*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 09:20:27 -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/t12608940942z0yzpd9ajtwfaq.htm/, Retrieved Tue, 15 Dec 2009 17:21:47 +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/t12608940942z0yzpd9ajtwfaq.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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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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = -99.7566815768365 + 0.382755975409948X[t] + 1.02354018150553`Y(t-1)`[t] + 0.104099099931590`Y(t-4)`[t] + 0.00510435475890887`Y(t-5)`[t] + 4.0412038052447M1[t] + 67.3233309731705M2[t] + 21.7024053188966M3[t] + 3.10372942599408M4[t] -8.80075905104333M5[t] -17.7562793542933M6[t] + 4.46357831698049M7[t] + 4.86982749424487M8[t] + 1.88816508082327M9[t] -4.80574981277827M10[t] -2.9185475181839M11[t] -0.470059344867009t + 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) -99.7566815768365 41.611218 -2.3974 0.020291 0.010146 X 0.382755975409948 0.240011 1.5947 0.117072 0.058536 `Y(t-1)` 1.02354018150553 0.074896 13.6661 0 0 `Y(t-4)` 0.104099099931590 0.174031 0.5982 0.55243 0.276215 `Y(t-5)` 0.00510435475890887 0.159946 0.0319 0.974669 0.487334 M1 4.0412038052447 3.873252 1.0434 0.301799 0.1509 M2 67.3233309731705 5.603693 12.0141 0 0 M3 21.7024053188966 6.548458 3.3141 0.001715 0.000858 M4 3.10372942599408 7.725819 0.4017 0.689591 0.344796 M5 -8.80075905104333 7.69218 -1.1441 0.258023 0.129011 M6 -17.7562793542933 9.010969 -1.9705 0.054329 0.027164 M7 4.46357831698049 4.085768 1.0925 0.279863 0.139931 M8 4.86982749424487 4.134005 1.178 0.244376 0.122188 M9 1.88816508082327 4.263069 0.4429 0.65974 0.32987 M10 -4.80574981277827 5.161882 -0.931 0.356323 0.178162 M11 -2.9185475181839 3.500481 -0.8338 0.408384 0.204192 t -0.470059344867009 0.172063 -2.7319 0.008679 0.004339 Multiple Linear Regression - Regression Statistics Multiple R 0.995733449711533 R-squared 0.99148510287443 Adjusted R-squared 0.988760335794248 F-TEST (value) 363.878846777648 F-TEST (DF numerator) 16 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.38069327362718 Sum Squared Residuals 1447.59300524284 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 431 440.995986249429 -9.99598624942896 2 484 487.759982902138 -3.75998290213822 3 510 500.578636629142 9.42136337085846 4 513 509.779093282283 3.22090671771729 5 503 503.134090215368 -0.134090215368056 6 471 486.247271849365 -15.2472718493649 7 471 473.780922453283 -2.78092245328301 8 476 476.267280673961 -0.267280673961445 9 475 476.86930629062 -1.86930629062032 10 470 469.088861281805 0.911138718195271 11 461 464.114971643031 -3.11497164303055 12 455 457.451062109505 -2.45106210950471 13 456 455.835829288319 0.164170711680960 14 517 511.988300698301 5.01169930169949 15 525 529.705664547819 -4.70566454781862 16 523 523.819505405741 -0.819505405740737 17 519 511.65305925204 7.34694074795991 18 509 500.316428196518 8.68357180348152 19 512 509.26225018614 2.73774981386033 20 519 514.168579468475 4.83142053152457 21 517 517.301931481317 -0.301931481317353 22 510 510.24461865493 -0.244618654929754 23 509 503.724792952717 5.27520704728275 24 501 504.133070221441 -3.13307022144072 25 507 501.793063755847 5.20693624415297 26 569 563.46234307939 5.53765692061006 27 580 581.647909688873 -1.64790968887270 28 578 580.61706320411 -2.61706320411020 29 565 567.506431133992 -2.5064311339919 30 547 