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*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: Fri, 20 Nov 2009 09:54:01 -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/20/t1258736155pr5jmnpftdev2q2.htm/, Retrieved Fri, 20 Nov 2009 17:56:08 +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/20/t1258736155pr5jmnpftdev2q2.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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3956.2 3977.7 3142.7 3983.4 3884.3 4152.9 3892.2 4286.1 3613 4348.1 3730.5 3949.3 3481.3 4166.7 3649.5 4217.9 4215.2 4528.2 4066.6 4232.2 4196.8 4470.9 4536.6 5121.2 4441.6 4170.8 3548.3 4398.6 4735.9 4491.4 4130.6 4251.8 4356.2 4901.9 4159.6 4745.2 3988 4666.9 4167.8 4210.4 4902.2 5273.6 3909.4 4095.3 4697.6 4610.1 4308.9 4718.1 4420.4 4185.5 3544.2 4314.7 4433 4422.6 4479.7 5059.2 4533.2 5043.6 4237.5 4436.6 4207.4 4922.6 4394 4454.8 5148.4 5058.7 4202.2 4768.9 4682.5 5171.8 4884.3 4989.3 5288.9 5202.1 4505.2 4838.4 4611.5 4876.5 5104 5875.5 4586.6 5717.9 4529.3 4778.8 4504.1 6195.9 4604.9 4625.4 4795.4 5549.8 5391.1 6397.6 5213.9 5856.7 5415 6343.8 5990.3 6615.5 4241.8 5904.6 5677.6 6861 5164.2 6553.5 3962.3 5481 4011 5435.3 3310.3 5278 3837.3 4671.8 4145.3 4891.5 3796.7 4241.6 3849.6 4152.1 4285 4484.4 4189.6 4124.7

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 1538.6271090769 + 0.577774022625357X[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) 1538.6271090769 343.692243 4.4768 3.5e-05 1.8e-05 X 0.577774022625357 0.06964 8.2966 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.733797991705475 R-squared 0.538459492630988 Adjusted R-squared 0.530636772167106 F-TEST (value) 68.8327666976111 F-TEST (DF numerator) 1 F-TEST (DF denominator) 59 p-value 1.72734049286305e-11 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 396.36172774853 Sum Squared Residuals 9269054.5342042 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 3956.2 3836.83883887379 119.361161126214 2 3142.7 3840.13215080275 -697.432150802746 3 3884.3 3938.06484763775 -53.7648476377447 4 3892.2 4015.02434745144 -122.824347451443 5 3613 4050.84633685422 -437.846336854215 6 3730.5 3820.43005663122 -89.9300566312224 7 3481.3 3946.03812914997 -464.738129149975 8 3649.5 3975.62015910839 -326.120159108393 9 4215.2 4154.90343832904 60.2965616709584 10 4066.6 3983.88232763194 82.7176723680641 11 4196.8 4121.79698683261 75.0030131673918 12 4536.6 4497.52343374588 39.0765662541222 13 4441.6 3948.40700264274 493.192997357261 14 3548.3 4080.02392499680 -531.723924996795 15 4735.9 4133.64135429643 602.258645703571 16 4130.6 3995.20669847539 135.393301524607 17 4356.2 4370.81759058414 -14.6175905841375 18 4159.6 4280.28040123874 -120.680401238744 19 3988 4235.04069526718 -247.040695267178 20 4167.8 3971.2868539387 196.513146061297 21 4902.2 4585.57619479398 316.623805206017 22 3909.4 3904.78506393452 4.61493606547546 23 4697.6 4202.22313078206 495.376869217942 24 4308.9 4264.6227252256 44.2772747744026 25 4420.4 3956.90028077533 463.499719224668 26 3544.2 4031.54868449853 -487.348684498528 27 4433 4093.89050153980 339.109498460196 28 4479.7 4461.70144434311 17.9985556568938 29 4533.2 4452.68816959015 80.511830409849 30 4237.5 4101.97933785656 135.520662143441 31 4207.4 4382.77751285248 -175.377512852483 32 4394 4112.49482506834 281.505174931659 33 5148.4 4461.41255733179 686.987442668206 34 4202.2 4293.97364557496 -91.773645574965 35 4682.5 4526.75879929072 155.741200709279 36 4884.3 4421.31504016159 462.984959838406 37 5288.9 4544.26535217627 744.63464782373 38 4505.2 4334.12894014743 171.071059852573 39 4611.5 4356.14213040945 255.357869590547 40 5104 4933.33837901219 170.661620987815 41 4586.6 4842.28119304643 -255.681193046428 42 4529.3 4299.69360839896 229.606391601044 43 4504.1 5118.45717586135 -614.357175861349 44 4604.9 4211.06307332823 393.836926671774 45 4795.4 4745.15737984311 50.2426201568932 46 5391.1 5234.99419622488 