Workshop 7

*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: Thu, 19 Nov 2009 11:27:50 -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/Nov/19/t1258655661wqqcw85b65x5r9h.htm/, Retrieved Thu, 19 Nov 2009 19:34:33 +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/Nov/19/t1258655661wqqcw85b65x5r9h.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:

Workshop 7 link 1

Dataseries X:

» Textbox « » Textfile « » CSV «

449 0 452 0 462 0 455 0 461 0 461 0 463 0 462 0 456 0 455 0 456 0 472 0 472 0 471 0 465 0 459 0 465 0 468 0 467 0 463 0 460 0 462 0 461 0 476 0 476 0 471 0 453 0 443 0 442 0 444 0 438 0 427 0 424 0 416 0 406 0 431 0 434 0 418 0 412 0 404 0 409 0 412 1 406 1 398 1 397 1 385 1 390 1 413 1 413 1 401 1 397 1 397 1 409 1 419 1 424 1 428 1 430 1 424 1 433 1 456 1 459 1

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 9 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 449.780487804878 -35.2304878048781X[t] + 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) 449.780487804878 3.202545 140.4447 0 0 X -35.2304878048781 5.593004 -6.299 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.634108968973948 R-squared 0.402094184533203 Adjusted R-squared 0.391960187660885 F-TEST (value) 39.6777490263036 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 4.10346300272479e-08 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 20.5062927052310 Sum Squared Residuals 24809.9743902439 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 449 449.780487804878 -0.780487804877668 2 452 449.780487804878 2.21951219512193 3 462 449.780487804878 12.2195121951219 4 455 449.780487804878 5.21951219512194 5 461 449.780487804878 11.2195121951219 6 461 449.780487804878 11.2195121951219 7 463 449.780487804878 13.2195121951219 8 462 449.780487804878 12.2195121951219 9 456 449.780487804878 6.21951219512194 10 455 449.780487804878 5.21951219512194 11 456 449.780487804878 6.21951219512194 12 472 449.780487804878 22.2195121951219 13 472 449.780487804878 22.2195121951219 14 471 449.780487804878 21.2195121951219 15 465 449.780487804878 15.2195121951219 16 459 449.780487804878 9.21951219512194 17 465 449.780487804878 15.2195121951219 18 468 449.780487804878 18.2195121951219 19 467 449.780487804878 17.2195121951219 20 463 449.780487804878 13.2195121951219 21 460 449.780487804878 10.2195121951219 22 462 449.780487804878 12.2195121951219 23 461 449.780487804878 11.2195121951219 24 476 449.780487804878 26.2195121951219 25 476 449.780487804878 26.2195121951219 26 471 449.780487804878 21.2195121951219 27 453 449.780487804878 3.21951219512194 28 443 449.780487804878 -6.78048780487806 29 442 449.780487804878 -7.78048780487806 30 444 449.780487804878 -5.78048780487806 31 438 449.780487804878 -11.7804878048781 32 427 449.780487804878 -22.7804878048781 33 424 449.780487804878 -25.7804878048781 34 416 449.780487804878 -33.7804878048781 35 406 449.780487804878 -43.7804878048781 36 431 449.780487804878 -18.7804878048781 37 434 449.780487804878 -15.7804878048781 38 418 449.780487804878 -31.7804878048781 39 412 449.780487804878 -37.7804878048781 40 404 449.780487804878 -45.7804878048781 41 409 449.780487804878 -40.7804878048781 42 412 414.55 -2.55 43 406 414.55 -8.55 44 398 414.55 -16.55 45 397 414.55 -17.55 46 385 414.55 -29.55 47 390 414.55 -24.55 48 413 414.55 -1.55 49 413 414.55 -1.55 50 401 414.55 -13.55 51 397 414.55 -17.55 52 397 414.55 -17.55 53 409 414.55 -5.55 54 419 414.55 4.45 55 424 414.55 9.45 56 428 414.55 13.45 57 430 414.55 15.45 58 424 414.55 9.45 59 433 