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 11:29:20 -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/t1259260256s9njq955vmvd47o.htm/, Retrieved Thu, 26 Nov 2009 19:31:08 +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/t1259260256s9njq955vmvd47o.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}, }

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Dataseries X:

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7.8 9.5 7.8 7.8 7.6 7.5 7.7 8.1 7.5 9.1 7.8 7.8 7.8 7.6 7.5 7.7 7.5 8.9 7.5 7.8 7.8 7.8 7.6 7.5 7.1 9 7.5 7.5 7.8 7.8 7.8 7.6 7.5 10.1 7.1 7.5 7.5 7.8 7.8 7.8 7.5 10.3 7.5 7.1 7.5 7.5 7.8 7.8 7.6 10.2 7.5 7.5 7.1 7.5 7.5 7.8 7.7 9.6 7.6 7.5 7.5 7.1 7.5 7.5 7.7 9.2 7.7 7.6 7.5 7.5 7.1 7.5 7.9 9.3 7.7 7.7 7.6 7.5 7.5 7.1 8.1 9.4 7.9 7.7 7.7 7.6 7.5 7.5 8.2 9.4 8.1 7.9 7.7 7.7 7.6 7.5 8.2 9.2 8.2 8.1 7.9 7.7 7.7 7.6 8.2 9 8.2 8.2 8.1 7.9 7.7 7.7 7.9 9 8.2 8.2 8.2 8.1 7.9 7.7 7.3 9 7.9 8.2 8.2 8.2 8.1 7.9 6.9 9.8 7.3 7.9 8.2 8.2 8.2 8.1 6.6 10 6.9 7.3 7.9 8.2 8.2 8.2 6.7 9.8 6.6 6.9 7.3 7.9 8.2 8.2 6.9 9.3 6.7 6.6 6.9 7.3 7.9 8.2 7 9 6.9 6.7 6.6 6.9 7.3 7.9 7.1 9 7 6.9 6.7 6.6 6.9 7.3 7.2 9.1 7.1 7 6.9 6.7 6.6 6.9 7.1 9.1 7.2 7.1 7 6.9 6.7 6.6 6.9 9.1 7.1 7.2 7.1 7 6.9 6.7 7 9.2 6.9 7.1 7.2 7.1 7 6.9 6.8 8.8 7 6.9 7.1 7.2 7.1 7 6.4 8.3 6.8 7 6.9 7.1 7.2 7.1 6.7 8.4 6.4 6.8 7 6.9 7.1 7.2 6.6 8.1 6.7 6.4 6.8 7 6.9 7.1 6.4 7.7 6.6 6.7 6.4 6.8 7 6.9 6.3 7.9 6.4 6.6 6.7 6.4 6.8 7 6.2 7. 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 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = + 0.0591462227848196 + 0.0660454365791428X[t] + 1.52512703254502`Y-1`[t] -0.620545370718424`Y-2`[t] -0.441317537172667`Y-3`[t] + 0.498373687006295`Y-4`[t] + 0.162625204980074`Y-5`[t] -0.231593669035930`Y-6`[t] + 0.111512253826071M1[t] + 0.193313373161585M2[t] -0.0219414502710726M3[t] -0.0762588111162378M4[t] + 0.453205779017736M5[t] -0.292372779576293M6[t] -0.0892150900311018M7[t] + 0.181115372158409M8[t] + 0.0796701373967479M9[t] + 0.213160532284749M10[t] + 0.225129629065532M11[t] + 0.00219583945301820t + 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) 0.0591462227848196 0.786612 0.0752 0.940503 0.470252 X 0.0660454365791428 0.077628 0.8508 0.400835 0.200417 `Y-1` 1.52512703254502 0.170155 8.9632 0 0 `Y-2` -0.620545370718424 0.313336 -1.9804 0.055792 0.027896 `Y-3` -0.441317537172667 0.337024 -1.3095 0.199161 0.09958 `Y-4` 0.498373687006295 0.342229 1.4563 0.154495 0.077248 `Y-5` 0.162625204980074 0.333675 0.4874 0.62912 0.31456 `Y-6` -0.231593669035930 0.201016 -1.1521 0.257311 0.128655 M1 0.111512253826071 0.153467 0.7266 0.472432 0.236216 M2 0.193313373161585 0.163651 1.1813 0.245699 0.12285 M3 -0.0219414502710726 0.163436 -0.1343 0.893996 0.446998 M4 -0.0762588111162378 0.165896 -0.4597 0.648673 0.324337 M5 0.453205779017736 0.167597 2.7041 0.010618 0.005309 M6 -0.292372779576293 0.169298 -1.727 0.09325 0.046625 M7 -0.0892150900311018 0.181994 -0.4902 0.627135 0.313568 M8 0.181115372158409 0.204158 0.8871 0.381239 0.19062 M9 0.0796701373967479 0.194521 0.4096 0.68469 0.342345 M10 0.213160532284749 0.170468 1.2504 0.219675 0.109838 M11 0.225129629065532 0.157307 1.4311 0.161517 0.080759 t 0.00219583945301820 0.00389 0.5645 0.57611 0.288055 Multiple Linear Regression - Regression Statistics Multiple R 0.966793725165996 R-squared 0.934690107020344 