Paper: Multiple Regression

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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 15:42:29 +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/t1292773304pskov1ajctzm9th.htm/, Retrieved Sun, 19 Dec 2010 16:41:44 +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/t1292773304pskov1ajctzm9th.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}, }

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Dataseries X:

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695 0 641 696 729 839 627 608 638 0 695 641 696 729 696 651 762 0 638 695 641 696 825 691 635 0 762 638 695 641 677 627 721 0 635 762 638 695 656 634 854 0 721 635 762 638 785 731 418 0 854 721 635 762 412 475 367 0 418 854 721 635 352 337 824 0 367 418 854 721 839 803 687 0 824 367 418 854 729 722 601 0 687 824 367 418 696 590 676 0 601 687 824 367 641 724 740 0 676 601 687 824 695 627 691 0 740 676 601 687 638 696 683 0 691 740 676 601 762 825 594 0 683 691 740 676 635 677 729 0 594 683 691 740 721 656 731 0 729 594 683 691 854 785 386 0 731 729 594 683 418 412 331 0 386 731 729 594 367 352 706 0 331 386 731 729 824 839 715 0 706 331 386 731 687 729 657 0 715 706 331 386 601 696 653 0 657 715 706 331 676 641 642 0 653 657 715 706 740 695 643 0 642 653 657 715 691 638 718 0 643 642 653 657 683 762 654 0 718 643 642 653 594 635 632 0 654 718 643 642 729 721 731 0 632 654 718 643 731 854 392 1 731 632 654 718 386 418 344 1 392 731 632 654 331 367 792 1 344 392 731 632 706 824 8 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 5 seconds R Server 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation faillissement[t] = -122.703425274455 + 78.463476286789crisis[t] + 0.0523803521122521`t-1`[t] + 0.0444563774906133`t-2`[t] + 0.063446088726049`t-3`[t] -0.00750219719536323`t-4`[t] + 0.668851325427979`t-12`[t] + 0.345558412581939`t-24`[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) -122.703425274455 118.622544 -1.0344 0.305735 0.152867 crisis 78.463476286789 20.476574 3.8319 0.000344 0.000172 `t-1` 0.0523803521122521 0.072402 0.7235 0.472633 0.236317 `t-2` 0.0444563774906133 0.082172 0.541 0.590807 0.295404 `t-3` 0.063446088726049 0.072969 0.8695 0.388572 0.194286 `t-4` -0.00750219719536323 0.075037 -0.1 0.920745 0.460372 `t-12` 0.668851325427979 0.143201 4.6707 2.2e-05 1.1e-05 `t-24` 0.345558412581939 0.154023 2.2436 0.029146 0.014573 Multiple Linear Regression - Regression Statistics Multiple R 0.899985961342426 R-squared 0.80997473061345 Adjusted R-squared 0.784394405888338 F-TEST (value) 31.6639737500396 F-TEST (DF numerator) 7 F-TEST (DF denominator) 52 p-value 1.11022302462516e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 73.883945471156 Sum Squared Residuals 283859.544716007 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 695 611.241170290506 83.7588297094942 2 638 671.365882501669 -33.3658825016687 3 762 767.643041926765 -5.64304192676461 4 635 654.337167140086 -19.3371671400856 5 721 637.548938578817 83.4510614211828 6 854 764.503616161986 89.4963838380138 7 418 428.361027731063 -10.3610277310628 8 367 330.026894628637 36.9731053713627 9 824 802.526472673378 21.4735273266224 10 687 693.9728542089 -6.97285420890017 11 601 639.462713734943 -38.4627137349425 12 676 667.762958729284 8.23704127071627 13 740 658.346423952423 81.653576047577 14 691 646.323437103487 44.6765628965134 15 683 779.52025319878 -96.52025319878 16 594 644.333969382177 -50.3339693821765 17 729 685.991955378767 43.0080446212331 18 731 822.500985775012 -91.5009857750121 19 386 402.508207341752 -16.5082073417520 20 331 338.913893794667 -7.91389379466694 21 706 793.7616223983 -87.7616223983001 22 715 659.411191705887 55.5888082941131 23 657 606.727837984335 50.2721620156654 24 653 659.452925792345 -6.45292579234464 25 642 715.889264446529 -73.8892644465287 26 643 658.917317679321 -15.9173176793208 27 718 696.1604535182 21.8395464818002 28 654 596.051851755908 57.9481482440921 29 632 716.192660205219 -84.1926602052187 30 731 764.243010280862 -33.243010280862 31 392 460.873711495473 -68.8737114954731 32 344 392.191978229303 -48.1919782293028 33 792 789.792662066212 2.20733793378826 34 852 747.552371497606 104.447628502394 35 649 701.598081973644 -52.598081973644 36 629 745.657682000875 -116.657682000875 37 685 750.789585133611 -65.7895851336115 38 617 723.240558567889 -106.240558567889 39 715 769.82165812631 -54.8216581263104 40 715 702.073740329332 12.9262596706678 41 629 733.631664786247 -104.631664786247 42 916 802.762218651564 113.237781348436 43 531 467.278664257589 63.7213357424113 44 357 403.304269091188 -46.3042690911879 45 917 825.158397422909 91.8416025770912 46 828 863.417215406721 -35.4172154067214 47 708 719.680454953725 -11.6804549537245 48 858 731.514126943283 126.485873056717 49 775 757.843013820791 17.1569861792091 50 785 708.081734405807 76.9182655941932 51 1006 806.797146403147 199.202853596853 52 789 790.31067464605 -1.31067464604990 53 734 724.902641853983 9.09735814601649 54 906 952.491865452292 -46.4918654522917 55 532 568.978336264838 -36.9783362648382 56 387 422.206088986301 -35.2060889863012 57 991 938.676511931538 52.3234880684625 58 841 900.05459156188 -59.0545915618802 59 892 762.24481282967 129.755187170330 60 782 891.073540910223 -109.073540910223 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 11 0.366296828505451 0.732593657010903 0.633703171494549 12 0.235518118772881 0.471036237545761 0.76448188122712 13 0.20249548598751 0.40499097197502 0.79750451401249 14 0.118222352995701 0.236444705991402 0.8817776470043 15 0.306266616760127 0.612533233520254 0.693733383239873 16 0.270302153525888 0.540604307051776 0.729697846474112 17 0.193403678967360 0.386807357934719 0.80659632103264 18 0.238940330303897 0.477880660607795 0.761059669696103 19 0.180822116827093 0.361644233654186 0.819177883172907 20 0.140580468939363 0.281160937878726 0.859419531060637 21 0.136927740931019 0.273855481862037 0.863072259068981 22 0.163412149817833 0.326824299635666 0.836587850182167 23 0.163148443514637 0.326296887029274 0.836851556485363 24 0.112990413127758 0.225980826255516 0.887009586872242 25 0.106348845636183 0.212697691272367 0.893651154363817 26 0.0717534596194758 0.143506919238952 0.928246540380524 27 0.0487262992485758 0.0974525984971515 0.951273700751424 28 0.0459305936469611 0.0918611872939222 0.954069406353039 29 0.0417330384204142 0.0834660768408284 0.958266961579586 30 0.0258565842315159 0.0517131684630319 0.974143415768484 31 0.0163387602299168 0.0326775204598335 0.983661239770083 32 0.0100357595279463 0.0200715190558925 0.989964240472054 33 0.00843733915742236 0.0168746783148447 0.991562660842578 34 0.0162944310138086 0.0325888620276173 0.983705568986191 35 0.0102447807704118 0.0204895615408236 0.989755219229588 36 0.0173329480357047 0.0346658960714095 0.982667051964295 37 0.0156897867476944 0.0313795734953889 0.984310213252306 38 0.0233407503504210 0.0466815007008421 0.976659249649579 39 0.0189020707652204 0.0378041415304407 0.98109792923478 40 0.0119895130056501 0.0239790260113002 0.98801048699435 41 0.076502631909536 0.153005263819072 0.923497368090464 42 0.135110608252103 0.270221216504205 0.864889391747897 43 0.129816586035655 0.259633172071310 0.870183413964345 44 0.101910414285684 0.203820828571367 0.898089585714316 45 0.0826698677694323 0.165339735538865 0.917330132230568 46 0.0639520081351108 0.127904016270222 0.93604799186489 47 0.0634643227980983 0.126928645596197 0.936535677201902 48 0.0521377261665826 0.104275452333165 0.947862273833417 49 0.0245357262656658 0.0490714525313315 0.975464273734334 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 11 0.282051282051282 NOK 10% type I error level 15 0.384615384615385 NOK

Charts produced by software:

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Parameters (Session):

par1 = 48 ; par2 = 1 ; par3 = 0 ; par4 = 0 ; par5 = 12 ; par6 = White Noise ; par7 = 0.95 ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ; par4 = 0 ; par5 = 12 ; par6 = White Noise ; par7 = 0.95 ;

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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