Paper - multiple regression analysis (3)

*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, 04 Dec 2009 04:11:41 -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/04/t1259925171k7kb8z515tllrxy.htm/, Retrieved Fri, 04 Dec 2009 12:13:03 +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/04/t1259925171k7kb8z515tllrxy.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:

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No (this computation is public)

User-defined keywords:

Dataseries X:

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216234 562325 213587 560854 209465 555332 204045 543599 200237 536662 203666 542722 241476 593530 260307 610763 243324 612613 244460 611324 233575 594167 237217 595454 235243 590865 230354 589379 227184 584428 221678 573100 217142 567456 219452 569028 256446 620735 265845 628884 248624 628232 241114 612117 229245 595404 231805 597141 219277 593408 219313 590072 212610 579799 214771 574205 211142 572775 211457 572942 240048 619567 240636 625809 230580 619916 208795 587625 197922 565742 194596 557274 194581 560576 185686 548854 178106 531673 172608 525919 167302 511038 168053 498662 202300 555362 202388 564591 182516 541657 173476 527070 166444 509846 171297 514258 169701 516922 164182 507561 161914 492622 159612 490243 151001 469357 158114 477580 186530 528379 187069 533590 174330 517945 169362 506174 166827 501866 178037 516141 186412 528222 189226 532638 191563 536322 188906 536535 186005 523597 195309 536214 223532 586570 226899 596594 2141 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Werkl[t] = -176351.657431817 + 0.682260307575196X[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) -176351.657431817 12756.565583 -13.8244 0 0 X 0.682260307575196 0.02278 29.9503 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.964623948671983 R-squared 0.930499362351528 Adjusted R-squared 0.929462039401551 F-TEST (value) 897.019932290117 F-TEST (DF numerator) 1 F-TEST (DF denominator) 67 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 7594.04708356363 Sum Squared Residuals 3863859924.19455 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 216234 207300.370025404 8933.62997459586 2 213587 206296.765112962 7290.23488703831 3 209465 202529.323694531 6935.6763054686 4 204045 194524.363505752 9520.63649424837 5 200237 189791.523752102 10445.4762478975 6 203666 193926.021216008 9739.97878399182 7 241476 228590.302923289 12885.6970767113 8 260307 240347.694803732 19959.3051962679 9 243324 241609.876372746 1714.12362725380 10 244460 240730.442836282 3729.55716371823 11 233575 229024.902739214 4550.09726078587 12 237217 229902.971755063 7314.02824493659 13 235243 226772.079203601 8470.92079639916 14 230354 225758.240386544 4595.75961345591 15 227184 222380.369603739 4803.6303962607 16 221678 214651.724839527 7026.27516047252 17 217142 210801.047663573 6340.95233642693 18 219452 211873.560867081 7578.43913291872 19 256446 247151.194590872 9294.80540912806 20 265845 252710.933837302 13134.0661626978 21 248624 252266.100116763 -3642.10011676318 22 241114 241271.475260189 -157.475260188899 23 229245 229868.858739685 -623.85873968465 24 231805 231053.944893943 751.055106057238 25 219277 228507.067165765 -9230.06716576456 26 219313 226231.046779694 -6918.0467796937 27 212610 219222.186639974 -6612.18663997372 28 214771 215405.622479398 -634.622479398071 29 211142 214429.990239566 -3287.99023956554 30 211457 214543.927710931 -3086.9277109306 31 240048 246354.314551624 -6306.31455162411 32 240636 250612.983391508 -9976.98339150848 33 230580 246592.423398968 -16012.4233989679 34 208795 224561.555807057 -15766.5558070572 35 197922 209631.653496389 -11709.6534963892 36 194596 203854.273211842 -9258.27321184243 37 194581 206107.096747456 -11526.0967474557 38 185686 198109.641422059 -12423.6414220593 39 178106 186387.72707761 -8281.72707760984 40 172608 182462.001267822 -9854.00126782216 41 167302 172309.285630796 -5007.28563079567 42 168053 163865.632064245 4187.36793575496 43 202300 202549.791503759 -249.791503758653 44 202388 208846.37188237 -6458.37188237014 45 182516 193199.413988441 -10683.4139884406 46 173476 183247.282881841 -9771.28288184121 47 166444 171496.031344166 -5052.03134416603 48 171297 174506.163821188 -3209.1638211878 49 169701 176323.705280568 -6622.70528056812 50 164182 169937.066541357 -5755.06654135672 51 161914 159744.779806491 2169.22019350914 52 159612 158121.682534769 1490.31746523053 53 151001 143871.993750754 7129.00624924607 54 158114 149482.220259945 8631.77974005523 55 186530 