*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: Sat, 12 Dec 2009 14:45:38 -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/12/t1260654391abj9ywma9v3tzi6.htm/, Retrieved Sat, 12 Dec 2009 22:46:44 +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/12/t1260654391abj9ywma9v3tzi6.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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547344 0 565464 577992 554788 0 547344 565464 562325 0 554788 547344 560854 0 562325 554788 555332 0 560854 562325 543599 0 555332 560854 536662 0 543599 555332 542722 0 536662 543599 593530 1 542722 536662 610763 1 593530 542722 612613 1 610763 593530 611324 1 612613 610763 594167 1 611324 612613 595454 1 594167 611324 590865 1 595454 594167 589379 1 590865 595454 584428 1 589379 590865 573100 1 584428 589379 567456 1 573100 584428 569028 1 567456 573100 620735 1 569028 567456 628884 1 620735 569028 628232 1 628884 620735 612117 1 628232 628884 595404 1 612117 628232 597141 1 595404 612117 593408 1 597141 595404 590072 1 593408 597141 579799 1 590072 593408 574205 1 579799 590072 572775 1 574205 579799 572942 1 572775 574205 619567 1 572942 572775 625809 1 619567 572942 619916 1 625809 619567 587625 1 619916 625809 565742 1 587625 619916 557274 1 565742 587625 560576 1 557274 565742 548854 1 560576 557274 531673 1 548854 560576 525919 1 531673 548854 511038 1 525919 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 15281.0179858574 + 639.031849385553X[t] + 1.21329152111263Y1[t] -0.258949144967362Y2[t] -30.8517960553214M1[t] + 18948.0579225308M2[t] + 13586.0496020729M3[t] + 7223.28254784197M4[t] + 4475.86728372419M5[t] + 7396.19831256279M6[t] + 2278.67422706786M7[t] + 16154.5210622411M8[t] + 61280.2391484256M9[t] + 9312.77487688442M10[t] + 1391.16702241957M11[t] -61.7258250201235t + 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) 15281.0179858574 22689.40533 0.6735 0.503458 0.251729 X 639.031849385553 3843.731993 0.1663 0.868568 0.434284 Y1 1.21329152111263 0.131046 9.2585 0 0 Y2 -0.258949144967362 0.132899 -1.9485 0.056468 0.028234 M1 -30.8517960553214 4396.580505 -0.007 0.994427 0.497213 M2 18948.0579225308 4502.965195 4.2079 9.6e-05 4.8e-05 M3 13586.0496020729 4723.01371 2.8766 0.005711 0.002855 M4 7223.28254784197 4657.365051 1.5509 0.126652 0.063326 M5 4475.86728372419 4445.350881 1.0069 0.318409 0.159205 M6 7396.19831256279 4442.406881 1.6649 0.101618 0.050809 M7 2278.67422706786 4514.840106 0.5047 0.61578 0.30789 M8 16154.5210622411 4589.936171 3.5196 0.000876 0.000438 M9 61280.2391484256 4964.276187 12.3442 0 0 M10 9312.77487688442 8984.857863 1.0365 0.304506 0.152253 M11 1391.16702241957 4880.973129 0.285 0.776701 0.38835 t -61.7258250201235 75.912614 -0.8131 0.419657 0.209829 Multiple Linear Regression - Regression Statistics Multiple R 0.98761175325378 R-squared 0.975376975165005 Adjusted R-squared 0.968661604755461 F-TEST (value) 145.245446740925 F-TEST (DF numerator) 15 F-TEST (DF denominator) 55 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 7057.9828257733 Sum Squared Residuals 2739831686.2901 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 547344 551590.582861239 -4246.5828612385 2 554788 551767.039280395 3020.96071960472 3 562325 560067.205724888 2257.79427511161 4 560854 560859.673605126 -5.67360512606075 5 555332 554314.080982812 1017.91901718752 6 543599 550853.804599294 -7254.80459929401 7 536662 532868.922450074 3793.07754992579 8 542722 541304.690496171 1417.30950382899 9 593530 596156.591443302 -2626.59144330212 10 610763 604203.085132929 6559.91486707082 11 612613 603971.716079276 8641.2839207236 12 611324 600300.941930673 11023.0580693274 13 594167 598165.375620693 -3998.37562069333 14 595454 596599.902334393 -1145.90233439286 15 590865 597180.464856792 -6315.4648567918 16 589379 584854.909637582 4524.09036241811 17 584428 581431.134974326 2996.86502567415 18 573100 578667.532286537 -5567.53228653717 19 567456 561026.173241592 6429.82675840836 20 569028 570925.852820775 -1897.85282077531 21 620735 619358.648327325 1376.35167267541 22 628884 629658.054857045 -774.054857045401 23 628232 618172.35034428 10059.6496557201 24 612117 613818.214842736 -1701.21484273570 25 595404 594342.279201449 1061.72079855107 26 597141 597154.687373809 -13.6873738085848 27 593408 