*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: Tue, 17 Nov 2009 11:34:34 -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/17/t12584829407fhp66ulqmsb7ne.htm/, Retrieved Tue, 17 Nov 2009 19:35:52 +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/17/t12584829407fhp66ulqmsb7ne.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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8.9 1.4 8.8 1.2 8.3 1 7.5 1.7 7.2 2.4 7.4 2 8.8 2.1 9.3 2 9.3 1.8 8.7 2.7 8.2 2.3 8.3 1.9 8.5 2 8.6 2.3 8.5 2.8 8.2 2.4 8.1 2.3 7.9 2.7 8.6 2.7 8.7 2.9 8.7 3 8.5 2.2 8.4 2.3 8.5 2.8 8.7 2.8 8.7 2.8 8.6 2.2 8.5 2.6 8.3 2.8 8 2.5 8.2 2.4 8.1 2.3 8.1 1.9 8 1.7 7.9 2 7.9 2.1 8 1.7 8 1.8 7.9 1.8 8 1.8 7.7 1.3 7.2 1.3 7.5 1.3 7.3 1.2 7 1.4 7 2.2 7 2.9 7.2 3.1 7.3 3.5 7.1 3.6 6.8 4.4 6.4 4.1 6.1 5.1 6.5 5.8 7.7 5.9 7.9 5.4 7.5 5.5 6.9 4.8 6.6 3.2 6.9 2.7

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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 8.42354151735435 -0.263310125934268X[t] + 0.456805569775769M1[t] + 0.432604177331831M2[t] + 0.238935189925258M3[t] -0.0400000000000012M4[t] -0.211539367257091M5[t] -0.270474557182349M6[t] + 0.494791645336336M7[t] + 0.563194430224223M8[t] + 0.412662025186853M9[t] + 0.112662025186853M10[t] -0.134733797481316M11[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) 8.42354151735435 0.36092 23.3391 0 0 X -0.263310125934268 0.076988 -3.4202 0.001304 0.000652 M1 0.456805569775769 0.430801 1.0604 0.294398 0.147199 M2 0.432604177331831 0.430628 1.0046 0.320239 0.160119 M3 0.238935189925258 0.430449 0.5551 0.58147 0.290735 M4 -0.0400000000000012 0.430405 -0.0929 0.92635 0.463175 M5 -0.211539367257091 0.43087 -0.491 0.62574 0.31287 M6 -0.270474557182349 0.4312 -0.6273 0.533524 0.266762 M7 0.494791645336336 0.431296 1.1472 0.257097 0.128549 M8 0.563194430224223 0.430801 1.3073 0.197464 0.098732 M9 0.412662025186853 0.43068 0.9582 0.342883 0.171442 M10 0.112662025186853 0.43068 0.2616 0.79478 0.39739 M11 -0.134733797481316 0.430407 -0.313 0.755637 0.377818 Multiple Linear Regression - Regression Statistics Multiple R 0.576416949296871 R-squared 0.332256499436712 Adjusted R-squared 0.161768797165234 F-TEST (value) 1.94885903798293 F-TEST (DF numerator) 12 F-TEST (DF denominator) 47 p-value 0.0520596025542661 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.68052947892171 Sum Squared Residuals 21.7666574690283 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 8.9 8.51171291082218 0.388287089177822 2 8.8 8.54017354356507 0.259826456434935 3 8.3 8.39916658134535 -0.0991665813453451 4 7.5 7.9359143032661 -0.4359143032661 5 7.2 7.58005784785502 -0.380057847855022 6 7.4 7.62644670830347 -0.226446708303470 7 8.8 8.36538189822873 0.434618101771272 8 9.3 8.46011569571004 0.839884304289957 9 9.3 8.36224531585953 0.937754684140474 10 8.7 7.82526620251868 0.874733797481315 11 8.2 7.68319443022422 0.516805569775775 12 8.3 7.92325227807925 0.376747721920754 13 8.5 8.35372683526159 0.14627316473841 14 8.6 8.25053240503737 0.349467594962629 15 8.5 7.92520835466366 0.574791645336336 16 8.2 7.75159721511211 0.448402784887887 17 8.1 7.60638886044845 0.49361113955155 18 7.9 7.44212962014948 0.457870379850517 19 8.6 8.20739582266817 0.392604177331831 20 8.7 8.2231365823692 0.476863417630797 21 8.7 8.0462731647384 0.653726835261594 22 8.5 7.95692126548582 0.543078734514181 23 8.4 7.68319443022422 0.716805569775776 24 8.5 7.6862731647384 0.813726835261594 25 8.7 8.14307873451417 0.556921265485824 26 