WS 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: Wed, 25 Nov 2009 11:27:00 -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/25/t12591738493fkqtxn0hvahkis.htm/, Retrieved Wed, 25 Nov 2009 19:31:01 +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/25/t12591738493fkqtxn0hvahkis.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:

WS 7

Dataseries X:

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9.5 7.8 9.2 9.2 10.9 9.6 7.8 9.5 9.2 10 9.5 7.8 9.6 9.5 9.2 9.1 7.5 9.5 9.6 9.2 8.9 7.5 9.1 9.5 9.5 9 7.1 8.9 9.1 9.6 10.1 7.5 9 8.9 9.5 10.3 7.5 10.1 9 9.1 10.2 7.6 10.3 10.1 8.9 9.6 7.7 10.2 10.3 9 9.2 7.7 9.6 10.2 10.1 9.3 7.9 9.2 9.6 10.3 9.4 8.1 9.3 9.2 10.2 9.4 8.2 9.4 9.3 9.6 9.2 8.2 9.4 9.4 9.2 9 8.2 9.2 9.4 9.3 9 7.9 9 9.2 9.4 9 7.3 9 9 9.4 9.8 6.9 9 9 9.2 10 6.6 9.8 9 9 9.8 6.7 10 9.8 9 9.3 6.9 9.8 10 9 9 7 9.3 9.8 9.8 9 7.1 9 9.3 10 9.1 7.2 9 9 9.8 9.1 7.1 9.1 9 9.3 9.1 6.9 9.1 9.1 9 9.2 7 9.1 9.1 9 8.8 6.8 9.2 9.1 9.1 8.3 6.4 8.8 9.2 9.1 8.4 6.7 8.3 8.8 9.1 8.1 6.6 8.4 8.3 9.2 7.7 6.4 8.1 8.4 8.8 7.9 6.3 7.7 8.1 8.3 7.9 6.2 7.9 7.7 8.4 8 6.5 7.9 7.9 8.1 7.9 6.8 8 7.9 7.7 7.6 6.8 7.9 8 7.9 7.1 6.4 7.6 7.9 7.9 6.8 6.1 7.1 7.6 8 6.5 5.8 6.8 7.1 7.9 6.9 6.1 6.5 6.8 7.6 8.2 7.2 6.9 6.5 7.1 8.7 7.3 8.2 6.9 6.8 8.3 6.9 8.7 8.2 6.5 7.9 6.1 8.3 8.7 6.9 7.5 5.8 7.9 8.3 8.2 7.8 6.2 7.5 7.9 8.7 8.3 7.1 7.8 7.5 8.3 8.4 7.7 8.3 7.8 7.9 8.2 7.9 8.4 8.3 7.5 7.7 7 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][t] = + 0.96516966726725 + 0.0728233212314704`X[t]`[t] + 1.51008889291274Y1[t] -0.842097557192121Y2[t] + 0.195263037470142Y4[t] -0.252072399613067M1[t] -0.377998832307196M2[t] -0.311690799845318M3[t] -0.29134313680935M4[t] -0.361762877792294M5[t] -0.114910553279112M6[t] + 0.454982972646780M7[t] -0.508263771434458M8[t] -0.288690459113628M9[t] -0.0566750074272165M10[t] -0.223600327907046M11[t] -0.00450287437187023t + 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) 0.96516966726725 0.702637 1.3736 0.177399 0.088699 `X[t]` 0.0728233212314704 0.052408 1.3896 0.172547 0.086274 Y1 1.51008889291274 0.137715 10.9653 0 0 Y2 -0.842097557192121 0.150614 -5.5911 2e-06 1e-06 Y4 0.195263037470142 0.075751 2.5777 0.01384 0.00692 M1 -0.252072399613067 0.136919 -1.841 0.073233 0.036617 M2 -0.377998832307196 0.13793 -2.7405 0.009207 0.004603 M3 -0.311690799845318 0.14187 -2.197 0.034026 0.017013 M4 -0.29134313680935 0.139356 -2.0906 0.043127 0.021563 M5 -0.361762877792294 0.132293 -2.7346 0.009347 0.004674 M6 -0.114910553279112 0.131148 -0.8762 0.386293 0.193147 M7 0.454982972646780 0.135468 3.3586 0.00176 0.00088 M8 -0.508263771434458 0.177234 -2.8678 0.006637 0.003318 M9 -0.288690459113628 0.161826 -1.784 0.082213 0.041106 M10 -0.0566750074272165 0.168711 -0.3359 0.738725 0.369362 M11 -0.223600327907046 0.13873 -1.6118 0.115077 0.057539 t -0.00450287437187023 0.00352 -1.2793 0.208343 0.104171 Multiple Linear Regression - Regression Statistics Multiple R 0.984567048731084 R-squared 0.969372273447037 Adjusted R-squared 0.956807052297104 F-TEST (value) 77.1472512803463 F-TEST (DF numerator) 16 F-TEST (DF denominator) 39 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.190041825426013 Sum Squared Residuals 1.40851992103880 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 9.5 9.550503695942 -0.0505036959419945 2 9.6 9.69736432302671 -0.0973643230267106 3 9.5 9.50133867327424 -0.00133867327424250 4 9.1 9.26011782055841 -0.160117820558413 5 8.9 8.72394831499876 0.176051685001243 6 9 