598166.257660343 -4758.2576603427 28 590072 586762.75286797 3309.24713203011 29 579799 580872.728422563 -1073.72842256341 30 574205 572131.044177603 2073.95582239707 31 572775 562824.826064234 9950.17393576647 32 572942 576352.501716143 -3410.50171614299 33 619567 621989.410938637 -2422.41093863658 34 625809 626486.693506742 -677.693506742114 35 619916 614003.221617939 5912.77838206108 36 587625 603784.041273696 -16159.0412736962 37 565742 566039.054455665 -297.054455665445 38 557274 566767.506832865 -9493.50683286486 39 560576 556736.204225926 3839.79577407417 40 548854 556510.781308972 -7656.78130897231 41 531673 538624.38693267 -6951.38693266992 42 525919 523672.83238956 2246.16761044032 43 511038 515961.308326247 -4923.3083262468 44 498662 513210.431590865 -14548.4315908650 45 555362 547112.150212999 8249.84978700113 46 564591 567081.34398164 -2490.34398163978 47 541657 555613.061230854 -13956.0612308538 48 527070 523944.698979313 3125.30102068672 49 509846 512092.57763045 -2246.57763044937 50 514258 513889.31954201 368.68045798966 51 516922 518278.767660599 -1356.76766059908 52 507561 513943.999765996 -6382.99976599607 53 492622 499087.39622553 -6465.39622552979 54 490243 486244.662341486 3998.33765851388 55 469357 482047.433178912 -12690.4331789115 56 477580 471136.787494984 6443.21250501644 57 528379 531586.087776046 -3207.08777604549 58 533590 539061.554841418 -5471.55484141809 59 517945 524246.325663254 -6301.325663254 60 506174 502462.102973582 3711.89702641773 61 501866 492139.130230504 9726.86976949558 62 516141 508877.544636528 7263.45536347194 63 528222 521889.099871452 6332.90012854778 64 532638 526425.882814354 6212.11718564623 65 536322 525846.272462099 10475.7275379014 66 536535 532031.12420552 4503.8757944799 67 523597 526156.336738942 -2559.33673894227 68 536214 524217.735881062 11996.2641189379 69 586570 587940.111301692 -1370.11130169234 70 596594 593740.267680225 2853.73231977456 71 580523 584879.325064397 -4356.32506439698 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.0704426446334585 0.140885289266917 0.929557355366541 20 0.0469136868021406 0.093827373604281 0.95308631319786 21 0.0159081342475844 0.0318162684951688 0.984091865752416 22 0.0260358698229200 0.0520717396458401 0.97396413017708 23 0.0155310005033157 0.0310620010066314 0.984468999496684 24 0.0559424734626664 0.111884946925333 0.944057526537334 25 0.0383243941379119 0.0766487882758238 0.961675605862088 26 0.0195773618003229 0.0391547236006459 0.980422638199677 27 0.0111834329186336 0.0223668658372673 0.988816567081366 28 0.00573549842303096 0.0114709968460619 0.99426450157697 29 0.00321032234895991 0.00642064469791983 0.99678967765104 30 0.00285390440372860 0.00570780880745719 0.997146095596271 31 0.00628100111570191 0.0125620022314038 0.993718998884298 32 0.00385497065459460 0.00770994130918921 0.996145029345405 33 0.00184316080489719 0.00368632160979438 0.998156839195103 34 0.00122758206163809 0.00245516412327619 0.998772417938362 35 0.0159188454208922 0.0318376908417844 0.984081154579108 36 0.228540009195388 0.457080018390776 0.771459990804612 37 0.167330897210627 0.334661794421253 0.832669102789373 38 0.195246544931806 0.390493089863613 0.804753455068194 39 0.185484585098708 0.370969170197416 0.814515414901292 40 0.151885856340917 0.303771712681834 0.848114143659083 41 0.123555267746533 0.247110535493067 0.876444732253467 42 0.107754519356311 0.215509038712621 0.89224548064369 43 0.136235213514960 0.272470427029921 0.86376478648504 44 0.301452464051464 0.602904928102928 0.698547535948536 45 0.815967030043305 0.368065939913391 0.184032969956695 46 0.83280000057082 0.334399998858359 0.167199999429180 47 0.843694592047082 0.312610815905836 0.156305407952918 48 0.9130334093079 0.173933181384202 0.086966590692101 49 0.86526813522931 0.269463729541381 0.134731864770690 50 0.847845654884616 0.304308690230768 0.152154345115384 51 0.899644030835822 0.200711938328356 0.100355969164178 52 0.974054843719595 0.0518903125608097 0.0259451562804048 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 5 0.147058823529412 NOK 5% type I error level 12 0.352941176470588 NOK 10% type I error level 16 0.470588235294118 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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