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 11:26:06 -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/t1259260027uzo06mp1lnzhd9u.htm/, Retrieved Thu, 26 Nov 2009 19:27:19 +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/t1259260027uzo06mp1lnzhd9u.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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7.7 10 8.1 7.5 9.2 7.7 7.6 9.2 7.5 7.8 9.5 7.6 7.8 9.6 7.8 7.8 9.5 7.8 7.5 9.1 7.8 7.5 8.9 7.5 7.1 9 7.5 7.5 10.1 7.1 7.5 10.3 7.5 7.6 10.2 7.5 7.7 9.6 7.6 7.7 9.2 7.7 7.9 9.3 7.7 8.1 9.4 7.9 8.2 9.4 8.1 8.2 9.2 8.2 8.2 9 8.2 7.9 9 8.2 7.3 9 7.9 6.9 9.8 7.3 6.6 10 6.9 6.7 9.8 6.6 6.9 9.3 6.7 7 9 6.9 7.1 9 7 7.2 9.1 7.1 7.1 9.1 7.2 6.9 9.1 7.1 7 9.2 6.9 6.8 8.8 7 6.4 8.3 6.8 6.7 8.4 6.4 6.6 8.1 6.7 6.4 7.7 6.6 6.3 7.9 6.4 6.2 7.9 6.3 6.5 8 6.2 6.8 7.9 6.5 6.8 7.6 6.8 6.4 7.1 6.8 6.1 6.8 6.4 5.8 6.5 6.1 6.1 6.9 5.8 7.2 8.2 6.1 7.3 8.7 7.2 6.9 8.3 7.3 6.1 7.9 6.9 5.8 7.5 6.1 6.2 7.8 5.8 7.1 8.3 6.2 7.7 8.4 7.1 7.9 8.2 7.7 7.7 7.7 7.9 7.4 7.2 7.7 7.5 7.3 7.4 8 8.1 7.5 8.1 8.5 8

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 5 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Y[t] = -0.88950701685654 + 0.158263453991724X[t] + 0.876561943359466Y1[t] -0.0350224769283598M1[t] + 0.092789946679895M2[t] + 0.376979718036177M3[t] + 0.488008591198092M4[t] + 0.305502721955216M5[t] + 0.144327901969933M6[t] + 0.107961277896045M7[t] + 0.0473536395035613M8[t] + 0.0293919203823168M9[t] + 0.447288199200505M10[t] + 0.0349018923450704M11[t] + 0.00764007758049221t + 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) -0.88950701685654 0.85323 -1.0425 0.302866 0.151433 X 0.158263453991724 0.087169 1.8156 0.076251 0.038126 Y1 0.876561943359466 0.077701 11.2812 0 0 M1 -0.0350224769283598 0.190307 -0.184 0.854834 0.427417 M2 0.092789946679895 0.194913 0.4761 0.636389 0.318194 M3 0.376979718036177 0.192001 1.9634 0.055936 0.027968 M4 0.488008591198092 0.189646 2.5733 0.013522 0.006761 M5 0.305502721955216 0.192867 1.584 0.120353 0.060177 M6 0.144327901969933 0.198284 0.7279 0.470539 0.23527 M7 0.107961277896045 0.202531 0.5331 0.596673 0.298337 M8 0.0473536395035613 0.207577 0.2281 0.820605 0.410303 M9 0.0293919203823168 0.201912 0.1456 0.884927 0.442464 M10 0.447288199200505 0.188756 2.3697 0.022255 0.011128 M11 0.0349018923450704 0.191065 0.1827 0.855896 0.427948 t 0.00764007758049221 0.00399 1.9149 0.062024 0.031012 Multiple Linear Regression - Regression Statistics Multiple R 0.92753981229745 R-squared 0.860330103396789 Adjusted R-squared 0.815889681750313 F-TEST (value) 19.3591795829643 F-TEST (DF numerator) 14 F-TEST (DF denominator) 44 p-value 2.48689957516035e-14 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.280712364207064 Sum Squared Residuals 3.46717498242365 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 7.7 7.7658968649245 -0.065896864924496 2 7.5 7.42411382557608 0.0758861744239167 3 7.6 7.54063128584097 0.0593687141590346 4 7.8 7.79443546711684 0.00556453288316393 5 7.8 7.81070840952552 -0.0107084095255172 6 7.8 7.64134732172155 0.158652678278446 7 7.5 7.54931539363147 -0.0493153936314696 8 7.5 7.2017265590133 0.298273440986706 9 7.1 7.20723126287171 -0.107231262871714 10 7.5 7.4562326413175 0.0437673586824969 11 7.5 7.4337638801847 0.0662361198153081 12 7.6 7.39067572002094 0.209324279979058 13 7.7 7.35599144261399 0.344008557386014 14 7.7 7.51579475654199 0.184205243458009 15 7.9 7.82345095087794 0.0765490491220631 16 8.1 8.13325863569141 -0.0332586356914102 17 8.2 