Multiple regression mini tutorial

*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: Fri, 03 Dec 2010 17:38:13 +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/03/t12913979983leqheslkfcfbdq.htm/, Retrieved Fri, 03 Dec 2010 18:39:58 +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/03/t12913979983leqheslkfcfbdq.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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Dataseries X:

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162556 1081 213118 230380558 6282929 29790 309 81767 25266003 4324047 87550 458 153198 70164684 4108272 84738 588 -26007 -15292116 -1212617 54660 299 126942 37955658 1485329 42634 156 157214 24525384 1779876 40949 481 129352 62218312 1367203 42312 323 234817 75845891 2519076 37704 452 60448 27322496 912684 16275 109 47818 5212162 1443586 25830 115 245546 28237790 1220017 12679 110 48020 5282200 984885 18014 239 -1710 -408690 1457425 43556 247 32648 8064056 -572920 24524 497 95350 47388950 929144 6532 103 151352 15589256 1151176 7123 109 288170 31410530 790090 20813 502 114337 57397174 774497 37597 248 37884 9395232 990576 17821 373 122844 45820812 454195 12988 119 82340 9798460 876607 22330 84 79801 6703284 711969 13326 102 165548 16885896 702380 16189 295 116384 34333280 264449 7146 105 134028 14072940 450033 15824 64 63838 4085632 541063 26088 267 74996 20023932 588864 11326 129 31080 4009320 -37216 8568 37 32168 1190216 783310 14416 361 49857 17998377 467359 3369 28 87161 2440 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 R Framework error message The field 'Names of X columns' contains a hard return which cannot be interpreted. Please, resubmit your request without hard returns in the 'Names of X columns'. Multiple Linear Regression - Estimated Regression Equation Wealth [t] = + 715947.084583941 + 21.495781857674Costs[t] -3855.32416995473Trades[t] -1.17140535691399Dividends[t] + 0.0299110528431907TrDiv[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) 715947.084583941 312229.455963 2.293 0.026684 0.013342 Costs 21.495781857674 5.729245 3.7519 0.00051 0.000255 Trades -3855.32416995473 1100.628291 -3.5028 0.00107 0.000535 Dividends -1.17140535691399 2.1403 -0.5473 0.586932 0.293466 TrDiv 0.0299110528431907 0.006901 4.3345 8.4e-05 4.2e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.812965176413633 R-squared 0.66091237806125 Adjusted R-squared 0.630086230612272 F-TEST (value) 21.4399927579395 F-TEST (DF numerator) 4 F-TEST (DF denominator) 44 p-value 7.21631754352359e-10 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 721742.058513831 Sum Squared Residuals 22920110357222.4 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 6282929 6683887.45004589 -400958.450045888 2 4324047 824961.706658465 3499085.29334154 3 4108272 2751406.9293653 1356865.07063470 4 -1212617 -156412.514936801 -1056204.4850632 5 1485329 1724757.74742663 -239428.747426634 6 1779876 1580386.41283274 199489.587167257 7 1367203 1451258.52244419 -84055.5224441936 8 2519076 2373771.46159587 145304.538404128 9 912684 530253.029575272 382430.970424728 10 1443586 745447.591444877 698138.408555123 11 1220017 1384808.97951899 -164791.979518985 12 984885 666151.722151661 318733.277848339 13 1457425 171528.377322739 1285896.62267726 14 -572920 902912.652251892 -1475832.65225189 15 929144 632773.413245609 296370.586754391 16 1151176 748255.658595311 402920.341404689 17 790090 1050789.34519181 -260699.345191809 18 774497 810840.98934077 -36343.9893407696 19 990576 804647.361222844 185928.638777156 20 454195 887636.107061598 -433441.107061598 21 876607 732979.46088039 143627.539119610 22 711969 979123.626249426 -267154.626249426 23 702380 920307.921818147 -217927.921818147 24 264449 