SHw WS7

*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: Wed, 18 Nov 2009 08:00:59 -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/18/t1258557137pptwss21ig35fq6.htm/, Retrieved Wed, 18 Nov 2009 16:12:30 +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/18/t1258557137pptwss21ig35fq6.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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627 356 696 386 825 444 677 387 656 327 785 448 412 225 352 182 839 460 729 411 696 342 641 361 695 377 638 331 762 428 635 340 721 352 854 461 418 221 367 198 824 422 687 329 601 320 676 375 740 364 691 351 683 380 594 319 729 322 731 386 386 221 331 187 707 344 715 342 657 365 653 313 642 356 643 337 718 389 654 326 632 343 731 357 392 220 344 228 792 391 852 425 649 332 629 298 685 360 617 326 715 325 715 393 629 301 916 426 531 265 357 210 917 429 828 440 708 357 858 431

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] = -3.62513082855646 + 1.92818078915383X[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) -3.62513082855646 30.084793 -0.1205 0.904506 0.452253 X 1.92818078915383 0.085665 22.5083 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.947247051747586 R-squared 0.897276977044494 Adjusted R-squared 0.895505890441813 F-TEST (value) 506.625128147974 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 46.8069096162617 Sum Squared Residuals 127071.433693844 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 627 682.807230110208 -55.807230110208 2 696 740.652653784822 -44.6526537848222 3 825 852.487139555745 -27.4871395557445 4 677 742.580834573976 -65.5808345739762 5 656 626.889987224746 29.1100127752536 6 785 860.19986271236 -75.1998627123598 7 412 430.215546731056 -18.2155467310558 8 352 347.303772797441 4.69622720255887 9 839 883.338032182206 -44.3380321822058 10 729 788.857173513668 -59.8571735136681 11 696 655.812699062054 40.1873009379461 12 641 692.448134055977 -51.4481340559766 13 695 723.299026682438 -28.2990266824379 14 638 634.602710381362 3.39728961863826 15 762 821.636246929283 -59.6362469292833 16 635 651.956337483746 -16.9563374837462 17 721 675.094506953592 45.9054930464078 18 854 885.26621297136 -31.2662129713596 19 418 422.502823574440 -4.50282357444045 20 367 378.154665423902 -11.1546654239024 21 824 810.06716219436 13.9328378056397 22 687 630.746348803054 56.2536511969459 23 601 613.39272170067 -12.3927217006696 24 676 719.44266510413 -43.4426651041303 25 740 698.232676423438 41.7673235765619 26 691 673.166326164438 17.8336738355617 27 683 729.0835690499 -46.0835690498994 28 594 611.464540911516 -17.4645409115158 29 729 617.249083278977 111.750916721023 30 731 740.652653784822 -9.6526537848224 31 386 422.502823574440 -36.5028235744404 32 331 356.94467674321 -25.9446767432102 33 707 659.669060640362 47.3309393596385 34 715 655.812699062054 59.1873009379461 35 657 700.160857212592 -43.1608572125920 36 653 599.895456176593 53.1045438234072 37 642 682.807230110207 -40.8072301102075 38 643 646.171795116285 -3.17179511628472 39 718 746.437196152284 -28.4371961522839 40 654 624.961806435593 29.0381935644074 41 632 657.740879851208 -25.7408798512077 42 731 684.735410899361 46.2645891006387 43 392 420.574642785287 -28.5746427852866 44 344 436.000089098517 -92.0000890985173 45 792 750.293557730591 41.7064422694085 46 852 815.851704561822 36.1482954381783 47 649 636.530891170516 12.4691088294844 48 629 570.972744339285 58.0272556607146 49 685 690.519953266823 -5.51995326682281 50 617 624.961806435593 -7.9618064355926 51 715 623.033625646439 91.9663743535612 52 715 754.149919308899 -39.1499193088992 53 629 576.757286706747 52.2427132932532 54 916 817.779885350976 98.2201146490244 55 531 507.342778297209 23.657221702791 56 357 401.292834893748 -44.2928348937483 57 917 823.564427718437 93.435572281563 58 828 844.774416399129 -16.7744163991292 59 708 684.735410899361 23.2645891006387 60 858 827.420789296745 30.5792107032553 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.451158655941605 0.90231731188321 0.548841344058395 6 0.321904923128455 0.64380984625691 0.678095076871545 7 0.246571427382399 0.493142854764798 0.753428572617601 8 0.146255789829684 0.292511579659368 0.853744210170316 9 0.0932291090972962 0.186458218194592 0.906770890902704 