414.55 18.45 60 456 414.55 41.45 61 459 414.55 44.45 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.0365457494890923 0.0730914989781845 0.963454250510908 6 0.0122322307553652 0.0244644615107304 0.987767769244635 7 0.00483501617383041 0.00967003234766082 0.99516498382617 8 0.00153598142740105 0.00307196285480211 0.9984640185726 9 0.000386865171535913 0.000773730343071825 0.999613134828464 10 9.85118513328107e-05 0.000197023702665621 0.999901488148667 11 2.20062579253031e-05 4.40125158506062e-05 0.999977993742075 12 0.000102863650815365 0.000205727301630731 0.999897136349185 13 0.000170381865178536 0.000340763730357072 0.999829618134821 14 0.000164456647733504 0.000328913295467009 0.999835543352267 15 6.87033687295073e-05 0.000137406737459015 0.99993129663127 16 2.35871625770702e-05 4.71743251541404e-05 0.999976412837423 17 9.79013755920998e-06 1.95802751184200e-05 0.99999020986244 18 5.75944391306233e-06 1.15188878261247e-05 0.999994240556087 19 3.00640034926704e-06 6.01280069853408e-06 0.99999699359965 20 1.15355885451280e-06 2.30711770902559e-06 0.999998846441146 21 4.29430240532961e-07 8.58860481065923e-07 0.99999957056976 22 1.67359084995566e-07 3.34718169991133e-07 0.999999832640915 23 6.67136756015768e-08 1.33427351203154e-07 0.999999933286324 24 4.41801586339043e-07 8.83603172678087e-07 0.999999558198414 25 2.74269899978629e-06 5.48539799957258e-06 0.999997257301 26 6.96312430953272e-06 1.39262486190654e-05 0.99999303687569 27 1.04245235985521e-05 2.08490471971042e-05 0.999989575476401 28 5.74433731216712e-05 0.000114886746243342 0.999942556626878 29 0.000221632014986371 0.000443264029972742 0.999778367985014 30 0.00053246281788267 0.00106492563576534 0.999467537182117 31 0.00181408011495241 0.00362816022990483 0.998185919885048 32 0.0108494892536173 0.0216989785072346 0.989150510746383 33 0.0365189702372532 0.0730379404745064 0.963481029762747 34 0.109158963081623 0.218317926163245 0.890841036918377 35 0.286527285880989 0.573054571761977 0.713472714119011 36 0.296161604950793 0.592323209901587 0.703838395049207 37 0.315456285991737 0.630912571983474 0.684543714008263 38 0.361922791732531 0.723845583465062 0.638077208267469 39 0.418781180330679 0.837562360661357 0.581218819669321 40 0.497201765170375 0.99440353034075 0.502798234829625 41 0.518711398161075 0.96257720367785 0.481288601838925 42 0.437367061714888 0.874734123429777 0.562632938285112 43 0.367314875375452 0.734629750750903 0.632685124624548 44 0.330375351200115 0.66075070240023 0.669624648799885 45 0.302621093085024 0.605242186170049 0.697378906914976 46 0.384625411015590 0.769250822031181 0.615374588984410 47 0.452049478453219 0.904098956906438 0.547950521546781 48 0.382438881565008 0.764877763130015 0.617561118434993 49 0.314013154140693 0.628026308281387 0.685986845859307 50 0.313008936122782 0.626017872245564 0.686991063877218 51 0.38621180529533 0.77242361059066 0.61378819470467 52 0.552001067190711 0.895997865618578 0.447998932809289 53 0.616347662574256 0.767304674851489 0.383652337425744 54 0.590353691894803 0.819292616210393 0.409646308105197 55 0.524699671305205 0.950600657389589 0.475300328694795 56 0.422697301574483 0.845394603148966 0.577302698425517 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 25 0.480769230769231 NOK 5% type I error level 27 0.519230769230769 NOK 10% type I error level 29 0.557692307692308 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No 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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