*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 10:49:39 -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/t1258566690bw95lzphawi4wvy.htm/, Retrieved Wed, 18 Nov 2009 18:51:42 +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/t1258566690bw95lzphawi4wvy.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:

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Dataseries X:

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7.2 2.4 7.5 8.3 8.9 7.4 2 7.2 7.5 8.8 8.8 2.1 7.4 7.2 8.3 9.3 2 8.8 7.4 7.5 9.3 1.8 9.3 8.8 7.2 8.7 2.7 9.3 9.3 7.4 8.2 2.3 8.7 9.3 8.8 8.3 1.9 8.2 8.7 9.3 8.5 2 8.3 8.2 9.3 8.6 2.3 8.5 8.3 8.7 8.5 2.8 8.6 8.5 8.2 8.2 2.4 8.5 8.6 8.3 8.1 2.3 8.2 8.5 8.5 7.9 2.7 8.1 8.2 8.6 8.6 2.7 7.9 8.1 8.5 8.7 2.9 8.6 7.9 8.2 8.7 3 8.7 8.6 8.1 8.5 2.2 8.7 8.7 7.9 8.4 2.3 8.5 8.7 8.6 8.5 2.8 8.4 8.5 8.7 8.7 2.8 8.5 8.4 8.7 8.7 2.8 8.7 8.5 8.5 8.6 2.2 8.7 8.7 8.4 8.5 2.6 8.6 8.7 8.5 8.3 2.8 8.5 8.6 8.7 8 2.5 8.3 8.5 8.7 8.2 2.4 8 8.3 8.6 8.1 2.3 8.2 8 8.5 8.1 1.9 8.1 8.2 8.3 8 1.7 8.1 8.1 8 7.9 2 8 8.1 8.2 7.9 2.1 7.9 8 8.1 8 1.7 7.9 7.9 8.1 8 1.8 8 7.9 8 7.9 1.8 8 8 7.9 8 1.8 7.9 8 7.9 7.7 1.3 8 7.9 8 7.2 1.3 7.7 8 8 7.5 1.3 7.2 7.7 7.9 7.3 1.2 7.5 7.2 8 7 1.4 7.3 7.5 7.7 7 2.2 7 7.3 7.2 7 2.9 7 7 7.5 7.2 3.1 7 7 7.3 7.3 3.5 7.2 7 7 7.1 3.6 7.3 7.2 7 6.8 4.4 7.1 7.3 7 6.4 4.1 6.8 7.1 7.2 6.1 5.1 6.4 6.8 7.3 6.5 5.8 6.1 6.4 7.1 7.7 5.9 6.5 6.1 6.8 7.9 5.4 7.7 6.5 6.4 7.5 5. 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)[t] = + 2.27326564125142 + 0.0272063663525248`X(t)`[t] + 1.21401509190151`Y(t-1)`[t] -0.67019074567934`Y(t-2)`[t] + 0.196910902114001`Y(t-4) `[t] -0.0109050644084100t + 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) 2.27326564125142 1.061935 2.1407 0.0372 0.0186 `X(t)` 0.0272063663525248 0.041235 0.6598 0.512418 0.256209 `Y(t-1)` 1.21401509190151 0.113851 10.6632 0 0 `Y(t-2)` -0.67019074567934 0.11649 -5.7532 1e-06 0 `Y(t-4) ` 0.196910902114001 0.088435 2.2266 0.030508 0.015254 t -0.0109050644084100 0.003926 -2.7774 0.007693 0.003847 Multiple Linear Regression - Regression Statistics Multiple R 0.940926354067522 R-squared 0.8853424037788 Adjusted R-squared 0.87387664415668 F-TEST (value) 77.2162013645198 F-TEST (DF numerator) 5 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.264159428317082 Sum Squared Residuals 3.48901017844037 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.2 7.62269288502648 -0.422692885026483 2 7.4 7.75316225283869 -0.353162252838691 3 8.8 8.09038261609263 0.709617383907366 4 9.3 9.48481117288402 -0.184811172884023 5 9.3 9.07813206657059 0.221867933429413 6 8.7 8.79599953946258 -0.0959995394625806 7 8.2 8.32147813633185 -0.121478136331853 8 8.3 8.19325287789628 0.106747122103719 9 8.5 8.64156533215295 -0.141565332152948 10 8.6 8.69645958019426 -0.0964595801942612 11 8.5 8.5880656079594 -0.0880656079593962 12 8.2 8.3975485034633 -0.197548503463293 13 8.1 8.12611952983991 -0.0261195298399093 14 7.9 8.22544381669756 -0.325443816697561 15 8.6 8.01906371826538 0.580936281734617 16 8.7 8.9383753699602 -0.238375369960204 17 8.7 8.56276783919026 0.137232160809741 18 8.5 8.4236964267091 0.0763035732909052 19 8.4 8.31054661203544 0.0894533879645642 20 8.5 8.3455724609604 0.154427539039594 21 8.7 8.52308798031008 0.176912019689919 22 8.7 8.64858467929124 0.0514153207087615 23 8.6 8.46762655572405 0.132373444275954 24 8.5 8.3658936188779 0.134106381122106 25 8.3 8.34542957354057 -0.0454295735405719 26 8 8.15057865541404 -0.150578655414038 27 8.2 7.88709548572439 0.312904514275611 28 8.1 8.29763893655343 -0.197638936553430 29 8.1 7.98102948685519 0.118970513144808 