mini tutorial multiple linear 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, 30 Nov 2010 13:18:26 +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/Nov/30/t12911230108qddma2w1z8jygc.htm/, Retrieved Tue, 30 Nov 2010 14:17:07 +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/Nov/30/t12911230108qddma2w1z8jygc.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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No (this computation is public)

User-defined keywords:

Dataseries X:

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2284 41 76403 194493 3160 90 108094 530670 4150 136 134759 518365 7285 97 188873 491303 1134 63 146216 527021 4658 114 156608 233773 2384 77 61348 405972 3748 6 50350 652925 5371 47 87720 446211 1285 51 99489 341340 9327 85 87419 387699 5565 43 94355 493408 1528 32 60326 146494 3122 25 94670 414462 7561 77 82425 364304 2675 54 59017 355178 13253 251 90829 357760 880 15 80791 261216 2053 44 100423 397144 1424 73 131116 374943 4036 85 100269 424898 3045 49 27330 202055 5119 38 39039 378525 1431 35 106885 310768 554 9 79285 325738 1975 34 118881 394510 1765 20 77623 247060 1012 29 114768 368078 810 11 74015 236761 1280 52 69465 312378 666 13 117869 339836 1380 29 60982 347385 4677 66 90131 426280 876 33 138971 352850 814 15 39625 301881 514 15 102725 377516 5692 68 64239 357312 3642 100 90262 458343 540 13 103960 354228 2099 45 106611 308636 567 14 103345 386212 2001 36 95551 393343 2949 40 82903 378509 2253 68 63593 452469 6533 29 126910 364839 1889 43 37527 358649 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 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Wealth[t] = + 281385.616456098 + 9.22576570117291Costs[t] -138.707662852302Orders[t] + 0.755073191162331Dividends[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) 281385.616456098 39429.959907 7.1363 0 0 Costs 9.22576570117291 7.325235 1.2594 0.214087 0.107043 Orders -138.707662852302 456.693876 -0.3037 0.762681 0.38134 Dividends 0.755073191162331 0.401226 1.8819 0.066048 0.033024 Multiple Linear Regression - Regression Statistics Multiple R 0.346675455408413 R-squared 0.120183871382630 Adjusted R-squared 0.0640253950879047 F-TEST (value) 2.14008426353829 F-TEST (DF numerator) 3 F-TEST (DF denominator) 47 p-value 0.107746189670541 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 88335.758374865 Sum Squared Residuals 366750691760.139 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 194493 354460.108165008 -159967.108165008 2 530670 379674.227940598 150995.772059402 3 518365 402561.210135897 115803.789864103 4 491303 477753.615126872 13549.3848731278 5 527021 393512.833720524 133508.166279476 6 233773 426797.061848549 -193024.061848549 7 405972 339021.581979494 66950.4180205063 8 652925 353149.475502004 299775.524497996 9 446211 390652.964211799 55558.0357882009 10 341340 361288.111292187 -19948.1112921870 11 387699 421651.925106712 -33952.9251067117 12 493408 398007.504032598 95400.4959674021 13 146494 336594.486566275 -190100.486566275 14 414462 378203.54441119 36258.4555888099 15 364304 402698.048664594 -38394.0486645942 16 355178 343136.480435539 12041.5195644615 17 357760 437421.608797898 -79661.608797898 18 261216 348426.793517542 -87210.7935175415 19 397144 370049.691351199 27094.3086488005 20 374943 383399.623958790 -8456.62395879042 21 424898 382541.089288242 42356.9107117581 22 202055 323317.547850873 -121262.547850873 23 378525 352818.722201801 25706.2777981992 24 310768 370438.917012032 -59670.9170120316 25 325738 345114.299650183 -19376.2996501825 26 394510 384654.299217505 9855.70078249468 27 247060 353505.985979216 -106445.985979216 28 368078 373357.809126287 -5279.80912628667 29 236761 343219.444626553 -106458.444626553 30 312378 338432.957309371 -26054.957309371 31 