*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 04:51:37 -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/t12608780129sikaqttky6pf4a.htm/, Retrieved Tue, 15 Dec 2009 12:53:45 +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/t12608780129sikaqttky6pf4a.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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106.2 431 436 443 448 460 467 81 484 431 436 443 448 460 94.7 510 484 431 436 443 448 101 513 510 484 431 436 443 109.4 503 513 510 484 431 436 102.3 471 503 513 510 484 431 90.7 471 471 503 513 510 484 96.2 476 471 471 503 513 510 96.1 475 476 471 471 503 513 106 470 475 476 471 471 503 103.1 461 470 475 476 471 471 102 455 461 470 475 476 471 104.7 456 455 461 470 475 476 86 517 456 455 461 470 475 92.1 525 517 456 455 461 470 106.9 523 525 517 456 455 461 112.6 519 523 525 517 456 455 101.7 509 519 523 525 517 456 92 512 509 519 523 525 517 97.4 519 512 509 519 523 525 97 517 519 512 509 519 523 105.4 510 517 519 512 509 519 102.7 509 510 517 519 512 509 98.1 501 509 510 517 519 512 104.5 507 501 509 510 517 519 87.4 569 507 501 509 510 517 89.9 580 569 507 501 509 510 109.8 578 580 569 507 501 509 111.7 565 578 580 569 507 501 98.6 547 565 578 580 569 507 96.9 555 547 565 578 580 569 95.1 562 555 547 565 578 580 97 561 562 555 547 565 578 112.7 555 561 562 555 547 565 102 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] = + 302.521504071985 -1.72123794877019X[t] + 0.866754928207082`Y(t-1)`[t] + 0.134467604277825`Y(t-2)`[t] -0.125706630975887`Y(t-3)`[t] -0.380598880377657`Y(t-4)`[t] + 0.226892520847359`Y(t-5)`[t] + 0.764686012256499t + 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) 302.521504071985 46.185592 6.5501 0 0 X -1.72123794877019 0.223008 -7.7183 0 0 `Y(t-1)` 0.866754928207082 0.094569 9.1653 0 0 `Y(t-2)` 0.134467604277825 0.14831 0.9067 0.368272 0.184136 `Y(t-3)` -0.125706630975887 0.137991 -0.911 0.366013 0.183006 `Y(t-4)` -0.380598880377657 0.141786 -2.6843 0.009419 0.004709 `Y(t-5)` 0.226892520847359 0.094162 2.4096 0.019108 0.009554 t 0.764686012256499 0.226221 3.3803 0.001289 0.000645 Multiple Linear Regression - Regression Statistics Multiple R 0.975983400744419 R-squared 0.95254359852864 Adjusted R-squared 0.94691317801509 F-TEST (value) 169.178056281247 F-TEST (DF numerator) 7 F-TEST (DF denominator) 59 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 11.6937803278994 Sum Squared Residuals 8067.9254030729 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 431 432.531768903009 -1.53176890300867 2 484 475.004075426774 8.99592457322623 3 510 497.513705283016 12.4862947169842 4 513 519.255266091415 -6.2552660914153 5 503 505.310271144082 -2.31027114408167 6 471 485.457024453743 -14.4570244537435 7 471 478.859849748493 -7.85984974849263 8 476 471.86923891628 4.13076108372011 9 475 485.649101921996 -10.6491019219959 10 470 479.089354298221 -9.08935429822093 11 461 472.488294294603 -11.4882942946028 12 455 464.895921904341 -9.89592190434128 13 456 456.746122086669 -0.746122086669211 14 517 492.565368603292 24.4346313967077 15 525 538.882188457978 -13.8821884579780 16 523 529.424970078703 -6.42497007870345 17 519 510.310772265787 8.68922773421258 18 509 502.105704768258 6.89429523174182 19 512 521.408045175023 -9.40804517502284 20 519 517.212799457201 1.78720054279898 21 517 527.462344748823 -10.4623447488234 22 510 515.4976941924 -5.4976941924004 23 509 510.282834693893 -1.28283469389343 24 501 515.425085774191 -14.4250857741912 25 507 501.334733707902 5.66526629209798 26 569 537.993491131073 31.0065088689265 27 580 588.798697728164 -8.79869772816376 28 578 575.245702971715 2.75429702828488 29 565 560.5931261024 4.40687389759962 30 547 548.750731569233 -1.75073156923332 31 555 545.224016401394 9.77598359860627 32 562 558.491754962855 3.50824503714482 33 561 569.885833964901 -8.88583396490128 34 555 546.597126511179 8.40287348882148 35 544 550.886144412758 -6.88614441275765 36 537 550.053183942416 -13.0531839424165 37 543 521.897196839549 21.1028031604509 38 594 571.670323663741 22.3296763362587 39 611 