10 0.0640324358489574 0.128064871697915 0.935967564151043 11 0.180486721999649 0.360973443999298 0.81951327800035 12 0.146707221913183 0.293414443826366 0.853292778086817 13 0.100209213043219 0.200418426086438 0.899790786956781 14 0.0755448679170392 0.151089735834078 0.92445513208296 15 0.0650545041365753 0.130109008273151 0.934945495863425 16 0.0415560214110757 0.0831120428221514 0.958443978588924 17 0.106156650672287 0.212313301344573 0.893843349327713 18 0.0930434760021965 0.186086952004393 0.906956523997803 19 0.0643147375540308 0.128629475108062 0.93568526244597 20 0.0463313958036079 0.0926627916072158 0.953668604196392 21 0.0545860149694612 0.109172029938922 0.945413985030539 22 0.109843584741281 0.219687169482562 0.890156415258719 23 0.0775635010869767 0.155127002173953 0.922436498913023 24 0.0730596874523704 0.146119374904741 0.92694031254763 25 0.0962895141607117 0.192579028321423 0.903710485839288 26 0.0791547375954568 0.158309475190914 0.920845262404543 27 0.083502298289148 0.167004596578296 0.916497701710852 28 0.0612000502787968 0.122400100557594 0.938799949721203 29 0.371848644080493 0.743697288160986 0.628151355919507 30 0.324348426279734 0.648696852559467 0.675651573720266 31 0.310081399031943 0.620162798063886 0.689918600968057 32 0.271017854289999 0.542035708579999 0.728982145710001 33 0.284704653002800 0.569409306005601 0.7152953469972 34 0.334570581002841 0.669141162005682 0.665429418997159 35 0.354030846812446 0.708061693624891 0.645969153187554 36 0.38381056366486 0.76762112732972 0.61618943633514 37 0.39985026801972 0.79970053603944 0.60014973198028 38 0.332039207918557 0.664078415837113 0.667960792081443 39 0.344561551050676 0.689123102101352 0.655438448949324 40 0.296182180215101 0.592364360430202 0.7038178197849 41 0.276013200007192 0.552026400014384 0.723986799992808 42 0.258389077853678 0.516778155707356 0.741610922146322 43 0.211222597967745 0.422445195935491 0.788777402032255 44 0.431058387222583 0.862116774445167 0.568941612777417 45 0.379818788559927 0.759637577119854 0.620181211440073 46 0.320222735455876 0.640445470911753 0.679777264544124 47 0.247831125871329 0.495662251742658 0.752168874128671 48 0.247626550072655 0.495253100145311 0.752373449927345 49 0.200029130506323 0.400058261012646 0.799970869493677 50 0.154613125995873 0.309226251991747 0.845386874004127 51 0.265867838311786 0.531735676623572 0.734132161688214 52 0.403056302165284 0.806112604330567 0.596943697834716 53 0.373455198433878 0.746910396867756 0.626544801566122 54 0.465324916991568 0.930649833983136 0.534675083008432 55 0.381382967994474 0.762765935988948 0.618617032005526 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 0 0 OK 10% type I error level 2 0.0392156862745098 OK

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/10770l1258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/10770l1258556454.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/1ivc51258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/1ivc51258556454.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/27t7j1258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/27t7j1258556454.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/3edyw1258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/3edyw1258556454.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/4q7871258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/4q7871258556454.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/5rz6i1258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/5rz6i1258556454.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/66vyh1258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/66vyh1258556454.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/7f8bk1258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/7f8bk1258556454.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/8n9471258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/8n9471258556454.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/9c4l31258556454.png (open in new window) http://www.freestatistics.org/blog/date/2009/Nov/18/t1258557137pptwss21ig35fq6/9c4l31258556454.ps (open in new window)

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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