30 8 7.97262895311001 0.0273710468899900 31 7.9 7.88786646984001 0.0121335301599935 32 7.9 7.80560851723323 0.094391482766768 33 8 7.85083998085175 0.149160019148254 34 8 7.94436597205734 0.0556340279426604 35 7.9 7.8467507428696 0.0532492571304045 36 8 7.71444416927103 0.285555830728965 37 7.7 7.89804759565585 -0.198047595655847 38 7.2 7.45591892910905 -0.25591892910905 39 7.5 7.01937245224229 0.480627547757714 40 7.3 7.72473774182015 -0.424737741820147 41 7 7.21634043796394 -0.216340437963938 42 7 6.89857863714596 0.101421362854038 43 7 7.16684852352232 -0.166848523522322 44 7.2 7.12200255196162 0.0779974480383834 45 7.3 7.30570978184032 -0.00570978184031929 46 7.1 7.28488871412144 -0.184888714121445 47 6.8 6.98592664984682 -0.185926649846818 48 6.4 6.77607547752087 -0.376075477520865 49 6.1 6.52751905661958 -0.427519056619578 50 6.5 6.40014803893642 0.099851961063583 51 7.7 7.01955360099347 0.680446399006533 52 7.9 8.10502280457327 -0.205022804573272 53 7.5 7.47633922973101 0.0236607702689912 54 6.9 6.90550988382496 -0.00550988382495796 55 6.6 6.62703495892014 -0.0270349589201394 56 6.9 6.67981881159542 0.220181188404584 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 9 0.838470885850243 0.323058228299515 0.161529114149757 10 0.86465015782121 0.270699684357579 0.135349842178789 11 0.839325888053428 0.321348223893144 0.160674111946572 12 0.90251585357371 0.194968292852580 0.097484146426290 13 0.86257853205662 0.274842935886762 0.137421467943381 14 0.873264478081811 0.253471043836378 0.126735521918189 15 0.950959256529224 0.0980814869415527 0.0490407434707764 16 0.93880557439588 0.122388851208240 0.0611944256041202 17 0.911208654092325 0.177582691815350 0.0887913459076752 18 0.901266850281167 0.197466299437665 0.0987331497188325 19 0.85523390758216 0.28953218483568 0.14476609241784 20 0.806287023691956 0.387425952616088 0.193712976308044 21 0.752362915787165 0.49527416842567 0.247637084212835 22 0.676448273647045 0.64710345270591 0.323551726352955 23 0.60929199388164 0.781416012236721 0.390708006118361 24 0.529328016028845 0.94134396794231 0.470671983971155 25 0.461311862245071 0.922623724490142 0.538688137754929 26 0.476620770634217 0.953241541268435 0.523379229365783 27 0.420614041316863 0.841228082633727 0.579385958683137 28 0.470629054185345 0.94125810837069 0.529370945814655 29 0.397495099147312 0.794990198294623 0.602504900852688 30 0.339944940072246 0.679889880144492 0.660055059927754 31 0.275082034545592 0.550164069091184 0.724917965454408 32 0.210812230792277 0.421624461584555 0.789187769207723 33 0.163284684054704 0.326569368109409 0.836715315945296 34 0.122297550851157 0.244595101702313 0.877702449148843 35 0.0925332497892195 0.185066499578439 0.90746675021078 36 0.146159636845776 0.292319273691553 0.853840363154224 37 0.143209088050192 0.286418176100385 0.856790911949807 38 0.136762413954193 0.273524827908385 0.863237586045807 39 0.694824512518896 0.610350974962207 0.305175487481104 40 0.704288205892427 0.591423588215146 0.295711794107573 41 0.662623428849257 0.674753142301485 0.337376571150743 42 0.61593066904754 0.76813866190492 0.38406933095246 43 0.574110992467296 0.851778015065408 0.425889007532704 44 0.525234861274874 0.949530277450252 0.474765138725126 45 0.398562796857261 0.797125593714522 0.601437203142739 46 0.288547842392929 0.577095684785858 0.711452157607071 47 0.436385016268223 0.872770032536446 0.563614983731777 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 1 0.0256410256410256 OK

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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