*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 05:01:39 -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/t1260878552pozu0bk64s44d39.htm/, Retrieved Tue, 15 Dec 2009 13:02:45 +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/t1260878552pozu0bk64s44d39.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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106.2 431 436 460 467 81 484 431 448 460 94.7 510 484 443 448 101 513 510 436 443 109.4 503 513 431 436 102.3 471 503 484 431 90.7 471 471 510 484 96.2 476 471 513 510 96.1 475 476 503 513 106 470 475 471 503 103.1 461 470 471 471 102 455 461 476 471 104.7 456 455 475 476 86 517 456 470 475 92.1 525 517 461 470 106.9 523 525 455 461 112.6 519 523 456 455 101.7 509 519 517 456 92 512 509 525 517 97.4 519 512 523 525 97 517 519 519 523 105.4 510 517 509 519 102.7 509 510 512 509 98.1 501 509 519 512 104.5 507 501 517 519 87.4 569 507 510 517 89.9 580 569 509 510 109.8 578 580 501 509 111.7 565 578 507 501 98.6 547 565 569 507 96.9 555 547 580 569 95.1 562 555 578 580 97 561 562 565 578 112.7 555 561 547 565 102.9 544 555 555 547 97.4 537 544 562 555 111.4 543 537 561 562 87.4 594 543 555 561 96.8 611 594 544 555 114.1 613 611 537 544 110.3 611 613 543 537 103.9 594 611 594 543 101.6 595 594 611 594 94.6 591 595 613 611 95.9 589 591 611 613 104.7 584 589 594 611 102. 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] = + 295.147475775104 -1.66492981268395X[t] + 0.92094007036873`Y(t-1)`[t] -0.428433389858562`Y(t-4)`[t] + 0.233341475808244`Y(t-5)`[t] + 0.74824738479614t + 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) 295.147475775104 44.851297 6.5806 0 0 X -1.66492981268395 0.188413 -8.8366 0 0 `Y(t-1)` 0.92094007036873 0.055851 16.4893 0 0 `Y(t-4)` -0.428433389858562 0.083759 -5.1151 3e-06 2e-06 `Y(t-5)` 0.233341475808244 0.084726 2.7541 0.007746 0.003873 t 0.74824738479614 0.223406 3.3493 0.001393 0.000697 Multiple Linear Regression - Regression Statistics Multiple R 0.975558514327225 R-squared 0.951714414876342 Adjusted R-squared 0.94775658003014 F-TEST (value) 240.463397756413 F-TEST (DF numerator) 5 F-TEST (DF denominator) 61 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 11.6005175936534 Sum Squared Residuals 8208.8925148804 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 431 432.501157601142 -1.50115760114177 2 484 474.108746261375 9.89125373862475 3 510 500.199348181538 9.8006518184625 4 513 516.23530592598 -3.23530592598051 5 503 506.269739713973 -3.26973971397268 6 471 485.755911023593 -14.7559110235926 7 471 477.575092065238 -6.57509206523761 8 476 473.947803681711 2.05219631828935 9 475 484.451602725629 -9.45160272562922 10 470 479.172558611877 -9.17255861187697 11 461 472.677474875749 -11.6774748757491 12 455 464.826517471886 -9.82651747188625 13 456 457.148954709123 -1.14895470912309 14 517 491.861155134963 25.1388448650375 15 525 541.319868084565 -16.3198680845649 16 523 525.265201861465 -2.26520186146551 17 519 512.852986928518 6.14701307148239 18 509 502.16411368453 6.8358863154701 19 512 520.659142454107 -8.65914245410753 20 519 517.9031876477 1.09681235230044 21 517 527.011038057968 -10.0110380579682 22 510 515.282962870834 -5.28296287083424 23 509 510.461225329638 -1.46122532963783 24 501 515.648200480826 -14.6482004808263 25 507 500.86363361187 6.13636638812996 26 569 538.140171993168 30.8598280068323 27 580 590.619422268316 -10.6194222683160 28 578 571.560032797818 6.43996720218226 29 565 562.86570125216 2.13429874784043 30 547 548.289486951941 -1.28948695194066 31 555 545.045597963329 9.95440203667066 32 562 559.581862587514 2.41813741248573 33 561 568.716274937337 -7.71627493733682 34 555 547.082546024573 7.91745397542696 35 544 550.993851468043 -6.99385146804268 36 537 549.636570126001 -12.6365701260006 37 543 522.691043361157 20.3089566388435 38 594 571.260505535923 22.7394944640769 39 611 606.63907470389 4.36092529610998 40 613 594.672295020641 18.3277049793586 41 611 599.385165164565 11.6148348354351 42 594 588.497029181864 5.50297081813633 43 595 582.035681578189 12.9643184218106 44 