WS 10 - Multiple Regression

*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: Tue, 14 Dec 2010 08:45:57 +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/14/t1292316410xkhfs02wzrnjnbs.htm/, Retrieved Tue, 14 Dec 2010 09:46:58 +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/14/t1292316410xkhfs02wzrnjnbs.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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100.44 1576.23 29.29 710.45 100.88 1546.37 28.99 720 101.42 1545.05 28.91 720 99.97 1552.34 29.29 720 100.56 1594.3 30.96 754.78 99.51 1605.78 30.57 802.73 98.96 1673.21 30.59 845.24 100.85 1612.94 31.39 893.91 100.66 1566.34 31.28 931.43 100.22 1530.17 31.1 940 100.30 1582.54 31.7 947.73 100.73 1702.16 32.57 960 101.46 1701.93 32.49 996.96 101.35 1811.15 32.46 1000 101.14 1924.2 32.3 1000 101.68 2034.25 32.97 1000 101.47 2011.13 32.9 1013.04 100.59 2013.04 32.93 1095.24 101.18 2151.67 33.72 1159.09 100.87 1902.09 33.33 1200 99.79 1944.01 33.44 1200 100.74 1916.67 33.89 1282.61 99.34 1967.31 34.34 1513.64 100.07 2119.88 33.56 1669.05 103.68 2216.38 32.67 1700 103.52 2522.83 32.57 1700 104.68 2647.64 33.23 1700 103.75 2631.23 32.85 1665.91 103.70 2693.41 32.61 1650 102.98 3021.76 32.57 1650 106.30 2953.67 32.98 1619.57 107.21 2796.8 31.33 1599.05 106.83 2672.05 29.8 1572.73 105.60 2251.23 28.06 1470 104.30 2046.08 25.47 1268 104.43 2420.04 24.65 1217.39 104.36 2608.89 23.94 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 6 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Multiple Linear Regression - Estimated Regression Equation Chocolade[t] = + 116.668997738777 + 0.00156233966989654Cacao[t] -0.749158502206502Suiker[t] + 0.0048421384532872Grondnoten[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) 116.668997738777 6.05139 19.2797 0 0 Cacao 0.00156233966989654 0.001032 1.5138 0.136009 0.068004 Suiker -0.749158502206502 0.182389 -4.1075 0.000139 7e-05 Grondnoten 0.0048421384532872 0.00195 2.4827 0.016241 0.008121 Multiple Linear Regression - Regression Statistics Multiple R 0.758359579084536 R-squared 0.575109251189275 Adjusted R-squared 0.551058831445272 F-TEST (value) 23.9126492306927 F-TEST (DF numerator) 3 F-TEST (DF denominator) 53 p-value 6.38400776686865e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 2.9574051191386 Sum Squared Residuals 463.55098705148 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 100.44 100.628849131168 -0.188849131167517 2 100.88 100.853187641515 0.0268123584847969 3 101.42 100.911058033327 0.508941966672553 4 99.97 100.637767258683 -0.667767258682527 5 100.56 99.6206379079519 0.939362092048148 6 99.51 100.162925922058 -0.65292592205792 7 98.96 100.459130621604 -1.49913062160416 8 100.85 100.001308486456 0.848691513544215 9 100.66 100.192587927849 0.467412072151346 10 100.22 100.31242375893 -0.0924237589303361 11 100.3 99.9821781163628 0.31782188363717 12 100.73 99.576710329578 1.15328967042198 13 101.46 99.815249108864 1.64475089113603 14 101.35 100.023082703574 1.32691729642574 15 101.14 100.319570563609 0.8204294363909 16 101.68 99.9895698478029 1.69043015219715 17 101.47 100.06903113522 1.40096886477983 18 100.59 100.447564229784 0.14243577021632 19 101.18 100.381486701721 0.798513298279315 20 100.87 100.481821666892 0.388178333107574 21 99.79 100.464907510612 -0.674907510611772 22 100.74 100.48508087567 0.254919124330061 23 99.34 101.345755677424 -2.00575567742351 24 100.07 102.920982209606 -2.85098220960607 25 103.68 103.888363239844 -0.208363239844097 26 103.52 104.442058081905 -0.922058081904553 27 104.68 104.142609084648 0.53739091535196 28 103.75 104.236582821631 -0.486582821630952 29 103.7 104.436488720043 -0.736488720042878 30 102.98 104.979449290742 -1.99944929074166 31 106.3 104.41856832358 1.88143167641978 32 107.21 105.310234947143 1.89976505285717 33 106.83 106.134100497609 0.69589950239134 34 105.6 106.282739628256 -0.682739628255924 35 104.3 106.924434198127 -2.62443419812747 36 104.43 107.87793608577 -3.44793608577044 37 104.36 108.398379104904 -4.03837910490394 38 106.21 107.692823180668 -1.48282318066791 39 107.34 107.288175094723 0.0518249052770176 40 106.92 106.808352652138 0.111647347862366 41 104.8 105.987760324182 -1.1877603241816 42 103.85 105.274412481251 -1.42441248125139 43 103.39 105.43534814944 -2.04534814944043 44 103.38 105.581573521157 -2.20157352115725 45 103.93 106.078038572441 -2.14803857244138 46 104.41 107.014240201553 -2.60424020155318 47 104.47 106.364182185619 -1.89418218561891 48 103.84 107.582360055091 -3.74236005509077 49 103.65 107.967230223254 -4.31723022325357 50 103.17 108.289647365043 -5.11964736504314 51 103.4 108.620288410106 -5.22028841010626 52 112.72 108.485983721232 4.23401627876775 53 114.77 109.299411927909 5.4705880720908 54 116.18 108.983223396045 7.19677660395545 55 116.93 108.377258860062 8.5527411399378 56 115.19 107.823405473398 7.36659452660232 57 114.55 108.111472814225 6.43852718577467 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.0110919263314323 0.0221838526628646 0.988908073668568 8 0.00307341589620816 0.00614683179241632 0.996926584103792 9 0.000810860924750158 0.00162172184950032 0.99918913907525 10 0.000333032041551853 0.000666064083103707 0.999666967958448 11 5.94661321646689e-05 0.000118932264329338 0.999940533867835 12 4.89564065201209e-05 9.79128130402419e-05 0.99995104359348 13 4.24256990492723e-05 8.48513980985447e-05 0.99995757430095 14 1.40157723615351e-05 2.80315447230702e-05 0.999985984227639 15 3.03142571801483e-06 6.06285143602967e-06 0.999996968574282 16 7.28735107506242e-07 1.45747021501248e-06 0.999999271264893 17 1.57785564996345e-07 3.1557112999269e-07 0.999999842214435 18 5.74893316062774e-08 1.14978663212555e-07 0.999999942510668 19 1.28931498096247e-08 2.57862996192494e-08 0.99999998710685 20 2.72384101857145e-09 5.44768203714291e-09 0.99999999727616 21 1.6794062612227e-09 3.3588125224454e-09 0.999999998320594 22 4.8523449229751e-10 9.7046898459502e-10 0.999999999514765 23 1.24095322983146e-10 2.48190645966291e-10 0.999999999875905 24 3.43281005865454e-11 6.86562011730907e-11 0.999999999965672 25 2.43002797781207e-09 4.86005595562414e-09 0.999999997569972 26 6.05611122770546e-10 1.21122224554109e-09 0.999999999394389 27 2.32774084593878e-10 4.65548169187756e-10 0.999999999767226 28 5.09279563882809e-11 1.01855912776562e-10 0.999999999949072 29 1.2393172920087e-11 2.4786345840174e-11 0.999999999987607 30 3.06175388300055e-11 6.12350776600111e-11 0.999999999969382 31 4.66852046271487e-11 9.33704092542974e-11 0.999999999953315 32 1.30462374706155e-10 2.6092474941231e-10 0.999999999869538 33 6.34258773306336e-11 1.26851754661267e-10 0.999999999936574 34 1.71678736716874e-11 3.43357473433749e-11 0.999999999982832 35 6.97109256835058e-12 1.39421851367012e-11 0.999999999993029 36 3.4478032232141e-11 6.8956064464282e-11 0.999999999965522 37 9.50461915210832e-10 1.90092383042166e-09 0.999999999049538 38 4.12772994249908e-10 8.25545988499817e-10 0.999999999587227 39 2.6050983495391e-10 5.21019669907819e-10 0.99999999973949 40 1.00898982527744e-10 2.01797965055488e-10 0.9999999998991 41 5.89293304080852e-11 1.1785866081617e-10 0.99999999994107 42 3.6146951492443e-11 7.2293902984886e-11 0.999999999963853 43 4.02151343994447e-11 8.04302687988895e-11 0.999999999959785 44 3.75891694918603e-11 7.51783389837206e-11 0.99999999996241 45 2.0980305991169e-11 4.19606119823379e-11 0.99999999997902 46 1.38080906927148e-11 2.76161813854297e-11 0.999999999986192 47 4.0625829147808e-12 8.1251658295616e-12 0.999999999995937 48 3.2416436180385e-12 6.48328723607701e-12 0.999999999996758 49 1.65136635712565e-12 3.3027327142513e-12 0.999999999998349 50 4.26550611620233e-06 8.53101223240466e-06 0.999995734493884 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 43 0.977272727272727 NOK 5% type I error level 44 1 NOK 10% type I error level 44 1 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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