WS 75

*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 13:57:24 -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/t12585779078fsekstwvqotd1x.htm/, Retrieved Wed, 18 Nov 2009 21:58:39 +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/t12585779078fsekstwvqotd1x.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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103.91 89.00 103.88 103.77 103.91 86.40 103.91 103.88 103.92 84.50 103.91 103.91 104.05 82.70 103.92 103.91 104.23 80.80 104.05 103.92 104.30 81.80 104.23 104.05 104.31 81.80 104.30 104.23 104.31 82.90 104.31 104.30 104.34 83.80 104.31 104.31 104.55 86.20 104.34 104.31 104.65 86.10 104.55 104.34 104.73 86.20 104.65 104.55 104.75 88.80 104.73 104.65 104.75 89.60 104.75 104.73 104.76 87.80 104.75 104.75 104.94 88.30 104.76 104.75 105.29 88.60 104.94 104.76 105.38 91.00 105.29 104.94 105.43 91.50 105.38 105.29 105.43 95.40 105.43 105.38 105.42 98.70 105.43 105.43 105.52 99.90 105.42 105.43 105.69 98.60 105.52 105.42 105.72 100.30 105.69 105.52 105.74 100.20 105.72 105.69 105.74 100.40 105.74 105.72 105.74 101.40 105.74 105.74 105.95 103.00 105.74 105.74 106.17 109.10 105.95 105.74 106.34 111.40 106.17 105.95 106.37 114.10 106.34 106.17 106.37 121.80 106.37 106.34 106.36 127.60 106.37 106.37 106.44 129.90 106.36 106.37 106.29 128.00 106.44 106.36 106.23 123.50 106.29 106.44 106 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 R Framework error message The field 'Names of X columns' contains a hard return which cannot be interpreted. Please, resubmit your request without hard returns in the 'Names of X columns'. Multiple Linear Regression - Estimated Regression Equation Y[t] = + 2.85133191081378 -0.00468183176077428X[t] + 1.26651149270306Y1[t] -0.289998539166879`Y2 `[t] + 0.00645926897734081t + 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) 2.85133191081378 3.467109 0.8224 0.414757 0.207379 X -0.00468183176077428 0.001887 -2.4805 0.016529 0.008265 Y1 1.26651149270306 0.128257 9.8748 0 0 `Y2 ` -0.289998539166879 0.133035 -2.1799 0.034003 0.017002 t 0.00645926897734081 0.003566 1.8115 0.07608 0.03804 Multiple Linear Regression - Regression Statistics Multiple R 0.995797803529525 R-squared 0.991613265514226 Adjusted R-squared 0.990942326755364 F-TEST (value) 1477.94899671055 F-TEST (DF numerator) 4 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.0808895041753397 Sum Squared Residuals 0.327155594286615 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 103.91 103.913173605729 -0.00317360572922217 2 103.91 103.937901142757 -0.0279011427570598 3 103.92 103.944555935905 -0.0245559359048573 4 104.05 103.972107616979 0.0778923830213673 5 104.23 104.149208874961 0.0807911250388404 6 104.3 104.341258570773 -0.0412585707725991 7 104.31 104.384173907189 -0.0741739071891 8 104.31 104.377848378415 -0.0678483784149466 9 104.34 104.377194013416 -0.0371940134159191 10 104.55 104.410412230948 0.139587769051499 11 104.65 104.674607140395 -0.0246071403945386 12 104.73 104.746349682241 -0.0163496822410774 13 104.75 104.81295725414 -0.0629572541399611 14 104.75 104.817801404429 -0.0678014044293888 15 104.76 104.826887999793 -0.0668879997927819 16 104.94 104.843671467817 0.09632853218322 17 105.29 105.073798270561 0.216201729439249 18 105.38 105.460100428708 -0.0801004287082893 19 105.43 105.47670532744 -0.0467053274400829 20 105.43 105.502131158661 -0.0721311586605553 21 105.42 105.478640455869 -0.058640455868999 22 105.52 105.466816411806 0.0531835881936207 23 105.69 105.608913196735 0.0810868032653062 24 105.72 105.793720451562 -0.073720451561553 25 105.74 105.789343496838 -0.0493434968376984 26 105.74 105.811496673142 -0.0714966731419339 27 105.74 105.807474139575 -0.0674741395751641 28 105.95 105.806442477735 0.143557522264742 29 106.17 