591 598.469316031165 -7.46931603116504 45 589 594.692944109331 -5.6929441093307 46 584 585.76461367775 -1.76461367774963 47 573 580.676288876203 -7.67628887620293 48 567 581.0664406418 -14.0664406418001 49 569 549.906657440461 19.0933425595385 50 621 609.114952782242 11.8850472177580 51 629 636.657182507893 -7.65718250789253 52 628 615.64052283893 12.3594771610700 53 612 625.364902151384 -13.3649021513836 54 595 584.737958632468 10.2620413675321 55 597 589.358558226602 7.64144177339846 56 593 599.738119330317 -6.73811933031672 57 590 600.593818514004 -10.5938185140040 58 580 577.488188474635 2.51181152536500 59 574 588.844324515009 -14.8443245150091 60 573 570.929993711951 2.07000628804879 61 573 563.284065173107 9.71593482689325 62 620 616.232572559435 3.76742744056469 63 626 635.361748661103 -9.36174866110294 64 620 616.85552468174 3.14447531826002 65 588 609.014409486953 -21.0144094869528 66 566 566.149952622259 -0.149952622259350 67 557 573.181702441035 -16.1817024410347 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 9 0.176015020488895 0.35203004097779 0.823984979511105 10 0.0791278541664545 0.158255708332909 0.920872145833545 11 0.0403070078996487 0.0806140157992974 0.959692992100351 12 0.0213207233934196 0.0426414467868393 0.97867927660658 13 0.0316930795450825 0.063386159090165 0.968306920454918 14 0.141719589267491 0.283439178534982 0.85828041073251 15 0.282773239665363 0.565546479330727 0.717226760334637 16 0.257899479040741 0.515798958081482 0.742100520959259 17 0.320178847461062 0.640357694922124 0.679821152538938 18 0.337963082085929 0.675926164171858 0.662036917914071 19 0.275802471762547 0.551604943525095 0.724197528237453 20 0.235518174691753 0.471036349383506 0.764481825308247 21 0.199069692055039 0.398139384110078 0.800930307944961 22 0.158463602083447 0.316927204166893 0.841536397916553 23 0.119796615630315 0.239593231260630 0.880203384369685 24 0.186240450741195 0.372480901482391 0.813759549258805 25 0.178770692330820 0.357541384661640 0.82122930766918 26 0.421352600630157 0.842705201260313 0.578647399369843 27 0.436133661828312 0.872267323656623 0.563866338171688 28 0.454037610913579 0.908075221827157 0.545962389086421 29 0.417890459252035 0.83578091850407 0.582109540747965 30 0.384153868833564 0.768307737667127 0.615846131166436 31 0.384903801707382 0.769807603414764 0.615096198292618 32 0.319381870690545 0.63876374138109 0.680618129309455 33 0.310472128030313 0.620944256060625 0.689527871969687 34 0.279716783470794 0.559433566941589 0.720283216529206 35 0.348652077688386 0.697304155376772 0.651347922311614 36 0.67858319423312 0.642833611533761 0.321416805766880 37 0.702820769317969 0.594358461364063 0.297179230682031 38 0.683539000113318 0.632921999773364 0.316460999886682 39 0.620607226446943 0.758785547106114 0.379392773553057 40 0.621170676939902 0.757658646120196 0.378829323060098 41 0.552467258017151 0.895065483965698 0.447532741982849 42 0.49119844916687 0.98239689833374 0.50880155083313 43 0.467506667338823 0.935013334677647 0.532493332661177 44 0.428662372699190 0.857324745398379 0.57133762730081 45 0.378398983138663 0.756797966277326 0.621601016861337 46 0.313986358158856 0.627972716317713 0.686013641841144 47 0.363426142131972 0.726852284263945 0.636573857868028 48 0.703521504348107 0.592956991303786 0.296478495651893 49 0.636746444641612 0.726507110716776 0.363253555358388 50 0.555162956242792 0.889674087514415 0.444837043757208 51 0.515103250001496 0.969793499997008 0.484896749998504 52 0.485043887125869 0.970087774251737 0.514956112874131 53 0.63697326733911 0.72605346532178 0.36302673266089 54 0.530619829330585 0.93876034133883 0.469380170669415 55 0.492349152330015 0.98469830466003 0.507650847669985 56 0.399496859498894 0.798993718997788 0.600503140501106 57 0.295175435019488 0.590350870038977 0.704824564980512 58 0.24483851195475 0.4896770239095 0.75516148804525 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 1 0.02 OK 10% type I error level 3 0.06 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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