*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 04:40:48 -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/t1260877317w2dxeuefg3j72p9.htm/, Retrieved Tue, 15 Dec 2009 12:42:10 +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/t1260877317w2dxeuefg3j72p9.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 335.571157600404 -1.70477710726246X[t] + 0.89890147954134`Y(t-1)`[t] + 0.0936947988015468`Y(t-2)`[t] -0.208005305165498`Y(t-3)`[t] -0.142243480596650`Y(t-4)`[t] + 0.96939388495551t + 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) 335.571157600404 45.260139 7.4143 0 0 X -1.70477710726246 0.227642 -7.4888 0 0 `Y(t-1)` 0.89890147954134 0.096674 9.2983 0 0 `Y(t-2)` 0.0936947988015468 0.151481 0.6185 0.538533 0.269267 `Y(t-3)` -0.208005305165498 0.138109 -1.5061 0.137204 0.068602 `Y(t-4)` -0.142243480596650 0.10325 -1.3777 0.17334 0.08667 t 0.96939388495551 0.216996 4.4673 3.5e-05 1.7e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.975397615345982 R-squared 0.951400508022629 Adjusted R-squared 0.946620230123215 F-TEST (value) 199.026192209315 F-TEST (DF numerator) 6 F-TEST (DF denominator) 61 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 12.0710486924909 Sum Squared Residuals 8888.32320872567 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 436 440.902243740456 -4.90224374045607 2 431 431.272079739549 -0.272079739548576 3 484 472.798434031188 11.2015659688121 4 510 499.752940507473 10.2470594925274 5 513 520.355232311236 -7.35523231123551 6 503 501.824203931862 1.1757960681376 7 471 493.242542473447 -22.2425424734472 8 471 479.9632090583 -8.96320905829952 9 476 470.211417901527 5.78858209847298 10 475 483.924401466178 -8.92440146617794 11 470 472.137865882794 -2.13786588279427 12 461 472.422884656475 -11.4228846564752 13 455 466.205733951722 -11.2057339517219 14 456 457.517837587031 -1.5178375870309 15 517 493.286561213999 23.7134387860008 16 525 541.31172295184 -16.3117229518399 17 523 530.60246579095 -7.6024657909504 18 519 507.975818500148 11.0241814998525 19 509 513.403392580776 -4.40339258077575 20 512 520.823393181115 -8.82339318111535 21 519 515.463255319317 3.5367446806826 22 517 526.336981774414 -9.33698177441351 23 510 512.642727481362 -2.64272748136249 24 509 509.852552023586 -0.852552023585738 25 501 516.529461776951 -15.5294617769509 26 507 501.043899637646 5.95610036235386 27 569 537.012542212967 31.9874577870325 28 580 591.82033977606 -11.8203397760598 29 578 574.451579040923 3.54842095907682 30 565 557.664946445974 7.33505355402648 31 547 557.986657450614 -10.9866574506140 32 555 543.307245725519 11.6927542744807 33 562 555.838499789795 6.16150021020511 34 561 566.203946658889 -5.20394665888889 35 555 541.061642281309 13.9383577186910 36 544 550.656763160455 -6.65676316045531 37 537 549.764647008579 -12.7646470085793 38 543 520.934483559844 22.0655164401564 39 594 570.697592545136 23.3024074548642 40 611 605.068941293963 5.93105870603661 41 613 596.353123647544 16.6468763524564 42 611 595.729553631785 15.2704463682149 43 594 595.208599943498 -1.20859994349809 44 595 581.796116644877 13.2038833551227 45 591 594.136563829723 -3.13656382972284 46 589 593.20841350488 -4.20841350488008 47 584 579.213320556615 4.78667944338537 48 573 579.429671690124 -6.42967169012437 49 567 579.040112242969 -12.0401122429686 50 569 547.974489656133 21.0255103438671 51 621 609.436438006827 11.5635619931732 52 629 634.918107355607 -5.91810735560679 53 628 618.554693510729 9.44530648927072 54 612 620.718854359773 -8.71885435977258 55 595 593.207572729854 1.79242727014585 56 597 587.54763333938 9.45236666061956 57 593 597.818111421003 -4.81811142100308 58 590 