Adjusted R-squared 0.898193402119949 F-TEST (value) 25.6102601473540 F-TEST (DF numerator) 19 F-TEST (DF denominator) 34 p-value 7.7715611723761e-15 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.211400014540384 Sum Squared Residuals 1.51945884902094 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.8 7.61611765500372 0.183882344996277 2 7.5 7.69538272704505 -0.195382727045054 3 7.5 7.17383253769253 0.32616746230747 4 7.1 7.32384484526625 -0.223844845266247 5 7.5 7.4041809694169 0.0958190305830996 6 7.5 7.38276419279521 0.117235807204791 7 7.6 7.46103448322318 0.138965516776820 8 7.7 7.54004783721192 0.159952162788082 9 7.7 7.63913782626477 0.0608621737352328 10 7.9 7.832929863081 0.0670701369190068 11 8.1 8.0717928968507 0.0282071031492954 12 8.2 8.09587532880215 0.104124671197852 13 8.2 8.1296176100361 0.0703823899638934 14 8.2 8.1266028075001 0.073397192499899 15 7.9 8.00161184820047 -0.101611848200468 16 7.3 7.52799589293427 -0.227995892934273 17 6.9 7.35352385016391 -0.453523850163913 18 6.6 6.49486252199998 0.105137478000016 19 6.7 6.59296541840794 0.107034581592062 20 6.9 6.9998605574022 -0.0998605574021935 21 7 7.02871716462899 -0.0287171646289862 22 7.1 7.0730692876912 0.0269307123088047 23 7.2 7.18972070115202 0.0102792988479844 24 7.1 7.19852868261494 -0.098528682614943 25 6.9 7.11274082464347 -0.212740824643470 26 7 6.93602085932694 0.0639791406730605 27 6.8 7.06023775412614 -0.260237754126141 28 6.4 6.63954286319189 -0.239542863191892 29 6.7 6.50863771904235 0.191362280957651 30 6.6 6.57993282902121 0.0200671709787864 31 6.4 6.55962540069074 -0.159625400690735 32 6.3 6.21496077635801 0.0850392236419927 33 6.2 6.26222026357286 -0.0622202635728567 34 6.5 6.49390404323143 0.00609595676857139 35 6.8 6.87977347774989 -0.0797734777498863 36 6.8 6.89332926663582 -0.0933292666358247 37 6.4 6.63567461386656 -0.235674613866564 38 6.1 6.11382082001902 -0.0138208200190193 39 5.8 5.893087278089 -0.0930872780890019 40 6.1 5.75184590843284 0.348154091567157 41 7.2 6.87663481219026 0.323365187809742 42 7.3 7.5755840089807 -0.2755840089807 43 6.9 6.98637469767815 -0.0863746976781471 44 6.1 6.24513082902788 -0.145130829027881 45 5.8 5.76992474553339 0.0300752544666105 46 6.2 6.30009680599638 -0.100096805996383 47 7.1 7.0587129242474 0.0412870757526068 48 7.7 7.61226672194709 0.0877332780529151 49 7.9 7.70584929645014 0.194150703549864 50 7.7 7.62817278610889 0.0718272138911134 51 7.4 7.27123058189186 0.128769418108140 52 7.5 7.15677049017474 0.343229509825256 53 8 8.15702264918658 -0.15702264918658 54 8.1 8.0668564472029 0.0331435527971063 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 23 0.597193244950406 0.805613510099189 0.402806755049594 24 0.476981287181768 0.953962574363536 0.523018712818232 25 0.350147394197845 0.700294788395689 0.649852605802155 26 0.366210128277344 0.732420256554688 0.633789871722656 27 0.237093948121046 0.474187896242091 0.762906051878954 28 0.415235068579751 0.830470137159503 0.584764931420249 29 0.883742024685328 0.232515950629344 0.116257975314672 30 0.91128209278357 0.177435814432859 0.0887179072164294 31 0.92270232914616 0.154595341707680 0.0772976708538401 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 0 0 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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