604.971858897687 6.02814110231254 40 613 596.965944641376 16.0340553586235 41 611 598.007914880129 12.9920851198708 42 594 588.13774861589 5.86225138411006 43 595 582.707437257085 12.2925627429148 44 591 597.448982921819 -6.4489829218187 45 589 595.995503021163 -6.99550302116352 46 584 585.232604104466 -1.23260410446644 47 573 580.929987158916 -7.92998715891604 48 567 581.578550603035 -14.5785506030352 49 569 548.950164640669 20.0498353593306 50 621 610.274389494018 10.7256105059823 51 629 634.711310205572 -5.71131020557207 52 628 618.817049253447 9.18295074655298 53 612 623.696460501156 -11.6964605011559 54 595 585.124000220637 9.8759997793629 55 597 589.069744123952 7.93025587604845 56 593 599.169121093612 -6.16912109361154 57 590 601.809320380471 -11.8093203804712 58 580 576.550036920107 3.44996307989297 59 574 589.602548388001 -15.6025483880012 60 573 570.339940115684 2.66005988431582 61 573 562.832540082349 10.1674599176508 62 620 617.602457621507 2.39754237849277 63 626 633.254306937835 -7.25430693783494 64 620 619.945041008271 0.0549589917285095 65 588 607.254794387329 -19.2547943873291 66 566 567.030586523259 -1.03058652325947 67 557 572.31978940308 -15.3197894030801 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 11 0.0240089362730382 0.0480178725460764 0.975991063726962 12 0.0129273052309199 0.0258546104618398 0.98707269476908 13 0.0303471166331473 0.0606942332662947 0.969652883366853 14 0.188290841918754 0.376581683837508 0.811709158081246 15 0.443666957688695 0.887333915377389 0.556333042311305 16 0.41447789204595 0.8289557840919 0.58552210795405 17 0.416828362236745 0.83365672447349 0.583171637763255 18 0.470383321904405 0.94076664380881 0.529616678095595 19 0.403143387443185 0.80628677488637 0.596856612556815 20 0.344176594911546 0.688353189823091 0.655823405088454 21 0.298689461675988 0.597378923351975 0.701310538324012 22 0.244542276084163 0.489084552168326 0.755457723915837 23 0.1885762780483 0.3771525560966 0.8114237219517 24 0.270351106103884 0.540702212207768 0.729648893896116 25 0.269157171605672 0.538314343211344 0.730842828394328 26 0.52410342315113 0.95179315369774 0.47589657684887 27 0.529844142575076 0.940311714849847 0.470155857424924 28 0.565626477235141 0.868747045529718 0.434373522764859 29 0.488363763878124 0.976727527756249 0.511636236121876 30 0.466225546073178 0.932451092146355 0.533774453926823 31 0.447401926000595 0.89480385200119 0.552598073999405 32 0.378163012475963 0.756326024951925 0.621836987524037 33 0.410165190000671 0.820330380001342 0.589834809999329 34 0.377331830714036 0.754663661428071 0.622668169285964 35 0.456052884859346 0.912105769718692 0.543947115140654 36 0.819292036150736 0.361415927698527 0.180707963849264 37 0.849260543282684 0.301478913434631 0.150739456717316 38 0.829696232642317 0.340607534715366 0.170303767357683 39 0.787682294873181 0.424635410253638 0.212317705126819 40 0.781828998071812 0.436342003856375 0.218171001928188 41 0.749479773457189 0.501040453085621 0.250520226542811 42 0.696579780793319 0.606840438413362 0.303420219206681 43 0.67812348640734 0.643753027185322 0.321876513592661 44 0.626335188077581 0.747329623844838 0.373664811922419 45 0.602454982586537 0.795090034826926 0.397545017413463 46 0.52118011566542 0.95763976866916 0.47881988433458 47 0.596211925297738 0.807576149404523 0.403788074702262 48 0.907081907163106 0.185836185673788 0.092918092836894 49 0.86558975515199 0.268820489696019 0.134410244848010 50 0.804783353409784 0.390433293180431 0.195216646590216 51 0.784985732929932 0.430028534140135 0.215014267070068 52 0.687731413220797 0.624537173558405 0.312268586779202 53 0.629876458636639 0.740247082726723 0.370123541363361 54 0.498381623410748 0.996763246821495 0.501618376589252 55 0.616609789631374 0.766780420737252 0.383390210368626 56 0.558445513790778 0.883108972418444 0.441554486209222 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 2 0.0434782608695652 OK 10% type I error level 3 0.0652173913043478 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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