546.245496172745 0.754503827255213 31 555 550.382446168155 4.61755383184498 32 562 557.665946399344 4.33405360065648 33 561 560.742745256244 0.257254743755970 34 555 556.624359239572 -1.62435923957166 35 544 548.890166955041 -4.89016695504074 36 537 538.744083804635 -1.74408380463454 37 543 540.440662033594 2.55933796640645 38 594 599.578128581498 -5.57812858149836 39 611 608.109882780192 2.89011721980806 40 613 612.278167400739 0.721832599260754 41 611 601.085091351565 9.91490864843475 42 594 592.502473322878 1.49752667712190 43 595 598.00175661179 -3.00175661178942 44 591 596.577167028587 -5.57716702858733 45 589 589.330877821964 -0.330877821964188 46 584 581.708182395737 2.29181760426274 47 573 577.297713153688 -4.29771315368823 48 567 566.27701420105 0.722985798949873 49 569 569.525846364973 -0.525846364972995 50 621 621.223343153339 -0.223343153338821 51 629 632.85062415551 -3.85062415551058 52 628 627.987697437522 0.0123025624782818 53 612 611.973062884929 0.0269371150711113 54 595 592.704194567373 2.29580543262719 55 597 595.664115214937 1.33588478506305 56 593 596.321026429632 -3.32102642963226 57 590 587.755139149854 2.2448608501459 58 580 581.333978427957 -1.33397842795659 59 574 566.972355295523 7.02764470447678 60 573 566.39476966337 6.6052303366301 61 573 570.408612307838 2.59138769216159 62 620 620.987901585334 -0.987901585334136 63 626 628.107282198465 -2.10728219846462 64 620 620.518473269605 -0.518473269605384 65 588 602.648265162106 -14.6482651621058 66 566 563.984135891121 2.01586410887912 67 557 559.908509365696 -2.90850936569592 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 20 0.991031090418966 0.0179378191620689 0.00896890958103446 21 0.980480739497157 0.0390385210056862 0.0195192605028431 22 0.979965199431541 0.0400696011369174 0.0200348005684587 23 0.963125401502 0.073749196995999 0.0368745984979995 24 0.947283747100046 0.105432505799908 0.052716252899954 25 0.913758035570376 0.172483928859248 0.086241964429624 26 0.890289125525815 0.219421748948369 0.109710874474185 27 0.872630686463213 0.254738627073574 0.127369313536787 28 0.865523499670982 0.268953000658036 0.134476500329018 29 0.85740535179969 0.285189296400619 0.142594648200310 30 0.797845219316633 0.404309561366735 0.202154780683367 31 0.74102738373671 0.517945232526581 0.258972616263290 32 0.715831903255322 0.568336193489355 0.284168096744678 33 0.630818268457685 0.738363463084631 0.369181731542315 34 0.542866406210782 0.914267187578437 0.457133593789219 35 0.534159790556587 0.931680418886827 0.465840209443413 36 0.5321923496078 0.9356153007844 0.4678076503922 37 0.432833040337926 0.865666080675852 0.567166959662074 38 0.462684875128174 0.925369750256348 0.537315124871826 39 0.369301831757289 0.738603663514577 0.630698168242711 40 0.28697433970062 0.57394867940124 0.71302566029938 41 0.802059801433982 0.395880397132036 0.197940198566018 42 0.721772522136285 0.556454955727429 0.278227477863715 43 0.609316034326797 0.781367931346405 0.390683965673203 44 0.542856323667931 0.914287352664137 0.457143676332069 45 0.434957579645048 0.869915159290097 0.565042420354952 46 0.298746710327182 0.597493420654363 0.701253289672819 47 0.333503329456493 0.667006658912986 0.666496670543507 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 3 0.107142857142857 NOK 10% type I error level 4 0.142857142857143 NOK

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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