*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, 19 Nov 2009 10:52:10 -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/19/t1258653270bienxf1h2gdp8wl.htm/, Retrieved Thu, 19 Nov 2009 18:54:41 +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/19/t1258653270bienxf1h2gdp8wl.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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2.11 0 2.09 0 2.05 0 2.08 0 2.06 0 2.06 0 2.08 0 2.07 0 2.06 0 2.07 0 2.06 0 2.09 0 2.07 0 2.09 0 2.28 0 2.33 0 2.35 0 2.52 0 2.63 0 2.58 0 2.70 0 2.81 0 2.97 0 3.04 0 3.28 0 3.33 0 3.50 0 3.56 0 3.57 0 3.69 0 3.82 0 3.79 0 3.96 0 4.06 0 4.05 0 4.03 0 3.94 0 4.02 0 3.88 0 4.02 0 4.03 0 4.09 0 3.99 0 4.01 0 4.01 0 4.19 0 4.30 0 4.27 0 3.82 0 3.15 1 2.49 1 1.81 1 1.26 1 1.06 1 0.84 1 0.78 1 0.70 1 0.36 1 0.35 1 0.36 1

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] = + 1.77078780452645 -3.52383385534748X[t] + 0.0536256306760848t + 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) 1.77078780452645 0.148037 11.9618 0 0 X -3.52383385534748 0.230361 -15.297 0 0 t 0.0536256306760848 0.005147 10.4189 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.896747302244183 R-squared 0.80415572408222 Adjusted R-squared 0.797283995102648 F-TEST (value) 117.023783457245 F-TEST (DF numerator) 2 F-TEST (DF denominator) 57 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.512376397337417 Sum Squared Residuals 14.9641856352628 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 2.11 1.82441343520253 0.285586564797467 2 2.09 1.87803906587862 0.211960934121378 3 2.05 1.93166469655471 0.118335303445291 4 2.08 1.98529032723079 0.0947096727692085 5 2.06 2.03891595790688 0.0210840420931237 6 2.06 2.09254158858296 -0.032541588582961 7 2.08 2.14616721925905 -0.0661672192590457 8 2.07 2.19979284993513 -0.129792849935131 9 2.06 2.25341848061122 -0.193418480611215 10 2.07 2.3070441112873 -0.23704411128730 11 2.06 2.36066974196338 -0.300669741963385 12 2.09 2.41429537263947 -0.32429537263947 13 2.07 2.46792100331555 -0.397921003315554 14 2.09 2.52154663399164 -0.431546633991639 15 2.28 2.57517226466772 -0.295172264667724 16 2.33 2.62879789534381 -0.298797895343809 17 2.35 2.68242352601989 -0.332423526019893 18 2.52 2.73604915669598 -0.216049156695978 19 2.63 2.78967478737206 -0.159674787372063 20 2.58 2.84330041804815 -0.263300418048148 21 2.7 2.89692604872423 -0.196926048724232 22 2.81 2.95055167940032 -0.140551679400317 23 2.97 3.0041773100764 -0.0341773100764017 24 3.04 3.05780294075249 -0.0178029407524866 25 3.28 3.11142857142857 0.168571428571428 26 3.33 3.16505420210466 0.164945797895344 27 3.5 3.21867983278074 0.281320167219259 28 3.56 3.27230546345683 0.287694536543174 29 3.57 3.32593109413291 0.244068905867089 30 3.69 3.37955672480900 0.310443275191005 31 3.82 3.43318235548508 0.38681764451492 32 3.79 3.48680798616116 0.303192013838835 33 3.96 3.54043361683725 0.419566383162751 34 4.06 3.59405924751333 0.465940752486666 35 4.05 3.64768487818942 0.402315121810581 36 4.03 3.70131050886550 0.328689491134497 37 3.94 3.75493613954159 0.185063860458411 38 4.02 3.80856177021767 0.211438229782326 39 3.88 3.86218740089376 0.017812599106242 40 4.02 3.91581303156984 0.104186968430157 41 4.03 3.96943866224593 0.0605613377540729 42 4.09 4.02306429292201 0.0669357070779878 43 3.99 4.0766899235981 -0.0866899235980966 44 4.01 4.13031555427418 -0.120315554274182 45 4.01 4.18394118495027 -0.173941184950267 46 4.19 4.23756681562635 -0.0475668156263509 47 4.3 4.29119244630244 0.00880755369756383 48 4.27 4.34481807697852 -0.0748180769785212 49 3.82 4.39844370765461 -0.578443707654606 50 3.15 0.928235482983213 2.22176451701679 51 2.49 0.981861113659298 1.50813888634070 52 1.81 1.03548674433538 