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: Fri, 20 Nov 2009 14:17:06 -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/20/t1258751925aiglb38cn45q5rf.htm/, Retrieved Fri, 20 Nov 2009 22:18:57 +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/20/t1258751925aiglb38cn45q5rf.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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107.11 107.56 107.57 107.70 107.81 107.67 108.75 107.67 109.43 107.72 109.62 108.35 109.54 108.25 109.53 108.26 109.84 108.31 109.67 108.33 109.79 108.36 109.56 108.36 110.22 108.97 110.40 109.62 110.69 109.60 110.72 109.64 110.89 109.65 110.58 109.64 110.94 109.93 110.91 109.81 111.22 109.77 111.09 110.10 111.00 110.40 111.06 110.50 111.55 111.89 112.32 112.10 112.64 111.92 112.36 112.15 112.04 112.16 112.37 112.17 112.59 112.32 112.89 112.38 113.22 112.34 112.85 113.14 113.06 113.18 112.99 113.21 113.32 113.76 113.74 113.99 113.91 113.95 114.52 113.93 114.96 114.01 114.91 114.10 115.30 114.11 115.44 114.10 115.52 114.12 116.08 114.68 115.94 114.71 115.56 114.73 115.88 115.81 116.66 116.01 117.41 116.12 117.68 116.49 117.85 116.51 118.21 116.60 118.92 117.01 119.03 117.01 119.17 117.12 118.95 117.22 118.92 118.38 118.90 118.80

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] = + 39.038520381192 + 0.637407184691893X[t] -0.272539382924652M1[t] -0.00150391792607000M2[t] + 0.304227031804597M3[t] + 0.470522460723327M4[t] + 0.608184536264328M5[t] + 0.540258492164768M6[t] + 0.694706519912123M7[t] + 0.735689325948949M8[t] + 0.888275102075635M9[t] + 0.522867621037696M10[t] + 0.269330499234351M11[t] + 0.0686660801794755t + 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) 39.038520381192 23.271399 1.6775 0.100222 0.050111 X 0.637407184691893 0.218535 2.9167 0.005453 0.002727 M1 -0.272539382924652 0.387002 -0.7042 0.484838 0.242419 M2 -0.00150391792607000 0.393409 -0.0038 0.996966 0.498483 M3 0.304227031804597 0.37954 0.8016 0.426925 0.213462 M4 0.470522460723327 0.376334 1.2503 0.217521 0.108761 M5 0.608184536264328 0.371008 1.6393 0.107978 0.053989 M6 0.540258492164768 0.370173 1.4595 0.15123 0.075615 M7 0.694706519912123 0.369284 1.8812 0.066279 0.03314 M8 0.735689325948949 0.36833 1.9974 0.05172 0.02586 M9 0.888275102075635 0.37138 2.3918 0.020909 0.010455 M10 0.522867621037696 0.367904 1.4212 0.162 0.081 M11 0.269330499234351 0.367935 0.732 0.467878 0.233939 t 0.0686660801794755 0.040555 1.6931 0.097191 0.048596 Multiple Linear Regression - Regression Statistics Multiple R 0.987885678348134 R-squared 0.975918113485352 Adjusted R-squared 0.969112362948604 F-TEST (value) 143.39610425269 F-TEST (DF numerator) 13 F-TEST (DF denominator) 46 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.581204328880177 Sum Squared Residuals 15.5387297078166 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 107.11 107.394163863906 -0.284163863906472 2 107.57 107.823102414942 -0.253102414941710 3 107.81 108.178377229311 -0.368377229311087 4 108.75 108.413338738409 0.336661261590705 5 109.43 108.651537253364 0.778462746635643 6 109.62 109.053843815800 0.566156184199835 7 109.54 109.213217205258 0.326782794742190 8 109.53 109.329240163321 0.200759836678962 9 109.84 109.582362378862 0.257637621138211 10 109.67 109.298369121697 0.371630878302837 11 109.79 109.132620295614 0.657379704385954 12 109.56 108.931955876559 0.628044123440825 13 110.22 109.116900956476 1.10309904352394 14 110.4 109.870917171704 0.529082828296159 15 110.69 110.232566057920 0.457433942079853 16 110.72 110.493023854406 0.226976145593968 17 110.89 110.705726081973 0.184273918026572 18 110.58 110.700092046206 -0.120092046206423 19 110.94 111.108054237694 -0.168054237693908 20 110.91 111.141214261747 -0.231214261747181 21 111.22 111.336969830666 -0.116969830665660 22 111.09 111.250572800756 -0.160572800755516 23 111 111.256923914539 -0.256923914539225 24 111.06 111.120000213954 -0.0600002139535328 25 111.55 111.802122897930 -0.252122897930094 26 112.32 112.275679951893 0.0443200481065509 27 112.64 112.535343688559 0.104656311440953 28 112.36 112.916908850136 -0.556908850136393 29 112.04 113.129611077704 -1.08961107770377 30 112.37 113.136725185631 -0.766725185630614 31 112.59 113.455450371261 -0.865450371261225 32 112.89 113.603343688559 -0.713343688559044 33 113.22 113.799099257478 -0.579099257477537 34 112.85 114.012283604373 -1.16228360437259 35 113.06 113.852908850136 -0.792908850136393 36 112.99 113.671366646622 -0.681366646622274 37 113.32 113.818067295458 -0.498067295457647 38 113.74 114.304372493115 -0.564372493114832 39 113.91 114.653273235637 -0.743273235637301 40 114.52 114.875486601042 -0.355486601041673 41 114.96 115.132807331537 -0.172807331537501 42 114.91 115.190914014240 -0.280914014239678 43 115.3 115.420402194013 -0.120402194013431 44 115.44 115.523677008383 -0.0836770083828088 45 115.52 115.757677008383 -0.237677008382816 46 116.08 115.817883630952 0.262116369048187 47 115.94 115.652134804869 0.287865195131308 48 115.56 115.464218529508 0.0957814704923437 49 115.88 115.948744986230 -0.0687449862297302 50 116.66 116.415927968346 0.244072031653832 51 117.41 116.860439788572 0.549560211427583 52 117.68 117.331241956007 0.348758043993392 53 117.85 117.550318255421 0.29968174457906 54 118.21 117.608424938123 0.60157506187688 55 118.92 118.092875991774 0.827124008226373 56 119.03 118.20252487799 0.827475122010071 57 119.17 118.493891524612 0.676108475387803 58 118.95 118.260890842223 0.689109157777082 59 118.92 118.815412134842 0.104587865158356 60 118.9 118.882458733357 0.0175412666426384 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.633714297600288 0.732571404799424 0.366285702399712 18 0.90525716021546 0.189485679569079 0.0947428397845394 19 0.890753582182928 0.218492835634143 0.109246417817072 20 0.853905320408213 0.292189359183575 0.146094679591787 21 0.813280553597438 0.373438892805123 0.186719446402562 22 0.796238745769839 0.407522508460323 0.203761254230161 23 0.829209460270261 0.341581079459477 0.170790539729739 24 0.868909749822972 0.262180500354055 0.131090250177028 25 0.876419112761868 0.247161774476263 0.123580887238132 26 0.950117758810563 0.0997644823788732 0.0498822411894366 27 0.996312501355267 0.0073749972894665 0.00368749864473325 28 0.998312556075217 0.00337488784956679 0.00168744392478340 29 0.998743015233992 0.00251396953201578 0.00125698476600789 30 0.998103863807995 0.00379227238401088 0.00189613619200544 31 0.996172086508782 0.00765582698243605 0.00382791349121802 32 0.991956301593036 0.0160873968139280 0.00804369840696402 33 0.990020938706614 0.0199581225867714 0.00997906129338571 34 0.984804172078043 0.0303916558439132 0.0151958279219566 35 0.971450543845248 0.0570989123095035 0.0285494561547517 36 0.966881730679948 0.066236538640105 0.0331182693200525 37 0.97498584549975 0.0500283090004985 0.0250141545002493 38 0.962402390807337 0.0751952183853261 0.0375976091926631 39 0.93909075358029 0.121818492839421 0.0609092464197106 40 0.88815966161468 0.223680676770639 0.111840338385320 41 0.87882963765185 0.242340724696299 0.121170362348150 42 0.780028043183241 0.439943913633518 0.219971956816759 43 0.693213423959328 0.613573152081344 0.306786576040672 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 5 0.185185185185185 NOK 5% type I error level 8 0.296296296296296 NOK 10% type I error level 13 0.481481481481481 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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