WS 7: Multiple Regression

*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: Thu, 26 Nov 2009 11:21:11 -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/26/t1259259867n1ytaddg16j1e5p.htm/, Retrieved Thu, 26 Nov 2009 19:24:39 +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/26/t1259259867n1ytaddg16j1e5p.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:

2 lags

Dataseries X:

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7.5 9.2 7.7 8.1 7.6 9.2 7.5 7.7 7.8 9.5 7.6 7.5 7.8 9.6 7.8 7.6 7.8 9.5 7.8 7.8 7.5 9.1 7.8 7.8 7.5 8.9 7.5 7.8 7.1 9 7.5 7.5 7.5 10.1 7.1 7.5 7.5 10.3 7.5 7.1 7.6 10.2 7.5 7.5 7.7 9.6 7.6 7.5 7.7 9.2 7.7 7.6 7.9 9.3 7.7 7.7 8.1 9.4 7.9 7.7 8.2 9.4 8.1 7.9 8.2 9.2 8.2 8.1 8.2 9 8.2 8.2 7.9 9 8.2 8.2 7.3 9 7.9 8.2 6.9 9.8 7.3 7.9 6.6 10 6.9 7.3 6.7 9.8 6.6 6.9 6.9 9.3 6.7 6.6 7 9 6.9 6.7 7.1 9 7 6.9 7.2 9.1 7.1 7 7.1 9.1 7.2 7.1 6.9 9.1 7.1 7.2 7 9.2 6.9 7.1 6.8 8.8 7 6.9 6.4 8.3 6.8 7 6.7 8.4 6.4 6.8 6.6 8.1 6.7 6.4 6.4 7.7 6.6 6.7 6.3 7.9 6.4 6.6 6.2 7.9 6.3 6.4 6.5 8 6.2 6.3 6.8 7.9 6.5 6.2 6.8 7.6 6.8 6.5 6.4 7.1 6.8 6.8 6.1 6.8 6.4 6.8 5.8 6.5 6.1 6.4 6.1 6.9 5.8 6.1 7.2 8.2 6.1 5.8 7.3 8.7 7.2 6.1 6.9 8.3 7.3 7.2 6.1 7.9 6.9 7.3 5.8 7.5 6.1 6.9 6.2 7.8 5.8 6.1 7.1 8.3 6.2 5.8 7.7 8.4 7.1 6.2 7.9 8.2 7.7 7.1 7.7 7.7 7.9 7.7 7.4 7.2 7.7 7.9 7.5 7.3 7.4 7.7 8 8.1 7.5 7.4 8.1 8.5 8 7.5