0.774513255664618 53 1.26 1.08911237501147 0.170887624988533 54 1.06 1.14273800568755 -0.0827380056875516 55 0.84 1.19636363636364 -0.356363636363636 56 0.78 1.24998926703972 -0.469989267039721 57 0.7 1.30361489771581 -0.603614897715806 58 0.36 1.35724052839189 -0.99724052839189 59 0.35 1.41086615906798 -1.06086615906798 60 0.36 1.46449178974406 -1.10449178974406 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 6 6.99618293716003e-05 0.000139923658743201 0.999930038170628 7 5.21805273291234e-06 1.04361054658247e-05 0.999994781947267 8 2.03274914330383e-07 4.06549828660767e-07 0.999999796725086 9 6.42583077302468e-09 1.28516615460494e-08 0.99999999357417 10 2.40809172621059e-10 4.81618345242119e-10 0.999999999759191 11 6.93666250979949e-12 1.38733250195990e-11 0.999999999993063 12 8.35204811944443e-13 1.67040962388889e-12 0.999999999999165 13 2.73324180663265e-14 5.4664836132653e-14 0.999999999999973 14 1.92623081124881e-15 3.85246162249761e-15 0.999999999999998 15 7.79926840072244e-11 1.55985368014449e-10 0.999999999922007 16 4.43675156387278e-10 8.87350312774556e-10 0.999999999556325 17 4.27247226093449e-10 8.54494452186898e-10 0.999999999572753 18 3.58073344947624e-09 7.16146689895247e-09 0.999999996419267 19 1.8286734102155e-08 3.657346820431e-08 0.999999981713266 20 1.21693705508033e-08 2.43387411016067e-08 0.99999998783063 21 1.48002648609049e-08 2.96005297218097e-08 0.999999985199735 22 2.53166078005328e-08 5.06332156010655e-08 0.999999974683392 23 8.42255707846836e-08 1.68451141569367e-07 0.99999991577443 24 1.78646364634147e-07 3.57292729268294e-07 0.999999821353635 25 1.05709974259606e-06 2.11419948519213e-06 0.999998942900257 26 2.49507200014264e-06 4.99014400028529e-06 0.999997504928 27 6.95378152702759e-06 1.39075630540552e-05 0.999993046218473 28 1.16816955983223e-05 2.33633911966447e-05 0.999988318304402 29 1.30093587940959e-05 2.60187175881917e-05 0.999986990641206 30 1.51820306655954e-05 3.03640613311908e-05 0.999984817969334 31 1.82870817809078e-05 3.65741635618156e-05 0.999981712918219 32 1.65092053743668e-05 3.30184107487336e-05 0.999983490794626 33 1.64043956829378e-05 3.28087913658755e-05 0.999983595604317 34 1.51321857914557e-05 3.02643715829115e-05 0.999984867814208 35 1.12546633650036e-05 2.25093267300072e-05 0.999988745336635 36 8.00251112009772e-06 1.60050222401954e-05 0.99999199748888 37 7.94353854538233e-06 1.58870770907647e-05 0.999992056461455 38 7.83529808616074e-06 1.56705961723215e-05 0.999992164701914 39 2.24556157433078e-05 4.49112314866157e-05 0.999977544384257 40 4.52678258841343e-05 9.05356517682685e-05 0.999954732174116 41 0.000121553705967838 0.000243107411935675 0.999878446294032 42 0.000288467423630803 0.000576934847261606 0.99971153257637 43 0.00156709051927494 0.00313418103854989 0.998432909480725 44 0.0075370814501137 0.0150741629002274 0.992462918549886 45 0.0309491287940536 0.0618982575881072 0.969050871205946 46 0.0365409744900518 0.0730819489801036 0.963459025509948 47 0.0236749423828376 0.0473498847656751 0.976325057617162 48 0.0160726204324550 0.0321452408649100 0.983927379567545 49 0.0231348026763358 0.0462696053526715 0.976865197323664 50 0.209906516577804 0.419813033155609 0.790093483422196 51 0.806455916415761 0.387088167168478 0.193544083584239 52 0.985601367381717 0.0287972652365664 0.0143986326182832 53 0.981867361681927 0.0362652766361454 0.0181326383180727 54 0.960462985778262 0.0790740284434769 0.0395370142217385 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 38 0.775510204081633 NOK 5% type I error level 44 0.897959183673469 NOK 10% type I error level 47 0.959183673469388 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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