138.52 3992.02121212121 146.498787878788 12 4199.75 3992.02121212121 207.728787878788 13 4290.89 3992.02121212121 298.868787878788 14 4443.91 3992.02121212121 451.888787878788 15 4502.64 3992.02121212121 510.618787878788 16 4356.98 3992.02121212121 364.958787878788 17 4591.27 3992.02121212121 599.248787878789 18 4696.96 3992.02121212121 704.938787878788 19 4621.4 3992.02121212121 629.378787878788 20 4562.84 3992.02121212121 570.818787878788 21 4202.52 3992.02121212121 210.498787878789 22 4296.49 3992.02121212121 304.468787878788 23 4435.23 3992.02121212121 443.208787878788 24 4105.18 3992.02121212121 113.158787878788 25 4116.68 3992.02121212121 124.658787878788 26 3844.49 3992.02121212121 -147.531212121212 27 3720.98 3992.02121212121 -271.041212121212 28 3674.4 3992.02121212121 -317.621212121212 29 3857.62 3992.02121212121 -134.401212121212 30 3801.06 3992.02121212121 -190.961212121212 31 3504.37 3992.02121212121 -487.651212121212 32 3032.6 3992.02121212121 -959.421212121212 33 3047.03 3992.02121212121 -944.991212121212 34 2962.34 2322.6537037037 639.686296296296 35 2197.82 2322.6537037037 -124.833703703703 36 2014.45 2322.6537037037 -308.203703703704 37 1862.83 2322.6537037037 -459.823703703704 38 1905.41 2322.6537037037 -417.243703703704 39 1810.99 2322.6537037037 -511.663703703704 40 1670.07 2322.6537037037 -652.583703703704 41 1864.44 2322.6537037037 -458.213703703704 42 2052.02 2322.6537037037 -270.633703703704 43 2029.6 2322.6537037037 -293.053703703704 44 2070.83 2322.6537037037 -251.823703703704 45 2293.41 2322.6537037037 -29.2437037037039 46 2443.27 2322.6537037037 120.616296296296 47 2513.17 2322.6537037037 190.516296296296 48 2466.92 2322.6537037037 144.266296296296 49 2502.66 2322.6537037037 180.006296296296 50 2539.91 2322.6537037037 217.256296296296 51 2482.6 2322.6537037037 159.946296296296 52 2626.15 2322.6537037037 303.496296296296 53 2656.32 2322.6537037037 333.666296296296 54 2446.66 2322.6537037037 124.006296296296 55 2467.38 2322.6537037037 144.726296296296 56 2462.32 2322.6537037037 139.666296296296 57 2504.58 2322.6537037037 181.926296296296 58 2579.39 2322.6537037037 256.736296296296 59 2649.24 2322.6537037037 326.586296296296 60 2636.87 2322.6537037037 314.216296296296 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.157540945542743 0.315081891085486 0.842459054457257 6 0.0666454543829469 0.133290908765894 0.933354545617053 7 0.0433052759776753 0.0866105519553506 0.956694724022325 8 0.0178490643265518 0.0356981286531036 0.982150935673448 9 0.0086489909407389 0.0172979818814778 0.99135100905926 10 0.00674734338732934 0.0134946867746587 0.99325265661267 11 0.0131422570595305 0.026284514119061 0.98685774294047 12 0.0220101328043126 0.0440202656086251 0.977989867195687 13 0.0394323184086035 0.078864636817207 0.960567681591396 14 0.0916265116079284 0.183253023215857 0.908373488392072 15 0.166641531295609 0.333283062591219 0.83335846870439 16 0.173066273153845 0.346132546307691 0.826933726846155 17 0.278165952630901 0.556331905261802 0.721834047369099 18 0.456468868518545 0.912937737037089 0.543531131481455 19 0.579194064905773 0.841611870188454 0.420805935094227 20 0.668411831886576 0.663176336226847 0.331588168113424 21 0.626009513523494 0.747980972953013 0.373990486476506 22 0.618583596263775 0.76283280747245 0.381416403736225 23 0.692539775617003 0.614920448765994 0.307460224382997 24 0.671596216866418 0.656807566267164 0.328403783133582 25 0.67068318391449 0.65863363217102 0.32931681608551 26 0.649861595780132 0.700276808439735 0.350138404219868 27 0.63945576984284 0.721088460314321 0.360544230157161 28 0.633418439460025 0.733163121079949 0.366581560539975 29 0.639049246679757 0.721901506640485 0.360950753320243 30 0.67692418057127 0.646151638857459 0.32307581942873 31 0.733666799369742 0.532666401260516 0.266333200630258 32 0.851560792909242 0.296878414181515 0.148439207090758 33 0.902486204415075 0.19502759116985 0.097513795584925 34 0.935157439642207 0.129685120715585 0.0648425603577926 35 0.926490045738136 0.147019908523728 0.0735099542618641 36 0.923011789916161 0.153976420167678 0.0769882100838392 37 0.938022634511362 0.123954730977276 0.0619773654886381 38 0.94504053230585 0.109918935388299 0.0549594676941494 39 0.967246084242814 0.0655078315143711 0.0327539157571856 40 0.994964302343284 0.0100713953134318 0.00503569765671589 41 0.999006680728048 0.00198663854390438 0.000993319271952192 42 0.999572241774368 0.000855516451264202 0.000427758225632101 43 0.999946349468839 0.000107301062322376 5.36505311611882e-05 44 0.999999287714874 1.42457025170788e-06 7.12285125853942e-07 45 0.999999859279817 2.81440365061556e-07 1.40720182530778e-07 46 0.999999646369343 7.07261313850974e-07 3.53630656925487e-07 47 0.99999845004669 3.09990662063267e-06 1.54995331031633e-06 48 0.999995087612457 9.82477508639392e-06 4.91238754319696e-06 49 0.999980041471908 3.99170561843931e-05 1.99585280921965e-05 50 0.999911272580272 0.000177454839456408 8.87274197282038e-05 51 0.999703915549716 0.000592168900567883 0.000296084450283941 52 0.999039482947308 0.00192103410538301 0.000960517052691506 53 0.997871045095119 0.00425790980976188 0.00212895490488094 54 0.993924600692434 0.0121507986151326 0.00607539930756628 55 0.98136092143177 0.0372781571364606 0.0186390785682303 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 13 0.254901960784314 NOK 5% type I error level 21 0.411764705882353 NOK 10% type I error level 24 0.470588235294118 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal 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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