Workshop 7

*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, 24 Dec 2010 10:50:01 +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/24/t1293187901u9thvsokagkjwio.htm/, Retrieved Fri, 24 Dec 2010 11:51:51 +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/24/t1293187901u9thvsokagkjwio.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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6282929 213118 1081 162556 4324047 81767 309 29790 4108272 153198 458 87550 -1212617 -26007 588 84738 1485329 126942 299 54660 1779876 157214 156 42634 1367203 129352 481 40949 2519076 234817 323 42312 912684 60448 452 37704 1443586 47818 109 16275 1220017 245546 115 25830 984885 48020 110 12679 1457425 -1710 239 18014 -572920 32648 247 43556 929144 95350 497 24524 1151176 151352 103 6532 790090 288170 109 7123 774497 114337 502 20813 990576 37884 248 37597 454195 122844 373 17821 876607 82340 119 12988 711969 79801 84 22330 702380 165548 102 13326 264449 116384 295 16189 450033 134028 105 7146 541063 63838 64 15824 588864 74996 267 26088 -37216 31080 129 11326 783310 32168 37 8568 467359 49857 361 14416 688779 87161 28 3369 608419 106113 85 11819 696348 80570 44 6620 597793 102129 49 4519 821730 301670 22 2220 377934 102313 155 18562 651939 88577 91 10327 697458 112477 81 5336 700368 191778 79 2365 225986 79804 145 4069 348695 128294 816 7710 373683 96448 61 13718 50 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 7 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Wealth[t] = -216241.33931436 + 5.27817321004937Dividends[t] -230.700757875951Trades[t] + 28.3041837527339Costs[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) -216241.33931436 260735.548091 -0.8294 0.411189 0.205594 Dividends 5.27817321004937 1.794062 2.942 0.005092 0.002546 Trades -230.700757875951 834.369671 -0.2765 0.783405 0.391702 Costs 28.3041837527339 6.437106 4.397 6.4e-05 3.2e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.721039475305495 R-squared 0.519897924948823 Adjusted R-squared 0.488586920054181 F-TEST (value) 16.6043193662491 F-TEST (DF numerator) 3 F-TEST (DF denominator) 46 p-value 1.89261485483705e-07 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 843234.15063263 Sum Squared Residuals 32708016308484.1 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 6282929 5260259.75371044 1022669.24628956 2 4324047 987234.149362023 3336812.85063798 3 4108272 2964734.58056345 1143537.41943655 4 -1212617 1909277.08721999 -3121894.08721999 5 1485329 1931907.68163525 -446578.681635252 6 1779876 1784292.63561575 -4416.63561574885 7 1367203 1514561.87770431 -147358.877704313 8 2519076 2146253.73750155 372822.262498454 9 912684 1065717.87653985 -153033.876539853 10 1443586 471654.555211046 971931.444788954 11 1220017 1784359.45889780 -564342.458897803 12 984885 370708.200666769 614176.799333231 13 1457425 229467.069485852 1227957.93051415 14 -572920 1131914.39998605 -1704834.39998605 15 929144 866506.001951546 62637.9980484542 16 1151176 743741.482584666 407434.517415334 17 790090 1481234.15288781 -691144.15288781 18 774497 860532.346994978 -86035.3469949775 19 990576 990655.58317345 -79.583173450461 20 454195 850508.046470685 -396313.046470685 21 876607 558524.791194374 318082.208805626 22 711969 817615.720557757 -105646.720557757 23 702380 1011199.75464848 -308819.754648477 24 264449 788213.278763628 -523764.278763628 25 450033 669219.777202197 -219186.777202197 26 541063 553827.237267971 -12764.2372679711 27 588864 856402.982134945 -267538.982134945 28 -37216 238617.071471440 -275833.071471440 29 783310 187521.254858521 595788.745141479 30 467359 371662.681805265 95696.3181947352 31 688779 332706.689689186 356072.310310814 32 608419 658759.037877714 -50340.0378777138 33 696348 386243.939315873 310104.060684127 34 597793 439415.481697454 158377.518302546 35 821730 1433785.04421903 -612055.044219029 36 377934 813408.037672894 -435474.037672894 37 651939 522586.945759953 129352.054240047 38 697458 509776.111948998 187681.888051002 39 700368 844710.197265502 -144342.197265502 40 225986 286696.109338281 -60710.1093382805 41 348695 490890.052802516 -142195.052802516 42 373683 667031.956938051 -293348.956938051 43 501709 354847.427894736 146861.572105264 44 413743 574247.434951195 -160504.434951195 45 379825 265480.157138787 114344.842861213 46 336260 418892.833142698 -82632.8331426981 47 636765 924979.609921747 -288214.609921747 48 481231 644531.66585773 -163300.665857730 49 469107 546613.479365427 -77506.4793654269 50 211928 244060.559134379 -32132.559134379 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.9999999959842 8.031599993387e-09 4.0157999966935e-09 8 0.999999999972723 5.4554394701337e-11 2.72771973506685e-11 9 0.999999999874883 2.50235015477770e-10 1.25117507738885e-10 10 0.99999999997513 4.97396308670504e-11 2.48698154335252e-11 11 0.999999999994918 1.01644405554525e-11 5.08222027772623e-12 12 0.999999999989315 2.13694855513371e-11 1.06847427756685e-11 13 0.999999999999702 5.95564203285812e-13 2.97782101642906e-13 14 1 8.54398056652114e-17 4.27199028326057e-17 15 1 1.61287117539092e-16 8.06435587695458e-17 16 1 3.35283701050966e-17 1.67641850525483e-17 17 1 7.28514433037711e-17 3.64257216518855e-17 18 1 2.65667961995887e-16 1.32833980997944e-16 19 1 5.45953629383485e-16 2.72976814691742e-16 20 0.999999999999998 3.31301813495944e-15 1.65650906747972e-15 21 0.999999999999998 3.73430084398991e-15 1.86715042199495e-15 22 0.99999999999999 1.89370901551996e-14 9.4685450775998e-15 23 0.999999999999944 1.12319236372707e-13 5.61596181863533e-14 24 0.999999999999838 3.23248861417049e-13 1.61624430708524e-13 25 0.999999999999054 1.89219820308593e-12 9.46099101542965e-13 26 0.999999999993876 1.22481942924936e-11 6.12409714624678e-12 27 0.999999999968538 6.29236179496501e-11 3.14618089748251e-11 28 0.999999999989076 2.18477857889081e-11 1.09238928944541e-11 29 0.999999999993036 1.39278135215315e-11 6.96390676076577e-12 30 0.99999999995875 8.24994804329799e-11 4.12497402164899e-11 31 0.999999999897176 2.05648437832462e-10 1.02824218916231e-10 32 0.99999999940928 1.18144140057883e-09 5.90720700289415e-10 33 0.999999999249994 1.50001231948556e-09 7.5000615974278e-10 34 0.99999999689505 6.20990099601257e-09 3.10495049800629e-09 35 0.999999980739143 3.85217139968214e-08 1.92608569984107e-08 36 0.999999909920896 1.80158207267686e-07 9.00791036338429e-08 37 0.99999983333779 3.33324420138176e-07 1.66662210069088e-07 38 0.999999934560217 1.30879567015064e-07 6.54397835075318e-08 39 0.999999501278378 9.97443244377578e-07 4.98721622188789e-07 40 0.99999673524599 6.52950802144258e-06 3.26475401072129e-06 41 0.99998725520848 2.54895830394685e-05 1.27447915197342e-05 42 0.999839458393447 0.000321083213105298 0.000160541606552649 43 0.999299853996546 0.00140029200690877 0.000700146003454385 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 37 1 NOK 5% type I error level 37 1 NOK 10% type I error level 37 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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