106.050309986440 0.119690013560471 30 106.34 106.263733877537 0.0762661224632882 31 106.37 106.409059475903 -0.03905947590277 32 106.37 106.368164233445 0.00183576655512804 33 106.36 106.338768922035 0.0212310779652797 34 106.44 106.321794863035 0.118205136964755 35 106.29 106.441370517166 -0.151370517165961 36 106.23 106.255721422028 -0.0257214220279908 37 106.23 106.227348866438 0.00265113356221255 38 106.23 106.235289819779 -0.00528981977850895 39 106.23 106.240812722404 -0.0108127224036950 40 106.34 106.243526525972 0.096473474027583 41 106.44 106.375256563865 0.0647434361352238 42 106.44 106.459144365289 -0.0191443652891944 43 106.48 106.441753795287 0.0382462047133056 44 106.5 106.449714290484 0.0502857095159608 45 106.57 106.456794718819 0.113205281181402 46 106.4 106.529255227163 -0.129255227163008 47 106.37 106.310407674513 0.0595923254871302 48 106.25 106.362348722221 -0.11234872222114 49 106.21 106.237231147651 -0.0272311476510567 50 106.21 106.223616133036 -0.013616133035597 51 106.24 106.214520719367 0.0254792806328769 52 106.19 106.248207120076 -0.0582071200757735 53 106.08 106.179831759186 -0.0998317591864931 54 106.13 106.049301928347 0.080698071653169 55 106.09 106.121022887999 -0.0310228879987143 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 8 0.0465051922971893 0.0930103845943787 0.95349480770281 9 0.0157527027918181 0.0315054055836363 0.984247297208182 10 0.0317969130893807 0.0635938261787614 0.96820308691062 11 0.0389038082890316 0.0778076165780632 0.961096191710968 12 0.0349181746922135 0.069836349384427 0.965081825307786 13 0.0163647054213892 0.0327294108427784 0.98363529457861 14 0.00904133244438548 0.0180826648887710 0.990958667555615 15 0.0102999955086691 0.0205999910173382 0.98970000449133 16 0.00682084438361271 0.0136416887672254 0.993179155616387 17 0.0988442277214873 0.197688455442975 0.901155772278513 18 0.0650645450744863 0.130129090148973 0.934935454925514 19 0.282858853391818 0.565717706783637 0.717141146608182 20 0.248402935746490 0.496805871492981 0.75159706425351 21 0.210577172043988 0.421154344087977 0.789422827956011 22 0.160419036506981 0.320838073013962 0.839580963493019 23 0.127177751855763 0.254355503711527 0.872822248144237 24 0.150412330852787 0.300824661705574 0.849587669147213 25 0.128239490420713 0.256478980841425 0.871760509579287 26 0.169209118201504 0.338418236403009 0.830790881798496 27 0.352493072034138 0.704986144068276 0.647506927965862 28 0.305721588575387 0.611443177150774 0.694278411424613 29 0.241953669835749 0.483907339671498 0.758046330164251 30 0.213860805114185 0.427721610228369 0.786139194885815 31 0.158781175513222 0.317562351026445 0.841218824486778 32 0.113115162979296 0.226230325958593 0.886884837020704 33 0.0854094935325607 0.170818987065121 0.91459050646744 34 0.0730768732463889 0.146153746492778 0.926923126753611 35 0.536687753852254 0.926624492295492 0.463312246147746 36 0.573330132740862 0.853339734518275 0.426669867259138 37 0.602238744260342 0.795522511479317 0.397761255739658 38 0.678537219326828 0.642925561346344 0.321462780673172 39 0.859139732360346 0.281720535279308 0.140860267639654 40 0.893333420997179 0.213333158005643 0.106666579002821 41 0.880633123871401 0.238733752257198 0.119366876128599 42 0.962619955809162 0.0747600883816756 0.0373800441908378 43 0.935881921972426 0.128236156055147 0.0641180780275735 44 0.978036619935195 0.0439267601296100 0.0219633800648050 45 0.981675428664667 0.0366491426706665 0.0183245713353332 46 0.959295031950771 0.0814099360984575 0.0407049680492287 47 0.906090927913122 0.187818144173756 0.093909072086878 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 7 0.175 NOK 10% type I error level 13 0.325 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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