184140.361624457 2389.63837554286 56 187069 187695.620087231 -626.620087231485 57 174330 177021.657575218 -2691.65757521755 58 169362 168990.77149475 371.228505250083 59 166827 166051.594089716 775.405910284033 60 178037 175790.859980352 2246.14001964811 61 186412 184033.246756168 2378.75324383216 62 189226 187046.10827442 2179.89172558010 63 191563 189559.555247527 2003.44475247308 64 188906 189704.876693040 -798.87669304044 65 186005 180877.792833633 5127.20716636744 66 195309 189485.871134309 5823.1288656912 67 223532 223841.771182565 -309.771182565367 68 226899 230680.748505699 -3781.74850569913 69 214126 219716.143102658 -5590.14310265816 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.00491966394577273 0.00983932789154546 0.995080336054227 6 0.0006567240485604 0.0013134480971208 0.99934327595144 7 0.00443748092864668 0.00887496185729335 0.995562519071353 8 0.0168383846579393 0.0336767693158786 0.98316161534206 9 0.231858331024210 0.463716662048420 0.76814166897579 10 0.233451713827529 0.466903427655058 0.766548286172471 11 0.194739137194513 0.389478274389025 0.805260862805487 12 0.144270665913464 0.288541331826928 0.855729334086536 13 0.108804047988061 0.217608095976123 0.891195952011939 14 0.0893313260607567 0.178662652121513 0.910668673939243 15 0.0717984469328172 0.143596893865634 0.928201553067183 16 0.0543947937942393 0.108789587588479 0.94560520620576 17 0.0418849262913311 0.0837698525826622 0.958115073708669 18 0.0333192086392894 0.0666384172785787 0.96668079136071 19 0.0392576926799665 0.0785153853599331 0.960742307320033 20 0.133533198027726 0.267066396055453 0.866466801972274 21 0.336648304132395 0.67329660826479 0.663351695867605 22 0.415538202805836 0.831076405611673 0.584461797194164 23 0.495624628631763 0.991249257263525 0.504375371368237 24 0.561153155359583 0.877693689280834 0.438846844640417 25 0.822426106093126 0.355147787813748 0.177573893906874 26 0.894433227022836 0.211133545954328 0.105566772977164 27 0.931167821981663 0.137664356036674 0.0688321780183372 28 0.932944856248818 0.134110287502364 0.0670551437511819 29 0.936618457099105 0.12676308580179 0.063381542900895 30 0.937380885997054 0.125238228005892 0.062619114002946 31 0.948591875577706 0.102816248844587 0.0514081244222936 32 0.960039236268569 0.0799215274628622 0.0399607637314311 33 0.98090922532091 0.0381815493581801 0.0190907746790900 34 0.994425387307547 0.0111492253849055 0.00557461269245276 35 0.997220355878917 0.00555928824216534 0.00277964412108267 36 0.99780504111435 0.00438991777129975 0.00219495888564988 37 0.998786555759978 0.00242688848004422 0.00121344424002211 38 0.999588836527082 0.000822326945835043 0.000411163472917521 39 0.999667943405856 0.000664113188288777 0.000332056594144389 40 0.999848793722908 0.000302412554183457 0.000151206277091728 41 0.999810887679184 0.000378224641631663 0.000189112320815831 42 0.999683238884373 0.000633522231253034 0.000316761115626517 43 0.999409680030634 0.00118063993873141 0.000590319969365706 44 0.999094953076174 0.00181009384765228 0.000905046923826138 45 0.999664670603207 0.000670658793586074 0.000335329396793037 46 0.9999174243196 0.000165151360802194 8.25756804010972e-05 47 0.999931748203431 0.000136503593137172 6.82517965685861e-05 48 0.999905859445324 0.000188281109352024 9.41405546760121e-05 49 0.99997314104329 5.37179134199009e-05 2.68589567099505e-05 50 0.999997019192907 5.96161418561084e-06 2.98080709280542e-06 51 0.999992536816315 1.49263673709942e-05 7.46318368549712e-06 52 0.999986028260543 2.79434789146822e-05 1.39717394573411e-05 53 0.999965177442685 6.96451146296929e-05 3.48225573148465e-05 54 0.999960223049072 7.95539018555393e-05 3.97769509277696e-05 55 0.999885316519607 0.000229366960786783 0.000114683480393392 56 0.999682439828571 0.000635120342857605 0.000317560171428803 57 0.999722282686254 0.000555434627493128 0.000277717313746564 58 0.999494974800023 0.00101005039995307 0.000505025199976536 59 0.99941175638941 0.00117648722118182 0.000588243610590908 60 0.998435693601795 0.00312861279640940 0.00156430639820470 61 0.994942554422003 0.0101148911559944 0.00505744557799722 62 0.984405659174114 0.0311886816517721 0.0155943408258860 63 0.955038601896549 0.089922796206902 0.044961398103451 64 0.94444668215872 0.11110663568256 0.05555331784128 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 29 0.483333333333333 NOK 5% type I error level 34 0.566666666666667 NOK 10% type I error level 39 0.65 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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