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: Mon, 14 Dec 2009 12:25:40 -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/Dec/14/t1260818776m8s89yrj0wf3s5t.htm/, Retrieved Mon, 14 Dec 2009 20:26:28 +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/Dec/14/t1260818776m8s89yrj0wf3s5t.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

441 1919 449 1911 452 1870 462 2263 455 1802 461 1863 461 1989 463 2197 462 2409 456 2502 455 2593 456 2598 472 2053 472 2213 471 2238 465 2359 459 2151 465 2474 468 3079 467 2312 463 2565 460 1972 462 2484 461 2202 476 2151 476 1976 471 2012 453 2114 443 1772 442 1957 444 2070 438 1990 427 2182 424 2008 416 1916 406 2397 431 2114 434 1778 418 1641 412 2186 404 1773 409 1785 412 2217 406 2153 398 1895 397 2475 385 1793 390 2308 413 2051 413 1898 401 2142 397 1874 397 1560 409 1808 419 1575 424 1525 428 1997 430 1753 424 1623 433 2251 456 1890

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 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation wkl[t] = + 368.493200090368 + 0.0334516441425871bvg[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) 368.493200090368 22.133164 16.6489 0 0 bvg 0.0334516441425871 0.010555 3.1691 0.002424 0.001212 Multiple Linear Regression - Regression Statistics Multiple R 0.381398175443491 R-squared 0.145464568231624 Adjusted R-squared 0.130980916845719 F-TEST (value) 10.0433629860208 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 0.00242358095281825 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 24.3881493521553 Sum Squared Residuals 35092.127900559 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 441 432.686905199993 8.31309480000661 2 449 432.419292046852 16.5807079531475 3 452 431.047774637006 20.9522253629936 4 462 444.194270785043 17.8057292149569 5 455 428.773062835310 26.2269371646895 6 461 430.813613128008 30.1863868719917 7 461 435.028520289974 25.9714797100257 8 463 441.986462271632 21.0135377283676 9 462 449.078210829861 12.9217891701391 10 456 452.189213735121 3.81078626487854 11 455 455.233313352097 -0.233313352096891 12 456 455.40057157281 0.599428427190174 13 472 437.1694255151 34.8305744849002 14 472 442.521688577914 29.4783114220862 15 471 443.357979681478 27.6420203185215 16 465 447.405628622732 17.5943713772685 17 459 440.447686641073 18.5523133589266 18 465 451.252567699129 13.7474323008710 19 468 471.490812405394 -3.49081240539423 20 467 445.83340134803 21.1665986519701 21 463 454.296667316104 8.70333268389555 22 460 434.45984233955 25.5401576604497 23 462 451.587084140555 10.4129158594451 24 461 442.153720492345 18.8462795076547 25 476 440.447686641073 35.5523133589266 26 476 434.593648916121 41.4063510838794 27 471 435.797908105254 35.2020918947462 28 453 439.209975807798 13.7900241922023 29 443 427.769513511033 15.2304864889671 30 442 433.958067677411 8.04193232258852 31 444 437.738103465524 6.26189653447617 32 438 435.061971934117 2.93802806588314 33 427 441.484687609494 -14.4846876094936 34 424 435.664101528683 -11.6641015286834 35 416 432.586550267565 -16.5865502675654 36 406 448.67679110015 -42.6767911001498 37 431 439.209975807798 -8.20997580779766 38 434 427.970223375888 6.02977662411161 39 418 423.387348128354 -5.38734812835395 40 412 441.618494186064 -29.6184941860639 41 404 427.802965155175 -23.8029651551754 42 409 428.204384884887 -19.2043848848865 43 412 442.655495154484 -30.6554951544841 44 406 440.514589929359 -34.5145899293586 45 398 431.884065740571 -33.8840657405711 46 397 451.286019343272 -54.2860193432716 47 385 428.471998038027 -43.4719980380272 48 390 445.69959477146 -55.6995947714596 49 413 437.102522226815 -24.1025222268147 50 413 431.984420672999 -18.9844206729988 51 401 440.14662184379 -39.1466218437901 52 397 431.181581213577 -34.1815812135767 53 397 