8.13370523270092 0.0662947672990804 18 8.2 8.03617399383373 0.16382600616627 19 8.2 7.97579475654199 0.224205243458010 20 7.9 7.92282719573 -0.0228271957299975 21 7.3 7.64953697118141 -0.349536971181407 22 6.9 7.67574692475779 -0.775746924757786 23 6.6 6.9520286089374 -0.352028608937403 24 6.7 6.63014552036664 0.069854479633361 25 6.9 6.61128758835886 0.288712411641143 26 7 6.87457344202198 0.125426557978020 27 7.1 7.2540594852947 -0.154059485294701 28 7.2 7.47621097577223 -0.276210975772226 29 7.1 7.38900137844579 -0.28900137844579 30 6.9 7.14781044170505 -0.247810441705051 31 7 6.95959785193894 0.0404021480610638 32 6.8 6.9309811038662 -0.130981103866202 33 6.4 6.6662153466577 -0.266215346657694 34 6.7 6.75695327111176 -0.0569532711117606 35 6.6 6.56769658864714 0.0323034113528587 36 6.4 6.38947319794993 0.0105268020500740 37 6.3 6.21843110072851 0.081568899271489 38 6.2 6.26622740758131 -0.066227407581311 39 6.5 6.48622740758131 0.0137725924186891 40 6.8 6.85203859593239 -0.0520385959323855 41 6.8 6.89266235108032 -0.0926623510803243 42 6.4 6.65999588167967 -0.259995881679671 43 6.1 6.23316552164497 -0.133165521644974 44 5.8 5.86975034162762 -0.0697503416276246 45 6.1 5.65976549867572 0.440234501324278 46 7.2 6.55401292827148 0.645987071728518 47 7.3 7.19261656368781 0.107383436312185 48 6.9 7.1897055616625 -0.289705561662493 49 6.1 6.74839300337415 -0.648393003374151 50 5.8 6.11929056827864 -0.319290568278635 51 6.2 6.19563087040509 0.00436912959491395 52 7.1 6.74405632548714 0.355943674512858 53 7.7 7.37392262824745 0.326077371752551 54 7.9 7.71467236106 0.185327638940007 55 7.7 7.78212647624263 -0.0821264762426299 56 7.4 7.47471479976288 -0.0747147997628828 57 7.5 7.21725092061346 0.282749079386536 58 8 7.85705423454147 0.142945765458531 59 8.1 7.95389435854295 0.146105641457051 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 18 0.00096850457083687 0.00193700914167374 0.999031495429163 19 0.0213092970968957 0.0426185941937914 0.978690702903104 20 0.0150244384730460 0.0300488769460921 0.984975561526954 21 0.00951612242477907 0.0190322448495581 0.990483877575221 22 0.413075825983075 0.826151651966151 0.586924174016925 23 0.575070253728674 0.849859492542652 0.424929746271326 24 0.465251521153417 0.930503042306834 0.534748478846583 25 0.497002003976235 0.99400400795247 0.502997996023765 26 0.537226640149546 0.925546719700907 0.462773359850453 27 0.500286279688597 0.999427440622806 0.499713720311403 28 0.41946396518402 0.83892793036804 0.58053603481598 29 0.360578179102344 0.721156358204688 0.639421820897656 30 0.329389244054977 0.658778488109955 0.670610755945023 31 0.243585007648644 0.487170015297289 0.756414992351356 32 0.174556331257877 0.349112662515755 0.825443668742123 33 0.289804429667298 0.579608859334596 0.710195570332702 34 0.592892305549275 0.81421538890145 0.407107694450725 35 0.499586675339149 0.999173350678298 0.500413324660851 36 0.628821205906724 0.742357588186551 0.371178794093276 37 0.960800537911328 0.0783989241773443 0.0391994620886722 38 0.95126972266719 0.097460554665621 0.0487302773328105 39 0.945462279498959 0.109075441002083 0.0545377205010413 40 0.903836442381478 0.192327115237044 0.0961635576185218 41 0.877139935456774 0.245720129086452 0.122860064543226 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 1 0.0416666666666667 NOK 5% type I error level 4 0.166666666666667 NOK 10% type I error level 6 0.25 NOK

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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