*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 08:04:18 -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/t1260803138nvh8wzwoggg7tl1.htm/, Retrieved Mon, 14 Dec 2009 16:05:50 +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/t1260803138nvh8wzwoggg7tl1.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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Dataseries X:

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101.9 436 443 448 460 467 106.2 431 436 443 448 460 81 484 431 436 443 448 94.7 510 484 431 436 443 101 513 510 484 431 436 109.4 503 513 510 484 431 102.3 471 503 513 510 484 90.7 471 471 503 513 510 96.2 476 471 471 503 513 96.1 475 476 471 471 503 106 470 475 476 471 471 103.1 461 470 475 476 471 102 455 461 470 475 476 104.7 456 455 461 470 475 86 517 456 455 461 470 92.1 525 517 456 455 461 106.9 523 525 517 456 455 112.6 519 523 525 517 456 101.7 509 519 523 525 517 92 512 509 519 523 525 97.4 519 512 509 519 523 97 517 519 512 509 519 105.4 510 517 519 512 509 102.7 509 510 517 519 512 98.1 501 509 510 517 519 104.5 507 501 509 510 517 87.4 569 507 501 509 510 89.9 580 569 507 501 509 109.8 578 580 569 507 501 111.7 565 578 580 569 507 98.6 547 565 578 580 569 96.9 555 547 565 578 580 95.1 562 555 547 565 578 97 561 562 555 547 565 112.7 555 561 562 555 547 102.9 544 555 561 562 555 97.4 537 544 555 561 562 111.4 543 537 544 555 561 87.4 594 543 537 544 555 96.8 6 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 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = -92.3660600861995 + 0.362630771674071X[t] + 1.14703119955930`Y(t-1)`[t] -0.110653706293974`Y(t-2)`[t] -0.103815643244524`Y(t-3)`[t] + 0.183897872205844`Y(t-4)`[t] + 1.43317238272469M1[t] + 6.22529402508988M2[t] + 67.9438754681637M3[t] + 15.8729539109853M4[t] + 2.56701819620161M5[t] -2.37172747838859M6[t] -13.0528345870745M7[t] + 9.09459620655353M8[t] + 6.3060061917542M9[t] + 2.08728280715477M10[t] -2.22348237189695M11[t] -0.410683199025563t + 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) -92.3660600861995 40.930368 -2.2567 0.028432 0.014216 X 0.362630771674071 0.240401 1.5084 0.137735 0.068867 `Y(t-1)` 1.14703119955930 0.13991 8.1984 0 0 `Y(t-2)` -0.110653706293974 0.21447 -0.5159 0.608171 0.304085 `Y(t-3)` -0.103815643244524 0.236329 -0.4393 0.662349 0.331174 `Y(t-4)` 0.183897872205844 0.169614 1.0842 0.28347 0.141735 M1 1.43317238272469 3.430056 0.4178 0.677862 0.338931 M2 6.22529402508988 4.044369 1.5392 0.130049 0.065024 M3 67.9438754681637 5.705618 11.9082 0 0 M4 15.8729539109853 9.395397 1.6894 0.097362 0.048681 M5 2.56701819620161 9.354512 0.2744 0.784896 0.392448 M6 -2.37172747838859 8.773154 -0.2703 0.788012 0.394006 M7 -13.0528345870745 3.919804 -3.33 0.001637 0.000819 M8 9.09459620655353 4.469141 2.035 0.047172 0.023586 M9 6.3060061917542 4.976633 1.2671 0.210982 0.105491 M10 2.08728280715477 4.960015 0.4208 0.675688 0.337844 M11 -2.22348237189695 3.888194 -0.5719 0.569983 0.284991 t -0.410683199025563 0.162294 -2.5305 0.014585 0.007292 Multiple Linear Regression - Regression Statistics Multiple R 0.996010944770597 R-squared 0.992037802102818 Adjusted R-squared 0.989330654817775 F-TEST (value) 366.451359179527 F-TEST (DF numerator) 17 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.39666594775882 Sum Squared Residuals 1456.20016758498 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 436 442.295576143808 -6.29557614380812 2 431 440.718879653393 -9.71887965339332 3 484 486.240206147269 -2.24020614726883 4 510 499.879785212795 10.1202147872049 5 513 511.637698026276 1.36230197372448 6 503 503.476746416768 -0.476746416767555 7 471 485.055385018247 -14.0553850182473 8 471 471.457052086091 -0.457052086090865 9 476 475.383016766943 0.616983233056703 10 475 477.935624965714 -2.93562496571377 11 470 469.219189585594 0.78081041440638 12 461 463.836779012885 -2.83677901288505 13 455 455.713667087453 -0.713667087452525 14 456 455.523085117619 0.476914882381125 15 517 511.875592796857 5.12440720314337 16 525 530.432098224303 -5.43209822430274 17 