156.105803775116 47 5213.9 4922.47622738683 291.423772613171 48 5415 5203.90995380764 211.09004619236 49 5990.3 5360.89115575495 629.408844245051 50 4241.8 4950.15160307058 -708.351603070583 51 5677.6 5502.73467830947 174.865321690526 52 5164.2 5325.06916635218 -160.869166352177 53 3962.3 4705.40652708648 -743.106527086482 54 4011 4679.0022542525 -668.002254252503 55 3310.3 4588.11840049353 -1277.81840049353 56 3837.3 4237.87178797804 -400.571787978043 57 4145.3 4364.80874074883 -219.508740748834 58 3796.7 3989.31340344461 -192.613403444615 59 3849.6 3937.60262841965 -88.0026284196451 60 4285 4129.59693613805 155.403063861949 61 4189.6 3921.77162019971 267.828379800290 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.552890628089595 0.89421874382081 0.447109371910405 6 0.39283594283622 0.78567188567244 0.60716405716378 7 0.311567037505279 0.623134075010557 0.688432962494721 8 0.206194385074890 0.412388770149779 0.79380561492511 9 0.182443294327511 0.364886588655022 0.817556705672489 10 0.151537116052385 0.30307423210477 0.848462883947615 11 0.103474794496384 0.206949588992767 0.896525205503616 12 0.0627922199262303 0.125584439852461 0.93720778007377 13 0.173959280539783 0.347918561079566 0.826040719460217 14 0.213216810274546 0.426433620549093 0.786783189725454 15 0.361005120214768 0.722010240429537 0.638994879785232 16 0.300474242517537 0.600948485035073 0.699525757482463 17 0.230668199021836 0.461336398043672 0.769331800978164 18 0.177224962758482 0.354449925516964 0.822775037241518 19 0.146332579828194 0.292665159656387 0.853667420171806 20 0.122631900428141 0.245263800856283 0.877368099571859 21 0.0979944924255035 0.195988984851007 0.902005507574497 22 0.0689585558058437 0.137917111611687 0.931041444194156 23 0.0859888207970339 0.171977641594068 0.914011179202966 24 0.0584146636043414 0.116829327208683 0.941585336395659 25 0.076898142674929 0.153796285349858 0.923101857325071 26 0.094118518251762 0.188237036503524 0.905881481748238 27 0.0854335838177996 0.170867167635599 0.9145664161822 28 0.0602504251134638 0.120500850226928 0.939749574886536 29 0.0406486340976937 0.0812972681953875 0.959351365902306 30 0.027738988420387 0.055477976840774 0.972261011579613 31 0.0209477894547932 0.0418955789095863 0.979052210545207 32 0.0165271404510285 0.0330542809020569 0.983472859548971 33 0.0317113907917778 0.0634227815835556 0.968288609208222 34 0.0217612507099064 0.0435225014198127 0.978238749290094 35 0.0141492011087473 0.0282984022174945 0.985850798891253 36 0.0144696315666209 0.0289392631332418 0.985530368433379 37 0.0331028687676589 0.0662057375353179 0.96689713123234 38 0.0235047474354048 0.0470094948708097 0.976495252564595 39 0.0182977506265775 0.0365955012531551 0.981702249373422 40 0.0145348242091286 0.0290696484182573 0.985465175790871 41 0.0157113720786334 0.0314227441572668 0.984288627921367 42 0.0123506433727690 0.0247012867455379 0.987649356627231 43 0.0320975024519668 0.0641950049039337 0.967902497548033 44 0.0401103848543194 0.0802207697086388 0.95988961514568 45 0.0268427797370247 0.0536855594740495 0.973157220262975 46 0.0177637746008334 0.0355275492016669 0.982236225399167 47 0.0156266270764792 0.0312532541529585 0.98437337292352 48 0.0119562742216409 0.0239125484432818 0.98804372577836 49 0.0536193637154548 0.107238727430910 0.946380636284545 50 0.0772625450666616 0.154525090133323 0.922737454933338 51 0.160064842499907 0.320129684999815 0.839935157500093 52 0.65228542039746 0.695429159205079 0.347714579602540 53 0.619987263960278 0.760025472079444 0.380012736039722 54 0.641370473634938 0.717259052730123 0.358629526365062 55 0.860713333499748 0.278573333000504 0.139286666500252 56 0.821752201033874 0.356495597932251 0.178247798966126 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 13 0.25 NOK 10% type I error level 20 0.384615384615385 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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