Revieuw ws7

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R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Wed, 25 Nov 2009 13:43:40 -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/25/t1259182167rgbqgisboi4nf7u.htm/, Retrieved Wed, 25 Nov 2009 21:49:39 +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/25/t1259182167rgbqgisboi4nf7u.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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105.62 125.03 105.57 105.24 105.15 104.89 106.17 130.09 105.62 105.57 105.24 105.15 106.27 126.65 106.17 105.62 105.57 105.24 106.41 121.7 106.27 106.17 105.62 105.57 106.94 119.24 106.41 106.27 106.17 105.62 107.16 122.63 106.94 106.41 106.27 106.17 107.32 116.66 107.16 106.94 106.41 106.27 107.32 114.12 107.32 107.16 106.94 106.41 107.35 113.11 107.32 107.32 107.16 106.94 107.55 112.61 107.35 107.32 107.32 107.16 107.87 113.4 107.55 107.35 107.32 107.32 108.37 115.18 107.87 107.55 107.35 107.32 108.38 121.01 108.37 107.87 107.55 107.35 107.92 119.44 108.38 108.37 107.87 107.55 108.03 116.68 107.92 108.38 108.37 107.87 108.14 117.07 108.03 107.92 108.38 108.37 108.3 117.41 108.14 108.03 107.92 108.38 108.64 119.58 108.3 108.14 108.03 107.92 108.66 120.92 108.64 108.3 108.14 108.03 109.04 117.09 108.66 108.64 108.3 108.14 109.03 116.77 109.04 108.66 108.64 108.3 109.03 119.39 109.03 109.04 108.66 108.64 109.54 122.49 109.03 109.03 109.04 108.66 109.75 124.08 109.54 109.03 109.03 109.04 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 6 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 24.9913736952300 -0.00450103556579507X[t] + 0.832983649674458`Y(t-1)`[t] -0.0571738582741973`Y(t-2)`[t] -0.232786626019946`Y(t-3)`[t] + 0.228951294811920`Y(t-4)`[t] + 0.0226526075029203t + 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) 24.9913736952300 10.294385 2.4277 0.018839 0.009419 X -0.00450103556579507 0.004947 -0.9099 0.367258 0.183629 `Y(t-1)` 0.832983649674458 0.135743 6.1365 0 0 `Y(t-2)` -0.0571738582741973 0.178307 -0.3206 0.749813 0.374906 `Y(t-3)` -0.232786626019946 0.180137 -1.2923 0.202203 0.101102 `Y(t-4)` 0.228951294811920 0.13434 1.7043 0.09454 0.04727 t 0.0226526075029203 0.012086 1.8743 0.066741 0.03337 Multiple Linear Regression - Regression Statistics Multiple R 0.989693018832834 R-squared 0.979492271526448 Adjusted R-squared 0.977031344109622 F-TEST (value) 398.017537953093 F-TEST (DF numerator) 6 F-TEST (DF denominator) 50 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.253351340812442 Sum Squared Residuals 3.2093450945731 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 105.62 105.909556464123 -0.289556464122704 2 106.17 105.970792181225 0.199207818774814 3 106.27 106.407996695428 -0.137996695428173 4 106.41 106.568696767885 -0.158696767885347 5 106.94 106.596737168437 0.343262831563244 6 107.16 107.140252809085 0.0197471909147292 7 107.32 107.333035858797 -0.0130358587974443 8 107.32 107.396496501248 -0.0764965012481687 9 107.35 107.484678465875 -0.134678465874600 10 107.55 107.547694525346 0.00230547465391893 11 107.87 107.768305036109 0.101694963891410 12 108.37 108.031082197765 0.33891780223521 13 108.38 108.386001171749 -0.00600117174898755 14 107.92 108.366761851086 -0.446761851085839 15 108.03 107.974964200647 0.05503579935279 16 108.14 108.225937361696 -0.0859373616955538 17 108.3 108.441770055077 -0.141770055077426 18 108.64 108.450719550465 0.189280449535359 19 108.66 108.740985507442 -0.0809855074419642 20 109.04 108.766036424608 0.273963575391751 21 109.03 109.063004427526 -0.0330044275261714 22 109.03 109.116996126921 -0.0869961269214201 23 109.54 109.042387370762 0.497612629238232 24 109.75 109.572034351338 0.177965648662209 25 109.83 109.764226340530 0.0657736594696631 26 109.65 109.746870490776 -0.0968704907764538 27 109.82 109.679066221335 0.140933778664978 