8.7 8.11887734207024 0.581122657929763 27 8.6 8.08319443022422 0.516805569775775 28 8.5 7.69893518992526 0.801064810074742 29 8.3 7.47473379748132 0.825266202518686 30 8 7.49479164533634 0.505208354663663 31 8.2 8.28638886044845 -0.0863888604484498 32 8.1 8.38112265792976 -0.281122657929764 33 8.1 8.3359143032661 -0.235914303266100 34 8 8.08857632845295 -0.0885763284529536 35 7.9 7.7621874680045 0.137812531995496 36 7.9 7.8705902528924 0.0294097471076069 37 8 8.43271987304187 -0.432719873041870 38 8 8.3821874680045 -0.382187468004505 39 7.9 8.18851848059793 -0.288518480597932 40 8 7.90958329067267 0.090416709327327 41 7.7 7.86969898638272 -0.169698986382717 42 7.2 7.81076379645746 -0.610763796457458 43 7.5 8.57602999897614 -1.07602999897614 44 7.3 8.67076379645746 -1.37076379645746 45 7 8.46756936623323 -1.46756936623323 46 7 7.95692126548582 -0.95692126548582 47 7 7.52520835466366 -0.525208354663664 48 7.2 7.60728012695813 -0.407280126958125 49 7.3 7.95876164636019 -0.658761646360188 50 7.1 7.90822924132282 -0.808229241322822 51 6.8 7.50391215316883 -0.703912153168835 52 6.4 7.30397000102386 -0.903970001023856 53 6.1 6.8691205078325 -0.769120507832498 54 6.5 6.62586822975325 -0.125868229753252 55 7.7 7.36480341967851 0.33519658032149 56 7.9 7.56486126753353 0.335138732466469 57 7.5 7.38799784990273 0.112002150097266 58 6.9 7.27231493805672 -0.372314938056722 59 6.6 7.44621531688338 -0.846215316883383 60 6.9 7.71260417733183 -0.812604177331832 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.119686200902995 0.239372401805990 0.880313799097005 17 0.161740877364843 0.323481754729686 0.838259122635157 18 0.0997531127159773 0.199506225431955 0.900246887284023 19 0.0532367095131898 0.106473419026380 0.94676329048681 20 0.0455706117558529 0.0911412235117058 0.954429388244147 21 0.0379477084636609 0.0758954169273218 0.96205229153634 22 0.0236132422349685 0.047226484469937 0.976386757765032 23 0.0153700071023433 0.0307400142046867 0.984629992897657 24 0.0111657687815701 0.0223315375631402 0.98883423121843 25 0.00698238934255174 0.0139647786851035 0.993017610657448 26 0.0047997639928944 0.0095995279857888 0.995200236007106 27 0.00360719179859029 0.00721438359718058 0.99639280820141 28 0.00808944576762498 0.0161788915352500 0.991910554232375 29 0.0191897008347316 0.0383794016694633 0.980810299165268 30 0.0201834980250951 0.0403669960501901 0.979816501974905 31 0.0173834480945855 0.0347668961891711 0.982616551905414 32 0.0285144081424946 0.0570288162849892 0.971485591857505 33 0.0425279616339699 0.0850559232679398 0.95747203836603 34 0.0451902998244786 0.0903805996489572 0.954809700175521 35 0.0535658761393334 0.107131752278667 0.946434123860667 36 0.055898157687941 0.111796315375882 0.94410184231206 37 0.0527530182929485 0.105506036585897 0.947246981707051 38 0.0562130935649504 0.112426187129901 0.94378690643505 39 0.0627807188944748 0.125561437788950 0.937219281105525 40 0.162069036272537 0.324138072545075 0.837930963727463 41 0.643226369719047 0.713547260561905 0.356773630280953 42 0.906006446983474 0.187987106033053 0.0939935530165265 43 0.861306352119176 0.277387295761648 0.138693647880824 44 0.7952275927159 0.409544814568201 0.204772407284100 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 2 0.0689655172413793 NOK 5% type I error level 10 0.344827586206897 NOK 10% type I error level 15 0.517241379310345 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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