817233.378242163 -552784.378242163 25 450033 728682.238716216 -278649.238716216 26 541063 856780.969297827 -315717.969297827 27 588864 758443.660342362 -169579.660342362 28 -37216 545587.195872171 -582803.195872171 29 783310 755394.795401768 27915.2045982317 30 467359 114005.899149518 353353.100850482 31 688779 651314.598341962 37464.4016580384 32 608419 787787.72105513 -179368.721055130 33 696348 700271.842610513 -3923.84261051299 34 597793 654225.690650008 -56432.6906500078 35 821730 523984.615395247 297745.384604753 36 377934 871871.424980693 -493937.424980693 37 651939 724438.202882798 -72499.2028827978 38 697458 659119.90322267 38338.0967773304 39 700368 690730.492193417 9637.50780658263 40 225986 497026.724075425 -271040.724075425 41 348695 716776.189750495 -368081.189750495 42 373683 838648.276578457 -464965.276578457 43 501709 466174.313032287 35534.6869677131 44 413743 690219.442438361 -276476.442438361 45 379825 623640.588627346 -243815.588627346 46 336260 655776.642924459 -319516.642924459 47 636765 559996.41197262 76768.58802738 48 481231 794494.4236033 -313263.423603300 49 469107 660960.407339162 -191853.407339162 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.999999544831541 9.10336917402721e-07 4.55168458701361e-07 9 0.99999896131506 2.07736987972554e-06 1.03868493986277e-06 10 0.999999981572859 3.68542826753797e-08 1.84271413376898e-08 11 0.999999975032549 4.99349022652152e-08 2.49674511326076e-08 12 0.99999998110526 3.77894789084448e-08 1.88947394542224e-08 13 0.999999996862117 6.27576503541607e-09 3.13788251770803e-09 14 0.999999999999861 2.77044379890532e-13 1.38522189945266e-13 15 0.999999999999743 5.14462394796567e-13 2.57231197398284e-13 16 0.999999999999943 1.12926633765013e-13 5.64633168825065e-14 17 0.999999999999702 5.95001089092457e-13 2.97500544546228e-13 18 0.99999999999929 1.41925707076958e-12 7.09628535384791e-13 19 0.999999999999133 1.73471916365727e-12 8.67359581828634e-13 20 0.999999999996937 6.12646718667591e-12 3.06323359333795e-12 21 0.999999999997225 5.55083032034328e-12 2.77541516017164e-12 22 0.9999999999917 1.66015590504748e-11 8.3007795252374e-12 23 0.999999999960955 7.80905564356612e-11 3.90452782178306e-11 24 0.999999999921138 1.57724225776189e-10 7.88621128880944e-11 25 0.999999999672057 6.55886441583376e-10 3.27943220791688e-10 26 0.999999998528358 2.94328424053372e-09 1.47164212026686e-09 27 0.999999994901755 1.01964909135522e-08 5.09824545677611e-09 28 0.999999999136831 1.72633764939764e-09 8.63168824698822e-10 29 0.999999999311835 1.37633034144963e-09 6.88165170724813e-10 30 0.999999997375362 5.24927665137684e-09 2.62463832568842e-09 31 0.999999990664133 1.86717342542710e-08 9.33586712713552e-09 32 0.999999954789001 9.04219969326104e-08 4.52109984663052e-08 33 0.999999924360187 1.51279626529823e-07 7.56398132649113e-08 34 0.999999653620943 6.927581144794e-07 3.463790572397e-07 35 0.99999777938291 4.4412341817148e-06 2.2206170908574e-06 36 0.9999896051801 2.07896398017451e-05 1.03948199008725e-05 37 0.999982021717567 3.59565648659548e-05 1.79782824329774e-05 38 0.999989758799179 2.04824016423153e-05 1.02412008211576e-05 39 0.99994012713588 0.000119745728241537 5.98728641207684e-05 40 0.999937131187974 0.000125737624051992 6.2868812025996e-05 41 0.999827160805453 0.000345678389094785 0.000172839194547393 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 34 1 NOK 5% type I error level 34 1 NOK 10% type I error level 34 1 NOK

Charts produced by software:

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Parameters (Session):

par1 = 5 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 5 ; 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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