420.677764952804 -23.6777649528044 54 409 428.973772700166 -19.973772700166 55 419 421.179539614943 -2.1795396149432 56 424 419.506957407814 4.49304259218615 57 428 435.296133443115 -7.29613344311497 58 430 427.133932272324 2.86606772767629 59 424 422.785218533787 1.21478146621262 60 433 443.792851055332 -10.7928510553321 61 456 431.716807519858 24.2831924801419 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0336409293187162 0.0672818586374324 0.966359070681284 6 0.0212994771930986 0.0425989543861972 0.978700522806901 7 0.00850434187675478 0.0170086837535096 0.991495658123245 8 0.00259815926801178 0.00519631853602355 0.997401840731988 9 0.00073775728058086 0.00147551456116172 0.99926224271942 10 0.000325039926396449 0.000650079852792898 0.999674960073604 11 0.000120085339452238 0.000240170678904476 0.999879914660548 12 3.31735743509141e-05 6.63471487018282e-05 0.99996682642565 13 0.000107606184573103 0.000215212369146205 0.999892393815427 14 0.000147286840849637 0.000294573681699275 0.99985271315915 15 0.000135356176364944 0.000270712352729888 0.999864643823635 16 5.71804695993179e-05 0.000114360939198636 0.9999428195304 17 2.17755431717215e-05 4.35510863434431e-05 0.999978224456828 18 8.41784694151935e-06 1.68356938830387e-05 0.999991582153059 19 2.69343142157398e-06 5.38686284314796e-06 0.999997306568578 20 1.47202404279757e-06 2.94404808559515e-06 0.999998527975957 21 5.62851131113359e-07 1.12570226222672e-06 0.999999437148869 22 2.65425247576029e-07 5.30850495152058e-07 0.999999734574752 23 1.20208669614954e-07 2.40417339229908e-07 0.99999987979133 24 6.48087775551461e-08 1.29617555110292e-07 0.999999935191222 25 7.02513696394079e-07 1.40502739278816e-06 0.999999297486304 26 9.5252208575963e-06 1.90504417151926e-05 0.999990474779142 27 5.35363410628513e-05 0.000107072682125703 0.999946463658937 28 0.000110483968430645 0.000220967936861290 0.99988951603157 29 0.000288694578996074 0.000577389157992147 0.999711305421004 30 0.000842243078236206 0.00168448615647241 0.999157756921764 31 0.00236282267166079 0.00472564534332157 0.99763717732834 32 0.00698100969771356 0.0139620193954271 0.993018990302286 33 0.0376362731203514 0.0752725462407027 0.962363726879649 34 0.0935715919530939 0.187143183906188 0.906428408046906 35 0.192487901592005 0.38497580318401 0.807512098407995 36 0.466244675671870 0.932489351343739 0.53375532432813 37 0.513921133343217 0.972157733313567 0.486078866656783 38 0.527188069604148 0.945623860791704 0.472811930395852 39 0.515531307002674 0.968937385994652 0.484468692997326 40 0.584102511850901 0.831794976298197 0.415897488149099 41 0.630830462505103 0.738339074989794 0.369169537494897 42 0.620457855877232 0.759084288245536 0.379542144122768 43 0.631224931277608 0.737550137444784 0.368775068722392 44 0.642469406017758 0.715061187964483 0.357530593982241 45 0.674182662046863 0.651634675906274 0.325817337953137 46 0.728907332570965 0.54218533485807 0.271092667429035 47 0.84656944380258 0.306861112394839 0.153430556197419 48 0.913255135702342 0.173489728595316 0.086744864297658 49 0.878776798869146 0.242446402261708 0.121223201130854 50 0.828261502664661 0.343476994670677 0.171738497335339 51 0.86861602789517 0.262767944209661 0.131383972104830 52 0.924722273924615 0.15055545215077 0.075277726075385 53 0.94385746888849 0.112285062223019 0.0561425311115097 54 0.955160659318644 0.0896786813627115 0.0448393406813557 55 0.91003892307791 0.179922153844179 0.0899610769220897 56 0.808648451042643 0.382703097914715 0.191351548957357 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 24 0.461538461538462 NOK 5% type I error level 27 0.519230769230769 NOK 10% type I error level 30 0.576923076923077 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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