339836 374726.498765112 -34890.4987651121 32 347385 336140.524244461 11244.4757555388 33 426280 383435.318684884 42844.6813151162 34 352850 389823.31078522 -36973.3107852198 35 301881 316734.549993876 -14853.5499938756 36 377516 361611.938645867 15904.0613541332 37 357312 372971.700480295 -15659.7004802946 38 458343 369269.505235234 89073.4947647661 39 354228 363061.739270887 -8833.73927088741 40 308636 375007.761817514 -66371.7618175136 41 386212 362707.757269402 23504.2427305981 42 393343 367000.896250214 26342.103749786 43 378509 365641.925761696 12867.0742383044 44 452469 340756.51495247 111712.485047530 45 364839 433461.360249555 -68622.3602495552 46 358649 321184.290007713 37464.7099922865 47 376641 350899.995335569 25741.0046644309 48 429112 367966.150996534 61145.8490034660 49 330546 344373.337423343 -13827.3374233429 50 403560 400670.259217577 2889.74078242265 51 317892 348183.695585246 -30291.6955852465 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.997014403183511 0.00597119363297793 0.00298559681648897 8 0.999993775048847 1.24499023059513e-05 6.22495115297566e-06 9 0.999985168234437 2.96635311260023e-05 1.48317655630012e-05 10 0.999967894567776 6.4210864447534e-05 3.2105432223767e-05 11 0.999934956509957 0.000130086980085241 6.50434900426207e-05 12 0.999929893098663 0.000140213802673930 7.01069013369652e-05 13 0.999999533011842 9.33976316914071e-07 4.66988158457036e-07 14 0.999998969354279 2.06129144223396e-06 1.03064572111698e-06 15 0.999997320881894 5.35823621150409e-06 2.67911810575204e-06 16 0.999992496947504 1.50061049929911e-05 7.50305249649556e-06 17 0.999997270561781 5.45887643803713e-06 2.72943821901856e-06 18 0.999997514266527 4.97146694639639e-06 2.48573347319819e-06 19 0.999993416262209 1.31674755824159e-05 6.58373779120795e-06 20 0.99998767406572 2.46518685592314e-05 1.23259342796157e-05 21 0.99996907494527 6.18501094599347e-05 3.09250547299674e-05 22 0.999996047608897 7.90478220588834e-06 3.95239110294417e-06 23 0.999991787606252 1.64247874955791e-05 8.21239374778954e-06 24 0.999990939233894 1.81215322110067e-05 9.06076610550336e-06 25 0.999975744775143 4.85104497149074e-05 2.42552248574537e-05 26 0.99993740724187 0.000125185516260524 6.25927581302621e-05 27 0.99997338292368 5.32341526420876e-05 2.66170763210438e-05 28 0.99992879929751 0.000142401404981044 7.12007024905222e-05 29 0.999980143098715 3.97138025707882e-05 1.98569012853941e-05 30 0.999985330531563 2.93389368732703e-05 1.46694684366351e-05 31 0.99996573925228 6.85214954388671e-05 3.42607477194336e-05 32 0.99990311896725 0.000193762065501712 9.68810327508558e-05 33 0.999780899596397 0.000438200807206086 0.000219100403603043 34 0.9997829349054 0.000434130189198895 0.000217065094599448 35 0.999499383642325 0.00100123271535058 0.000500616357675290 36 0.99862736626319 0.00274526747361843 0.00137263373680922 37 0.99679047322029 0.00641905355942069 0.00320952677971035 38 0.993063438873858 0.0138731222522838 0.00693656112614191 39 0.984729829877842 0.0305403402443167 0.0152701701221583 40 0.99959273378154 0.00081453243692044 0.00040726621846022 41 0.998327743049988 0.00334451390002361 0.00167225695001181 42 0.99509839635481 0.0098032072903816 0.0049016036451908 43 0.984219865255467 0.0315602694890658 0.0157801347445329 44 0.947661516276428 0.104676967447145 0.0523384837235724 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 34 0.894736842105263 NOK 5% type I error level 37 0.973684210526316 NOK 10% type I error level 37 0.973684210526316 NOK

Charts produced by software:

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Parameters (Session):

par1 = 4 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 4 ; 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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