8.9915159846888 0.00848401531120342 7 10.1 9.8859380617181 0.214061938281910 8 10.3 10.4169712547617 -0.116971254761724 9 10.2 9.97598188301102 0.224018116988980 10 9.6 9.91087469546602 -0.310874695466022 11 9.2 9.13239226180305 0.0676077381969513 12 9.3 9.30632996422872 -0.00632996422872172 13 9.4 9.53264096291119 -0.132640962911189 14 9.4 9.35913529905831 0.0408647009416881 15 9.2 9.25862548644105 -0.0586254864410525 16 9 8.99197880026961 0.0080211997303855 17 9 8.78113722514825 0.218862774851748 18 9 9.1482121939891 -0.148212193989105 19 9.8 9.6454209095565 0.154579090443491 20 10 9.82484280157012 0.175157198429875 21 9.8 9.67553530447108 0.12446469552892 22 9.3 9.44717525601095 -0.147175256010945 23 9 8.85261488824056 0.147385111759439 24 9 9.08606939211515 -0.0860693921151495 25 9.1 9.05035310991697 0.0496468900830317 26 9.1 8.96601884128402 0.133981158715976 27 9.1 8.87047066816748 0.229529331832516 28 9.2 8.89359778895473 0.306402211045271 29 8.8 8.9746457023919 -0.174645702391907 30 8.3 8.49962051115633 -0.199620511156325 31 8.4 8.66865273550027 -0.268652735500265 32 8.1 8.28520475655836 -0.185204756558359 33 7.7 7.87036889167993 -0.170368891679933 34 7.9 7.6415613281288 0.258438671871202 35 7.9 8.12123390636036 -0.221233906360361 36 8 8.13517993358551 -0.135179933585512 37 7.9 7.97335533027323 -0.073355330273232 38 7.6 7.64675998569078 -0.0467599856907766 39 7.1 7.31061890313359 -0.210618903133586 40 6.8 6.82172781987652 -0.0217278198765233 41 6.5 6.67345401512749 -0.173454015127493 42 6.9 6.67867414968102 0.221325850318983 43 8.2 8.08320376017732 0.116796239822682 44 8.7 8.69043410051603 0.00956589948397417 45 8.3 8.47811392083797 -0.178113920837967 46 7.9 7.70038872039424 0.199611279605765 47 7.5 7.49375894359603 0.00624105640397059 48 7.8 7.57242071007062 0.227579289929383 49 8.3 8.09314690095662 0.206853099043384 50 8.4 8.43072155094018 -0.0307215509401769 51 8.2 8.15894626898364 0.0410537310163644 52 7.7 7.83257777034072 -0.132577770340720 53 7.2 7.24681474233359 -0.0468147423335921 54 7.3 7.18197716048476 0.118022839515244 55 8.1 8.31678453304782 -0.216784533047817 56 8.5 8.38254708659377 0.117452913406235 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 20 0.0477902396133455 0.095580479226691 0.952209760386655 21 0.0726054207734517 0.145210841546903 0.927394579226548 22 0.0370297621832048 0.0740595243664096 0.962970237816795 23 0.0198432881910530 0.0396865763821059 0.980156711808947 24 0.00912551194648553 0.0182510238929711 0.990874488053515 25 0.00306129419316981 0.00612258838633962 0.99693870580683 26 0.00117909018209122 0.00235818036418244 0.998820909817909 27 0.00307493365138852 0.00614986730277704 0.996925066348612 28 0.198052567647320 0.396105135294639 0.80194743235268 29 0.245556811944107 0.491113623888214 0.754443188055893 30 0.200698680774821 0.401397361549641 0.79930131922518 31 0.782883406616426 0.434233186767149 0.217116593383574 32 0.79951256010129 0.400974879797419 0.200487439898710 33 0.877313368291784 0.245373263416432 0.122686631708216 34 0.909330971139123 0.181338057721754 0.090669028860877 35 0.901459044338306 0.197081911323388 0.0985409556616939 36 0.804133955109746 0.391732089780508 0.195866044890254 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 3 0.176470588235294 NOK 5% type I error level 5 0.294117647058824 NOK 10% type I error level 7 0.411764705882353 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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