*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: Sun, 23 Nov 2008 07:06:19 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Nov/23/t122744922796iofpxlsouj2pi.htm/, Retrieved Sun, 23 Nov 2008 14:07:08 +0000

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/2008/Nov/23/t122744922796iofpxlsouj2pi.htm/}, year = {2008}, } @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 = {2008}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

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User-defined keywords:

Dataseries X:

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299,63 0 305,945 0 382,252 0 348,846 0 335,367 0 373,617 0 312,612 0 312,232 0 337,161 0 331,476 0 350,103 0 345,127 0 297,256 0 295,979 0 361,007 0 321,803 0 354,937 0 349,432 0 290,979 0 349,576 0 327,625 0 349,377 0 336,777 0 339,134 0 323,321 0 318,86 0 373,583 0 333,03 0 408,556 0 414,646 0 291,514 0 348,857 0 349,368 0 375,765 0 364,136 0 349,53 0 348,167 1 332,856 1 360,551 1 346,969 1 392,815 1 372,02 1 371,027 1 342,672 1 367,343 1 390,786 1 343,785 1 362,6 1 349,468 1 340,624 1 369,536 1 407,782 1 392,239 1 404,824 1 373,669 1 344,902 1 396,7 1 398,911 1 366,009 1 392,484 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 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Multiple Linear Regression - Estimated Regression Equation x[t] = + 340.539333333333 + 28.9914583333333y[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) 340.539333333333 4.581171 74.3346 0 0 y 28.9914583333333 7.243467 4.0024 0.00018 9e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.465211751280971 R-squared 0.216421973529908 Adjusted R-squared 0.202912007556285 F-TEST (value) 16.0194314295435 F-TEST (DF numerator) 1 F-TEST (DF denominator) 58 p-value 0.000180264528717888 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 27.4870263647888 Sum Squared Residuals 43821.1238659583 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 299.63 340.539333333334 -40.9093333333335 2 305.945 340.539333333333 -34.5943333333334 3 382.252 340.539333333333 41.7126666666667 4 348.846 340.539333333333 8.30666666666667 5 335.367 340.539333333333 -5.17233333333331 6 373.617 340.539333333333 33.0776666666667 7 312.612 340.539333333333 -27.9273333333333 8 312.232 340.539333333333 -28.3073333333333 9 337.161 340.539333333333 -3.37833333333333 10 331.476 340.539333333333 -9.06333333333333 11 350.103 340.539333333333 9.56366666666668 12 345.127 340.539333333333 4.58766666666668 13 297.256 340.539333333333 -43.2833333333334 14 295.979 340.539333333333 -44.5603333333333 15 361.007 340.539333333333 20.4676666666667 16 321.803 340.539333333333 -18.7363333333333 17 354.937 340.539333333333 14.3976666666667 18 349.432 340.539333333333 8.89266666666669 19 290.979 340.539333333333 -49.5603333333333 20 349.576 340.539333333333 9.0366666666667 21 327.625 340.539333333333 -12.9143333333333 22 349.377 340.539333333333 8.83766666666668 23 336.777 340.539333333333 -3.76233333333334 24 339.134 340.539333333333 -1.40533333333332 25 323.321 340.539333333333 -17.2183333333333 26 318.86 340.539333333333 -21.6793333333333 27 373.583 340.539333333333 33.0436666666667 28 333.03 340.539333333333 -7.50933333333336 29 408.556 340.539333333333 68.0166666666667 30 414.646 340.539333333333 74.1066666666667 31 291.514 340.539333333333 -49.0253333333333 32 348.857 340.539333333333 8.3176666666667 33 349.368 340.539333333333 8.82866666666667 34 375.765 340.539333333333 35.2256666666667 35 364.136 340.539333333333 23.5966666666667 36 349.53 340.539333333333 8.99066666666664 37 348.167 369.530791666667 -21.3637916666667 38 332.856 369.530791666667 -36.6747916666667 39 360.551 369.530791666667 -8.97979166666668 40 346.969 369.530791666667 -22.5617916666667 41 392.815 369.530791666667 23.2842083333333 42 372.02 369.530791666667 2.48920833333332 43 371.027 369.530791666667 1.49620833333332 44 342.672 369.530791666667 -26.8587916666666 45 367.343 369.530791666667 -2.18779166666664 46 390.786 369.530791666667 21.2552083333333 47 343.785 369.530791666667 -25.7457916666666 48 362.6 369.530791666667 -6.93079166666664 49 349.468 369.530791666667 -20.0627916666666 50 340.624 369.530791666667 -28.9067916666666 51 369.536 369.530791666667 0.00520833333333888 52 407.782 369.530791666667 38.2512083333333 53 392.239 369.530791666667 22.7082083333333 54 404.824 369.530791666667 35.2932083333334 55 373.669 369.530791666667 4.13820833333332 56 344.902 369.530791666667 -24.6287916666667 57 396.7 369.530791666667 27.1692083333333 58 398.911 369.530791666667 29.3802083333333 59 366.009 369.530791666667 -3.52179166666665 60 392.484 369.530791666667 22.9532083333333 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.88093279596251 0.238134408074981 0.119067204037490 6 0.88920027105633 0.221599457887341 0.110799728943671 7 0.865893177084854 0.268213645830292 0.134106822915146 8 0.836005648009875 0.327988703980250 0.163994351990125 9 0.754554731724346 0.490890536551308 0.245445268275654 10 0.661646510962864 0.676706978074273 0.338353489037137 11 0.584833365088483 0.830333269823034 0.415166634911517 12 0.491231349065952 0.982462698131904 0.508768650934048 13 0.582888915027505 0.83422216994499 0.417111084972495 14 0.666685349898506 0.666629300202988 0.333314650101494 15 0.655982161786555 0.688035676426889 0.344017838213445 16 0.597350177220022 0.805299645559956 0.402649822779978 17 0.553331258520409 0.893337482959183 0.446668741479591 18 0.487515460018435 0.97503092003687 0.512484539981565 19 0.644145875357568 0.711708249284864 0.355854124642432 20 0.585767141078757 0.828465717842486 0.414232858921243 21 0.526811278722531 0.946377442554938 0.473188721277469 22 0.465802677956315 0.93160535591263 0.534197322043685 23 0.397188051199106 0.794376102398211 0.602811948800894 24 0.332288789367189 0.664577578734378 0.667711210632811 25 0.305140635252748 0.610281270505496 0.694859364747252 26 0.307890188310963 0.615780376621925 0.692109811689037 27 0.342752985400962 0.685505970801924 0.657247014599038 28 0.305247391599550 0.610494783199101 0.69475260840045 29 0.633059560262328 0.733880879475343 0.366940439737672 30 0.911295659712412 0.177408680575175 0.0887043402875876 31 0.977056019929382 0.0458879601412362 0.0229439800706181 32 0.966344446722746 0.0673111065545085 0.0336555532772543 33 0.952721529561908 0.0945569408761844 0.0472784704380922 34 0.948639110083906 0.102721779832188 0.0513608899160941 35 0.933171335616973 0.133657328766055 0.0668286643830275 36 0.904145600107062 0.191708799785877 0.0958543998929385 37 0.885033621932746 0.229932756134507 0.114966378067254 38 0.907984947185605 0.184030105628789 0.0920150528143946 39 0.879529464819573 0.240941070360854 0.120470535180427 40 0.870417600365328 0.259164799269345 0.129582399634672 41 0.868677516553668 0.262644966892664 0.131322483446332 42 0.819955236689176 0.360089526621649 0.180044763310824 43 0.75875580188959 0.482488396220819 0.241244198110410 44 0.772660490275926 0.454679019448148 0.227339509724074 45 0.704186269091595 0.59162746181681 0.295813730908405 46 0.661663117356812 0.676673765286376 0.338336882643188 47 0.683467404123338 0.633065191753325 0.316532595876662 48 0.613564303606265 0.77287139278747 0.386435696393735 49 0.622801520844173 0.754396958311655 0.377198479155828 50 0.763992179553605 0.47201564089279 0.236007820446395 51 0.703118704728784 0.593762590542433 0.296881295271216 52 0.704524686278668 0.590950627442665 0.295475313721332 53 0.600872353233313 0.798255293533375 0.399127646766687 54 0.584969696093735 0.83006060781253 0.415030303906265 55 0.419418806321483 0.838837612642965 0.580581193678517 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 1 0.0196078431372549 OK 10% type I error level 3 0.0588235294117647 OK

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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