28 109.95 109.880193522839 0.069806477161267 29 110.12 110.042412723295 0.0775872767046766 30 110.15 110.110983271139 0.0390167288614961 31 110.21 110.166972455293 0.0430275447070031 32 109.99 110.227763735441 -0.237763735440703 33 110.14 110.098008168350 0.0419918316495721 34 110.14 110.251853089454 -0.111853089454194 35 110.81 110.328494204779 0.481505795220788 36 110.97 110.811175637796 0.158824362204449 37 110.99 110.946217944697 0.0437820553029272 38 109.73 110.791698761526 -1.06169876152646 39 109.81 109.884391056912 -0.0743910569119973 40 110.02 110.070451225203 -0.0504512252029452 41 110.18 110.547933579171 -0.367933579170848 42 110.21 110.381964856789 -0.171964856788717 43 110.25 110.367519921817 -0.117519921816746 44 110.36 110.450479682502 -0.0904796825019399 45 110.51 110.613502064465 -0.103502064464917 46 110.6 110.76452296484 -0.164522964840062 47 110.95 110.814659377530 0.135340622470497 48 111.18 111.094352748633 0.0856472513667392 49 111.19 111.266054379230 -0.0760543792303016 50 111.69 111.229633655535 0.460366344465301 51 111.7 111.674363677023 0.025636322977022 52 111.83 111.693332356689 0.136667643311365 53 111.77 111.699334938915 0.0706650610853777 54 111.73 111.749537452190 -0.0195374521897772 55 112.01 111.723240447188 0.286759552812021 56 111.86 112.068851352161 -0.208851352161338 57 112.04 111.957014625318 0.0829853746823928 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 10 0.343887050882068 0.687774101764135 0.656112949117932 11 0.196242850758297 0.392485701516594 0.803757149241703 12 0.114542358901213 0.229084717802426 0.885457641098787 13 0.2824376980459 0.5648753960918 0.7175623019541 14 0.536132660446009 0.927734679107981 0.463867339553991 15 0.419048359201166 0.838096718402332 0.580951640798834 16 0.40486546848887 0.80973093697774 0.59513453151113 17 0.371756543957006 0.743513087914013 0.628243456042993 18 0.282944096634609 0.565888193269219 0.71705590336539 19 0.236031448776554 0.472062897553109 0.763968551223446 20 0.187305799334444 0.374611598668888 0.812694200665556 21 0.151174015487692 0.302348030975384 0.848825984512308 22 0.113990647804410 0.227981295608820 0.88600935219559 23 0.198583095752682 0.397166191505364 0.801416904247318 24 0.159705287650408 0.319410575300815 0.840294712349593 25 0.113388983117610 0.226777966235220 0.88661101688239 26 0.105177652581467 0.210355305162934 0.894822347418533 27 0.0769462346036761 0.153892469207352 0.923053765396324 28 0.0553633046730464 0.110726609346093 0.944636695326954 29 0.0380159159246459 0.0760318318492917 0.961984084075354 30 0.0277570839664933 0.0555141679329866 0.972242916033507 31 0.0196627503197175 0.0393255006394349 0.980337249680282 32 0.0240029822852979 0.0480059645705958 0.975997017714702 33 0.0154650809415682 0.0309301618831364 0.984534919058432 34 0.010736817713614 0.021473635427228 0.989263182286386 35 0.0494832176992929 0.0989664353985858 0.950516782300707 36 0.092245672564766 0.184491345129532 0.907754327435234 37 0.659991608712508 0.680016782574985 0.340008391287493 38 0.938116116677975 0.123767766644049 0.0618838833220246 39 0.927453616026723 0.145092767946555 0.0725463839732774 40 0.882991167406407 0.234017665187186 0.117008832593593 41 0.956163381780883 0.087673236438233 0.0438366182191165 42 0.91865955185206 0.162680896295879 0.0813404481479394 43 0.903011464580366 0.193977070839269 0.0969885354196343 44 0.855042917075347 0.289914165849305 0.144957082924653 45 0.756375786452976 0.487248427094048 0.243624213547024 46 0.715817940718264 0.568364118563471 0.284182059281736 47 0.655616975448154 0.688766049103692 0.344383024551846 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 4 0.105263157894737 NOK 10% type I error level 8 0.210526315789474 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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