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 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.253786890021161 + 0.117297499777638X[t] + 1.38717974201476Y1[t] -0.594692239920104Y2[t] + 0.142919396207107M1[t] + 0.368637080481907M2[t] + 0.320542671321784M3[t] + 0.0997477098498515M4[t] + 0.076608910371553M5[t] + 0.147111616500441M6[t] + 0.104252609161747M7[t] + 0.121497196855735M8[t] + 0.54958481069371M9[t] -0.162273531176337M10[t] + 0.0200943483296912M11[t] + 0.00233209196489073t + 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) 0.253786890021161 0.737165 0.3443 0.732358 0.366179 X 0.117297499777638 0.073294 1.6004 0.117012 0.058506 Y1 1.38717974201476 0.128288 10.813 0 0 Y2 -0.594692239920104 0.129254 -4.601 3.8e-05 1.9e-05 M1 0.142919396207107 0.159059 0.8985 0.374026 0.187013 M2 0.368637080481907 0.157806 2.336 0.024339 0.012169 M3 0.320542671321784 0.162329 1.9746 0.054907 0.027454 M4 0.0997477098498515 0.166858 0.5978 0.553183 0.276591 M5 0.076608910371553 0.163819 0.4676 0.642456 0.321228 M6 0.147111616500441 0.164765 0.8929 0.37702 0.18851 M7 0.104252609161747 0.167459 0.6226 0.536942 0.268471 M8 0.121497196855735 0.163832 0.7416 0.462459 0.23123 M9 0.54958481069371 0.160308 3.4283 0.001372 0.000686 M10 -0.162273531176337 0.169222 -0.9589 0.343077 0.171538 M11 0.0200943483296912 0.166615 0.1206 0.90458 0.45229 t 0.00233209196489073 0.003531 0.6604 0.512608 0.256304 Multiple Linear Regression - Regression Statistics Multiple R 0.95192436469737 R-squared 0.90615999610449 Adjusted R-squared 0.872645708998951 F-TEST (value) 27.0380209267444 F-TEST (DF numerator) 15 F-TEST (DF denominator) 42 p-value 1.11022302462516e-16 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.234084331376249 Sum Squared Residuals 2.30140991622636 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.5 7.34245224630823 0.157547753691775 2 7.6 7.53094297011301 0.0690570298869905 3 7.8 7.77802632503656 0.0219736749634351 4 7.8 7.78925992991823 0.0107400700817726 5 7.8 7.63778502444304 0.162214975556965 6 7.5 7.66370082262576 -0.163700822625758 7 7.5 7.183560484692 0.316439515307999 8 7.1 7.39327458630467 -0.293274586304675 9 7.5 7.39784964505704 0.102150354942961 10 7.5 7.50453168788135 -0.00453168788135466 11 7.6 7.43962501340647 0.160374986593531 12 7.7 7.49020223137656 0.20979776862344 13 7.7 7.66778346984697 0.0322165301530307 14 7.9 7.84809377207241 0.0519062279275875 15 8.1 8.0914971532579 0.00850284674210331 16 8.2 8.03153178416978 0.168468215830215 17 8.2 8.0070451029183 0.192954897081696 18 8.2 7.99695117706454 0.203048822935455 19 7.9 7.95642426169074 -0.0564242616907414 20 7.3 7.5598470187452 -0.259847018745195 21 6.9 7.43020455113735 -0.530204551137346 22 6.6 6.54608124833388 0.0539187516661231 23 6.7 6.52904469321288 0.170955306787119 24 6.9 6.76975933313677 0.130240666863231 25 7 7.09778829578642 -0.0977882957864174 26 7.1 7.34561759824356 -0.245617598243563 27 7.2 7.39083378123556 -0.190833781235559 28 7.1 7.25161966193798 -0.151619661937984 29 6.9 7.03262575623109 -0.132625756231088 30 7 6.89922357989169 0.100776420108309 31 6.8 7.06943408679233 -0.269434086792328 32 6.4 6.69345684416743 -0.293456844167426 33 6.7 6.69967285112617 0.000327148873825966 34 6.6 6.6089881698602 -0.00898816986019411 35 6.4 6.42964349524255 -0.0296434952425499 36 6.3 6.21737401442234 0.0826259855776623 37 6.2 6.34284597637688 -0.142845976376879 38 6.5 6.50337675238487 -0.00337675238486883 39 6.8 6.92150783180831 -0.121507831808310 40 6.8 6.90560196299637 -0.105601962996373 41 6.4 6.64773883361812 -0.247738833618115 42 6.1 6.1305124849727 -0.0305124849726999 43 5.8 5.87651929302922 -0.0765192930292182 44 6.1 5.70526872197076 0.394731278029243 45 7.2 6.88273677206501 0.317263227934990 46 7.3 7.57934931628888 -0.279349316288876 47 6.9 7.2016867981381 -0.301686798138101 48 6.1 6.52266442106433 -0.422664421064333 49 5.8 5.74913001168151 0.050869988318491 50 6.2 6.07196890718615 0.128031092813853 51 7.1 6.81813490866167 0.281865091338331 52 7.7 7.62198666097763 0.0780133390223694 53 7.9 7.87480528278946 0.0251947172105426 54 7.7 7.80961193544531 -0.109611935445306 55 7.4 7.31406187379571 0.0859381262042888 56 7.5 7.04815282881195 0.451847171188052 57 8 7.88953618061443 0.110463819385569 58 8.1 7.8610495776357 0.238950422364302 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 19 0.138512653293719 0.277025306587438 0.86148734670628 20 0.150145391476079 0.300290782952158 0.849854608523921 21 0.788805895566057 0.422388208867885 0.211194104433943 22 0.67638197174038 0.64723605651924 0.32361802825962 23 0.587273530095083 0.825452939809835 0.412726469904917 24 0.569082476781707 0.861835046436585 0.430917523218293 25 0.504886189159857 0.990227621680287 0.495113810840143 26 0.424631904293524 0.849263808587048 0.575368095706476 27 0.319185476561977 0.638370953123953 0.680814523438023 28 0.229808320705273 0.459616641410545 0.770191679294727 29 0.171568336224479 0.343136672448958 0.828431663775521 30 0.174025041998721 0.348050083997442 0.825974958001279 31 0.143465439341176 0.286930878682352 0.856534560658824 32 0.209432645747977 0.418865291495954 0.790567354252023 33 0.288031191402264 0.576062382804529 0.711968808597736 34 0.241567273359368 0.483134546718735 0.758432726640632 35 0.391748783437454 0.783497566874909 0.608251216562546 36 0.898694919970128 0.202610160059744 0.101305080029872 37 0.96780678062852 0.0643864387429605 0.0321932193714802 38 0.948596096135548 0.102807807728905 0.0514039038644524 39 0.95908270677143 0.0818345864571392 0.0409172932285696 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 2 0.0952380952380952 OK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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