523 523.301585367332 -0.301585367332146 18 519 510.691003477078 8.30899652292194 19 509 501.66696543107 7.33303456892982 20 512 510.537311634153 1.46268836584718 21 519 513.891342077552 5.10865792244803 22 517 517.116705406912 -0.116705406912436 23 510 510.222291515929 -0.222291515928921 24 509 503.073060734859 5.92693926514102 25 501 503.549909505286 -2.54990950528584 26 507 501.545502755459 5.45449724454095 27 569 563.236362189392 5.76363781060809 28 580 582.759973771037 -2.75997377103660 29 578 579.922443781353 -1.92244378135339 30 565 566.41757755764 -1.41757755764060 31 547 546.144921960388 0.85507803961171 32 555 550.287641713652 4.71235828634762 33 562 558.585457038347 3.41454296165276 34 561 561.267046907192 -0.267046907191797 35 555 556.176607655119 -1.17660765511894 36 544 548.408565249458 -4.40856524945765 37 537 536.874264980246 0.125735019753701 38 543 539.959502586603 3.0404974133972 39 594 600.259610294342 -6.25961029434169 40 611 607.725236640082 3.27476335991787 41 613 612.228142482842 0.771857517158244 42 611 601.72245549675 9.2775445032501 43 594 593.408383985959 0.591616014041096 44 595 597.951490366801 -2.95149036680107 45 591 595.817372988715 -4.8173729887153 46 589 588.35767809448 0.642321905519389 47 584 581.745853462449 2.25414653755141 48 573 577.954966029115 -4.95496602911459 49 567 564.681055719929 2.31894428007122 50 569 569.278346399407 -0.278346399407539 51 621 621.799896529754 -0.799896529754448 52 629 632.709559424026 -3.70955942402556 53 628 627.450217363976 0.549782636023795 54 612 612.390705302925 -0.39070530292505 55 595 592.840862955161 2.15913704483890 56 597 596.066438062969 0.933561937031172 57 593 597.322811128442 -4.3228111284422 58 590 587.322944625701 2.67705537429862 59 580 581.63605778091 -1.63605778090994 60 574 567.726628973684 6.27337102631626 61 573 565.885526563278 7.11447343672159 62 573 571.974683487518 1.02531651248158 63 620 621.588332042386 -1.58833204238650 64 626 627.493346727758 -1.49334672775785 65 620 620.459912978221 -0.459912978220981 66 588 603.301511748839 -15.3015117488388 67 566 562.883480649174 3.11651935082574 68 557 560.700066136334 -3.70006613633404 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.95892248465378 0.0821550306924388 0.0410775153462194 22 0.96228136152731 0.0754372769453798 0.0377186384726899 23 0.928001956836039 0.143996086327923 0.0719980431639614 24 0.900706140343339 0.198587719313323 0.0992938596566614 25 0.870298752791565 0.259402494416869 0.129701247208435 26 0.8263292515386 0.347341496922799 0.173670748461399 27 0.807994731864986 0.384010536270029 0.192005268135014 28 0.798164107725607 0.403671784548785 0.201835892274393 29 0.771914047873933 0.456171904252134 0.228085952126067 30 0.757501189768281 0.484997620463438 0.242498810231719 31 0.696912596813477 0.606174806373046 0.303087403186523 32 0.615181836920694 0.769636326158612 0.384818163079306 33 0.56428753184188 0.87142493631624 0.43571246815812 34 0.493731894464299 0.987463788928598 0.506268105535701 35 0.409335992290908 0.818671984581815 0.590664007709092 36 0.370718822481043 0.741437644962086 0.629281177518957 37 0.39594667414585 0.7918933482917 0.60405332585415 38 0.300049913927194 0.600099827854388 0.699950086072806 39 0.379760108044419 0.759520216088838 0.620239891955581 40 0.282876353745788 0.565752707491576 0.717123646254212 41 0.213517943990979 0.427035887981958 0.786482056009021 42 0.684346671617592 0.631306656764816 0.315653328382408 43 0.579738478971633 0.840523042056733 0.420261521028367 44 0.495093142471813 0.990186284943626 0.504906857528187 45 0.420603296942968 0.841206593885936 0.579396703057032 46 0.284537259739542 0.569074519479084 0.715462740260458 47 0.165659659573057 0.331319319146114 0.834340340426943 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 0 0 OK 10% type I error level 2 0.0740740740740741 OK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; par2 = Include Monthly Dummies ; par3 = 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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