*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, 15 Dec 2009 09:14:46 -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/15/t1260893760dgmy7bl50g3chph.htm/, Retrieved Tue, 15 Dec 2009 17:16:13 +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/15/t1260893760dgmy7bl50g3chph.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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594 0 611 613 611 594 595 0 594 611 613 611 591 0 595 594 611 613 589 0 591 595 594 611 584 0 589 591 595 594 573 0 584 589 591 595 567 0 573 584 589 591 569 0 567 573 584 589 621 0 569 567 573 584 629 0 621 569 567 573 628 0 629 621 569 567 612 0 628 629 621 569 595 0 612 628 629 621 597 0 595 612 628 629 593 0 597 595 612 628 590 0 593 597 595 612 580 0 590 593 597 595 574 0 580 590 593 597 573 0 574 580 590 593 573 0 573 574 580 590 620 0 573 573 574 580 626 0 620 573 573 574 620 0 626 620 573 573 588 0 620 626 620 573 566 0 588 620 626 620 557 0 566 588 620 626 561 0 557 566 588 620 549 0 561 557 566 588 532 0 549 561 557 566 526 0 532 549 561 557 511 0 526 532 549 561 499 0 511 526 532 549 555 0 499 511 526 532 565 0 555 499 511 526 542 0 565 555 499 511 527 0 542 565 555 499 510 0 527 542 565 555 514 0 510 527 542 565 517 0 514 510 527 542 508 0 517 514 510 527 493 0 508 517 514 510 490 0 493 508 517 514 469 0 490 493 508 517 478 0 469 490 493 508 528 0 478 46 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 18.1060876871175 + 12.0059262813842X[t] + 0.882180375958156Y1[t] + 0.0827073052168757Y2[t] + 0.0159908284361135Y3[t] -0.0437113176740089Y4[t] + 5.76211827668037M1[t] + 22.5118645785743M2[t] + 23.9561811611634M3[t] + 17.4162490894352M4[t] + 11.6541761387199M5[t] + 15.0607192910846M6[t] + 9.38902965726225M7[t] + 22.0663759722488M8[t] + 72.1734533323831M9[t] + 35.0106605241267M10[t] + 8.61892947473618M11[t] -0.226257822163898t + 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) 18.1060876871175 25.602404 0.7072 0.483256 0.241628 X 12.0059262813842 4.002842 2.9994 0.004486 0.002243 Y1 0.882180375958156 0.143842 6.133 0 0 Y2 0.0827073052168757 0.198326 0.417 0.678733 0.339367 Y3 0.0159908284361135 0.198814 0.0804 0.936267 0.468134 Y4 -0.0437113176740089 0.141035 -0.3099 0.758109 0.379055 M1 5.76211827668037 7.296741 0.7897 0.434046 0.217023 M2 22.5118645785743 9.598605 2.3453 0.023694 0.011847 M3 23.9561811611634 10.751501 2.2282 0.031151 0.015575 M4 17.4162490894352 10.383859 1.6772 0.100751 0.050375 M5 11.6541761387199 7.980524 1.4603 0.151469 0.075734 M6 15.0607192910846 7.96088 1.8918 0.065259 0.032629 M7 9.38902965726225 8.888024 1.0564 0.296699 0.148349 M8 22.0663759722488 9.227786 2.3913 0.021236 0.010618 M9 72.1734533323831 9.272423 7.7837 0 0 M10 35.0106605241267 13.47285 2.5986 0.012771 0.006385 M11 8.61892947473618 10.48962 0.8217 0.415802 0.207901 t -0.226257822163898 0.118262 -1.9132 0.062397 0.031199 Multiple Linear Regression - Regression Statistics Multiple R 0.991373738052704 R-squared 0.982821888500591 Adjusted R-squared 0.976030542093848 F-TEST (value) 144.716795409636 F-TEST (DF numerator) 17 F-TEST (DF denominator) 43 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 6.32429195090337 Sum Squared Residuals 1719.85675325123 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 594 597.159609426116 -3.15960942611592 2 595 597.809506160538 -2.80950616053790 3 591 598.384316816014 -7.38431681601391 4 589 587.98769127544 1.01230872455958 5 584 580.663253758672 3.33674624132844 6 573 579.15954796723 -6.1595479672294 7 567 563.286943463443 3.71305653655711 8 569 569.542637836298 -0.542637836298484 9 621 620.734231770457 0.265768229543422 10 629 629.768854824092 -0.76885482409166 11 628 614.803338394396 13.1966616056037 12 612 606.481729606603 5.5182703933971 13 595 595.674934849013 -0.674934849012486 14 597 595.512358684156 1.48764131584432 15 593 596.876612070506 -3.87661207050647 16 590 587.174652282586 2.82534771741423 17 580 578.984025218295 1.01597478170509 18 574 572.942998924171 1.05700107582893 19 573 561.051768945655 11.9482310543451 20 573 572.095658899879 0.904341100120988 21 620 622.234939338756 -2.23493933875593 22 626 626.554643455977 -0.554643455976893 23 620 609.160691503038 10.8393084969615 24 588 596.270234718188 -8.27023471818812 25 566 571.12159235068 -5.12159235068064 26 557 565.23226591573 -8.23226591573049 27 561 556.441701973849 4.55829802615049 28 549 553.506831776812 -4.50683177681194 29 532 538.080897246205 -6.08089724620548 30 526 525.728993695326 0.271006304674322 31 511 512.765204582974 -1.76520458297425 32 499 511.740035333797 -12.7400353337975 33 555 550.441228211858 4.55877178814162 34 565 561.484196451995 3.51580354800536 35 542 548.783400256043 -6.78340025604353 36 527 521.895159568785 5.10484043121488 37 510 510.008120858558 -0.00812085855775108 38 514 509.489031137975 4.51096886202470 39 517 513.595285093507 3.40471490649321 40 508 510.190291230053 -2.19029123005287 41 493 497.317514703403 -4.31751470340342 42 490 486.393855861892 3.60614413810766 43 469 476.333706290831 -7.33370629083146 44 478 470.164424405407 7.83557559459338 45 528 526.855711197248 1.14428880275219 46 534 534.115371667551 -0.115371667550825 47 518 529.993611721053 -11.9936117210525 48 506 507.935921802863 -1.93592180286277 49 502 499.472679949328 2.52732005067240 50 516 510.956838101601 5.04316189839939 51 528 524.702084046123 3.29791595387669 52 533 530.140533435109 2.85946656489101 53 536 529.954309073425 6.04569092657538 54 537 535.774603551381 1.22539644861848 55 524 530.562376717097 -6.56237671709654 56 536 531.457243524618 4.54275647538163 57 587 590.733889481681 -3.73388948168130 58 597 599.076933600386 -2.07693360038598 59 581 586.258958125469 -5.25895812546921 60 564 564.416954303561 -0.416954303561106 61 558 551.563062566306 6.4369374336944 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 21 0.074310899095243 0.148621798190486 0.925689100904757 22 0.0255947008182866 0.0511894016365732 0.974405299181713 23 0.140615145221131 0.281230290442262 0.85938485477887 24 0.349476407083016 0.698952814166032 0.650523592916984 25 0.339035667078921 0.678071334157843 0.660964332921079 26 0.331922944002392 0.663845888004785 0.668077055997608 27 0.523688738602267 0.952622522795465 0.476311261397733 28 0.435416585622052 0.870833171244104 0.564583414377948 29 0.360414037657881 0.720828075315762 0.639585962342119 30 0.46902130747697 0.93804261495394 0.53097869252303 31 0.520889581406392 0.958220837187216 0.479110418593608 32 0.828538874303277 0.342922251393446 0.171461125696723 33 0.91285085795762 0.174298284084761 0.0871491420423804 34 0.95320584756662 0.0935883048667606 0.0467941524333803 35 0.960633156456812 0.078733687086376 0.039366843543188 36 0.97143496438148 0.0571300712370407 0.0285650356185204 37 0.939453052318693 0.121093895362613 0.0605469476813065 38 0.93868564946609 0.122628701067820 0.0613143505339099 39 0.933957224173573 0.132085551652855 0.0660427758264275 40 0.9330068363165 0.133986327366998 0.066993163683499 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 4 0.2 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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