598.29315378041 -8.29315378041001 59 580 572.962491403863 7.03750859613707 60 574 590.440021543954 -16.4400215439536 61 573 570.588099014714 2.41190098528601 62 573 564.59075371663 8.40924628336974 63 620 617.916410971808 2.0835890281924 64 626 636.453506264288 -10.4535062642883 65 620 622.983895416908 -2.98389541690800 66 588 606.4476787923 -18.4476787923001 67 566 576.293778706233 -10.2937787062328 68 557 573.465747896203 -16.4657478962033 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 10 0.236510475682015 0.47302095136403 0.763489524317985 11 0.165178093814837 0.330356187629675 0.834821906185162 12 0.203423241239741 0.406846482479482 0.796576758760259 13 0.151887457793023 0.303774915586047 0.848112542206977 14 0.0917150316631639 0.183430063326328 0.908284968336836 15 0.161503145696357 0.323006291392715 0.838496854303643 16 0.394973913338768 0.789947826677535 0.605026086661232 17 0.345993260746964 0.691986521493927 0.654006739253036 18 0.390604244612135 0.78120848922427 0.609395755387865 19 0.340779892732296 0.681559785464593 0.659220107267704 20 0.277482962140613 0.554965924281226 0.722517037859387 21 0.256513633080538 0.513027266161077 0.743486366919462 22 0.216300945344194 0.432601890688387 0.783699054655806 23 0.175588011856562 0.351176023713125 0.824411988143438 24 0.133748545586970 0.267497091173940 0.86625145441303 25 0.220596277813233 0.441192555626467 0.779403722186767 26 0.219488155827634 0.438976311655268 0.780511844172366 27 0.464391419873061 0.928782839746122 0.535608580126939 28 0.500222238837007 0.999555522325986 0.499777761162993 29 0.556968597064726 0.886062805870548 0.443031402935274 30 0.494832509055834 0.989665018111667 0.505167490944166 31 0.510423104258759 0.979153791482482 0.489576895741241 32 0.541466910214398 0.917066179571204 0.458533089785602 33 0.50019541379878 0.99960917240244 0.49980458620122 34 0.519887565068556 0.960224869862887 0.480112434931444 35 0.516921964370707 0.966156071258585 0.483078035629293 36 0.607098486422318 0.785803027155365 0.392901513577682 37 0.903944364820316 0.192111270359368 0.0960556351796842 38 0.90786055481905 0.184278890361899 0.0921394451809495 39 0.895293546750045 0.209412906499910 0.104706453249955 40 0.864322126320253 0.271355747359494 0.135677873679747 41 0.853872930855082 0.292254138289836 0.146127069144918 42 0.833125813006842 0.333748373986317 0.166874186993159 43 0.780629094100719 0.438741811798563 0.219370905899281 44 0.776744834405987 0.446510331188026 0.223255165594013 45 0.717295045786541 0.565409908426919 0.282704954213459 46 0.690860662865715 0.618278674268571 0.309139337134285 47 0.610082444139056 0.779835111721888 0.389917555860944 48 0.682961515556475 0.634076968887051 0.317038484443526 49 0.941115447533883 0.117769104932235 0.0588845524661173 50 0.909242526041199 0.181514947917603 0.0907574739588014 51 0.86706914548049 0.265861709039021 0.132930854519510 52 0.847585928554038 0.304828142891923 0.152414071445962 53 0.769619312580366 0.460761374839268 0.230380687419634 54 0.72170325599937 0.55659348800126 0.27829674400063 55 0.608947924779973 0.782104150440055 0.391052075220028 56 0.751008872272548 0.497982255454905 0.248991127727452 57 0.720536312537067 0.558927374925865 0.279463687462933 58 0.574999953159469 0.850000093681063 0.425000046840531 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 0 0 OK

Charts produced by software:

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Parameters (Session):

par1 = 2 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 2 ; par2 = Do not include Seasonal 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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