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 12:38:01 -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/t1258659573n2d7aiydnbuu54o.htm/, Retrieved Thu, 19 Nov 2009 20:39:45 +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/Nov/19/t1258659573n2d7aiydnbuu54o.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 3

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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 466.969012395042 -10.2662020905923X[t] -0.818501170960189t + 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) 466.969012395042 5.74343 81.3049 0 0 X -10.2662020905923 8.814711 -1.1647 0.248922 0.124461 t -0.818501170960189 0.23502 -3.4827 0.000951 0.000475 Multiple Linear Regression - Regression Statistics Multiple R 0.710988453927457 R-squared 0.505504581618156 Adjusted R-squared 0.488453015467058 F-TEST (value) 29.6456394174445 F-TEST (DF numerator) 2 F-TEST (DF denominator) 58 p-value 1.35115718613577e-09 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 18.8089221315243 Sum Squared Residuals 20518.9820014851 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 449 466.150511224081 -17.1505112240815 2 452 465.332010053122 -13.3320100531216 3 462 464.513508882161 -2.51350888216142 4 455 463.695007711201 -8.69500771120126 5 461 462.876506540241 -1.87650654024107 6 461 462.058005369281 -1.05800536928088 7 463 461.239504198321 1.76049580167931 8 462 460.421003027360 1.57899697263950 9 456 459.6025018564 -3.60250185640032 10 455 458.78400068544 -3.78400068544013 11 456 457.96549951448 -1.96549951447994 12 472 457.14699834352 14.8530016564802 13 472 456.32849717256 15.6715028274404 14 471 455.509996001599 15.4900039984006 15 465 454.691494830639 10.3085051693608 16 459 453.872993659679 5.127006340321 17 465 453.054492488719 11.9455075112812 18 468 452.235991317759 15.7640086822414 19 467 451.417490146798 15.5825098532016 20 463 450.598988975838 12.4010110241618 21 460 449.780487804878 10.2195121951219 22 462 448.961986633918 13.0380133660821 23 461 448.143485462958 12.8565145370423 24 476 447.324984291998 28.6750157080025 25 476 446.506483121037 29.4935168789627 26 471 445.687981950077 25.3120180499229 27 453 444.869480779117 8.13051922088307 28 443 444.050979608157 -1.05097960815674 29 442 443.232478437197 -1.23247843719655 30 444 442.413977266236 1.58602273376364 31 438 441.595476095276 -3.59547609527617 32 427 440.776974924316 -13.776974924316 33 424 439.958473753356 -15.9584737533558 34 416 439.139972582396 -23.1399725823956 35 406 438.321471411435 -32.3214714114354 36 431 437.502970240475 -6.50297024047523 37 434 436.684469069515 -2.68446906951505 38 418 435.865967898555 -17.8659678985549 39 412 435.047466727595 -23.0474667275947 40 404 434.228965556634 -30.2289655566345 41 409 433.410464385674 -24.4104643856743 42 412 422.325761124122 -10.3257611241218 43 406 421.507259953162 -15.5072599531616 44 398 420.688758782201 -22.6887587822014 45 397 419.870257611241 -22.8702576112412 46 385 419.051756440281 -34.051756440281 47 390 418.233255269321 -28.2332552693208 48 413 417.414754098361 -4.41475409836066 49 413 416.5962529274 -3.59625292740047 50 401 415.777751756440 -14.7777517564403 51 397 414.95925058548 -17.9592505854801 52 397 414.14074941452 -17.1407494145199 53 409 413.32224824356 -4.32224824355972 54 419 412.5037470726 6.49625292740047 55 424 411.685245901639 12.3147540983607 56 428 410.866744730679 17.1332552693208 57 430 410.048243559719 19.9517564402810 58 424 409.229742388759 14.7702576112412 59 433 408.411241217799 24.5887587822014 60 456 407.592740046838 48.4072599531616 61 459 406.774238875878 52.2257611241218 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 0.0153821869908169 0.0307643739816337 0.984617813009183 7 0.00291559344882224 0.00583118689764448 0.997084406551178 8 0.000683124759420981 0.00136624951884196 0.99931687524058 9 0.000742904950948234 0.00148580990189647 0.999257095049052 10 0.000421417056277284 0.000842834112554568 0.999578582943723 11 0.000137249114948865 0.00027449822989773 0.999862750885051 12 0.000180949951716847 0.000361899903433694 0.999819050048283 13 8.71276933881784e-05 0.000174255386776357 0.999912872306612 14 2.63892472603223e-05 5.27784945206445e-05 0.99997361075274 15 8.62057252838563e-06 1.72411450567713e-05 0.999991379427472 16 7.57122002568798e-06 1.51424400513760e-05 0.999992428779974 17 2.25060985254793e-06 4.50121970509586e-06 0.999997749390148 18 6.17887550475612e-07 1.23577510095122e-06 0.99999938211245 19 1.78367245554418e-07 3.56734491108836e-07 0.999999821632754 20 8.1132718553081e-08 1.62265437106162e-07 0.999999918867281 21 5.9515304037368e-08 1.19030608074736e-07 0.999999940484696 22 2.78414780236999e-08 5.56829560473997e-08 0.999999972158522 23 1.51324510907866e-08 3.02649021815731e-08 0.99999998486755 24 3.16177383050540e-08 6.32354766101081e-08 0.999999968382262 25 8.29022418079762e-08 1.65804483615952e-07 0.999999917097758 26 2.04132472712372e-07 4.08264945424744e-07 0.999999795867527 27 5.5490604728503e-06 1.10981209457006e-05 0.999994450939527 28 0.000299393506259262 0.000598787012518524 0.99970060649374 29 0.00297743532262446 0.00595487064524891 0.997022564677376 30 0.0144253384859235 0.0288506769718469 0.985574661514077 31 0.0614644411033239 0.122928882206648 0.938535558896676 32 0.192655907967932 0.385311815935863 0.807344092032068 33 0.348730681339598 0.697461362679196 0.651269318660402 34 0.497011091247092 0.994022182494183 0.502988908752908 35 0.646285622686308 0.707428754627385 0.353714377313692 36 0.687649074416439 0.624701851167122 0.312350925583561 37 0.792638448544899 0.414723102910202 0.207361551455101 38 0.80519707320312 0.389605853593759 0.194802926796880 39 0.800646967458137 0.398706065083727 0.199353032541863 40 0.795593146732716 0.408813706534568 0.204406853267284 41 0.755791444324741 0.488417111350518 0.244208555675259 42 0.858182972467385 0.283634055065229 0.141817027532615 43 0.913521689812013 0.172956620375974 0.086478310187987 44 0.917830701230008 0.164338597539983 0.0821692987699916 45 0.91369769931031 0.17260460137938 0.08630230068969 46 0.877911237983856 0.244177524032289 0.122088762016144 47 0.821722148516403 0.356555702967194 0.178277851483597 48 0.904990310413406 0.190019379173187 0.0950096895865935 49 0.979600778282388 0.0407984434352243 0.0203992217176121 50 0.975046365361994 0.0499072692760129 0.0249536346380064 51 0.948165146794998 0.103669706410003 0.0518348532050016 52 0.915944493402558 0.168111013194884 0.0840555065974421 53 0.84748614835644 0.305027703287119 0.152513851643559 54 0.77045081659276 0.459098366814479 0.229549183407239 55 0.691455377860369 0.617089244279262 0.308544622139631 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 23 0.46 NOK 5% type I error level 27 0.54 NOK 10% type I error level 27 0.54 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; 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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